Datasets:

mmosbach

/

pile-validation-1k

like 0

--- address: - 'Rice University, Houston, Texas, 77005-1892' - 'SUNY at Buffalo, Buffalo,N.Y.' author: - 'Tim D. Cochran' - Joseph Masters title: 'The Growth Rate of the First Betti Number in Abelian Covers of $3$-Manifolds ' --- [^1] Abstract {#abstract .unnumbered} ======== We give examples of closed hyperbolic 3-manifolds with first Betti number $2$ and $3$ for which no sequence of finite abelian covering spaces increases the first Betti number. For $3$-manifolds $M$ with first Betti number $2$ we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta _1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop \[KL\], by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz \[IK\]). Introduction {#introduction .unnumbered} ============ Motivated by Waldhausen’s work on Haken manifolds, and by W. Thurston’s [**Geometrization Conjecture**]{}, it has been variously conjectured that, if $M$ is an orientable, irreducible closed $3$-manifold with infinite fundamental group, then: $M$ is finitely covered by a Haken manifold; Some finite cover of $M$ has positive first Betti number; Either $\pi_1(M)$ is virtually solvable or $M$ has finite covers with arbitrarily large first Betti number; $M$ has a finite cover that fibers over the circle. There are easy implications VIBNC$\Longrightarrow$VPBNC$\Longrightarrow$VHC and VFC$\Longrightarrow$VPBNC $\Longrightarrow$VHC. Each implies, if $M$ is atoroidal, the long-standing conjecture of Thurston that such a manifold admits a geometric structure. It is interesting to note that even if $M$ is [**assumed**]{} to be hyperbolic, the conjectures above are open. In this paper, we restrict our attention to VIBNC. (We note in passing that the alternative “$\pi_1(M)$ is virtually solvable” is sometimes replaced by the a priori stronger alternative that “$M$ is finitely covered by the 3-torus, a nilmanifold or a solvmanifold.”) One rich source of finite covering spaces is the set of iterated (regular) finite [**abelian**]{} covering spaces. Thus specifically, in this paper we consider the question: Does there exist an integer $m$, such that, if $M$ is any closed, atoroidal $3$-manifold with $\beta_1(M) \geq m$ then $\b_1(M)$ can be increased in a finite abelian covering space? Note that some condition on $H_1(M)$ is necessary, for if $H_1(M)=0$, then $M$ admits no non-trivial abelian covering spaces. Counter-examples also exist for many manifolds with $\b_1(M)=1$. For if $M$ is zero-framed surgery on a knot in $S^3$, then it is easy to show that $H_1(\wt M;\BQ)\cong\BQ\op Q[t,t^{-1}]/\<\Delta_k,t^n-1\>$ where $\wt M$ is the $n$-fold cyclic cover and $\Delta_k$ is the Alexander polynomial of $K$. Thus $\b_1(\wt M)=\b_1(M)=1$ except when $\Delta_k$ has a cyclotomic factor. We begin this paper by observing that counter-examples also exist in the cases $\b_1(M)=2$ and $\b_1(M)=3$. \[mainthm1\] *There exist closed hyperbolic $3$-manifolds $M$ with $\b_1(M)=2$ (respectively $3$) for which no sequence of finite abelian covers increases the first Betti number. More generally, if a sequence of regular covers of M increases the first Betti number, then one of the covering groups contains a non-trivial perfect subgroup.* It is noteworthy that Question A is still open. If $\b_1 > 0$, then there is an epimorphism $\pi_1(M)\to \mathbb{Z}$, and a corresponding sequence of finite cyclic covers of $M$. Our second contribution is, in the case $\b_1(M)=2$, to give necessary and sufficient conditions, of a somewhat geometric flavor, for the Betti number of these covers to increase linearly with the covering degree. This is the content of Section 2. On abelian covers of hyperbolic 3-manifolds with $\b_1(M)=2$ and $3$ {#failure} ==================================================================== In this section, we observe that, if Question A has an affirmative answer, then the integer $m$ must be at least 4. \[failure23\] There exist closed hyperbolic $3$-manifolds $M$ with $\b_1(M)=2$ (respectively $3$) such that if $\wt M$ is obtained from $M$ by taking a sequence of finite abelian covering spaces, then $\b_1(\wt M)=\b_1(M)$. More generally, if a sequence of regular covers of M increases the first Betti number, then one of the covering groups contains a non-trivial perfect subgroup. Begin with a “seed” manifold $N$ whose fundamental group is nilpotent. Recall that the [**Heisenberg manifold**]{} with Euler class $e$ is the circle bundle over the torus with Euler class $e$. The fundamental group of such a $3$-manifold is the nilpotent group $\langle x,y,t : [x,y]=t^e, [x,t], [y,t]\rangle$, called the [**Heisenberg group**]{} of Euler class $e$. For our seed manifold with $\b_1(N)=2$, we shall take $N$ to be the Heisenberg manifold with Euler class $1$, that can also be described as $0$-framed surgery on a Whitehead link. Thus in this case $\pi_1(N)\cong F/F_3$ where $F$ is the free group of rank 2 and $F_3$ is the third term of the lower-central series of $F$. When $\b_1(N)=3$, we take our seed manifold $N$ to be $S^1\x S^1\x S^1$, the Heisenberg manifold of Euler class $0$. Note that each of the Heisenberg groups of non-zero Euler class has $\beta_1=2$ while the Heisenberg group of Euler class $0$ has $\beta_1=3$. First we claim that *no* finite cover of $N$ will increase the first Betti number, which follows immediately from the Lemma below (which surely is well-known to experts). \[nilpotent\] Suppose $A$ is a Heisenberg group with non-zero (respectively zero) Euler class. If $\wt A$ is any finite index subgroup of $A$, then $\wt A$ is a Heisenberg group of non-zero (respectively, zero) Euler class. Hence in all cases $\beta_1(\wt A) = \beta_1(A)$. The result is obvious for $A=\mathbb{Z}\x \mathbb{Z}\x \mathbb{Z}$ so we assume that $A$ is a Heisenberg group of non-zero Euler class $e$. Then $A$ is a central extension as shown below. $$1\lra \mathbb{Z}\overset{i}{\lra} A \overset{\pi}{\lra} \mathbb{Z}\times\mathbb{Z}\lra 1$$ Since $\wt A$ is a finite index subgroup of $A$, $\pi(\wt A)$ is a finite index subgroup of $\mathbb{Z}\times\mathbb{Z}$ which is hence isomorphic to $\mathbb{Z}\times\mathbb{Z}$. Moreover the kernel of the map $\pi:\wt A\to \pi(\wt A)$ is a finite index subgroup of kernel($\pi$)$=\mathbb{Z}$ which is contained in the center of $A$. It follows that $\wt A$ is also a central extension of the above form and hence is also a Heisenberg group. We claim that $\wt A$ has non-zero Euler class. Suppose not. Then $\wt A$ is abelian. But $A\cong \langle x,y,t : [x,y]=t^e, [x,t], [y,t]\rangle$ where $e\neq 0$. Consider the elements $\{x,y\}$. There is some positive integer $n$ such that both $x^n$ and $y^n$ lie in the subgroup $\wt A$ where they commute. Thus $[x^n,y^n]=1$ in $A$. However since $[x,y]=t^e$, and $t$ commutes with $x$ and $y$, it is easy to see that $x^ny^n=t^ky^nx^n$ where $k=n^2e$ and so $1=[x^n,y^n]=t^k$. This implies that $t$ is of finite order. However, any Heisenberg group is the fundamental group of a circle bundle over the torus, which is an aspherical 3-manifold. Thus $A$ has geometric dimension $3$ and cannot have torsion, for a contradiction. Next alter the seed manifold in a subtle way using the following result of A. Kawauchi \[Ka1 p. 450-452 , and Ka2 Corollary 4.3\] (see also Boileau-Wang \[BW section 4\]). \[hyperbolic\] (Kawauchi) For any closed $3$-manifold $N$, there exists a hyperbolic $3$-manifold $M$ and a degree 1 map $f:M\to N$ that induces an isomorphism on homology groups with local coefficients in $\pi_1(N)$. Equivalently, if $\wt N$ is any covering space of $N$ and $\wt f:\wt M \to \wt N$ is the pull-back, then $\wt f$ induces isomorphisms on homology groups. To the best of our knowledge, this result was first established by Kawauchi using his theory of [**almost identical imitations**]{}. We sketch a proof using the approach of Boileau and Wang (which overlaps substantially with Kawauchi’s approach). Recall that any $3$-manifold $N$ contains a knot $J$ whose exterior is hyperbolic. With more work, Boileau and Wang ensure that there exists such a knot $J$ which is “totally null-homotopic”, i.e., bounds a map of a 2-disk, $\phi:D^2\to N$, such that the inclusion map $\pi_1({\operatorname{image}}\phi)\to\pi_1(N)$ is trivial. Let $M_n$ be the result of $1/n$-Dehn surgery on $N$ along $J$. By work of W. Thurston, for almost all $n$, $M_n$ is hyperbolic. Choose such an $M_n$ and denote it by $M$. Since $J$ is null-homotopic there is a degree one map $f:M\to N$ that induces an isomorphism on $H_1$. Let $\wt N$ be a cover of $N$. Since $J$ is null-homotopic, it lifts to $\wt N$, and there is an induced cover $\wt M$ and an induced map $\tl f:\wt M\to\wt N$. Since $J$ is totally null-homotopic, the pre-images of $J$ bound disjoint Seifert surfaces in $\wt M$, and so $\tl f:\wt M\to\wt N$ is an isomorphism on homology. For any map $f: M \to N$ satisfying the conclusion of Proposition \[hyperbolic\], $ker(f_*)$ is a perfect group. Indeed, Proposition \[hyperbolic\] states that for [**any**]{} covering space $\wt M$ of $M$ that is “induced” from a cover $\wt N$ of $N$, the induced map $\tl f:\wt M\to\wt N$ is an isomorphism on homology, so $\b_1(\wt M)=\b_1(\wt N)$. Specifically, letting $\wt N$ be the universal cover, $H_1(\wt M)\cong H_1(\wt N)=0$ showing that that $\pi_1(\wt M)$ is a perfect group. But $\pi_1(\wt M)$ is kernel$(f_*)$. (Indeed, the condition that $f:M\to N$ induce an isomorphism on first homology with local coefficients in $\pi_1(N)$ is equivalent to the condition that the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ be a perfect group). Returning to the proof of our theorem, recall that $N$ is our seed Heisenberg manifold, and let $M$ be the manifold guaranteed by Proposition \[hyperbolic\]. We claim that the manifold $M$ satisfies the conclusion of the theorem. For suppose $\wt M\overset{p}{\lra}M$ is a regular finite covering space of $M$ corresponding to a surjection $\psi:\pi_1(M)\to F$, where $F$ is a finite group that contains no nontrivial perfect subgroup (for example if $F$ is abelian). Then, since the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ is a perfect group $P$, and the perfect subgroup $\psi(P)\subset F$ must be trivial, $\psi$ factors through $f_*:\pi_1(M)\to\pi_1(N)$ via a surjection $\phi:\pi_1(N)\to F$. Therefore there is a finite regular cover $\wt N$ of $N$ and a lift $\tl f:\wt M\to\wt N$. Notice that the only property of $M$ and $N$ needed for this argument is that the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ is a perfect group. Proceeding, by Proposition \[hyperbolic\] $H_1(\wt M)\cong H_1(\wt N)$ and by Lemma \[nilpotent\], $\beta_1(\wt N)=\beta_1(N)$. Since $\beta_1(M)=\beta_1(N)$ we conclude that $\beta_1(\wt M)=\beta_1(M)$. This shows that the first Betti number of $M$ cannot be increased by a *single* regular $F$-cover unless $F$ contains a nontrivial perfect subgroup. In particular, it shows that the first Betti number of $M$ cannot be increased by a single *abelian* cover. Now suppose that $M_k \to ... \to M_0 = M$ is a sequence of regular covers, with covering groups $F_1, ..., F_k$, where no $F_i$ contains a nontrivial perfect subgroup. In the last paragraph we showed that the cover $M_1 \to M_0$ is the pull-back of a corresponding cover $N_1 \to N_0$. We claim that the kernel, $P_1$, of the lift $(f_1)*:\pi_1(M_1)\to\pi_1(N_1)$ is *equal to* the kernel, $P_0$, of $f_*:\pi_1(M_0)\to\pi_1(N_0)$ (here we view $\pi_1(M_1)$ as a subgroup of $\pi_1(M_0)$). For, obviously $P_1 \subset P_0$ and since $F_0$ contains no perfect subgroups, $P_0\subset P_1$. Thus $P_1$ is a perfect group and thus $\wt f_1$ induces an isomorphism on homology (even with twisted coefficients). Thus we have recovered the inductive hypothesis of the previous paragraph and continuing inductively, we get a sequence of finite covers $N_k \to ... \to N_1$, with $\b_1(M_k) = \b_1(N_k)$. Therefore, to finish the proof we only need to observe that $\b_1(N)$ cannot be increased by any sequence of finite covers, which was shown in Lemma \[nilpotent\]. Linear Growth of Betti Numbers in Cyclic Covering Spaces {#lineargrowth} ======================================================== In this section we ask whether or not it is possible to increase the first Betti number with *linear growth rate* in some *compatible family* of cyclic covering spaces. If $M_\infty$ is a fixed infinite cyclic covering space corresponding to an epimorphism $\psi :\pi_1(M)\to \mathbb{Z}$ then by a *compatible family* we mean the usual family of finite cyclic covers $M_n$ associated to $\pi_1(M)\to \mathbb{Z}\to \mathbb{Z}_n$. By a *linear growth rate* we mean $\varinjlim (\beta_1(M_n)/n)$ is positive. It was already known that linear growth occurs precisely when $H_1(M_\infty)$ has positive rank as a $\mathbb{Z}[t,t^{-1}]$-module [@Lu2 Theorem 0.1][@Lu1 pg.35 Lemma 1.34,pg.453]. Therefore our contribution is to offer a more geometric way of viewing this criterion. We also point out an application to certain optimal systolic inequalities for such $3$-manifolds as have appeared in work of Katz \[IK\]\[KL\]. One should note from the outset that if $\pi_1(M)$ admits an epimorphism to $\mathbb{Z}\ast\mathbb{Z}$, then it is an easy exercise to show that $\beta_1(M)$ can be increased linearly in finite cyclic covers since the same is patently true of the wedge of two circles. Such manifolds arise, for example, as $0$-framed surgery on $2$-component boundary links. This condition is not necessary, however, as we shall see in Example \[example3\] below. Suppose $M$ is a closed, oriented $3$-manifold with $\b_1(M)=2$. Given any basis $\{x,y\}$ of $H^1(M,\BZ)$ we shall define a sequence of higher-order invariants $\b^n(x,y)$; $n\ge1$ taking values in sets of rational numbers. The invariants can be interpreted as certain Massey products in $M$. The invariant $\b^1(x,y)$ is always defined, is independent of basis, and essentially coincides with the invariant $\la$, an extension of Casson’s invariant, due to Christine Lescop \[Les\]. If $\b^i$ is defined for all $i<n$ and is zero, then $\b^n$ is defined (this is why the invariants are called higher-order). If $H_1(M)$ has no torsion, so that $M$ can be viewed as $0$-framed surgery on a 2-component link in a homology sphere (with Seifert surfaces dual to $\{x,y\}$) then $\b^n$, when defined, is the same as the sequence of link concordance invariants of the same name due to the first author \[C1\]. In this case $\b^1$ was previously known as the Sato-Levine invariant. After defining the invariants $\b^n(x,y)$, we show that their vanishing is equivalent to the linear growth of Betti numbers in the family corresponding to the infinite cyclic cover associated to $x$. \[linear\] Let $M$ be a closed oriented $3$-manifold with $\b_1(M)=2$. The following are equivalent. 1. There exists a compatible family $\{M_n|n\ge1\}$ of finite cyclic covers of $M$ such that $\b_1(M_n)$ grows linearly with $n$. 2. There exists a primitive class $x\in H^1(M;\BZ)$ such that for $\textbf{any}$ basis $\{x,y\}$ of $H^1(M;\BZ)$, $\b^n(x,y)=0$ for all $n\ge1$. 3. There exists a primitive class $x\in H^1(M;\BZ)$ such that for $\textbf{some}$ basis $\{x,y\}$ of $H^1(M;\BZ)$, $\b^n(x,y)$ can be defined and contains $0$ for each $n\ge1$. \[systole\] Let $M$ be a closed oriented $3$-manifold with $\b_1(M)=2$. Let $\wt M$ denote the universal torsion-free abelian ($\textbf{Z}\oplus \textbf{Z}$) cover of $M$. Let $[F]$ denote the class in $H_1(\wt M)$ of a lift of a typical fiber of the Abel-Jacobi map of $M$ (represented by a lift of the circle we called $c(x,y)$ below). If, for **some** $\{x,y\}$, and **some** $n$, $\b^n(x,y)\neq 0$ then $[F]$ is non-zero. The above Corollary generalizes an (independent) result of A. Marin (see Prop.12.1 of \[KL\]), which dealt with only the case $n=1$. The significance of this Corollary is that it has been previously shown by Ivanov and Katz (\[IK, Theorem 9.2 and Cor.9.3\]) that the conclusion of Corollary \[systole\] is sufficient to guarantee a certain optimal systolic inequality for $M$. The interested reader is referred to those works. Suppose $c$ and $d$ are disjointly embedded oriented circles in $M$ that are zero in $H_1(M;\BQ)$. Then the [**linking number of $c$ with $d$**]{}, ${\ensuremath{\ell k}}(c,d)\in\BQ$ is defined as follows. Choose an embedded oriented surface $V_d$ whose boundary is “$m$ times $d$” (i.e. a circle in a regular neighborhood $N$ of $d$ that is homotopic in $N$ to $md$) for some positive integer $m$, and set: $${\ensuremath{\ell k}}(c,d) = \f1m(V_d\cd c).$$ Given this, the invariants $\b^n(x,y)$ are defined as follows. Let $\{V_x,V_y\}$ be embedded, oriented connected surfaces that are Poincaré Dual to $\{x,y\}$ and meet transversely in an oriented circle that we call $c(x,y)$ (by the proof of \[C1, Theorem 4.1\] we may assume that $c(x,y)$ is connected). Let $c^+(x,y)$ denote a parallel of $c(x,y)$ in the direction given by $V_y$. Note that $\{V_x,V_y\}$ induce two maps $\psi_x$, $\psi_y$ from $M$ to $S^1$ wherein the surfaces arise as inverse images of a regular value. The product of these maps yields a map $\psi:M\to S^1\x S^1$ that induces an isomorphism on $H_1$/torsion. Since $c(x,y)$ and $c^+(x,y)$ are mapped to points under $\psi$, they represent the zero class in $H_1(M;\BQ)$. Therefore we may define $\b^1(x,y)={\ensuremath{\ell k}}(c(x,y)$, $c^+(x,y))$. In fact, $-\b^1(x,y)\cd|{\operatorname{Tor}}H_1(M;\BZ)|$ is precisely Lescop’s invariant of $M$ \[Les; p.90-94\]. An example is shown in Figure \[satolevine\] of a manifold with $\b^1(x,y)= -k$. (105,92) (10,10)[![Example of $\b^1(x,y)= -k$[]{data-label="satolevine"}](satolevine.eps "fig:")]{} (11,63)[$0$]{} (92,20)[$0$]{} (76,61)[$k$]{} (39,0)[$M$]{} The idea of the higher invariants is to iterate this process as long as possible (compare \[C1\]). Since $c^+(x,y)$ is rationally null-homologous, there is a surface $V_{c(x,y)}$ whose boundary is “$k$ times $c^+(x,y)$” (in the sense above). We could then define $c(x,x,y)$ to be $V_x\cap V_{c(x,y)}$, an embedded oriented circle on $V_x$. If $c(x,x,y)$ is rationally null-homologous, then $\b^2(x,y)$ is defined as ${\ensuremath{\ell k}}(c(x,x,y),c^+(x,x,y))$ and we may also continue and define $c(x,x,x,y)$. In general $c(x,x\dots,x,y)=c(x^n,y)$ will be able to be defined using the chosen surfaces if $c(x^{n-1},y)$ is defined and is also rationally null-homologous (but to do so involves one more choice of a bounding surface). Once $c(x^{n},y)$ is defined and is rationally null-homologous, we may define $\b^n(x,y)$. In general, we do not claim that the value of $\b^n(x,y)$ is independent of the choices of surfaces. Therefore the invariants can be thought of as taking values in a set, just like Massey products. This indeterminacy will not concern us here, for we are only interested in the first non-vanishing value (if it exists) and we shall see that this is independent of the surfaces. Much of the time it is convenient to abbreviate $c(\overbrace{x\dots x}^n,y)$ as $c(n)$ so $c(x,y)=c(1)$. If $c(n)$ is defined and rationally null-homologous then $\b^n(x,y)$ is defined to be the set of rational numbers ${\ensuremath{\ell k}}(c(n),c^+(n))$, ranging over all possible ways of defining such a $c(n)$. If no such $c(n)$ exists then $\b^n(x,y)$ is undefined. \[Example3\] Consider the manifold $M$, shown in Figure \[example3\], obtained from $0$-framed surgery on a two component link $\{ L_x,L_y\}$. Use a genus one Seifert surface for $L_y$ obtained from the obvious twice-punctured disk and a tube that goes up to avoid $L_x$. Let $V_y$ be this surface capped-off in $M$. Similarly use the fairly obvious Seifert surface for $L_x$ in the complement of $L_y$. Then $c^+(x,y)$ is shown. Since it has self-linking zero with respect to $V_x$, $\b^1(x,y)=\b^1(y,x)=0$. Furthermore $\b^2(x,y)=-1$ (note the link $\{c^+(x,y),L_x\}$ is very similar to that of Figure \[satolevine\]). This means that $\pi_1(M)$ does **not** admit an epimorphism to $\mathbb{Z}\ast \mathbb{Z}$ since that would imply that $\{ L_x,L_y\}$ were a homology boundary link. But $\b^2(x,y)=-1$ precludes this by [@C2]. Nonetheless, further $c(yy...y,x)$ may be taken to be empty since $c^+(x,y)$ and $L_y$ form a boundary link in the complement of $L_x$. Thus $\b^n(y,x)= 0$ for all $n$, indicating, by Theorem \[linear\], that the first Betti numbers will grow linearly in the family of finite cyclic covers corresponding to the map $\pi_1(M)\to \mathbb{Z}$ that sends a meridian of $L_x$ to zero and a meridian of $L_y$ to one. (123,73) (10,10)[![Example with linear growth in cyclic covers but no map to $\mathbb{Z}\ast\mathbb{Z}$[]{data-label="example3"}](example3.eps "fig:")]{} (72,39)[$c^+(x,y)$]{} (-5,39)[$L_y$]{} (117,17)[$L_x$]{} (65,0)[$M$]{} (36,46)[$0$]{} (119,68)[$0$]{} \[Example2\] Consider the family of manifolds $M_k$, shown in Figure \[example2\] and Figure \[example2b\], obtained from $0$-framed surgery on a two component link. (135,98) (10,10)[![Example with $\b^1(x,y)=0$,$\b^2(x,y)=-k$, $\b^2(y,x)= -1$[]{data-label="example2"}](example2.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (65,0)[$M$]{} (135,98) (10,10)[![The circle $c(x,y)$[]{data-label="example2b"}](example2b.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (69,61)[$c(x,y)$]{} (65,0)[$M$]{} If $V_x$ denotes the capped-off Seifert surface (obtained using Seifert’s algorithm) for the link component, $L_x$, on the right-hand side and $V_y$ denotes the capped-off Seifert surface for the link component, $L_y$, on the left-hand side, then the dashed circle in Figure \[example2b\] is $c(x,y)=V_x\cap V_y$. The circle $c(y,x)$ is merely this circle with opposite orientation. Since it lies on an untwisted band of $V_x$, $\b^1(x,y)=0=\b^1(y,x)$. Therefore the Lescop invariant of $M$ vanishes. But the link $\{c(x,y),L_x\}$ is the link of Figure \[satolevine\] so $\b^2(x,y)=-k$, whereas the link $\{c(y,x),L_y\}$ is a Whitehead link so $\b^2(y,x)= -1$. We claim further that, as long as $k\neq 0$, for $\textbf{any}$ basis $\{X,Y\}$ of $H^1(M)$, $\b^2(X,Y)\neq 0$. It will then follow from Theorem \[linear\] that the first Betti number of $M$ will grow sub-linearly in **any** family of finite cyclic covers. A general basis, $\{V_X,V_Y\}$, of $H_2(M)$ can be represented as follows. Represent $V_X$ by $p$ parallel copies of $V_x$ together with $q$ parallel copies $V_y$, and represent $V_Y$ by $r$ parallel copies of $V_x$ together with $s$ parallel copies $V_y$, where $ps-qr=\pm 1$. Thus $c(X,Y)=-c(Y,X)$ is represented by $ps-qr$ parallel copies of $c(x,y)$. It follows that $\b^1(X,Y)=\b^1(x,y)=0$, reinforcing our above claim that $\b^1$ is independent of basis. Hence $V_{c(X,Y)}=\pm V_{c(x,y)}$ so $c(X,X,Y)$ is represented by $\pm pc(x,x,y)\mp qc(y,y,x)$. Since $\b^2(X,Y)$ is the self-linking number of this class, it can be evaluated to be $$p^2\b^2(x,y) + q^2\b^2(y,x) - 2pq{\ensuremath{\ell k}}(c(x,x,y),c(y,y,x))$$ but the latter mixed linking number is easily seen to be zero in this case. Hence $\b^2(X,Y)=-kp^2-q^2$ which is non-zero if $k$ is non-zero. \[equivalence\] Suppose $c(1),\dots,c(n)$ have been defined as embedded oriented curves on $V_x$ arising as $c(1)=V_x\cap V_y$ and $$c(j) = V_x\cap V_{c(j-1)}\qquad2\le j\le n$$ where $V_{c(j)}$, $1\le j\le n-1$, is an embedded, oriented connected surface whose boundary is a positive multiple $k_j$ of $c^+(j)$ (in the sense above). Then $\b^j$ is defined for $1\le j\le n-1$ and the following are equivalent: 1. $\b^1,\dots,\b^n$ are defined using the given system of surfaces. 2. $\b^j$ is defined for $1\le j\le n$ and is [**zero**]{} for $1\le j\le\[\f n2\]$ 3. $c(n+1)$ exists 4. For all $s$, $t$ such that $1\le s\le t$ and $s+t\le n$ , ${\ensuremath{\ell k}}(c(s),c^+(t))=0$. Assume $1\le j\le n-1$. The hypotheses imply that a positive multiple of $c^+(j)$ is (homotopic to) the boundary of a surface so $c^+(j)$ and $c(j)$ are rationally null-homologous. Thus their linking number is well-defined, establishing the first claim. [**B1$\Longleftrightarrow$B3**]{}: $\b^n$ is defined precisely when $[c(n)]=0$ in $H_1(M;\BQ)$ which is precisely the condition under which $c(n+1)$ can be defined. [**B1$\Longrightarrow$B4**]{}: If $n=1$ the implication is true since B4 is vacuous. Thus assume by induction that the implication is true for $n-1$, that is our inductive assumption is that, for all $s+t<n$, ${\ensuremath{\ell k}}(c(s),c^+(t))=0$. Now consider the case that $s+t=n$. Since $\b^n$ is defined $[c(n)]=0$ in $H_1(M;\BQ)$. We claim this is true precisely when $c(n)\cd c(1)=0$ (here we refer to oriented intersection number on the surface $V_x$). For suppose $\psi_x:M\to S^1$ and $\psi_y:M\to S^1$ are maps such that $\psi^{-1}_x(*)=V_x$ and $\psi^{-1}_y(*)=V_y$. Then $(\psi_x)_*([c(n)])=0$ since $c(n)\subset V_x$; and $(\psi_y)_*([c(n)])=0$ precisely when $c(n)\cd V_y=c(n)\cd c(1)=0$. But the map $\psi_x \times \psi_y$ completely detects $H_1(M)$/Torsion. Therefore, once $c(n)$ exists, $\b^n$ is defined if and only if: $$\begin{aligned} 0 = c(n)\cd c(1) &= \pm(c(1)\cd V_{c(n-1)})\\ &= \pm k_{n-1}{\ensuremath{\ell k}}(c(1),c^+(n-1))\end{aligned}$$ which establishes B4 in the case $s=1$. But we claim that, if B4 is true for $s+t<n$, then for $s+t=n$ and $s< t$, $$k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1))=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).$$ This equality can then be applied, successively decreasing $s$, to establish B4 in generality. This claimed equality is established as follows. $$\begin{aligned} \pm k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1)) &= \pm V_{c(t-1)}\cd c(s+1)\\ &= \pm V_{c(t-1)}\cd(V_x\cap V_{c(s)})\\ &= V_{c(s)}\cd(V_x\cap V_{c(t-1)})\\ &= V_{c(s)}\cd c(t)\\ &= k_s{\ensuremath{\ell k}}(c(t),c^+(s))\\ &=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).\end{aligned}$$ The last step is justified by verifying that $c(s)\cd c(t)=0$ if $s< t$. For $$\begin{aligned} c(s)\cd c(t) &= c(s)\cd V_{c(t-1)}\\ &= \pm k_{t-1}{\ensuremath{\ell k}}(c(s),c^+(t-1))\end{aligned}$$ which vanishes by our inductive assumption since $s+(t-1)<n$. [**B4$\Longrightarrow$B1**]{}: Since $\b^1$ is always defined we may assume $n>1$. It follows from B4 that ${\ensuremath{\ell k}}(c(1),c^+(n-1))=0$ if $n>1$. But we saw in the proof of B1$\Longrightarrow$B4 that once $c(n)$ was defined, this was equivalent to $\b^n$ being defined. [**B2$\Longrightarrow$B1**]{}: This is obvious. [**B4$\Longrightarrow$B2**]{}: Since B4$\Longrightarrow$B1, we have $\b^j$ defined for $j\le n$. Now suppose $1\le j\le\[\f n2\]$. Since $\b^j={\ensuremath{\ell k}}(c(j),c^+(j)$ and $2j\le n$, this vanishes by B4. This completes the proof of Lemma \[equivalence\]. The proof shows slightly more, namely that there is a correspondence between the infinite cyclic cover implicit in part $A$ and the class $x$ in parts $B$ and $C$. Suppose $\{M_n\}$ is a family of $n$-fold cyclic covers of $M$ corresponding to the infinite cyclic cover $M_\infty$. Note that $H_1(M_\infty;\BQ)$ is a finitely generated $\La=\BQ[t,t^{-1}]$ module (this involves a choice of generator of the infinite cyclic group of deck translations of $M_\infty$). Throughout this proof, homology will be taken with rational coefficients unless specified otherwise. [**Step 1**]{}: $\b_1(M_n)$ grows linearly $\Longleftrightarrow H_1(M_\infty;\BQ)$ has positive rank as a $\La$-module. As remarked above, this fact was previously known. We present a quick proof for the convenience of the reader. We are indebted to Shelly Harvey for showing us this elementary proof. Since $\La$ is a PID, $$H_1(M_\infty)\cong\La^{r_1}\oplus_j\f\La{\<p_j(t)\>}$$ where $p_j(t)\neq0$. By examining the “Wang sequence” with $\BQ$-coefficients $$H_2(M_\infty)\lra H_2(M_n)\overset{\p_*}{\lra}H_1(M_\infty) \overset{t^n-1}{\lra}H_1(M_\infty)\overset{\pi}{\lra}H_1(M_n) \overset{\p_*}{\lra}H_0(M_\infty)\overset{t^n-1}{\lra}$$ it is easily seen that $$\begin{aligned} H_1(M_n) &\cong\f{H_1(M_\infty)}{\<t^n-1\>}\op\BQ\\ &\cong\(\f\La{\<t^n-1\>}\)^{r_1}\oplus_j\f\La{\<p_j(t),t^n-1\>}\op\BQ.\end{aligned}$$ The first summand contributes $nr_1$ to $\b_1(M_n)$. The $\BQ$-rank of the second summand is bounded above by the sum of the degrees of the $p_j$, a number that is [**independent**]{} of $n$. Therefore $\b_1(M_n)$ grows linearly with $n$ if $r_1\neq0$ and otherwise is bounded above by a constant (independent of $n$). [**Step 2**]{}: $H_1(M_\infty)$ has positive $\La$-rank $\Longleftrightarrow H_1(M_\infty)$ has no $(t-1)$-torsion (equivalently $t-1$ acts injectively). To verify Step 2, consider the “Wang sequence” with $\BQ$-coefficients $$H_2(M_\infty)\lra H_2(M)\overset{\p_*}{\lra}H_1(M_\infty) \overset{t-1}{\lra}H_1(M_\infty)\overset{\pi}{\lra}H_1(M) \overset{\p_*}{\lra}H_0(M_\infty)\overset{t-1}{\lra}H_0(M_\infty)$$ associated to the exact sequence of chain complexes $$0\lra C_*(M_\infty;\BQ)\overset{t-1}{\lra} C_*(M_\infty;\BQ)\overset{\pi}{\lra}C_*(M;\BQ)\lra0.$$ Since $H_0(M_\infty)\cong\BQ$, ${\operatorname{image}}\p_*\cong\BQ$ on $H_1(M)$. If $\b_1(M)=2$ then it follows that $\BQ\cong{\operatorname{ker}}\p_*={\operatorname{image}}(\pi)\cong{\operatorname{cokernel}}(t-1)$. It follows that $H_1(M_\infty)$ contains at most one summand of the form $\La/\<(t-1)^m\>$ since each such summand contributes precisely one $\BQ$ to ${\operatorname{cokernel}}(t-1)$. Similarly each $\La$ summand of $H_1(M_\infty)$ contributes one $\BQ$ to the cokernel. Therefore $H_1(M_\infty)$ has positive $\La$ rank if and only if it has no summand of the form $\La/\<(t-1)^m\>$. The latter is equivalent to saying that it has no $(t-1)$-torsion, or that $t-1$ acting on $H_1(M_\infty)$ is injective. This completes Step 2. [**Step 3**]{}: $(t-1):H_1(M_\infty)\to H_1(M_\infty)$ is injective $\Longleftrightarrow$ For any surface $V_x$, dual to $x$, and for each surface $V_y$ such that $\{[V_y],[V_x]\}$ generates $H_2(M;\BZ)$, the class $[\tl c(x,y)]\in H_1(M_\infty;\BQ)$ is zero. Moreover the latter statement is equivalent to one where “for each” is replaced by “for some”. Suppose that $t-1$ is injective. Note that the injectivity of $t-1$ is equivalent to $\p_*:H_2(M)\to H_1(M_\infty)$ being the zero map. Then, for [**any**]{} $[V_y]$ as above, $\p_*([V_y])=0$. But we claim that $\p_*([V_y])$ is represented by $[\tl c(x,y)]$, since $V_x$ is Poincaré Dual to the class $x$ defining $M_\infty$. For if $Y=M-{\operatorname{int}}(V_x\x[-1,1])$ then a copy of $Y$, denoted $\wt Y$, can be viewed as a fundamental domain in $M_\infty$, as shown in Figure \[cover\]. (177,84) (10,10)[![Fundamental Domain of $M_\infty$[]{data-label="cover"}](cover.eps "fig:")]{} (15,62)[$\wt c(x,y)$]{} (80,33)[$\wt V_y$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(x,y)$]{} (87,-5)[$\wt Y$]{} Moreover if $\wt V_y$ denotes $p^{-1}(V_y)\cap \wt Y$ then $\wt V_y$ is a compact surface in $\wt M$ whose boundary is $t_*(\tl c(x,y))-\tl c(x,y)$. Thus $\wt V_y$ is a 2-chain in $M_\infty$ such that $\pi_\#(\wt V_y)$ gives the chain representing $[V_y]$. Since $\p\wt V_y$ is $(t-1)\tl c(x,y)$ in $C_*(M_\infty;\BQ)$, it follows from the explicit construction of $\p_*$ in the proof of the Zig-Zag Lemma \[Mu,Section 24\] that $\p_*([V_y])=[\tl c(x,y)]$. Conversely, if $\p_*([V_y])=0$ for [**some**]{} $[V_y]$ then $\p_*$ is the zero map (note that since $V_x$ lifts to $M_\infty$, $[V_x]$ lies in the image of $H_2(M_\infty)\lra H_2(M)$ so $\p_*([V_x])=0$ t). Therefore the injectivity statement implies the “for each” statement which clearly implies the “for some” statement. Conversely, the “for some” statement implies the injectivity statement. [**Step 4**]{}: The class $[\tl c(x,y)]$ from Step 3 is 0 if and only if it is divisible by $(t-1)^k$ for every positive $k$. In fact it suffices that it be divisible by $(t-1)^N$ where $N$ is the largest nonnegative integer such that $\La/\<(t-1)^N\>$ is a summand of $H_1(M_\infty,\BQ)$. One implication is immediate, so assume that there exists a class $[V_y]$ as in Step 3 such that $\p_*([V_y])=[\tl c_{1}]=(t-1)^N\b$ for some $\b\in H_1(M_\infty)$. Since $[\tl c_{1}]\in{\operatorname{image}}\p_*$, it is $(t-1)$-torsion so $\b$ is $(t-1)^{N+1}$-torsion. Moreover $\b$ lies in the submodule $A\subset H_1(M_\infty,\BQ)$ consisting of elements annihilated by some power of $t-1$, so, by choice of $N$, $(t-1)^N\b=0=[\tl c_{1}]=0$ as desired. This completes the verification of Step 4. [**Step 5**]{}: C$\Rightarrow$A Let $\{x,y\}$ be as in the hypotheses of C and let $M_\infty$ correspond to the class $x$. Let $N$ be the positive integer as above. If $\b^{(N+1}$ can be defined, we know in particular that there exists some system of surfaces $\{V_x,V_y,...,V_{c(N)}\}$ that defines $\{c(j)\}$, $1\leq j\leq (N+1)$. Choose a preferred lift $\wt V_x$, of $V_x$ to $M_\infty$ and a preferred fundamental domain $\wt Y$ as above lying on the positive side of $\wt V_x$. Consider any $m, 1\leq m \leq N$. Since $c(m)$ and $c(m+1)$ lie on $V_x$, they lift to oriented 1-manifolds $\tl c(m)$ and $\tl c(m+1)$ in $\wt V_x$. Similarly $c^+(m)$ lifts to $\tl c^+(m)$, which is a push-off of $\tl c(m)$ lying in $\wt Y$. Recall that $c(m+1)=V_{c(m)}\cap V_x$ where $\p V_{c(m)}=k_mc^+_{(m)}$ for some positive integer $k_m$. Letting $\wt V_{c(m)}$ be $V_{c(m)}$ cut open along $c(m+1)$ we observe that $\wt V_{c(m)}$ can be lifted to $\wt Y$ and viewed as a 2-chain showing that $k_m[\tl c^+(m)]=(t-1)[\tl c(m+1)]$ in $H_1(M_\infty;\BQ)$, as in Figure \[stepfive\]. (177,84) (10,10)[![[]{data-label="stepfive"}](stepfive.eps "fig:")]{} (15,64)[$\wt c(m+1)$]{} (108,72)[$k_m\wt c^+(m)$]{} (65,34)[$\wt V_{c(m)}$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(m+1)$]{} (83,-5)[$\wt Y$]{} Thus $[\tl c^+(1)]=(t-1)(1/k_1)[\tl c(2)]=(t-1)^2(1/k_1)(1/k_2)[\tl c(3)]$, et cetera, showing that $[\tl c(1)]$ is divisible by $(t-1)^N$. By Steps 1 through 4, this implies A of Theorem \[linear\], completing Step 5. [**Step 6**]{}: A$\Rightarrow$B We assume that there is a primitive class $x\in H^1(M;\BZ)$ corresponding to $M_\infty$ and $\{M_n\}$ where $\b_1(M_n)$ grows linearly. By Steps 1, 2, and 3, for any $\{x,y\}$ generating $H^1(M;\BZ)$, $H_1(M_\infty;\BQ)$ has no $(t-1)$-torsion, and for any surfaces dual to ${x,y}$, $[\tl c(1)]=0$. Recall that $c(1)$ and $c(2)$ are always defined. We shall establish inductively that for all $m\ge2$, $c(m)$ is defined and that for *any* system of surfaces used to define $c(m-1)$, $[\tl c(m-1)]=0$ in $H_1(M_\infty;\BQ)$. This has already been shown for $m=2$. Suppose it has been established for $m$ (and all lesser values). We now establish it for $m+1$. Since $c(m)$ and $c(m-1)$ exist, the argument in Step 5 (Figure \[stepfive\]) shows that $k_{m-1}[\tl c(m-1)]=(t-1)[\tl c(m)]$ in $H_1(M_\infty;\BQ)$. But $[\tl c(m-1)]=0$ so $[\tl c(m)]$ is $(t-1)$-torsion. Since there is no non-trivial $(t-1)$-torsion, $[\tl c(m)]=0$ in $H_1(M_\infty;\BQ)$. Hence $[c(m)]=0$ in $H_1(M;\BQ)$ so $c(m+1)$ is defined. Since this holds for any system of defining surfaces, this completes the inductive step. Since $c(m)$ is defined for all $m\ge1$, by Lemma \[equivalence\], $\b^n(x,y)=0$ for all $n\ge1$. This completes the proof of Step 6. Since B clearly implies C, this completes the proof of Theorem \[linear\]. Assume some $\b^m(x,y)\neq 0$. If $[F]$ were zero then certainly, for the fixed infinite cyclic cover, $M_\infty$, corresponding to $x$, $[\wt c(x,y)]=0$ in $H_1(M_\infty;\mathbb{Q})$ so by Steps $1-3$ of the above proof, $\b_1(M_n)$ grows linearly with $n$. By Theorem \[linear\], this would imply that $\b^m(x,y)= 0$ for all $m$, a contradiction. [Str]{} T. Cochran, [*Geometric Invariants of Link Cobordism*]{}, Comment. Math. Helvetici, [**60**]{} (1985), 291-311. T. Cochran, [*Derivatives of links: Milnor’s concordance invariants and Massey’s products*]{}, Memoirs of AMS \#427, American Math. Soc., Providence, RI, 1990. M. Boileau and S. Wang, [*Non-zero degree maps and surface bundles over $S^1$*]{}, J. Diff. Geometry [**43**]{} (1996), 789–806. S. Ivanov and M. Katz, *Generalized degree and optimal Loewner-type inequalities*, Israel J. Math., 141 (2004),221-233. M. Katz and C. Lescop, [*Filling Area Conjecture, Optimal Systolic Inequalities, and the fiber class in Abelian covers*]{}, **Contemporary Math.** vol. 387, preprint math.DG/0412011. Akio Kawauchi, [*An imitation theory of manifolds*]{}, Osaka J. Math. [**26**]{} (1989), 447-464. , [*Almost identical imitations of $(3,1)$-manifold pairs*]{}, Osaka Journal Math. [**26**]{} (1989), 743-758. D. Kraines, [*Massey Higher Products*]{}, Trans. Amer. Math. Soc. [**124**]{} (1966), 431-449. C. Lescop, **Global Surgery Formula for the Casson-Walker Invariant**, Annals of Math Studies 140, Princeton Univ. Press, Princeton, N.J., 1996. W. Lück, **$L^2$-invariants: Theory and Applications to Geometry and K-theory**, A series of Modern Surveys in Mathematics volume[**44**]{}, Springer-Verlag, Berlin Heidelberg New York,(2002). , [*Approximating $L^2$-invariants by their finite-dimensional analogues*]{}, Geom.Funct.Anal. [**4**]{}(4) (1994), 455-481. [^1]: The first author was partially supported by the National Science Foundation. The second author was supported by a National Science Foundation Postdoctoral Research Fellowship

ArXiv

--- author: - 'H. Stiele' - 'W. Pietsch' - 'F. Haberl' - 'D. Hatzidimitriou R. Barnard' - 'B. F. Williams' - 'A. K. H. Kong' - 'U. Kolb' bibliography: - 'papers2.bib' - '/Users/apple/work/papers/my1990.bib' - '/Users/apple/work/papers/my2000.bib' - '/Users/apple/work/papers/my2001.bib' - '/Users/apple/work/papers/catalog.bib' - '/Users/apple/work/papers/my2007.bib' - '/Users/apple/work/papers/my2008.bib' - '/Users/apple/work/papers/my2010.bib' date: 'Received / Accepted ' title: 'The deep XMM-Newton Survey of M 31 [^1] [^2] ' --- Introduction {#Sec:Intro} ============ Our nearest neighbouring large spiral galaxy, the Andromeda galaxy, also known as 31 or , is an ideal target for an X-ray source population study of a galaxy similar to the Milky Way. Its proximity [distance 780 kpc, @1998AJ....115.1916H; @1998ApJ...503L.131S] and the moderate Galactic foreground absorption [= 7[$\times 10^{20}$ cm$^{-2}$]{}, @1992ApJS...79...77S] allow a detailed study of source populations and individual sources. After early detections of 31 with X-ray detectors mounted on rockets [  @1974ApJ...190..285B] and the [*Uhuru*]{} satellite [@1974ApJS...27...37G], the imaging X-ray optics flown on the  X-ray observatory permitted the resolution of individual X-ray sources in 31 for the first time. In the entire set of  imaging observations of 31, @1991ApJ...382...82T [hereafter TF91] found 108 individual X-ray sources brighter than $\sim6.4$[$\times 10^{36}$ ]{}, of which 16 sources showed variability [@1979ApJ...234L..45V; @1990ApJ...356..119C]. In July 1990, the bulge region of 31 was observed with the  High Resolution Imager (HRI) for $\sim 48$ks. @1993ApJ...410..615P [hereafter PFJ93] reported 86 sources brighter than $\sim1.8$[$\times 10^{36}$ ]{} in this observation. Of the  HRI sources located within 75 of the nucleus, 18 sources were found to vary when compared to previous  observations and about three of the sources may be “transients". Two deep PSPC (Position Sensitive Proportional Counter) surveys of 31 were performed with , the first in July 1991 , the second in July/August 1992 . In total 560 X-ray sources were detected in the field of 31; of these, 491 sources were not detected in previous  observations. In addition, a comparison with the results of the  survey revealed long term variability in 18 sources, including 7 possible transients. Comparing the two  surveys, 34 long term variable sources and 8 transient candidates were detected. The derived luminosities of the detected 31 sources ranged from 5[$\times 10^{35}$ ]{} to 5[$\times 10^{38}$ ]{}. Another important result obtained with  was the establishment of supersoft sources (SSSs) as a new class of 31 X-ray sources and the identification of the first SSS with an optical nova in 31 . @2000ApJ...537L..23G reported on first observations of the nuclear region of 31 with . They found that the nuclear source has an unusual X-ray spectrum compared to the other point sources in 31. @2002ApJ...577..738K report on eight  ACIS-I observations taken between 1999 and 2001, which cover the central $\sim 17\arcmin\!\times\!17\arcmin$ region of 31. They detected 204 sources, of which $\sim$50% are variable on timescales of months and 13 sources were classified as transients. @2002ApJ...578..114K detected 142 point sources ($L_X=2\!\times\!10^{35}$ to 2[$\times 10^{38}$ ]{} in the 0.1–10keV band) in a 47ks /HRC observation of the central region of 31. A comparison with a  observation taken 11yr earlier, showed that 46$\pm$26% of the sources with $L_X>5$[$\times 10^{36}$ ]{} are variable. Three different 31 disc fields, consisting of different stellar population mixtures, were observed by . @2002ApJ...570..618D investigated bright X-ray binaries (XRBs) in these fields, while @2004ApJ...610..247D examined the populations of supersoft sources (SSSs) and quasisoft sources (QSSs), including observations of the central field. Using  HRC observations, @2004ApJ...609..735W measured the mean fluxes and long-term time variability of 166 sources detected in these data. used  data to examine the low mass X-ray binaries (LMXBs) in the bulge of 31. Good candidates for LMXBs are the so-called transient sources. Studies of transient sources in 31 are presented in numerous papers, e.g. @2006ApJ...643..356W, @2006ApJ...645..277T [ hereafter TPC06], @2005ApJ...632.1086W, @2006ApJ...637..479W [ hereafter WGM06], and . Using  and  data, @2004ApJ...616..821T detected 43 X-ray sources coincident with globular cluster candidates from various optical surveys. They studied their spectral properties, time variability and logN-logS relations. used  Performance Verification observations to study the variability of X-ray sources in the central region of 31. They found 116 sources brighter than a limiting luminosity of 6[$\times 10^{35}$ ]{} and examined the $\sim60$ brightest sources for periodic and non-periodic variability. At least 15% of these sources appear to be variable on a time scale of several months. used  to study the X-ray binary RX J0042.6+4115 and suggested it as a Z-source. @2006ApJ...643..844O studied the population of SSSs and QSSs with . Recently, @2008ApJ...676.1218T reported the discovery of 217s pulsations in the bright persistent SSS XMMU J004252.5+411540. presented the results of a complete spectral survey of the 335 X-ray point sources they detected in five  observations located along the major axis of 31. They obtained background subtracted spectra and lightcurves for each of the 335 X-ray sources. Sources with more than 50 source counts were individually spectrally fitted. In addition, they selected 18 HMXB candidates, based on a power law photon index of $0.8\!\le\!\Gamma\!\le\!1.2$. prepared a catalogue of 31point-like X-ray sources analysing all observations available at that time in the  archive which overlap at least in part with the optical ${\mathrm}{D}_{25}$ extent of the galaxy. In total, they detected 856 sources. The central part of the galaxy was covered four times with a separation of the observations of about half a year starting in June 2000. PFH2005 only gave source properties derived from an analysis of the combined observations of the central region. Source identification and classification were based on hardness ratios, and correlations with sources in other wavelength regimes. In follow-up work, (i) searched for X-ray burst sources in globular cluster (GlC) sources and candidates and identified two X-ray bursters and a few more candidates, while (ii) searched for correlations with optical novae. They identified 7 SSSs and 1 symbiotic star from the catalogue of PFH2005 with optical novae, and identified anadditional  source with an optical nova. This work was continued and extended on archival  HRC-I and ACIS-I observations by . presented a time variability analysis of all of the 31 central sources. They detected 39 sources not reported at all in PFH2005. 21 sources were detected in the July 2004 monitoring observations of the low mass X-ray binary RX J0042.6+4115 (PI Barnard), which became available in the meantime. Six sources, which were classified as “hard" sources by PFH2005, show distinct time variability and hence are classified as XRB candidates in SPH2008. The SNR classifications of three other sources from PFH2005 had to be rejected due to the distinct time variability found by SPH2008. reported on the first two SSSs ever discovered in the 31 globular cluster system, and discussed the very short supersoft X-ray state of the classical nova M31N 2007-11a. A comparative study of supersoft sources detected with ,  and , examining their long-term variability, was presented by @2010AN....331..212S. An investigation of the logN-logS relation of sources detected in the 2.0–10.0keV range will be presented in a forthcoming paper (Stiele et al. 2011 in prep.). In this work the contribution of background objects and the spatial dependence of the logN-logS relations for sources of 31 is studied. In this paper we report on the large  survey of 31, which covers the entire ${\mathrm}{D}_{25}$ ellipse of 31, for the first time, down to a limiting luminosity of $\sim$[$10^{35}$ erg s$^{-1}$]{} in the 0.2–4.5keV band. In Sect.\[sec:obsana\] information about the observations used is provided. The analysis of the data is presented in Sect.\[Sec:analys\]. Section \[Sec:coim\] presents the combined colour image of all observations used. The source catalogue of the deep  survey of 31 is described in Sect.\[Sec:srccat\]. The results of the temporal variability analysis are discussed in Sect.\[Sec:var\]. Cross-correlations with other 31 X-ray catalogues are discussed in Sect.\[Sec:CrossX-ray\], while Sect.\[SEC:CCow\] discusses cross-correlations with catalogues at other wavelengths. Our results related to foreground stars and background sources in the field of 31 are presented in Sect.\[Sec:fgback\]. Individual source classes belonging to M31 are discussed in Sect.\[Sec:Srcsm31\]. We draw our conclusions in Sect.\[Sec:Concl\]. [lcrcll]{} Paper & S$^{+}$ & \#ofSrc$^{*}$ & & field & comments\ & & & erg cm$^{-2}$ s$^{-1}$&\ & E & 108 & $6.4\!\times\!10^{36}$–$1.3\!\times\!10^{38}$ & entire set of  & 16 sources showed variability\ & & & (0.2–4keV) &imaging observations &\ & R (HRI) & 86 & $\ga1.8\!\times\!10^{36}$ & bulge region & 18 sources variable; $\sim$3 transients\ & & & (0.2–4keV) & &\ & R (PSPC) & 560 & $5\!\times\!10^{35}$–$5\!\times\!10^{38}$ & whole galaxy & two deep surveys\ (SPH97, SHL2001) & & & (0.1–2.4keV) & & 491 sources not detected with\ & & & & & 11 sources variable, 7 transients compared to\ & & & & & 34 sources variable, 8 transients between  surveys\ & X & 116 & $\ga6\!\times\!10^{35}$ & centre & examined the $\sim60$ brightest sources for variability\ & & & (0.3–12keV) & &\ & C (ACIS-I) & 204 & $\ga2\!\times\!10^{35}$ & central $\sim 17\arcmin\!\times\!17\arcmin$ & observations between 1999 and 2001\ & & & (0.3–7keV) & & $\sim$50% of the sources are variable, 13 transients\ & C (HRC) & 142 & $2\!\times\!10^{35}$–$2\!\times\!10^{38}$ & centre & one 47ks observation; 46$\pm$26% of the sources\ & & & (0.1–10keV) & & with $L_X>5$[$\times 10^{36}$ ]{} are variable\ & C (ACIS-I/S) & 28 & $5\!\times\!10^{35}$–$3\!\times\!10^{38}$ & 3 disc fields & bright X-ray binaries\ & & & (0.3–7keV) & &\ & C (ACIS-S S3) & 33 & & 3 disc fields + centre & supersoft sources and quasisoft sources\ & C (HRC) & 166 & $1.4\!\times\!10^{36}$–$5\!\times\!10^{38}$ & major axis + centre & $\ga$25% showed significant variability\ & & & (0.1–10keV) & &\ & C, X & 43 & $\sim10^{35}$–$\sim10^{39}$ & major axis + centre & globular cluster study\ & & & (0.3–10keV) & &\ & X & 856 & $4.4\!\times\!10^{34}$–$2.8\!\times\!10^{38}$ & major axis + centre & source catalogue\ & & & (0.2–4.5keV) & &\ & C, R, X & 21 & $\sim10^{35}$–$\sim10^{38}$ & centre & correlations with optical novae\ & & & (0.2–1keV) & &\ & C, X & 42 & $6\!\times\!10^{35}$–$\sim10^{39}$ & major axis + centre & supersoft sources and quasisoft sources\ & & & (0.2–2keV) & &\ & & & (0.3–10keV) & &\ & C, X & 46 & $\sim10^{35}$–$\sim10^{38}$ & centre & correlations with optical novae\ & & & (0.2–1keV) & &\ & C & 263 & $5\!\times\!10^{33}$–$1.5\!\times\!10^{38}$ & bulge region & low mass X-ray binary study\ & & & (0.5–8keV) & &\ & X & 39 & $7\!\times\!10^{34}$–$6\!\times\!10^{37}$ & centre & re-analysis of archival and new 2004 observations\ & & 300 & $4.4\!\times\!10^{34}$–$2.8\!\times\!10^{38}$ & & time variability analysis; 149 sources with a significance\ & & & (0.2–4.5keV) & & for variability $>$3; 6 new X-ray binary candidates,\ & & & & & 3 supernova remnant classifications were rejected\ & X & 335 & $\sim10^{34}$–$\sim10^{39}$ & 5 fields along & background subtracted spectra and lightcurves for\ & & & (0.3–10keV) & major axis & each source; 18 HMXB candidates, selected from their\ & & & & & power law photon index\ & X & 40 & & whole galaxy & supersoft sources; comparing ,  and\ & & & & &  catalogues\ \[Tab:VarSNRs1\] Notes:\ $^{ +~}$: X-ray satellite(s) on which the study is based: E for , R for , C for , and X for  (EPIC)\ $^{ *~}$: Number of sources\ $^{ \dagger~}$: observed luminosity range in the indicated energy band, assuming a distance of 780kpc to 31 Observations {#sec:obsana} ============ Figure \[fig:deepsurveyfields\] shows the layout of the individual  observations over the field of 31. The observations of the “Deep  Survey of 31” (PI Pietsch) mainly point at the outer parts of 31, while the area along the major axis is covered by archival  observations (PIs Watson, Mason, Di Stefano). To treat all data in the same way, we re-analysed all archival  observations of 31, which were used in . In addition we included an  target of opportunity (ToO) observation of source CXOM31 J004059.2+411551 and the four observations of source RX J0042.6+4115 (PI Barnard). All observations of the “Deep  Survey of 31” and the ToO observation were taken between June 2006 and February 2008. All other observations were available via the  Data Archive[^3] and were taken between June 2000 and July 2004. The journal of observations is given in Table \[tab:observations\]. It includes the 31 field name (Column 1), the identification number (2) and date (3) of the observation and the pointing direction (4, 5), while col. 6 contains the systematic offset (see Sect.\[SubSec:AstCorr\]). For each EPIC camera the filter used and the exposure time after screening for high background is given (see Sect.\[sec:Screening\]). [llllrrrlrlrlr]{} & & & & & & &\ & & & & & & & & & & &\ & & & & & & & & & & &\ Centre 1 & (c1) & 0112570401 & 2000-06-25 & 0:42:36.2 & 41:16:58 & $-1.9,+0.1$ & medium & 23.48(23.48) & medium & 29.64(29.64) &medium & 29.64(29.64)\ Centre 2 & (c2) & 0112570601 & 2000-12-28 & 0:42:49.8 & 41:14:37 & $-2.1,+0.2$ & medium & 5.82( 5.82) & medium & 6.42( 6.42) &medium & 6.42( 6.42)\ Centre 3 & (c3) & 0109270101 & 2001-06-29 & 0:42:36.3 & 41:16:54 & $-3.2,-1.7$ & medium & 21.71(21.71) & medium & 23.85(23.85) &medium & 23.86(23.86)\ N1 & (n1) & 0109270701 & 2002-01-05 & 0:44:08.2 & 41:34:56 & $-0.3,+0.7$ & medium & 48.31(48.31) & medium & 55.68(55.68) &medium & 55.67(55.67)\ Centre 4 & (c4) & 0112570101 & 2002-01-06/07 & 0:42:50.4 & 41:14:46 & $-1.0,-0.8$ & thin & 47.85(47.85) & thin & 52.87(52.87) &thin & 52.86(52.86)\ S1 & (s1) & 0112570201 & 2002-01-12/13 & 0:41:32.7 & 40:54:38 & $-2.1,-1.7$ & thin & 46.75(46.75) & thin & 51.83(51.83) &thin & 51.84(51.84)\ S2 & (s2) & 0112570301 & 2002-01-24/25 & 0:40:06.0 & 40:35:24 & $-1.1,-0.3$ & thin & 22.23(22.23) & thin & 24.23(24.23) &thin & 24.24(24.24)\ N2 & (n2) & 0109270301 & 2002-01-26/27 & 0:45:20.0 & 41:56:09 & $-0.3,-1.5$ & medium & 22.73(22.73) & medium & 25.22(25.22) &medium & 25.28(25.28)\ N3 & (n3) & 0109270401 & 2002-06-29/30 & 0:46:38.0 & 42:16:20 & $-2.3,-1.7$ & medium & 39.34(39.34) & medium & 43.50(43.50) &medium & 43.63(43.63)\ H4 & (h4) & 0151580401 & 2003-02-06 & 0:46:07.0 & 41:20:58 & $+0.3,+0.0$ & medium & 10.14(10.14) & medium & 12.76(12.76) &medium & 12.76(12.76)\ RX 1 & (b1)$^{\ddagger}$ & 0202230201 & 2004-07-16 & 0:42:38.6 & 41:16:04 & $-1.3,-1.2$ & medium & 16.32(16.32) & medium & 19.21(19.21) &medium & 19.21(19.21)\ RX 2 & (b2) & 0202230301 & 2004-07-17 & 0:42:38.6 & 41:16:04 & $-1.0,-0.9$ & medium & 0.0(0.0) & medium & 0.0(0.0) &medium & 0.0(0.0)\ RX 3 & (b3)$^{\ddagger}$ & 0202230401 & 2004-07-18 & 0:42:38.6 & 41:16:04 & $-1.7,-1.5$ & medium & 12.30(12.30) & medium & 17.64(17.64) &medium & 17.68(17.68)\ RX 4 & (b4)$^{\ddagger}$ & 0202230501 & 2004-07-19 & 0:42:38.6 & 41:16:04 & $-1.4,-1.8$ & medium & 7.94(7.94) & medium & 10.12(10.12) &medium & 10.13(10.13)\ S3 & (s3) & 0402560101 & 2006-06-28 & 0:38:52.8 & 40:15:00 & $-3.1,-3.0$ & thin & 4.99(4.99) & medium & 6.96(6.96) &medium & 6.97(6.97)\ SS1 & (ss1) & 0402560201 & 2006-06-30 & 0:43:28.8 & 40:55:12 & $-4.4,-3.7$ & thin & 14.07(9.57) & medium & 24.56(10.65) &medium & 24.58(10.66)\ SN1 & (sn1) & 0402560301 & 2006-07-01 & 0:40:43.2 & 41:17:60 & $-2.7,-1.5$ & thin & 41.23(35.42) & medium & 47.60(39.40) &medium & 47.64(39.44)\ SS2 & (ss2) & 0402560401 & 2006-07-08 & 0:42:16.8 & 40:37:12 & $-1.2,-1.3$ & thin & 21.64(9.92) & medium & 25.59(11.04) &medium & 25.64(11.05)\ SN2 & (sn2) & 0402560501 & 2006-07-20 & 0:39:40.8 & 40:58:48 & $-0.8,-0.7$ & thin & 48.79(21.45) & medium & 56.13(23.85) &medium & 56.17(23.86)\ SN3 & (sn3) & 0402560701 & 2006-07-23 & 0:39:02.4 & 40:37:48 & $-0.9,-2.0$ & thin & 23.80(15.43) & medium & 28.02(17.16) &medium & 28.04(17.17)\ SS3 & (ss3) & 0402560601 & 2006-07-28 & 0:40:45.6 & 40:21:00 & $-1.8,-1.7$ & thin & 27.77(20.22) & medium & 31.92(22.49) &medium & 31.94(22.5)\ S2 & (s21) & 0402560801 & 2006-12-25 & 0:40:06.0 & 40:35:24 & $-1.6,-0.7$ & thin & 39.12(39.12) & medium & 45.19(45.19) &medium & 45.21(45.21)\ NN1 & (nn1) & 0402560901 & 2006-12-26 & 0:41:52.8 & 41:36:36 & $-1.5,-1.5$ & thin & 37.9(37.9) & medium & 43.08(43.08) &medium & 43.1(43.1)\ NS1 & (ns1) & 0402561001 & 2006-12-30 & 0:44:38.4 & 41:12:00 & $-1.0,-1.3$ & thin & 45.11(45.11) & medium & 50.9(50.9) &medium & 50.93(50.93)\ NN2 & (nn2) & 0402561101 & 2007-01-01 & 0:43:09.6 & 41:55:12 & $-0.0,-1.2$ & thin & 41.73(41.73) & medium & 46.45(46.45) &medium & 46.47(46.47)\ NS2 & (ns2) & 0402561201 & 2007-01-02 & 0:45:43.2 & 41:31:48 & $-2.3,-1.7$ & thin & 34.96(34.96) & medium & 40.55(40.55) &medium & 40.58(40.58)\ NN3 & (nn3) & 0402561301 & 2007-01-03 & 0:44:45.6 & 42:09:36 & $-1.4,-0.7$ & thin & 31.04(31.04) & medium & 34.81(34.81) &medium & 34.81(34.81)\ NS3 & (ns3) & 0402561401 & 2007-01-04 & 0:46:38.4 & 41:53:60 & $-2.1,+0.3$ & thin & 39.41(39.41) & medium & 45.50(45.50) &medium & 45.52(45.52)\ N2 & (n21) & 0402561501 & 2007-01-05 & 0:45:20.0 & 41:56:09 & $-2.6,-1.3$ & thin & 37.18(37.18) & medium & 41.98(41.98) &medium & 42.03(42.03)\ SS1 & (ss11) & 0505760201 & 2007-07-22 & 0:43:28.8 & 40:55:12 & $-2.5,-2.6$ & thin & 30.07(23.90) & medium & 34.01(26.70) &medium & 34.02(26.72)\ S3 & (s31) & 0505760101 & 2007-07-24 & 0:38:52.8 & 40:15:00 & $-1.8,-1.0$ & thin & 21.86(15.74) & medium & 24.74(17.65) &medium & 24.74(17.65)\ CXOM31& (sn11)$^{\diamond}$ & 0410582001 & 2007-07-25 & 0:40:59.2 & 41:15:51 & $-1.2,-0.3$ & thin & 11.27(11.27) & medium & 14.01(14.01) &medium & 14.02(14.02)\ SS3 & (ss31) & 0505760401 & 2007-12-25 & 0:40:45.6 & 40:21:00 & $-1.0,+0.1$ & thin & 23.56(22.82) & medium & 28.18(25.8) &medium & 28.2(25.82)\ SS2 & (ss21) & 0505760301 & 2007-12-28 & 0:42:16.8 & 40:37:12 & $+1.3,-0.1$ & thin & 35.28(35.28) & medium & 40.00(40.00) &medium & 40.01(40.01)\ SN3 & (sn31) & 0505760501 & 2007-12-31 & 0:39:02.4 & 40:37:48 & $-1.6,-1.3$ & thin & 24.26(24.26) & medium & 28.77(28.77) &medium & 28.78(28.78)\ S3 & (s32) & 0511380101 & 2008-01-02 & 0:38:52.8 & 40:15:00 & $-1.7,-3.3$ & thin & 38.31(38.31) & medium & 44.92(44.92) &medium & 44.95(44.95)\ SS1 & (ss12) & 0511380201 & 2008-01-05 & 0:43:28.8 & 40:55:12 & $-0.9,-1.4$ & thin & 8.85( 8.85) & medium & 11.28(11.28) &medium & 11.29(11.29)\ SN2 & (sn21) & 0511380301 & 2008-01-06 & 0:39:40.8 & 40:58:48 & $-0.2,-0.4$ & thin & 24.79(24.79) & medium & 29.28(29.28) &medium & 29.29(29.29)\ SS1 & (ss13) & 0511380601 & 2008-02-09 & 0:43:28.8 & 40:55:12 & $-0.8,-1.8$ & thin & 13.35(13.35) & medium & 15.07(15.07) &medium & 15.08(15.08)\ Data analysis {#Sec:analys} ============= In this section, the basic concepts of the X-ray data reduction and source detection processes are described. Screening for high background {#sec:Screening} ----------------------------- The first step was to exclude times of increased background, due to soft proton flares. Most of these times are located at the start or end of an orbit window. We selected good time intervals (GTIs) – intervals where the intensity was lower than a certain threshold – using 7–15keV light curves constructed from source-free regions of each observation. The GTIs with PN and MOS data were determined from the higher statistic PN light curves. Outside the PN time coverage, GTIs were determined from the combined MOS light curves. For each observation, the limiting thresholds for the count rate were adjusted individually; this way we avoided cutting out short periods (up to a few hundred seconds) of marginally increased background. Short periods of low background, which were embedded within longer periods of high background, were omitted. For most observations, the PN count rate thresholds were 2–8ctsks$^{-1}$arcmin$^{-2}$. As many of the observations were affected by strong background flares, the net exposure which can be used for our analysis was strongly reduced. The GTIs of the various observations ranged over 6–56ks, apart from observation b2 (ObsID 0202230301) which had to be rejected, because it showed high background throughout the observation. The exposures for all three EPIC instruments are given in Cols. 8, 10 and 12 of Table \[tab:observations\]. The observations obtained during the summer visibility window of 31 were affected more strongly by background radiation than those taken during the winter window. The most affected observations of the deep survey were reobserved. After screening for times of enhanced particle background, the second step was to examine the influence of solar wind charge exchange. This was done by producing soft energy ($<\!2$keV) background light curves. These lightcurves varied only for 10 observations, for which additional screening was necessary. The screening of enhanced background due to solar wind charge exchange was applied to the observations only for the creation of colour images, in order to avoid that these observations will appear in the mosaic image with a tinge of red. The screening was not used for source detection. The third and last step includes the study of the background due to detector noise. The processing chains take into account all known bad or hot pixels and columns and flag the affected pixels in the event lists. We selected data with [(FLAG & 0xfa0000)=0]{}, excluded rows and columns near edges, and searched by eye for additional warm or hot pixels and columns in each observation. To avoid background variability over the PN images, we omitted the energy range from 7.2–9.2keV where strong fluorescence lines cause higher background in the outer detector area [@2004SPIE.5165..112F]. An additional background component can occur during the EPIC PN offset map calculation. If this period is affected by high particle background, the offset calculation will lead to a slight underestimate of the offset in some pixels which can then result in blocks of pixels ($\approx 4\!\times\!4$) with enhanced low energy signal.[^4] These blocks will be found by the [SAS]{} detection tools and appear as sources with extremely soft spectrum (so called supersoft sources). To reduce the number of false detections in this source class, we decided to include the task [epreject]{} in [epchain]{}, which locates the pixels with a slight underestimate of the offset and corrects this underestimate. To ensure that [epreject]{} produces reliable results, difference images of the event lists obtained with and without [epreject]{}, were created. Only events with energies above 200eV were used. We checked whether [epreject]{} removed all pixels with an enhanced low energy signal. Only in observation ns1 the difference image still shows a block of pixels with enhanced signal. As this block is also visible at higher energies (PHA$>30$) it cannot be corrected with [epreject]{}. Additionally, we ascertained that almost all pixels not affected during the offset map calculation have a value consistent with zero in the difference images, with two exceptions discussed in Sect.\[Sec:srccat\]. Images {#Sec:Images} ------ For each observation, the data were split into five energy bands: (0.2–0.5)keV, (0.5–1.0)keV, (1.0–2.0)keV, (2.0–4.5)keV, and (4.5–12)keV. For the PN data, we used only single-pixel events (PATTERN$=$0) in the first energy band, while for the other bands, single-pixel and double-pixel events were selected (PATTERN$\le$4). In the MOS cameras, single-pixel to quadruple-pixel events (PATTERN$\le$12) were used. We created images, background images and exposure maps (with and without vignetting correction) for PN, MOS1 and MOS2 in each of the five energy bands and masked them for the acceptable detector area. The image bin size is 2. The same procedure was applied in our previous 31 and M 33 studies . To create background images, the [SAS]{} task [eboxdetect]{} was run in local mode, in which it determines the background from the surrounding pixels of a sliding box, with box sizes of $5\times5$, $10\times10$ and $20\times20$ pixels (10$\times$10, 20$\times$20and 40$\times$40). The detection threshold is set to [likemin=15]{}, which is a good compromise between cutting out most of the sources and leaving sufficient area to derive the appropriate background. For the background calculation, a two dimensional spline is fitted to a rebinned and exposure corrected image (task [esplinemap]{}). The number of bins used for rebinning is controlled by the parameter [nsplinenodes]{}, which is set to 16 for all but the observations of the central region, where it was set to 20 (maximum value). For PN, the background maps contain the contribution from the “out of time (OoT)" events. Source detection {#Sec:SrcDet} ---------------- For each observation, source detection was performed simultaneously on 5 energy bands for each EPIC camera, using the XMM-[SAS]{} detection tasks [eboxdetect]{} and [emldetect]{}, as such fitting provides the most statistically robust measurements of the source positions by including all of the data. This method was also used to generate the 2XMM catalog . In the following we describe the detection procedure used. The source detection procedure consists of two consecutive detection steps. An initial source list is created with the task [eboxdetect]{} ( Sect.\[Sec:Images\]). To select source candidates down to a low statistical significance level, a low likelihood threshold of four was used at this stage. The background was estimated from the previously created background images (see Sect.\[Sec:Images\]). This list is the starting point for the XMM-[SAS]{} task [emldetect]{} (v. 4.60.1). The [emldetect]{} task performs a Maximum Likelihood fit of the distribution of source counts [based on Cash C-statistics approach; @1979ApJ...228..939C], using a point spread function model obtained from ray tracing calculations. If $P$ is the probability that a Poissonian fluctuation in the background is detected as a spurious source, the likelihood of the detection is then defined as $\mathcal{L}=-\ln{\left}( P {\right})$.[^5] The fit is performed simultaneously in all energy bands for all three cameras by summing the likelihood contribution of each band and each camera. Sources exceeding the detection likelihood threshold in the full band (combination of the 15 bands) are regarded as detections; the catalogue is thus full band selected. The detection threshold used is 7, as in PFH2005. Some other parameters differ from the values used in PFH2005, as in this work a parameter setting optimised for the detection of extended sources was used (G. Lamer; private communication). The parameters in question are the event cut-out ([ecut=30.0]{}) and the source selection radius ([scut=0.9]{}) for multi-source fitting, the maximum number of sources into which one input source can be split ([nmulsou=2]{}), and the maximum number of sources that can be fitted simultaneously ([nmaxfit=2]{}). Multi-PSF fitting was performed in a two stage process for objects with a detection likelihood larger than ten. All of the sources were also fitted with a convolution of a $\beta$-model cluster brightness profile with the  point spread function, in order to detect any possible extension in the detected signal. Sources which have a core radius significantly larger than the PSF are flagged as extended. The free parameters of the fit were the source location, the source extent and the source counts in each energy band of each telescope. To derive the X-ray flux of a source from its measured count rate, one uses the so-called energy conversion factors (ECF): $${\mathrm}{Flux}=\frac{{\mathrm}{Rate}}{{\mathrm}{ECF}}$$ These factors were calculated using the detector response, and depended on the used filter, the energy band in question, and the spectrum of the source. As we wanted to apply the conversion factors to all sources found in the survey, we assumed a power law model with photon index $\Gamma\!=\!1.7$ and the Galactic foreground absorption of $N_{{\mathrm}{H}}\!=\!7\times10^{20}$cm$^{-2}$ [@1992ApJS...79...77S see also PFH2005] to be the universal source spectrum for the ECF calculation. The ECFs (see Table \[tab:ECFvalues\]) were derived with [XSPEC]{}[^6](v 11.3.2) using response matrices (V.7.1) available from the  calibration homepage[^7]. As all necessary corrections of the source parameters ( vignetting corrections) were included in the image creation and source detection procedure[^8], the *on axis* ECF values were derived . The fluxes determined with the ECFs given in Table \[tab:ECFvalues\] are absorbed ( observed) fluxes and hence correspond to the observed count rates, which are derived in the [emldetect]{} task. During the mission lifetime, the MOS energy distribution behaviour has changed. Near the nominal boresight positions, where most of the detected photons hit the detectors, there has been a decrease in the low energy response of the MOS cameras [@2006ESASP.604..925R]. To take this effect into account, different response matrices for observations obtained before and after the year 2005 were used (see Table \[tab:ECFvalues\]). [lrrrrrr]{} & & & & & &\ & &\ EPIC PN & thin & $11.33$ & $8.44$ & $5.97$ & $1.94$ & $0.58$\ & medium & $10.05$ & $8.19$ & $5.79$ & $1.94$ & $0.58$\ EPIC MOS1 & thin & $2.25$ & $1.94$ & $2.06$ & $0.76$ & $0.14$\ & medium & $2.07$ & $1.90$ & $2.07$ & $0.75$ & $0.15$\ EPIC MOS2 & thin & $2.29$ & $1.98$ & $2.09$ & $0.78$ & $0.15$\ & medium & $2.06$ & $1.90$ & $2.04$ & $0.75$ & $0.15$\ EPIC MOS1 & thin & $2.59$ & $2.04$ & $2.12$ & $0.76$ & $0.15$\ OLD & medium & $2.33$ & $1.98$ & $2.09$ & $0.76$ & $0.15$\ EPIC MOS2 & thin & $2.58$ & $2.04$ & $2.13$ & $0.76$ & $0.15$\ OLD & medium & $2.38$ & $1.99$ & $2.09$ & $0.75$ & $0.16$\ \[tab:ECFvalues\] For most sources, band 5 just adds noise to the total count rate. If converted to flux, this noise often dominates the total flux due to the small ECF. To avoid this problem we calculated count rates and fluxes for detected sources in the “XID" (0.2–4.5)keV band (bands 1 to 4 combined). While for most sources this is a good solution, for extremely hard or soft sources there may still be bands just adding noise. This, then, may lead to rate and flux errors that seem to falsely indicate a lower source significance. A similar effect occurs in the combined rates and fluxes, if a source is detected primarily by one instrument ( soft sources in PN). Sources are entered in the  catalogue from the observation in which the highest source detection likelihood is obtained (either combined or single observations). For variable sources this means that the source properties given in the  catalogue (see Sect.\[Sec:srccat\] and Table 5) are those observed during their brightest state. We rejected spurious detections in the vicinity of bright sources. In regions with a highly structured background, the [SAS]{} detection task [emldetect]{} registered some extended sources. We also rejected these “sources" as spurious detections. In an additional step we checked whether an object had visible contours in at least one image out of the five energy bands. The point-like or extended nature, which was determined with [emldetect]{}, was taken into account. In this way, “sources" that are fluctuations in the background, but which were not fully modelled in the background images, were detected. In addition, objects located on hot pixels, or bright pixels at the rim or in the corners of the individual CCD chips (which were missed during the background screening) were recognised and excluded from the source catalogue, especially if they were detected with a likelihood larger than six in one detector only. To allow for a statistical analysis, the source catalogue only contains sources detected by the [SAS]{} tasks [eboxdetect]{} and [emldetect]{} as described above,  the few sources that were not detected by the analysis program, despite being visible on the X-ray images, have not been added by hand as it was done in previous studies (SPH2008; PFH2005). To classify the source spectra, we computed four hardness ratios. The hardness ratios and errors are defined as: $${\mathrm}{HR}i = \frac{B_{i+1} - B_{i}}{B_{i+1} + B_{i}}\; \mbox{and}\;\; {\mathrm}{EHR}i = 2 \frac{\sqrt{(B_{i+1} EB_{i})^2 + (B_{i} EB_{i+1})^2}}{(B_{i+1} + B_{i})^2}, \label{Eq:hardr}$$ for [*i*]{} = 1 to 4, where $B_{i}$ and $EB_{i}$ denote count rates and corresponding errors in energy band [*i*]{}. Astrometrical corrections {#SubSec:AstCorr} ------------------------- To obtain astrometrically-corrected positions for the sources of the five central fields we used the [SAS]{}-task [eposcorr]{} with  source lists [@2002ApJ...577..738K; @2002ApJ...578..114K; @2004ApJ...609..735W]. For the other fields we selected sources from the USNO-B1 [@2003AJ....125..984M], 2MASS [@2006AJ....131.1163S] and Local Group Galaxy Survey [LGGS; @2006AJ....131.2478M] catalogues[^9]. ### Astrometry of optical/infrared catalogues In a first step, we examined the agreement between the positions given by the various optical catalogues.[^10] A close examination of the shifts obtained, showed significant differences between the positions given in the individual catalogues. In summary, between the USNO-B1 and LGGS catalogues we found an offset of: $-$0197 in R.A. and 0067 in Dec[^11]; and between the USNO-B1 and 2MASS catalogues we found an offset of: $-$0108 in R.A. and 0204 in Dec. We chose the USNO-B1 catalogue as a reference, since it covers the entire field observed in the Deep  survey, and in addition it provides values for the proper motion of the optical sources. Since the optical catalogues, as well as the Deep  catalogue, are composed of individual observations of sub-fields of 31, we searched for systematic drifts in the positional zero points from region to region. However no systematic offsets were found. Finally, we applied the corrections found to the sources in the LGGS and 2MASS catalogues, to bring all catalogues to the USNO-B1 reference frame. The offsets found between the USNO-B1 and 2MASS catalogues can be explained by the independent determination of the astrometric solutions for these catalogues. Given that the positions provided in the LGGS catalogue are corrected with respect to the USNO-B1 catalogue [see @2006AJ....131.2478M], the offset found in right ascension was totally unexpected and cannot be explained. ### Corrections of the X-ray observations From the positionally corrected catalogues, we selected sources which either correlate with globular clusters from the Revised Bologna Catalogue or with foreground stars, characterised by their optical to X-ray flux ratio [@1988ApJ...326..680M] and their hardness ratio . For sources selected from the USNO-B1 catalogue, we used the proper motion corrected positions. We then used the [SAS]{}-task [eposcorr]{} to derive the offset of the X-ray aspect solution. Four observations did not have enough optical counterparts to apply this method. The lack of counterparts is due to the very short exposure times resulting after the screening for high background (obs. s3, ss12, ss13) and the location of the observation (obs. sn11). In these cases, we used bright persistent X-ray sources, which we correlated with another observation of the same field. We checked for any residual systematic uncertainty in the source positions and found it to be well characterised by a conservative $1\sigma$ value of 05. This uncertainty is due to positional errors of the optical sources as well as inaccuracy in the process of the determination of the offset between optical and X-ray sources, and is called systematic positional error. The appropriate offset, given in Col. 6 of Table \[tab:observations\], was applied to the event file of each pointing, and images and exposure maps were then reproduced with the corrected astrometry.\ Fields that were observed at least twice are treated in a special way, which is described in the following section. Multiple observations of selected fields ---------------------------------------- The fields that were observed more than once were the central field, the fields pointing on RX J0042.6+4115[^12], two fields located on the major axis of 31  (S2, N2) and all fields of the “Large Survey" located in the southern part of the galaxy (SS1, SS2, SS3, S3, SN3, SN2, SN1). To reach higher detection sensitivity we merged the images, background images and exposure maps of observations which have the same pointing direction and were obtained with the same filter setting. Subsequently, source detection, as described in Sect. \[Sec:SrcDet\], was repeated on the merged data. For the S2 field, there are two observations with different filter settings. In this case, source detection was performed simultaneously on all 15 bands of both observations,  on 30 bands simultaneously. The N2 field was treated in the same way. For the central field images, background images and exposure maps of observations c1, c2 and c3 were merged. These merged data were used together with the data of observation c4 to search for sources simultaneously; in this way it was possible to take into account the different ECFs for the different filters. One field was observed twice with slightly different pointing direction in observations sn1 and sn11; simultaneous source detection was used for these observations also. Variability calculation {#Sec:DefVar} ----------------------- To examine the time variability of each source listed in the total source catalogue, we determined the XID flux at the source position in each observation or at least an upper limit for the XID flux. We used the task [emldetect]{} with fixed source positions when calculating the total flux. To get fluxes and upper limits for all sources in the input list we set the detection likelihood threshold to 0. A starting list was created from the full source catalogue, which only contains the identification number and position of each source located in the field examined. To give correct results, the task [emldetect]{} has to process the sources from the brightest one to the faintest one. We, therefore, had to first order the sources in each observation by the detection likelihood. For sources not visible in the observation in question we set the detection likelihood to 0. This list was used as input for a first [emldetect]{} run. In this way we achieved an output list in which a detection likelihood was allocated to every source. For a final examination of the sources in order of detection likelihood, a second [emldetect]{} run was necessary. We only accepted XID fluxes for detections $\ge$ 3 $\sigma$; otherwise we used a 3 $\sigma$ upper limit. To compare the XID fluxes between the different observations, we calculated the significance of the difference $$S=\frac{F_{{\mathrm}{max}}- F_{{\mathrm}{min}}}{\sqrt{\sigma_{{\mathrm}{max}}^2+\sigma_{{\mathrm}{min}}^2}}$$ and the ratio of the XID fluxes $V=F_{{\mathrm}{max}}/F_{{\mathrm}{min}}$, where $F_{{\mathrm}{max}}$ and $F_{{\mathrm}{min}}$ are the maximum and minimum (or upper limit) source XID flux, and $\sigma_{{\mathrm}{max}}$ and $\sigma_{{\mathrm}{min}}$ are the errors of the maximum and minimum flux, respectively. This calculation was not performed whenever $F_{{\mathrm}{max}}$ was an upper limit. Finally, the largest XID flux of each source was derived, excluding upper limits. Spectral analysis ----------------- To extract the X-ray spectrum of individual sources, we selected an extraction region and a corresponding background region which was at least as large as the source region, was located on the same CCD at a similar off axis angle as the source, and did not contain any point sources or extended emission. For EPIC PN, we only accepted single-pixel events for the spectra of supersoft sources, while for all other spectra single and double-pixel events were used. For the EPIC-MOS detectors, single-pixel through to quadruple-pixel events were always used. Additionally, we only kept events with FLAG$=$0 for all three detectors. For each extraction region, we produced the corresponding response matrix files and ancillary response files. For each source, the spectral fit was obtained by fitting all three EPIC spectra simultaneously, using the tool [XSPEC]{}. For the absorption, we used the [TBabs]{} model, with abundances from @2000ApJ...542..914W and photoelectric absorption cross-sections from @1992ApJ...400..699B with a new He cross-section based on @1998ApJ...496.1044Y. Cross correlations {#Sec:CrossCorr_Tech} ------------------ Sources were regarded as correlating if their positions overlapped within their 3$\sigma$ (99.73%) positional errors, defined as : $$\Delta{\mathrm}{pos}\le3.44\times\sqrt{\sigma_{{\mathrm}{stat}}^2 + \sigma_{{\mathrm}{syst}}^2}+3\times\sigma_{{\mathrm}{ccat}} \label{Eq:Cor}$$ where $\sigma_{{\mathrm}{stat}}$ is the statistical and $\sigma_{{\mathrm}{syst}}$ the systematic error of the X-ray sources detected in the present study. The statistical error was derived by [emldetect]{}. The determination of the systematic error is described in Sect.\[SubSec:AstCorr\]. We use a value of 05, for all sources. The positional error of the sources in the catalogue used for cross-correlation is given by $\sigma_{{\mathrm}{ccat}}$. The values of $\sigma_{{\mathrm}{ccat}}$ (68% error) used for the different X-ray catalogues can be found in Table \[Tab:XrayRefCat\]. Exceptions to Eq. \[Eq:Cor\] are sources that are listed in more than one catalogue or that are resolved into multiple sources with . The first case is restricted to catalogues with comparable spatial resolution and hence positional uncertainty. To identify the X-ray sources in the field of 31 we searched for correlations with catalogues in other wavelength regimes. The  source catalogue was correlated with the following catalogues and public data bases: Globular Clusters: : Bologna Catalogue , @2009AJ....137...94C [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2009AJ....138..770H [$\sigma_{{\mathrm}{ccat}}=0.\arcsec5$], @2008PASP..120....1K [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2007PASP..119....7K [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2005PASP..117.1236F, @1993PhDT........41M [$\sigma_{{\mathrm}{ccat}}=1\arcsec$] Novae: : Nova list of the 31 Nova Monitoring Project[^13] ($\sigma_{{\mathrm}{ccat}}$ is given for each individual source), PHS2007, @2010AN....331..187P Supernova Remnants: : , and , ; An X-ray source is considered as correlating with a SNR, if the X-ray source position (including 3$\sigma$ error) lies within the extent given for the SNR. Radio Catalogues: : @2005ApJS..159..242G [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2004ApJS..155...89G [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2008AJ....136..684K [$\sigma_{{\mathrm}{ccat}}=3\arcsec$], @1990ApJS...72..761B [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], NVSS [NRAO/VLA Sky Survey[^14]; @1998AJ....115.1693C $\sigma_{{\mathrm}{ccat}}$ is given for each individual source] H [II]{} Regions, H $\alpha$ Catalogue: : , @2007AJ....134.2474M [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$] Optical Catalogues: : USNO-B1 [@2003AJ....125..984M $\sigma_{{\mathrm}{ccat}}$ is given for each individual source], Local Group Survey [LSG; @2006AJ....131.2478M $\sigma_{{\mathrm}{ccat}}=0.\arcsec2$] Infrared catalogues: : 2MASS [@2006AJ....131.1163S $\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2008ApJ...687..230M [$\sigma_{{\mathrm}{ccat}}=0.\arcsec8$, for Table 2: $\sigma_{{\mathrm}{ccat}}=0.\arcsec5$] Data bases: : the SIMBAD catalogue[^15] (Centre de Données astronomiques de Strasbourg; hereafter SIMBAD) , the NASA Extragalactic Database[^16] (hereafter NED) [lrlr]{} & & &\ PFH2005 & $*$ & DKG2004 & 03\ SPH2008 & $*$ & WNG2006 & 03\ SHP97 & $*$ & VG2007 & 04\ SHL2001 & $*$ & OBT2001 & 3\ PFJ93 & $*$ & O2006 & 1\ TF91 & $*$ & SBK2009 & 3$^{+}$\ Ka2002 & 03 & D2002 & 05\ KGP2002 & $*$ & TP2004 & 1\ WGK2004 & 1$^{+}$ & ONB2010 & 1\ \[Tab:XrayRefCat\] Notes:\ $^{ {\dagger}~}$: $*$ indicates that the catalogue provides $\sigma_{{\mathrm}{ccat}}$ values for each source individually\ $^{ +~}$: value taken from indicated paper\ $^{ {\ddagger}~}$: TF91: @1991ApJ...382...82T, PFJ93: @1993ApJ...410..615P, SHP97: , SHL2001: , OBT2001: , D2002: @2002ApJ...570..618D, KGP2002: @2002ApJ...577..738K, Ka2002: @2002ApJ...578..114K, WGK2004: @2004ApJ...609..735W, DKG2004: @2004ApJ...610..247D, TP2004: @2004ApJ...616..821T, PFH2005: , O2006: @2006ApJ...643..844O, WNG2006: @2006ApJ...643..356W, VG2007: , SPH2008: , SBK2009: , ONB2010: @2010ApJ...717..739O Colour image {#Sec:coim} ============ Figure \[Fig:cimage\] shows the combined, exposure corrected EPIC PN, MOS1 and MOS2 RGB (red-green-blue) mosaic image of the Deep Survey and archival data. The colours represent the X-ray energies as follows: red: 0.2–1.0keV, green: 1.0–2.0keV and blue: 2.0–12keV. The optical extent of 31 is indicated by the $\mathrm{D_{25}}$ ellipse and the boundary of the observed field is given by the green contour. The image is smoothed with a 2D-Gaussian of 20 FWHM. In some observations, individual noisy MOS1 and MOS2 CCDs are omitted. The images have not been corrected for the background of the detector or for vignetting.\ The colour of the sources reflects their class. Supersoft sources appear in red. Thermal SNRs and foreground stars are orange to yellow. “Hard" sources (background objects, mainly AGN, and X-ray binaries or Crab-like SNRs) are blue to white. {width="12cm"} Logarithmically scaled  EPIC low background images made up of the combined images from the PN, MOS1 and MOS2 cameras in the (0.2–4.5) keV XID band for each 31 observation can be found in the Appendix. The images also show X-ray contours, and the sources from the  catalogue are marked with boxes. Source catalogue () {#Sec:srccat} =================== The source catalogue of the Deep  survey of 31 (hereafter  catalogue) contains 1897 X-ray sources. Of these sources 914 are detected for the first time in X-rays. The source parameters are summarised in Table 5, which gives the source number (Col. 1), detection field from which the source was entered into the catalogue (2), source position (3 to 9) with $3\sigma$ (99.73%) uncertainty radius (10), likelihood of existence (11), integrated PN, MOS1 and MOS2 count rate and error (12,13) and flux and error (14,15) in the (0.2–4.5) keV XID band, and hardness ratios and errors (16–23). Hardness ratios are calculated only for sources for which at least one of the two band count rates has a significance greater than $2\sigma$. Errors are the properly combined statistical errors in each band and can extend beyond the range of allowed values of hardness ratios as defined previously (–1.0 to 1.0; Eq. \[Eq:hardr\]). The “Val” parameter (Col 24) indicates whether the source is within the field of view (true or false, “T” or “F”) in the PN, MOS1 and MOS2 detectors respectively. Table 5 also gives the exposure time (25), source existence likelihood (26), the count rate and error (27, 28) and the flux and error (29, 30) in the (0.2–4.5)keV XID band, and hardness ratios and errors (31–38) for the EPIC PN. Columns 39 to 52 and 53 to 66 give the same information corresponding to Cols. 25 to 38, but for the EPIC MOS1 and MOS2 instruments. Hardness ratios for the individual instruments were again screened as described above. From the comparison between the hardness ratios derived from the integrated PN, MOS1 and MOS2 count rates (Cols. 16–23) and the hardness ratios from the individual instruments (Cols. 31–38, 45–52 and 59–66), it is clear that the combined count rates from all instruments yielded a significantly larger fraction of hardness ratios above the chosen significance threshold. Column 67 shows cross correlations with published 31 X-ray catalogues ( Sect.\[Sec:CrossCorr\_Tech\]). We discuss the results of the cross correlations in Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. In the remaining columns of Table 5, we give information extracted from the USNO-B1, 2MASS and LGGS catalogues ( Sect.\[Sec:CrossCorr\_Tech\]). The information from the USNO-B1 catalogue (name, number of objects within search area, distance, B2, R2 and I magnitude of the brightest[^17] object) is given in Cols. 68 to 73. The 2MASS source name, number of objects within search area, and the distance can be found in Cols. 74 to 76. Similar information from the LGGS catalogue is given in Cols. 77 to 82 (name, number of objects within search area, distance, V magnitude, V-R and B-V colours of the brightest[^18] object). To improve the reliability of source classifications we used the USNO-B1 B2 and R2 magnitudes to calculate $$\log{\left}(\frac{f_{{\mathrm}{x}}}{f_{{\mathrm}{opt}}}{\right}) = \log{\left}(f_{{\mathrm}{x}}{\right}) + \frac{m_{{\mathrm}{B2}} + m_{{\mathrm}{R2}}}{2\times2.5} + 5.37, \label{Eq:fxopt}$$ and the LGGS V magnitude to calculate $$\log{\left}(\frac{f_{{\mathrm}{x}}}{f_{{\mathrm}{opt}}}{\right}) = \log{\left}(f_{{\mathrm}{x}}{\right}) + \frac{m_{{\mathrm}{V}}}{2.5} + 5.37, \label{Eq:fxvopt}$$ following @1988ApJ...326..680M [ see Cols. 83–86]. The X-ray sources in the  catalogue are identified or classified based on properties in X-rays (HRs, variability, extent) and of the correlated objects in other wavelength regimes (Cols. 87 and 88 in Table 5). For classified sources the class name is given in angled brackets. Identification and classification criteria are summarised in Table \[Tab:class\], which provides, for each source class (Col.1), the classification criteria (2), and the numbers of identified (3) and classified (4) sources. The hardness ratio criteria are based on model spectra. Details on the definition of these criteria can be found in Sect.6 of PFH2005. As we have no clear hardness ratio criteria to discriminate between XRBs, Crab-like supernova remnants (SNRs) or AGN we introduced a $<$hard$>$ class for those sources. If such a source shows strong variability (i.e. V$\ge$10) on the examined time scales it is likely to be an XRB. Compared with SPH2008 the HR2 selection criterion for SNRs was tightened (from HR2$<\!-0.2$ to HR2$+$EHR2$<\!-0.2$) to exclude questionable SNR candidates from the class of SNRs. If we applied the former criterion to the survey data, $\sim$35 sources would be classified as SNRs in addition to those listed in Table \[Tab:class\]. Most of the 35 sources are located outside the D$_{25}$ ellipse, and none of them correlates with an optically identified SNR, a radio source, or an H[II]{} region. In addition, the errors in HR2 are of the same order as the HR2 values. It is therefore very likely that these sources do belong to other classes, since the strip between $-0.3\!<$HR2$<$0 is populated by foreground stars, XRBs, background objects, and candidates for these three classes. Outcomes of the identification and classification processes are discussed in detail in Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. The last column (89) of Table 5 contains the  source name as registered to the IAU Registry. Source names consist of the acronym XMMM31 and the source position as follows: XMMM31 Jhhmmss.s+ddmmss, where the right ascension is given in hours (hh), minutes (mm) and seconds (ss.s) truncated to decimal seconds and the declination is given in degrees (dd), arc minutes (mm) and arc seconds (ss) truncated to arc seconds, for equinox 2000. In the following, we refer to individual sources by their source number (Col.1 of Table 5), which is marked with a “" at the front of the number. Of the 1897 sources, 1247 can only be classified as $<$hard$>$ sources, while 123 sources remain without classification. Two of them ( 482,  768) are highly affected by optical loading; both “X-ray sources" coincide spatially with very bright optical foreground stars (USNO-B1 R2 magnitudes of 6.76 and 6.74 respectively). The spectrum of source  482 is dominated by optical loading. This becomes evident from the hardness ratios which indicate an SSS. For  768 the hardness ratios would allow a foreground star classification. The obtained count rates and fluxes of both sources are affected by the usage of [epreject]{}, which neutralises the corrections applied for optical loading. Therefore residuals are visible in the difference images created from event lists obtained with and without [epreject]{}. As we cannot exclude the possibility that some of the detected photons are true X-rays – especially for source  768 –, we decided to include them in the  catalogue, but without a classification. [llrr]{} & & &\ fg Star & ${\rm log}({{f}_{\rm x} \over {f}_{\rm opt}})\!<\!-1.0$ and HR2$-$EHR$2\!<\!0.3$ and HR3$-$EHR$3\!<\!-0.4$ or not defined & 40 & 223\ AGN & Radio source and not classification as SNR from HR2 or optical/radio & 11 & 49\ Gal & optical id with galaxy & 4 & 19\ GCl & X-ray extent and/or spectrum & 1 & 5\ SSS & HR$1\!<\!0.0$, HR2$-$EHR$2\!<\!-0.96$ or HR2 not defined, HR3, HR4 not defined & & 30\ SNR & HR$1\!>\!-0.1$ and HR2$+$EHR$2\!<\!-0.2$ and not a fg Star, or id with optical/radio SNR & 25 & 31\ GlC & optical id & 36 & 16\ XRB & optical id or X-ray variability & 10 & 26\ hard & HR2$-$EHR$2\!>\!-0.2$ or only HR3 and/or HR4 defined, and no other classification& & 1247\ \[Tab:class\] Flux distribution {#Sec:flux_dist} ----------------- The faintest source ( 526) has an XID band flux of 5.8[$\times 10^{-16}$ ]{}. The source with the highest XID Flux ( 966, XID band flux of 3.75[$\times 10^{-12}$ ]{}) is located in the centre of 31 and identified as a Z-source LMXB . This source has a mean absorbed XID luminosity of 2.74[$\times 10^{38}$ ]{}. Figure \[Fig:XIDfluxdist\] shows the distribution of the XID (0.2–4.5keV) source fluxes. Plotted are the number of sources in a certain flux bin. We see from the inlay that the number of sources starts to decrease in the bin from 2.4 to 2.6[$\times 10^{-15}$ ]{}. This XID flux roughly determines the completeness limit of the survey and corresponds to an absorbed 0.2–4.5keV limiting luminosity of $\sim\!2$[$\times 10^{35}$ ]{}. Previous X-ray studies [@2004ApJ...609..735W and references therein] noted a lack of bright sources ($L_{{\mathrm}{X}}\!\ga$[$10^{37}$ erg s$^{-1}$]{}; 0.1–10keV) in the northern half of the disc compared to the southern half. This finding is not supported in the present study. Excluding the pointings to the centre of 31, we found in the remaining observations 13 sources in each hemisphere that were brighter than $L_{{\mathrm}{X\,abs}}\!\ga$[$10^{37}$ erg s$^{-1}$]{}.[^19] The reason our survey does not support the old results is that we found several bright sources in the outer regions of the northern half of the disk, which have not been covered in @2004ApJ...609..735W [and references therein]. In the central field of 31, a total of 41 sources brighter than $L_{{\mathrm}{X}}\!\ga$[$10^{37}$ erg s$^{-1}$]{} (0.2–4.5keV) were found. Figure \[Fig:brightS\] shows the spatial distribution of the bright sources. Striking features are the two patches located north and south of the centre. The southern one seems to point roughly in the direction of M 32 ( 995), while the northern one ends in the globular cluster B116 ( 947). However there is no association to any known spatial structure of 31, like  the spiral arms. Exposure map {#Sec:ExpMap} ------------ Figure \[Fig:ExpMap\] shows the exposure map used to create the colour image of all  Large Survey and archival observations (Fig.\[Fig:cimage\]). The combined MOS exposure was weighted by a factor of 0.4, before being added to the PN exposure. However, this map does not quite represent the exposures used in source detection; overlapping regions were not combined during source detection. From Fig.\[Fig:ExpMap\] we see that the exposure for most of the surveyed area is rather homogeneous. Exceptions are the central area, overlapping regions and observation h4. Hardness ratio diagrams ----------------------- We plot X-ray colour/colour diagrams based on the HRs (see Fig.\[Fig:HR\_diagrams\]). Sources are plotted as dots if the error in both contributing HRs is below 0.2. Classified and identified sources are plotted as symbols in all cases. Symbols including a dot therefore mark the well-defined HRs of a class. From the HR1-HR2 diagram (upper panel in Fig.\[Fig:HR\_diagrams\]) we note that the class of SSSs is the only one that can be defined based on hardness ratios alone. In the part of the HR1-HR2 diagram that is populated by SNRs, most of the foreground stars and some background objects and XRBs are also found. Foreground star candidates can be selected from the HR2-HR3 diagram (middle panel in Fig.\[Fig:HR\_diagrams\]), where most of them are located in the lower left corner. The HR3-HR4 diagram (lower panel in Fig.\[Fig:HR\_diagrams\]) does not help to disentangle the different source classes. Thus, we need additional information from correlations with sources in other wavelengths or on the source variability or extent to be able to classify the sources. Extended sources {#Sec:ExtSrcs} ---------------- The  catalogue contains 12 sources which are fitted as extended sources with a likelihood of extension larger than 15. This value was chosen so as to minimise the number of spurious detections of extended sources (H. Brunner; private communication), as well as keeping all sources that can clearly be seen as extended sources in the X-ray images. A convolution of a $\beta$-model cluster brightness profile with the  point spread function was used to determine the extent of the sources ( Sect.\[Sec:SrcDet\]). This model describes the brightness profile of galaxy clusters, as $$f{\left}(x,y{\right})={\left}(1+\frac{{\left}(x-x_0{\right})^2+{\left}(y-y_0{\right})^2}{r_{\rm{c}}^2}{\right})^{-3/2},$$ where $r_{\rm{c}}$ denotes the core radius; this is also the extent parameter given by [emldetect]{}. Table \[Tab:ExtSrcs\] gives the source number (Col. 1), likelihood of detection (2), the extent found (3) and its associated error (4) in arcsec, the likelihood of extension (5), and the classification of the source (6, see Sect.\[SubSec:Gal\_GCl\_AGN\]) for each of the 12 extended sources. Additional comments taken from Table 5 are provided in the last column. [rrrrrrrcl]{} & & & & & & & &\ & & & & & & & &\ 141 & 65.08 & 11.22 & 1.29 & 23.68 & 1.45 & 0.20 & $<$GCl$>$ & GLG127(Gal), 37W 025A (IR, RadioS; NED)\ 199 & 275.16 & 17.33 & 1.05 & 174.73 & 4.31 & 0.29 & $<$hard$>$ &\ 252 & 222.05 & 14.64 & 1.12 & 81.60 & 4.40 & 0.49 & $<$GCl$>$ & 5 optical objects in error box\ 304 & 299.75 & 15.10 & 0.92 & 133.62 & 2.20 & 0.18 & $<$GCl$>$ & B242 \[CHM09\]; RBC3.5: $<$GlC$>$\ 442 & 33.76 & 11.60 & 1.71 & 15.44 & 1.62 & 0.28 & $<$hard$>$ &\ 618 & 271.08 & 6.20 & 0.73 & 42.86 & 3.15 & 0.21 & $<$hard$>$ &\ 718 & 77.75 & 7.18 & 1.23 & 21.47 & 0.58 & 0.07 & Gal & B052 \[CHM09\], RBC3.5\ 1130 & 168.31 & 10.80 & 0.97 & 44.23 & 3.27 & 0.31 & $<$hard$>$ &\ 1543 & 70.49 & 11.87 & 1.37 & 28.63 & 1.51 & 0.19 & $<$GCl$>$ & \[MLA93\] 1076 PN (SIM,NED)\ 1795 & 11416.36 & 18.79 & 0.29 & 4169.74 & 98.87 & 1.43 & GCl & GLG253 (Gal), \[B90\] 473, z=0.3 \[KTV2006\]\ 1859 & 107.09 & 13.73 & 1.40 & 43.89 & 1.23 & 0.19 & $<$hard$>$ &\ 1912 & 332.06 & 23.03 & 1.23 & 213.90 & 5.43 & 0.37 & $<$GCl$>$ & cluster of galaxies candidate\ \[Tab:ExtSrcs\] Notes:\ $^{ +~}$: Extent and error of extent in units of 1; 1 corresponds to 3.8pc at the assumed distance of 31\ $^{ *~}$: XID Flux and flux error in units of 1[$\times 10^{-14}$ ]{}\ $^{ \dagger~}$: Taken from Table 5 The extent parameter found for the sources ranges from 62 to 2303 (see Fig.\[Fig:extdist\]). The brightest source ( 1795), which has the highest likelihood of extension and the second largest extent, was identified from its X-ray properties as a galaxy cluster located behind 31 [@2006ApJ...641..756K]. The iron emission lines in the X-ray spectrum yield a cluster redshift of $z\!=\!0.29$. For further discussion see Sect.\[SubSec:Gal\_GCl\_AGN\]. Variability between *XMM-Newton* observations {#Sec:var} ============================================= To examine the long-term time variability of each source, we determined the XID flux at the source position in each observation or at least an upper limit for the XID flux. The XID fluxes were used to derive the variability factor and the significance of variability ( Sect.\[Sec:DefVar\]). The sources are taken from the  catalogue (Table 5). Table 8 contains all information necessary to examine time variability. Sources are only included in the table if they are observed at least twice. Column 1 gives the source number. Columns 2 and 3 contain the flux and the corresponding error in the (0.2–4.5) keV XID band. The hardness ratios and errors are given in columns 4 to 11. Column 12 gives the type of the source. All this information was taken from Table 5. The subsequent 140 columns provide information related to individual observations in which the position of the source was observed. Column 13 gives the name of one of these observations, which we will call observation 1. The EPIC instruments contributing to the source detection in observation 1, are indicated by three characters in the “obs1\_val" parameter (Col. 14, first character for PN, second MOS1, third MOS2), each one being either a “T" if the source is inside the FoV, or “F" if it lies outside the FoV. Then the count rate and error (15,16) and flux and error (17,18) in the (0.2–4.5) keV XID band, and hardness ratios and error (19–26) of observation 1 are given. Corresponding information is given for the remaining observations which cover the position of the source: obs. 2 (cols. 27–40), obs. 3 (41–54), obs. 4 (55–68), obs. 5 (69–82), obs. 6 (83–96), obs. 7 (97–110), obs. 8 (111–124), obs. 9 (125–138), obs. 10 (139–152). Whether the columns corresponding to obs. 3 – obs. 10 are filled in or not, depends on the number of observations in which the source has been covered in the combined EPIC FoV. This number is indicated in column 153. The maximum significance of variation and the maximum flux ratio (fvar\_max) are given in columns 154 and 155. As described in Sect.\[Sec:DefVar\], only detections with a significance greater than 3$\sigma$ were used, otherwise the 3$\sigma$ upper limit was adopted. Column 156 indicates the number of observations that provide only an upper limit. The maximum flux (fmax) and its error are given in columns 157 and 158. In a few cases a maximum flux value could not be derived, because each observation only yielded an upper limit. There can be two reasons for this: The first reason is that faint sources detected in merged observations may not be detected in the individual observations at the 3$\sigma$ limit. The second reason is that in cases where the significance of detection was not much above the 3$\sigma$ limit, it can become smaller than the 3$\sigma$ limit when the source position is fixed to the adopted final mean value from all observations. [llrrrcl]{} & & & & & &\ 966 & 1.63 & 49.01 & 46.73 & 0.59 & XRB & 1(sv,z), 2, 10(v), 12(v), 13, 14, 20, 22(v), 25(LMXB), 27, 28(1.56)\ 877 & 3.13 & 49.13 & 16.06 & 0.20 & $<$hard$>$ & 1(sv), 2, 10(v), 12(v), 13, 14, 20(v), 22(v), 27, 28(3.05)\ 745 & 2.43 & 26.89 & 12.65 & 0.18 & AGN & 13, 14\ 1157 & 1.32 & 11.10 & 9.87 & 0.25 & GlC & 1(sv), 2, 5, 10, 12, 13, 14, 20, 21, 22(v), 27, 28(1.37)\ 1060 & 2.13 & 30.00 & 9.04 & 0.14 & $<$XRB$>$ & 1(sv), 2, 10, 12, 13, 14, 20(v, NS-LMXB), 22(v), 27\ 1171 & 4.14 & 18.86 & 9.02 & 0.41 & GlC & 1(d,sv), 2(t, 53.4), 5, 10, 12, 13, 14, 16, 20, 22, 27, 28(2.47)\ 1116 & 3.76 & 51.98 & 8.16 & 0.10 & GlC & 1(sv), 2(t, 58.6), 3(t, 33), 5, 10, 12, 13, 14, 16, 20, 21, 22(v,t), 27\ \[Tab:varlist\_bright\] Notes:\ $^{ {\ddagger}~}$: maximum XID flux and error in units of 1[$\times 10^{-13}$ ]{} or maximum absorbed 0.2–4.5keV luminosity and error in units of 7.3[$\times 10^{36}$ ]{}\ $^{ {+}~}$: class according to Table \[Tab:class\]\ $^{{\dagger}~}$: for comment column see Table \[Tab:varlist\] Figure \[Fig:var\_fmax\] shows the variability factor plotted versus maximum detected XID flux. Apart from XRBs, or XRBs in GlCs, or candidates of these source classes, which were selected based on their variability, there are a few SSS candidates showing pronounced temporal variability. The sources classified or identified as AGN, background galaxies or galaxy clusters all show $F_{{\mathrm}{var}}\!<\!4$. Most of the foreground stars show $F_{{\mathrm}{var}}\!<\!4$. Out of the 1407 examined sources, we found 317 sources with a variability significance $>\!3.0$,  182 more than reported in SPH2008. For bright sources it is much easier to detect variability than for faint sources, because the difference between the maximum observed flux and the detection limit is larger. Therefore the significance of the variability declines with decreasing flux. This is illustrated by the distribution of the sources marked in green in Fig.\[Fig:var\_fmax\]. Table \[Tab:varlist\] lists all sources with a variability factor larger than five. There are 69 such sources (34 in addition to SPH2008). The sources are sorted in descending order with respect to their variability factors. Table \[Tab:varlist\] gives the source number (Col. 1), maxima of flux variability (2) and maxima of the significance parameter (3). The next columns (4, 5) indicate the maximum observed flux and its error. Column 6 contains the class of the source. Sources with $F_{{\mathrm}{var}}\!\ge\!10$ that were not already classified as SSS or foreground stars, were classified as XRB. Time variability can also be helpful to verify a SNR candidate classification. If there is significant variability, the SNR classification must be rejected, and if an optical counterpart is detected, the source has to be re-classified as foreground star candidate. Column 7 contains references to the individual sources in the literature. In some cases the reference provides information on the temporal behaviour and a more precise classification (see brackets). The numbers given in connection with and @2006ApJ...643..356W are the variability factors obtained in these papers from  data. From the 69 sources of Table \[Tab:varlist\], ten show a flux variability larger than 100. With a flux variability factor $>\!690$ source  523 is the most variable source in our sample. Source  57 has the largest significance of variability, with a value of $\approx 97$. The variability significance is below 10 for just 33 sources, 15 of which show significance values below 5. Thirty-five of the variable sources are classified as XRBs or XRB candidates, and eight of them are located in globular clusters. Nine of the variable sources are SSS candidates, while six variable sources are classified as foreground stars and foreground star candidates. Table \[Tab:varlist\_bright\] lists all “bright" sources with a maximum flux larger than 8[$\times 10^{-13}$ ]{} and a flux variability smaller than five (the description of the columns is the same as in Table \[Tab:varlist\]). All seven sources listed in Table \[Tab:varlist\_bright\] (three in addition to SPH2008) have a significance of variability $>\!10$. Apart from two sources, they are XRBs (three in globular clusters) or XRB candidates. The most luminous source in our sample is source  966 with an absorbed 0.2–4.5keV luminosity of $\approx 3.3$[$\times 10^{38}$ ]{} at maximum. Figure \[Fig:var\_hr\] shows the relationship between the variability factor and the hardness ratios HR1 and HR2, respectively. The hardness ratios are taken from Table 5. The HR1 plot shows that the sample of highly variable sources includes SSS and XRB candidates, which occupy two distinct regions in this plot . The SSSs marked by triangles, appear on the left hand side, while the XRBs or XRB candidates have much harder spectra, and appear on the right. It seems that foreground stars, SSSs and XRBs can be separated, on the HR2 diagram, although there is some overlap between foreground stars and XRBs. Individual sources are discussed in the Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. Cross-correlations with other 31 X-ray catalogues {#Sec:CrossX-ray} ================================================= Cross-correlations were determined by applying Eq.\[Eq:Cor\] to the sources of the  catalogue and to sources reported in earlier X-ray catalogues. The list of X-ray catalogues used is given in Table \[Tab:XrayRefCat\]. Previous *XMM-Newton* catalogues {#SubSec:prevXMM} -------------------------------- Previous source lists based on archival  observations were presented in , PFH2005, @2006ApJ...643..844O, SPH2008, and SBK2009. Of these four studies, PFH2005 covers the largest area of 31. Table \[Tab:CompXMM\] lists all sources from previous  studies that are not detected in the present investigation. [ll]{}\ \ 6 not detected, LH$>$100: & 327 ($<$SNR$>$,LH$=$2140.0), 384 (XRB,667.0), 332 ($<$SNR$>$,654.0),\ & 316 ($<$SNR$>$,259.0), 312 ($<$SNR$>$,241.0), 281($<$hard$>$,160.0)\ 10 not detected, 20$\le$LH$<$50: & 75 ($<$SSS$>$), 423 ($<$fg Star$>$), 120 ($<$hard$>$), 505 ($<$hard$>$),\ & 220 ($<$SNR$>$), 304 ($<$fg Star$>$), 819 ($<$hard$>$), 799 ($<$SSS$>$), 413 ($<$SNR$>$), 830 ($<$hard$>$)\ 14 not detected, 15$\le$LH$<$20: & 427($<$hard$>$), 734 ($<$hard$>$), 424 ($<$hard$>$), 518 ($<$SSS$>$),\ & 232 ($<$hard$>$), 339 ($<$hard$>$), 446 ($<$SSS$>$), 219 ($<$fg Star$>$), 567 ($<$hard$>$), 256 ($<$fg Star$>$),\ & 356 ($<$hard$>$), 248 ($<$hard$>$), 160 ($<$hard$>$), 399 ()\ 21 not detected, 10$\le$LH$<$15: & 375 ($<$hard$>$), 17 ($<$hard$>$), 195 ($<$hard$>$), 417 ($<$SNR$>$),\ & 783 ($<$hard$>$), 803 ($<$hard$>$), 829 ($<$hard$>$), 135 ($<$hard$>$), 151 ($<$hard$>$), 131 ($<$hard$>$),\ & 426 ($<$hard$>$), 593 ($<$fg Star$>$), 526 ($<$hard$>$), 250 ($<$hard$>$), 62 ($<$hard$>$), 67 ($<$hard$>$),\ & 188 ($<$hard$>$), 186 ($<$AGN$>$), 510 ($<$hard$>$), 529 ($<$hard$>$), 754 ($<$hard$>$)\ 52 not detected, LH$<$10: & 599 ($<$hard$>$), 439 ($<$hard$>$), 809 ($<$hard$>$), 14 ($<$SNR$>$), 743 ($<$hard$>$),\ & 433 ($<$hard$>$), 5 (), 210 ($<$hard$>$), 97 ($<$hard$>$), 708 ($<$hard$>$), 476 (), 534 ($<$hard$>$), 501 (),\ & 170 ($<$hard$>$), 146 (SNR), 769 (), 838 ($<$hard$>$), 571 ($<$hard$>$), 816 ($<$hard$>$, 554 (), 627 ($<$hard$>$),\ & 464 ($<$fg Star$>$), 811 ($<$hard$>$), 655 ($<$hard$>$), 184 ($<$hard$>$), 447 ($<$hard$>$), 380 ($<$hard$>$),\ & 566 ($<$hard$>$), 137 ($<$fg Star$>$), 63 (), 48 (), 152 ($<$fg Star$>$), 291 ($<$hard$>$), 559 ($<$hard$>$),\ & 102 ($<$hard$>$), 740 ($<$hard$>$), 540 ($<$fg Star$>$), 240 ($<$hard$>$), 485 (), 668 ($<$hard$>$), 44 (),\ & 560 ($<$hard$>$), 836 ($<$hard$>$), 436 ($<$hard$>$), 484 ($<$fg Star$>$), 216 ($<$hard$>$), 362 ($<$hard$>$), 527 ($<$$>$), 179 ($<$hard$>$),\ & 834 ($<$hard$>$), 86 ($<$hard$>$), 455 ()\ \ \ 3 not detected, 50$\le$LH$<$100: & 874 ($<$SNR$>$,LH$=$85.5), 895 ($<$hard$>$,75.9), 882 ( ,56.4)\ 6 not detected, 10$\le$LH$<$50: & 869 (), 885 ($<$SNR$>$), 863 ($<$hard$>$), 875 ($<$SSS$>$), 893 ($<$hard$>$), 866 ($<$hard$>$)\ 6 not detected, LH$<$10: & 870 ($<$SNR$>$), 891 ($<$hard$>$), 889 ($<$hard$>$), 872 ($<$SNR$>$), 867 ($<$hard$>$), 862 ($<$SNR$>$)\ \ \ & 4 ($<$hard$>$), 18 ($<$hard$>$), 29 ($<$hard$>$), 32 ($<$hard$>$), 34 ($<$hard$>$), 45 ($<$SSS$>$), 67 ($<$hard$>$),\ & 102 ($<$hard$>$), 106 ($<$hard$>$), 117 ($<$hard$>$), 149 ($<$hard$>$), 152 ($<$hard$>$), 179 ($<$hard$>$),\ & 183 ($<$hard$>$), 184 ($<$hard$>$), 188 ($<$hard$>$), 191 ($<$hard$>$), 192 ($<$AGN$>$), 202 ($<$hard$>$),\ & 204 ($<$fg star$>$), 217 ($<$hard$>$), 249 ($<$hard$>$), 250 ($<$hard$>$), 260 ($<$hard$>$), 274 ($<$hard$>$),\ & 279 ($<$hard$>$), 285 ($<$hard$>$), 295 ($<$hard$>$), 306 ($<$hard$>$), 325 ($<$hard$>$), 333 ($<$hard$>$)\ \[Tab:CompXMM\] In the ten observations covering the major axis, and a field in the halo of 31, PFH2005 detected 856 X-ray sources with a detection likelihood threshold of 7 ( Sect.\[Sec:SrcDet\]). Of these 856 sources, only 753 sources are also present in the  catalogue,  103 sources of PFH2005 were not detected. This can be due to: the search strategy; the parameter settings used in the [emldetect]{} run; the determination of the extent of a source for the  catalogue; the more severe screening for GTIs for the  catalogue, which led to shorter final exposure times; the use of the [epreject]{} task and last but not least due to the [SAS]{} versions and calibration files applied. The search strategy of PFH2005 was optimised to detect sources located close to each other in crowded fields. This point especially explains the non-detection of the bright PFH2005 sources \[PFH2005\] 281, 312, 316, 327, 332, 384 ($\mathcal{L}>50$) in the present study, as four of them (\[PFH2005\] 312, 316, 327, 332) are located in the innermost central region of 31 where source detection is complicated by the bright diffuse X-ray emission, while \[PFH2005\] 281 and 384 lie in the immediate vicinity of two bright sources (\[PFH2005\] 280 and 381 at distances of 7.7and 5.5, respectively). The changes in the [SAS]{} versions and the GTIs, in particular, affect sources with small detection likelihoods ($\mathcal{L}<10$). The improvements in the [SAS]{} detection tools and calibration files should reduce the number of spurious detections, which increase with decreasing detection likelihood. However, this does not necessarily imply that *all* undetected sources with $\mathcal{L}<10$ of PFH2005 are spurious detections. The changes in the [SAS]{} versions, calibration files and GTIs do not only affect the source detection tasks, but also can cause changes in the background images. These changes may increase the assumed background value at the position of a source, which would result in a lower detection likelihood. Going from [mlmin=7]{} to [mlmin=6]{}, but leaving everything else unchanged, we detected an additional nine sources of PFH2005. One of the previously undetected sources (\[PFH2005\] 75) was classified as $<$SSS$>$, but correlates with blocks of pixels with enhanced low energy signal in the PN offset map and was corrected by [epreject]{}. Another source classified as $<$SSS$>$ (\[PFH2005\] 799) is only detected in the MOS1 camera, but not in MOS2. From an examination by eye, it seems that source \[PFH2005\] 799 is the detection of some noisy pixels at the rim of the MOS1 CCD6 and not a real X-ray source. SPH2008 extended the source catalogue of PFH2005 by re-analysing the data of the central region of 31 and also including data of monitoring observations of LMXB RX J0042.6+4115. Of the 39 new sources presented in SPH2008, 24 are also listed in the  catalogue,  15 sources of SPH2008 were not detected. Differences between the two studies include the detection likelihood thresholds used for [eboxdetect]{} (SPH2008: [likemin]{}=5) and [emldetect]{} (SPH2008: [mlmin]{}=6), the lower limit for the likelihood of extention (SPH2008: [dmlextmin]{}=4; : 15), the screening for GTIs, the use of the [epreject]{} task and the [SAS]{} versions and calibration files used. Concerning the GTIs, images, background images and exposure maps SPH2008 followed the same procedures as in PFH2005. The arguments given above are therefore also valid here. From the 14 undetected sources, three sources were detected in SPH2008 with [mlmin]{} $<$ 7. One source (\[SPH2008\] 882) was added by hand to the final source list, as SPH2008 could not find any reason why [emldetect]{} did not automatically find it. The two extended sources (\[SPH2008\] 863, 869) detected with extent likelihoods between 4.7 and 5.1 in SPH2008, are neither detected as extended nor as pointlike sources in the present study, where the extent likelihood has to be larger than 15. SBK2009 re-analysed the  observations located along the major axis of 31, ignoring all observations pointing to the centre of the galaxy. They used a detection likelihood threshold of ten. Of the 335 sources detected by SBK2009, 304 sources are also contained in the  catalogue,  31 sources are not detected. Of the 304 re-detected sources, two (\[SBK2009\] 298, 233) are found with a detection likelihood below ten. Of the 31 undetected sources, 27 were also not detected in PFH2005. The remaining four sources correlate with PFH2005 sources, which were not detected in the present study. SBK2009 state that they find 34 sources not present in the source catalogue of PFH2005. A possible reason for this may be that SBK2009 used different energy bands for source detection. They also had five bands, but they combined bands 2 and 3 from PFH2005 into one band in the range 0.5–2keV, and on the other hand they split band 5 of PFH2005 into two bands from 4.5–7keV and from 7–12keV, respectively. This might also explain why most of the additional found sources were classified as $<$hard$>$. @2006ApJ...643..844O addressed the population of SSSs and QSSs based on the same archival observations as PFH2005. @2006ApJ...643..844O detected 15 SSSs, 18 QSSs and 10 SNRs of which one (\[O2006\] Table4, Src.3) is also listed as an SSS (\[O2006\] Table2, Src.13). Of these sources two SSSs, four QSSs and two SNRs (among them is the source \[O2006\] Table4, Src.3) are not contained in the  catalogue. These seven sources are also not present in the PFH2005 catalogue. The nine bright variable sources from were all detected. *Chandra* catalogues {#SubSec:Chcat} -------------------- The  catalogues used for cross-correlations were presented in Sect.\[Sec:Intro\] (see also Table \[Tab:XrayRefCat\]). Details of the comparison between the  catalogue and the different  catalogues can be found in Table \[Tab:CompChan\]. Here, we only give a few general remarks. A non-negligible number of  sources not reported in the  catalogue have already been classified as transient or variable sources. Thus, it is not surprising that these sources were not detected in the  observations . One  source (n1-66) lies outside the field of 31 covered by the  observations. For the innermost central region of 31, the point spread function of  causes source confusion and therefore only  observations are able to resolve the individual sources, especially if they are faint compared to the diffuse emission or nearby bright sources . This explains why a certain number of these sources are not detected in  observations. [ll]{}\ \ 5 transient: & r3-46,r3-43,r2-28,r1-23,r1-19\ 20 variable: & r3-53,r3-77,r3-106,r3-76,r2-52,r2-31,r2-23,r1-31,r2-20,r1-24,r1-28,r1-27,r1-33,r1-21,r1-20,r1-7,r2-15,r1-17,r1-16,r2-47\ 33 unclassified: & r3-102,r3-92,r3-51,r3-75,r3-91,r3-89,r3-101,r3-88,r2-44,r2-55,r2-54,r3-32,r2-53,r1-30,r3-99,r1-22,r1-26,r1-18,r3-26,\ &r2-41,r2-40,r3-71,r2-50,r2-49,r2-38,r3-97,r2-46,r3-12,r3-66,r3-104,r3-82,r3-5,r3-4\ \ \ 3 transient: & J004217.0+411508,J004243.8+411604,J004245.9+411619\ 7 variable: & J004232.7+411311,J004242.0+411532,J004243.1+411640,J004244.3+411605,J004245.2+411611,J004245.5+411608,\ &J004248.6+411624\ 16 unclassified: & J004207.3+410443,J004229.1+412857,J004239.5+411614,J004239.6+411700,J004242.5+411659,J004242.7+411503,\ &J004243.1+411604,J004244.2+411614,J004245.0+411523,J004246.1+411543,J004247.4+411507,J004249.1+411742,\ & J004251.2+411639,J004252.3+411734,J004252.5+411328,J004318.5+410950\ \ \ 12 transient: & s1-79,s1-80,s1-82,r3-46,r2-28,r1-23,r1-19,r2-69,r1-28,r1-35,r1-34,n1-85\ 7 variable: & r2-31,r1-31,r1-24,r1-20,r1-7,r1-17,r1-16\ 9 unclassified: & s1-81,r2-68,s1-85,r1-30,r1-22,r1-26,r1-18,n1-77,n1-84\ \ \ 11 transient: & 6,12,29,32,41,51,59,84,118,130,146\ 15 variable: & 3,5,8,9,18,22,24,27,44,63,92,96,99,149,169\ 78 unclassified: & 4,19,21,25,26,30,37,39,40,42,48,49,53,56,57,58,60,62,65,70,73,75,76,77,80,82,84,86,87,89,91,94,97,98,104,109,114,\ &115,117,119,122,124,129,133,138,141,143,144,145,150,152,158,162,164,167,171,173,182,183,188,189,191,193,194,\ &197,202,205,206,210,213,217,219,220,225,256,257,263\ \ \ 25 transient: & n1-26,n1-85,n1-86,n1-88,n1-89,r1-19,r1-23,r1-28,r1-34,r1-35,r2-28,r2-61,r2-62,r2-66,r2-69,r2-72,r3-43,r3-46,s1-18,\ & s1-27,s1-69,s1-79,s1-80,s1-82,s2-62\ \ \ 9 transient: & s2-62,s1-27,s1-69,s1-18,n1-26,r2-62,r1-35,r2-61,r2-66\ 5 unclassified: & s2-27,s2-10,n1-29,n1-46,r2-54\ 1 not in FoV: & n1-66\ \ \ 2 unclassified: & 17 ($\hat{=}$ r2-15), 28 ($\hat{=}$ r3-71)\ \[Tab:CompChan\] Notes:\ Variability information (transient, variable) is taken from the papers. “Unclassified" denotes sources which are not indicated as transient or variable sources in the papers. Of the 28 bright X-ray sources located in globular clusters [@2002ApJ...570..618D], two were not found in the  data (see Table \[Tab:CompChan\]). They are also not included in the source catalogue of PFH2005 and SPH2008. Hence, both objects are good candidates for being transient or at least highly variable sources ( Sect.\[SubSub:comp\_GlC\]). Another study of the globular cluster population of 31 is presented by @2004ApJ...616..821T. Their work is based on  and  data and contains 43 X-ray sources. Of these sources three were not found in the present study. One of them (\[TP2004\] 1) is located well outside the field of 31 covered by the Deep  Survey[^20]. The second source (\[TP2004\] 21) correlates with r3-71, which is discussed above (see @2002ApJ...570..618D in Table \[Tab:CompChan\]). The transient nature of the third source (\[TP2004\] 35), and the fact that it was not observed in any  observation taken before 2004 was already reported by @2004ApJ...616..821T. The source was first detected with  in the observation from 31 December 2006. *ROSAT* catalogues ------------------ Of the 86 sources detected with  HRI in the central $\sim$34 of 31 (PFJ93), all but eight sources (\[PFJ93\] 1,2,31,33,40,48,63,85) are detected in the  observations. Six of these eight sources (\[PFJ93\] 1,2,31,33,63,85) have already been discussed in PFH2005 and classified as transients. Sources \[PFJ93\] 40 and 48 correlate with \[PFH2005\] 312 and 332, respectively, which are discussed in Sect.\[SubSec:prevXMM\]. In addition to these eight sources, PFH2005 did not detect source \[PFJ93\] 51. This source was detected in the  observations centred on RX J0042.6+4115 and was thus classified as a recurrent transient (see SPH2008). In each of the two  PSPC surveys of 31, 396 individual X-ray sources were detected (SHP97 and SHL2001). From the SHP97 catalogue 130 sources were not detected. Of these sources 48 are located outside the FoV of our  31 survey. From the SHL2001 catalogue, 93 sources are not detected, 60 of which lie outside the  FoV. For information on individual sources see Table \[Tab:CompRos\]. [ll]{}\ \ 48 outside FoV: & 1,2,3,4,5,7,8,14,31,41,72,91,98,104,120,125,159,202,209,271,276,285,286,290,300,312,314,\ & 320,342,350,363,367,371,374,383,385,386,387,388,389,390,391,392,393,394,395,396\ 1 transient: & 69\ 21 not detected, LH$<$12: & 19,24,27,33,46,52,59,63,68,71,133,149,161,264,273,275,307,329,330,358,377\ 16 not detected, 12$\le$LH$<$15: & 12,15,49,82,93,113,114,128,196,230,262,283,334,364,372,376\ 44 not detected, LH$\ge$15: & 16(LH$=$26.6),32(30.2),43(18.2),45(51.2),60(20.1),66(36.2),67(4536.2),78(20.5),80(16.3),81(26.6),\ & 88(33.7),95(548.0),102(16.4),126(217.3),141(843.3),145(46.9),146(673.7),166(17.4),167(90.0),\ & 171(54.3),182(454.4),186(39.8),190(113.0),191(54.5),192(54.3),203(103.3),214(400.2),215(251.0),\ & 232(104.4),245(26.0),260(54.6),263(38.1),265(24.6),268(54.3),270(40.4),277(15.6),309(81.8),\ & 319(23.4),331(19.5),335(51.2),340(27.5),341(28.1),365(22.4),373(69.5)\ \ \ 60 outside FoV: & 1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,21,22,32,39,58,67,69,75,77,81,83,85,90,93,125,141,146,\ & 160,164,192,202,243,260,282,296,298,302,325,326,328,355,371,372,378,379,383,388,389,390,\ & 391,392,393,394,395,396\ 4 not detected, LH$<$12: & 62,96,238,269\ 2 not detected, 12$\le$LH$<$15: & 231,361\ 27 not detected, LH$\ge$15: & 51(LH$=$28.4),104(901.2),121(94.1),126(46.2),143(34.7),168(131.9),171(43.0),173(317.8),190(215.8),\ & 207(98.0),208(298.8),226(73.1),230(75.6),232(1165.6),240(218.4),246(39.9),248(219.6),256(60.0),\ & 267(22.2),271(52.8), 322(2703.3),324(147.7),344(40.7),356(15.3),365(19.0),380(17.4),384(15.8)\ \[Tab:CompRos\] Forty-four (out of 302) sources from SHP97 and 27 (out of 293) sources from SHL2001, have  detection likelihoods larger than 15, but are not listed in the  catalogue. These sources have to be regarded as transient or at least highly variable. *Einstein* catalogue -------------------- The list of  X-ray sources in the field of 31 reported by TF91 contains 108 sources, with 81 sources taken from the  HRI data with an assumed positional error of 3 [reported by @1984ApJ...284..663C], and 27 sources based on  IPC data with a 45 positional error. Applying the above mentioned correlation procedure to the  HRI sources, 64 of these sources are also detected in this work and listed in the  catalogue,  17 sources are not detected (\[TF91\] 29, 31, 35, 39, 40, 43, 46, 50, 53, 54, 65, 66, 72, 75, 78, 93, 96). For the  IPC sources only the 1 $\sigma$ positional error was used to search for counterparts among the  sources. Of the 27  IPC sources six remain without a counterpart in our catalogue (\[TF91\] 15, 99, 100, 106, 107, 108), of which \[TF91\] 15 and 108 are located outside the field of 31 covered by the  catalogue. Sources \[TF91\] 50 and 54 correlate with \[PFH2005\] 312 and 316, respectively. Both sources were already discussed in Sect.\[SubSec:prevXMM\]. Apart from \[TF91\] 106, which is suggested as a possible faint transient by SHL2001, the remaining 18 sources are also not detected by PFH2005. They classified those sources as transient. Cross-correlations with catalogues at other wavelengths {#SEC:CCow} ======================================================= The  catalogue was correlated with the catalogues and public data bases given in Sect.\[Sec:CrossCorr\_Tech\]. Two sources (from the  and from the reference catalogues) were be considered as correlating, if their positions matched within the uncertainty (see Eq. \[Eq:Cor\]). However, the correlation of an X-ray source with a source from the reference catalogue does not necessarily imply that the two sources are counterparts. To confirm this, additional information is needed, like corresponding temporal variability of both sources or corresponding spectral properties. We should also take into account the possibility that the counterpart of the examined X-ray source is not even listed in the reference catalogue used (due to faintness for example). The whole correlation process will get even more challenging if an X-ray source correlates with more than one source from the reference catalogue. In this case we need a method to decide which of the correlating sources is the most likely to correspond to the X-ray source in question. Therefore, the method used should indicate how likely the correlation is with each one of the sources from the reference catalogue. Based on these likelihoods one can define criteria to accept a source from the reference catalogue as being the most likely source to correspond to the X-ray source. The simplest method uses the spatial distance between the X-ray source and the reference sources to derive the likelihoods. In other words, the source from the reference catalogue that is located closest to the X-ray source is regarded as the most likely source corresponding to the X-ray source. An improved method is a “likelihood ratio" technique, were an additional source property ( an optical magnitude in deep field studies) is used to strengthen the correlation selection process. This technique was applied successfully to deep fields to find optical counterparts of X-ray sources [  @2007ApJS..172..353B]. A drawback of this method is that one a priori has to know the expected probability distribution of the optical magnitudes of the sources belonging to the studied object. In our case, this means that we have to know the distribution function for all optical sources of 31 that can have X-ray counterparts, *without* including foreground and background sources. Apart from the fact that such distribution functions are unknown, an additional challenge would be the time dependence of the magnitude of the optical sources ( of novae) and of the connection between optical and X-ray sources ( optical novae and SSSs). Therefore it is not possible to apply this “likelihood ratio" technique to the sources in the  survey. The whole correlation selection process becomes even more challenging if more than one reference catalogue is used. To be able to take all available information into account, we decided not to automate the selection process, but to select the class and most likely correlations for each source by hand (as it was done  in PFH2005). Therefore the source classification, and thus the correlation selection process, is based on the cross correlations between the different reference catalogues, on the X-ray properties (hardness ratios, extent and time variability), and on the criteria given in Table \[Tab:class\]. For reasons of completeness we give for each X-ray source the number of correlations found in the USNO-B1, 2MASS and LGGS catalogues in Table 5. The caveat of this method is that it cannot quantify the probability of the individual correlations. Foreground stars and background objects {#Sec:fgback} ======================================= Foreground stars {#Sec:fgStar} ---------------- X-ray emission has been detected from many late-type – spectral types F, G, K, and M – stars, as well as from hot OB stars [see review by @2000RvMA...13..115S]. Hence, X-ray observations of nearby galaxies also reveal a significant fraction of Galactic stars. With typical absorption-corrected luminosities of $L_{\mathrm{0.2-10\,keV}}\!<$[$10^{31}$ erg s$^{-1}$]{}, single stars in other galaxies are too faint to be detected with present instruments. However, concentrations of stars can be detected, but not resolved. Foreground stars (fg Stars) are a class of X-ray sources which are homogeneously distributed over the field of 31 (Fig.\[Fig:fgS\_spdist\]). The good positional accuracy of  and the available catalogues USNO-B1, 2MASS and LGGS allow us to efficiently select this type of source. The selection criteria are given in Table \[Tab:class\]. The optical follow-up observations of and have confirmed the foreground star nature of bright foreground star candidates selected in PFH2005, based on the same selection criteria as used in this paper. Somewhat different criteria were applied for very red foreground stars, with an LGGS colour ${\mathrm}{V}-{\mathrm}{R}\!>\!1$ or USNO-B1 colour ${\mathrm}{B2}-{\mathrm}{R2}\!>\!1$. These are classified as foreground star candidates, if $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}\!<\!-0.65$ and $f_{{\mathrm}{x}}/f_{{\mathrm}{opt,R}}\!<\!-1.0$. A misclassification of symbiotic systems in 31 as foreground objects by this criterion can be excluded, as symbiotic systems typically have X-ray luminosities below [$10^{33}$ erg s$^{-1}$]{}, which is more than a factor 100 below the detection limit of our survey. If the foreground star candidate lies within the field covered by the LGGS we checked its presence in the LGGS images (as the LGGS catalogue itself does not list bright stars, because of saturation problems). Otherwise DSS2 images were used. Correlations with bright optical sources from the USNO-B1 catalogue, with an $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ in the range expected for foreground stars, that were not visible in the optical images were rejected as spurious. We found 223 foreground star candidates. Fourty sources were identified as foreground stars, either because they are listed in the globular cluster catalogues as spectroscopically confirmed foreground stars or because they have a spectral type assigned to them in the literature . Two of the foreground star candidates close to the centre of 31 ( 826,  1110) have no entry in the USNO-B1 and LGGS catalogues, and one has no entry in the USNO-B1 R2 and B2 columns ( 976). However, they are clearly visible on LGGS images, they are 2MASS sources and they fulfil the X-ray hardness ratio selection criteria. Therefore, we also classify them as foreground stars. The following 19 sources were selected as very red foreground star candidates:  54,  118,  384,  391,  393,  585,  646,  651,  711,  1038,  1119,  1330,  1396,  1429,  1506,  1605,  1695,  1713 and  1747. A further 10 sources ( 210,  269,  278,  310,  484,  714,  978,  1591,  1908 and  1930) fulfil the hardness ratio criteria, but violate the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ criteria and are therefore marked as “foreground star candidates" in the comment column of Table 5. \ \ \ \ Six sources ( 473,  780,  1551,  1585,  1676,  1742), classified as foreground star candidates, have X-ray light curves that in a binning of 1000s showed flares (see Fig.\[Fig:fgS\_flare\]). These observations strengthen the foreground star classification. A seventh source ( 714) is classified as a foreground star candidate, since its hardness ratios and its $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ ratio in the quiescent state fulfil the selection criteria of foreground star candidates. In addition, the source shows a flare throughout observation ss3. Hence, the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ ratio for this observation, in which the source is brightest, is too high to be consistent with the range of values expected for foreground stars. [rrrrrr]{} & & & & &\ 473 & 12.984 & 12.681 & 12.558 & K0 & 0.4\ 714 & 14.310 & 13.618 & 13.458 & M0 & 0.2\ 780 & 14.251 & 13.595 & 13.351 & M3 & 0.1\ 1551 & 12.666 & 12.009 & 11.806 & M2 & 0.1\ 1585 & 13.488 & 12.899 & 12.650 & M2 & 0.1\ 1676 & 10.460 & 9.878 & 9.798 & K1 & 0.2\ 1742 & 13.722 & 13.138 & 12.896 & M1 & 0.2\ \[Tab:fgStar\_flare\] Notes:\ $^{ *~}$: spectral type\ $^{ +~}$: error (in subtypes) Table \[Tab:fgStar\_flare\] gives the J, H and K magnitudes taken from the 2MASS catalogue for each of the six flaring foreground stars. Using the standard calibration of spectral types for dwarf stars based on their near infrared colours (from the fourth edition of Allen’s astrophysical quantities, edt. A.Cox, p.151) we derived the spectral classification for the objects, using both H-K and J-K. The spectral types (and “error”) we give in Table \[Tab:fgStar\_flare\] are derived from averaging the two classes (derived from the two colours). The spectral types are entirely consistent with those expected for flare stars (usually K and M types). Figure \[Fig:fgS\_fldist\] shows the XID flux distribution for foreground stars and foreground star candidates, which ranges from 6.9[$\times 10^{-16}$ ]{} to 2.0[$\times 10^{-13}$ ]{}. Most of the foreground stars and candidates (257 sources) have fluxes below 5[$\times 10^{-14}$ ]{}. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues In the combined  PSPC survey (SHP97, SHL2001) 55 sources were classified as foreground stars. Of these, 14 sources remain without counterparts in the present  survey. Five of these 14 sources are located outside the field observed with . Forty-one  foreground star candidates have counterparts in the  catalogue. Of these counterparts, 16 were classified as foreground star candidates and four were identified as foreground stars . In addition 12 sources were listed as $<$hard$>$, two as AGN candidates and one as a globular cluster candidate in the  catalogue. The counterparts of three  sources remain without classification in the  catalogue. Another three  sources have more than one counterpart in the  data. Source \[SHP97\] 109 correlates with sources  597,  604,  606, and  645. The former three are classified as $<$hard$>$, while source  645 is classified as a foreground star candidate. However source  645 has the largest distance from the position of \[SHP97\] 109 compared to the other three  counterparts. Furthermore, this source had a flux below the  detection threshold (about a factor 2.6) in the  observations and is about a factor 3–34 fainter than the three other possible  counterparts. Thus it is very unlikely that \[SHP97\] 109 represents the X-ray emission of a foreground star. Source \[SHL2001\] 156 has two  counterparts and is discussed in Sect.\[Sec:SSS\_comp\]. The third source (\[SHL2001\] 374) correlates with sources  1922 and  1924. The two  sources are classified as $<$hard$>$ and as a foreground star candidate, respectively. In the source catalogue of SHL2001 source \[SHP97\] 369 is listed as the counterpart of \[SHL2001\] 374. The source in the first  survey has a smaller positional error and only correlates with source  1924. Although this seems to indicate that source  1924 is the counterpart of \[SHL2001\] 374, we cannot exclude the possibility that \[SHL2001\] 374 is a blend of both  sources, as these two sources have similar luminosities in the  observations. @2002ApJ...577..738K classified four sources as foreground stars. For two sources ( 960$\hat{=}$r2-42 and  976$\hat{=}$r3-33) the classification is confirmed by our study. The third source ( 1000$\hat{=}$r2-19) remained without classification in the  catalogue, as it is too soft to be classified as $<$hard$>$ and the optical counterpart found in the LGGS catalogue does not fulfil the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ criteria. The fourth source (r2-46) was not detected in the  observations. The foreground star classification of three sources (s1-74, s1-45, n1-82) in @2004ApJ...609..735W is confirmed by the  study ( 289,   603,  1449). For source  289 the spectral type F0 was determined . The source list of DKG2004 contains six sources (s2-46, s2-29, s2-37, s1-45, s1-20, r3-122) that are classified as foreground stars. All six sources are confirmed as foreground star candidates by our  study ( Table 5). For source  696 ($\hat{=}$s1-20) obtained the spectral type G0. Of the four sources listed as foreground stars in only one source ( 936$\hat{=}$ \[VG2007\] 168) was confirmed as a foreground star, based on the entry in the RBCV3.5 and @2009AJ....137...94C. The second source ( 1118$\hat{=}$\[VG2007\] 180) is listed in the RBCV3.5 and @2009AJ....137...94C as a globular cluster. The third source ( 829$\hat{=}$\[VG2007\] 181) does not have a counterpart in the USNO-B1, 2MASS or LGGS catalogues, nor does it fulfil the hardness ratio criteria for foreground stars. Hence, the source is classified as $<$hard$>$. The fourth source (\[VG2007\] 81) is not spatially resolved from its neighbouring source \[VG2007\] 79 in our  observations (source  1078). Hence source  1078 is classified as $<$hard$>$. Galaxies, galaxy clusters and AGN {#SubSec:Gal_GCl_AGN} --------------------------------- The majority of background sources belong to the class of active galactic nuclei (AGN). This was shown by the recent deepest available surveys of the X-ray background . The class of AGN is divided into many sub-sets. The common factor in all the sub-sets is that their emission emanates from a small, spatially unresolved galactic core. The small size of the emitting region is implied by the X-ray flux variability observed in many AGN, which is on time scales as short as several minutes (to years). The observed X-ray luminosities range from [$10^{39}$]{} to [$10^{46}$ erg s$^{-1}$]{}, sometimes even exceeding [$10^{46}$ erg s$^{-1}$]{}. Although AGN show many different properties, like the amount of radio emission or the emission line strengths and widths, they are believed to be only different facets of one underlying basic phenomenon [  @1995PASP..107..803U]: the accretion of galactic matter onto a supermassive black hole ($\sim\!10^{6}\!-\!10^{9}$M) in the centre of the galaxy. It is difficult and, to some extent, arbitrary to distinguish between active and normal galaxies, since most galaxies are believed to host a black hole at the position of their kinetic centre [@2005ApJ...631..280B]. In normal galaxies the accretion rate to the central supermassive BH is so low, that only weak activity can be detected – if at all. The overall thermal emission of the nuclear region is due to bremsstrahlung from hot gas. The total X-ray luminosity of a normal galaxy can reach some [$10^{41}$ erg s$^{-1}$]{}, at most. It consists of diffuse emission and emission of unresolved individual sources. Galaxy clusters (GCls) are by far the largest and most massive virialised objects in the Universe. Their masses lie in the range of $10^{14}$–$10^{15}$M and they have sizes of a few megaparsecs (Mpc). A mass-to-light ratio of $M/L\!\simeq\!200\,$M/L indicates that galaxy clusters are clearly dominated by their dark matter content. Furthermore, galaxy clusters allow us to study the baryonic matter component, as they define the only large volumes in the Universe from which the majority of baryons emit detectable radiation. This baryonic gas, the [*hot intracluster medium*]{} (ICM), is extremely thin, with electron densities of $n_{\mathrm{e}}\!\simeq\!10^2$–10$^5$m$^{-3}$, and fills the entire cluster volume. Owing to the plasma temperatures of $k_{\mathrm{B}}\,T\!\simeq\!2$–10keV, the thermal ICM emission gives rise to X-ray luminosities of $L_{\mathrm{X}}\!\simeq\!10^{43}$–$3\!\times\!10^{45}$ergs$^{-1}$. Therefore galaxy clusters are the most X-ray luminous objects in the Universe next to AGN. We identified four sources as background galaxies and 11 as AGN, and classified 19 galaxy and 49 AGN candidates. The classification is based on SIMBAD and NED correlations and correlations with sources listed as background objects in the globular cluster catalogues [RBCV3.5 and @2009AJ....137...94C]. Sources are classified as AGN candidates, if they have a radio counterpart [NVSS; @1990ApJS...72..761B; @2004ApJS..155...89G] with the additional condition of being neither a SNR nor a SNR candidate from X-ray hardness ratios, as well as not being listed as a “normal" background galaxy in @2004ApJS..155...89G. Most AGN will be classified as $<$hard$>$ ((HR2$-$EHR2)$>-$0.2, see Table \[Tab:class\]) because of their intrinsic power law component. Additional absorption in the line of sight by the interstellar medium of 31 will lead to an even higher HR2. Only the few AGN with a dominant component in the measured flux below 1keV may lead to a classification $<$SNR$>$ or $<$fg Star$>$ in our adapted scheme. One ( 995) of the four identified galaxies is M 32. An overview of previous X-ray observations of this galaxy is given in PFH2005. They also discuss the fact that  resolved the X-ray emission of M 32 into several distinct point sources (maximum separation of the three central  sources 83). Although M 32 is located closer to the centre of the FoV in the observations of field SS1, than it was in the s1 observation used in PFH2005,  still detects only one source. The remaining three sources ( 88,  403,  718) are identified as galaxies, because they are listed as background galaxies in both the RBCV3.5 and @2009AJ....137...94C. For source  403 (B007) NED gives a redshift of $0.139692\pm0.000230$ [@2007AJ....134..706K]. Eleven X-ray sources are identified as AGN. The first one ( 363) correlates with a BL Lac object located behind 31 (NED, see also PFH2005). The second source ( 745) correlates with a Seyfert 1 galaxy (5C 3.100), which has a redshift of $\approx 0.07$ (SIMBAD). The third source ( 1559) correlates with a quasar (Sharov 21) that showed a single strong optical flare, during which its UV flux has increased by a factor of $\sim$20 . The remaining sources were spectroscopically confirmed (from our optical follow-up observations) to be AGN (D. Hatzidimitriou, private communication; and Hatzidimitriou et al. (2010) in prep.). [rcllc]{} & & & &\ 141 & $1.19^{+1.63}_{-0.88}$ & $2.17^{+2.30}_{-0.68}$ & $0.24^{+1.24}_{-0.11}$ & 78.5/53\ 252 & $0.61^{+1.16}_{-0.43}$ & $1.95^{+0.64}_{-0.29}$ & $0.22^{+0.15}_{-0.07}$ & 56.4/151\ 304 & $2.68^{+2.64}_{-1.85}$ & $0.95^{+3.32}_{-1.95}$ & $0.12^{+0.07}_{-0.05}$ & 50.9/57\ 1543 & $2.74^{+6.91}_{-1.76}$ & $2.08^{+2.31}_{-1.11}$ & $0.61^{+1.11}_{-0.26}$ & 32.9/34\ \[Tab:spfit\_ext\] In Sect.\[Sec:ExtSrcs\] the 12 extended sources in the  catalogue were presented. @2006ApJ...641..756K showed that the brightest of these sources ( 1795) is a galaxy cluster located at a redshift of $z\!=\!0.29$. For the remaining 11 sources, X-ray spectra were created and fitted with the [MEKAL]{} model in [XSPEC]{}. Unfortunately, for most of the examined sources the spectral parameters (foreground absorption, temperature and redshift) are not very well constrained. Nevertheless four sources ( 141,  252,  304,  1543) with temperatures in the range of $\sim\!1$–2keV and proposed redshifts between 0.1–0.6 were found (Table \[Tab:spfit\_ext\]). Inspection of optical images (DSS2 images and if available LGGS images) revealed an agglomeration of optical sources at the positions of these four extended X-ray sources. Thus they are classified as galaxy cluster candidates. Although, B242 (the optical counterpart of source  304) is listed as a globular cluster candidate in the RBC3.5 catalogue, @2009AJ....137...94C classified this source as a background object. Our findings from the X-rays favour the background object classification. Hence a globular cluster classification for this source seems to be excluded. Source  1912 was already classified as a galaxy cluster candidate in PFH2005. The spectrum confirms this classification. The best fit parameters are $=\!1.29^{+0.53}_{-0.41}$[$\times 10^{21}$ cm$^{-2}$]{}, $T\!=\!2.8^{+0.8}_{-0.5}$keV and redshift of $0.06^{+0.03}_{-0.04}$. A plot of the spatial distribution of the classified/identified background sources is given in Fig.\[Fig:BG\_spdist\], which shows that these sources are rather homogeneously distributed over the observed field. However, in the fields located along the major axis of 31 we mainly see AGN, which are bright enough to be visible through 31, while most of the galaxies and galaxy clusters are detected in the outer fields. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues Of the ten  PSPC survey sources classified as background galaxies one is located outside the field of the Deep  Survey. The remaining objects are confirmed to be background sources and are classified or identified as galaxies or AGN. The only case which is worth discussing in more detail is the source pair \[SHP97\] 246 and \[SHL2001\] 252. From the  observations it is evident that this source pair is not one source, as indicated in the combined  PSPC source catalogue (SHL2001), but consists of three individual sources ( 1269,  1279 and  1280). \[SHL2001\] 252 correlates spatially with all three  sources, while \[SHP97\] 246 correlates only with source  1269, which is identified as a foreground star of type K2 (SIMBAD). The two other  counterparts of \[SHL2001\] 252 are classified as a galaxy candidate and an AGN candidate, respectively. In summary, \[SHL2001\] 252 is most likely a blend of both background sources and maybe even a blend of all three  sources, while \[SHP97\] 246 seems to be the X-ray counterpart of the foreground star mentioned above. @2002ApJ...577..738K classified source r3-83 ( 1132) as an extragalactic object, as it is listed in SIMBAD and NED as an emission line object. Following PFH2005, we classified source  1132 as $<$hard$>$. The BL Lac object ( 363) was also detected in  observations [@2004ApJ...609..735W]. M 31 sources {#Sec:Srcsm31} ============ Supersoft sources ----------------- Supersoft source (SSS) classification is assigned to sources showing extremely soft spectra with equivalent blackbody temperatures of $\sim$15–80eV. The associated bolometric luminosities are in the range of [$10^{36}$]{}–[$10^{38}$ erg s$^{-1}$]{} . Because of the phenomenological definition, this class is likely to include objects of several types. The favoured model for these sources is that they are close binary systems with a white dwarf (WD) primary, burning hydrogen on the surface . Close binary SSSs include post-outburst, recurrent, and classical novae, the hottest symbiotic stars, and other LMXBs containing a WD (cataclysmic variables, CVs). Symbiotic systems, which contain a WD in a wide binary system, may also be observed as SSSs . Because matter that is burned can be retained by the WD, some SSS binaries may be progenitors of type-Ia supernovae . The  catalogue contains 30 SSS candidates that were selected on the basis of their hardness ratios (see Fig.\[Fig:HR\_diagrams\] and Table \[Tab:class\]). ### Spatial and flux distribution Figure \[Fig:SSS\_spdist\] shows the spatial distribution of the SSSs. Clearly visible is a concentration of sources in the central field. There are two explanations for that central enhancement. The first is that the central region was observed more often than the remaining fields and therefore there is a higher chance of catching a transient SSS in outburst. The second reason is that the major class of SSSs in the centre of 31 are optical novae (PFF2005, PHS2007). Optical novae are part of the old stellar population which is much denser in the centre of 31. Figure \[Fig:SSS\_fldist\] gives the distribution of 0.2–1.0keV source fluxes for all SSSs (black) and for those correlating with optical novae (blue). The unabsorbed fluxes were determined assuming a 50eV blackbody model (PFF2005). The two brightest SSSs ($F_{{\mathrm}{X}}>$[$10^{-12}$ erg cm$^{-2}$ s$^{-1}$]{}) consist of a persistent source with 217s pulsations [ 1061; @2008ApJ...676.1218T] and the nova M31N 2001-11a [ 1416; @2006IBVS.5737....1S]. A large fraction of SSSs are rather faint, with fluxes below 5[$\times 10^{-14}$ ]{}. Four sources have absorption-corrected luminosities below [$10^{36}$ erg s$^{-1}$]{} (0.2–1.0keV), which was indicated as the limiting luminosity for SSSs. That does not necessarily imply that these sources are not SSSs, since it is possible that the blackbody fit chosen does not represent well the properties of these sources. A higher absorption or a lower temperature would lead to increased unabsorbed luminosities. We also have to take into account that we might have observed the source during a phase of rising or decaying luminosity,  not at maximum luminosity. ### Correlations with optical novae {#SubSec:opt_novae} By cross-correlating with the nova catalogue[^21] indicated in Sect.\[Sec:CrossCorr\_Tech\], 14 of the 30 SSSs can be classified as X-ray counterparts of optical novae. Of these 14 novae, eight ( 748,  993,  1006,  1046,  1051,  1076,  1100, and  1236) are already discussed in PFF2005 and PHS2007. Nova M31N 2001-11a was first detected as a supersoft X-ray source. Motivated by that SSS detection, @2006IBVS.5737....1S found an optical nova at the position of the SSS in archival optical plates which had been overlooked in previous nova searches. Nova M31N 2007-06b has been discussed in . The remaining four novae are discussed individually in more detail below. As was shown in the / 31 nova monitoring project[^22], it is absolutely necessary to have a homogeneous and dense sample of deep optical and X-ray observations in order to study optical novae and their connections to supersoft X-ray sources. In the optical, the outer regions of 31 are regularly observed down to a limiting magnitude of $\sim$17 mag (Texas Supernova Search (TSS); @Quimby2006), while in X-rays only “snapshots" are available. Hence, the correlations of optical novae with detected SSSs have to be regarded as lucky coincidences. That also means that the identified nova counterparts are detected at a random stage of their SSS evolution which does not allow us to constrain the exact start or end point of the SSS phase, nor the maximum luminosity of the SSS. We also cannot exclude the possibility that some of the SSSs observed in the outer parts of 31 correspond to the supersoft phase of optical novae for which the optical outburst was missed. In the outer regions of 31, the samples of optical novae and X-ray SSSs are certainly incomplete, due to the rather high luminosity limit in the optical monitoring, and the lack of complete monitoring in X-rays, respectively. So one should be cautious in deriving properties of the disc nova population of 31 from the available data. #### Nova M31N 1997-10c was detected on 2 October 1997 at a B-band magnitude of $16.6$ [ShA 58; @1998AstL...24..641S]. An upper limit of 19 mag on 29 September 1997 was reported by the same authors. They classified this source as a very fast nova. In the  observation c1 (25 June 2000), an SSS ( 871), located within $\sim$19 of the optical nova, was detected. The source was fitted with an absorbed blackbody model. The formal best fit parameters of the  EPIC PN spectrum are: absorption $N_{{\mathrm}{H}}\approx3.45$[$\times 10^{21}$ cm$^{-2}$]{} and $k_{{\mathrm}{B}}T\approx41$eV. The unabsorbed luminosity in the 0.2–1keV band is $\approx5.9$[$\times 10^{37}$ ]{}. Confidence contours for absorption column density and blackbody temperature are shown in Fig.\[Fig:M31N1997-10c\_ccont\]. In the subsequent  observation of that region taken about half a year later (c2; 27 December 2000) the source is not detected. Although the source position is covered in observations c3 (29 June 2001), c4 (6/7 January 2002) and b (16–19 July 2004) the source was not re-detected. Using the count rates derived for the variability study (see Sect.\[Sec:var\]) and assuming the same spectrum for the source as in observation c1, upper limits of the source luminosity can be derived, which are given in Table \[Tab:M31N1997-10c\_uplim\]. [cc]{} &\ c2 & 10.8$^{+}$\ c3 & 1.9\ c4 & 1.0\ \[Tab:M31N1997-10c\_uplim\] Notes:\ $^{ +~}$: The count rate detected in observation c2 gives a luminosity of 2.4$\pm$2.8[$\times 10^{37}$ ]{}, which results in the upper limit given in the Table. The fact that this upper limit is higher than the luminosity detected in observation c1 is, at least in part, attributed to the very short effective observing time of less than 6000s. #### Nova M31N 2005-01b was discovered on 19 January 2005 at a white light magnitude of 16.3 by R. Quimby.[^23] An SSS ( 764) that correlates with the optical nova (distance: 43; 3$\sigma$ error: 55) was found in observation ss2 taken on 8 July 2006, which is 535 days after the discovery of the optical nova. Due to the severe background screening applied to observation ss2, there is not enough statistics to obtain a spectrum of the X-ray source. To get an estimate of the spectral properties of that source we created a spectrum in the 0.2–0.8keV range of the *unscreened* data. Although the spectrum was background corrected, we cannot totally exclude a contribution from background flares. The spectrum is best fitted by an absorbed blackbody model with an absorption of $N_{{\mathrm}{H}}\approx1.03$[$\times 10^{21}$ cm$^{-2}$]{} and a blackbody temperature of $k_{{\mathrm}{B}}T\approx45$eV. The unabsorbed 0.2–1keV luminosity is $L_{{\mathrm}{X}}\sim$1.0[$\times 10^{37}$ ]{}. In another  observation taken 1073 days after the optical outburst (ss21; 28 December 2007) the X-ray source is no longer visible. The 3$\sigma$ upper limit of the unabsorbed source luminosity is $\sim3.3$[$\times 10^{35}$ ]{} in the 0.2–4.5keV band, assuming the spectral model used for source detection. #### Nova M31N 2005-01c was discovered on 29 January 2005 at a white light magnitude of 16.1 by R. Quimby.[^24] In the  observation from 02 January 2007 (ns2, 703 days after optical outburst) an SSS was detected ( 1675) at a position consistent with that of the optical nova (distance: 09). The X-ray spectrum (Fig.\[Fig:M31N2005-01c\_spec\]) can be well fitted by an absorbed blackbody model with the following best fit parameters: absorption $N_{{\mathrm}{H}}=1.58^{+0.65}_{-0.45}$[$\times 10^{21}$ cm$^{-2}$]{} and $k_{{\mathrm}{B}}T=40\pm6$eV. The unabsorbed 0.2–1keV luminosity is $L_{{\mathrm}{X}}\sim$1.2[$\times 10^{38}$ ]{}. Confidence contours for absorption column density and blackbody temperature are shown in Fig.\[Fig:M31N2005-01c\_ccont\]. #### Nova M31N 2005-09b was discovered in optical images taken on 01 and 02 September 2005 at white light magnitudes of $\sim$18.0 and $\sim$16.5 respectively. From 31 August 2005, an upper limit of $\sim$18.7mag was reported [@2005ATel..600....1Q]. The nova was spectroscopically confirmed [@2006ATel..850....1P] and classified as a possible Fe[II]{} or hybrid nova[^25]. An X-ray counterpart ( 92) was detected in the  observation s3 (299 days after the optical outburst). Its position is consistent with that of the optical nova (distance: 057). As observation s3 was heavily affected by background flares, we only could estimate the spectral parameters from the *unscreened* data (see also paragraph about Nova M31N 2005-01b). A blackbody fit of the 0.2–0.8keV gives $N_{{\mathrm}{H}}\approx2.7$[$\times 10^{21}$ cm$^{-2}$]{}, k$T\approx35$eV, and an unabsorbed 0.2–1keV luminosity of $L_{{\mathrm}{X}}\sim$5.4[$\times 10^{38}$ ]{}. The X-ray source was no longer visible in observation s31, which was taken 391 days after observation s3. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues {#Sec:SSS_comp} The results and a detailed discussion of a study of the long-term variability of the SSS population of 31 are presented in @2010AN....331..212S. In summary our comparative study of SSS candidates in 31 detected with ,  and  demonstrated that strict selection criteria have to be applied to securely select SSSs. It also underlined the high variability of the sources in this class and the connection between SSSs and optical novae. Supernova remnants {#Sec:SNR_Diss} ------------------ After an supernova explosion the interaction between the ejected material and the ISM forms a supernova remnant (SNR). The SNR X-ray luminosities typically vary between $10^{35}$ and [$10^{37}$ erg s$^{-1}$]{} (0.2–10keV). SNRs can be divided into two categories, (i) sources where the thermal components dominate the X-ray spectrum below 2keV, and (ii) the so-called “plerions" or Crab-like SNRs with power law spectra. The former are located in areas of the X-ray colour/colour diagrams that overlap only with foreground star locii. If we assume that we have identified all foreground star candidates from the optical correlation and inspection of the optical images, the remaining sources can be classified as SNR candidates using the criteria given in Table \[Tab:class\]. Similar criteria were used to select supernova remnant candidates in  observations of M 33 . @2005AJ....130..539G and @2010ApJS..187..495L confirmed the supernova remnant nature of many of these candidates based on optical and radio follow-up observations. They also used a hardness ratio criterion to select supernova remnant candidates from  data. An X-ray source is classified as a SNR candidate if it either fulfils the hardness ratio criterion given in Table \[Tab:class\] (these are 25 such sources), or if it correlates with a known optical or radio SNR candidate (six sources). The sources assigned the classification of a SNR candidate based on the latter criterion alone, are marked in the comment column of Table 5 with the flag ‘*only correlation*’. As these six SNR candidates would be classified as $<$hard$>$ on the basis of their hardness ratios, they are good candidates for being “plerions". SNRs are taken as identified when they coincide with SNR candidates from the optical or radio and fulfil the hardness ratio criterion. For a discussion of detection of SNRs in different wavelength bands see @2010ApJS..187..495L. All together, we identified 25 SNRs and 31 SNR candidates in the  catalogue. This number is in the range expected from an extrapolation of the X-ray detected SNRs in the Milky Way as shown below. Assuming that our own Galaxy contains about 1440 X-ray sources which are brighter than $\sim$1[$\times 10^{35}$ ]{} , and that it contains $\sim$110 SNRs detected in X-rays [@2009BASI...37...45G], we would expect to detect $\sim$50 SNRs in the  catalogue ($0.4\times{\left}(1\,897 {\mathrm}{sources} - 263 {\mathrm}{fg Stars} {\right})$). This number is in good agreement with the number of identified and classified SNRs. The XID fluxes for SNRs range between 5.9[$\times 10^{-14}$ ]{} for source   1234 and 1.5[$\times 10^{-15}$ ]{} for source  419. These fluxes correspond to luminosities of 4.3[$\times 10^{36}$ ]{} to 1.1[$\times 10^{35}$ ]{}. A diagram of the flux distribution of the detected SNRs and candidates is shown in Fig.\[Fig:SNR\_fldist\]. Among the 25 identified SNRs, there are 20 SNRs from the PFH2005 catalogue. Source \[PFH2005\] 146, which correlates with the radio source \[B90\] 11 and the SNR candidate BA146, was not found in the present study. Source \[SPH2008\] 858, which coincides with a source reported as a ring-like extended object in  observations that was also detected in the optical and radio wavelength regimes and identified as a SNR [@2003ApJ...590L..21K], was re-detected ( 1050). Of the 31 SNR candidates ten have been reported by PFH2005. In the following, we first discuss in more detail the remaining four identified SNRs, that appear in the new catalogue but were not included in PFH2005: #### XMMM31 J003923.5+404419 ( 182) was classified as a SNR candidate from its \[S[II]{}\]:H$\alpha$ ratio. It appears as an *‘irregular ring with southerly projection’* and correlates with a radio source [@1969MNRAS.144..101P]. X-ray radiation of that source was first detected in the present study. #### XMMM31 J004413.5+411954 ( 1410) was classified as a SNR candidate from its \[S[II]{}\]:H$\alpha$ ratio . From Fig.\[Fig:src1410\_opt\] we can see that the source *‘appears as a bright knot’*, as was already reported by . The source has counterparts in the radio [@1990ApJS...72..761B] and X-ray (SHP97) range. It was reported as a SNR by SHP97. #### XMMM31 J004510.5+413251 and XMMM31 J004512.3+420029 ( 1587 and  1593, respectively) are new X-ray detections and correlate with the radio sources: \#354 and \#365 in the list of @1990ApJS...72..761B. Source  1587 also correlates with source 37W209 from the catalogue of . No optical counterparts were reported in the literature. In the following, we discuss two SNR candidates in more detail: #### XMMM31 J004434.8+412512 ( 1481) lies in the periphery of a super-shell with \[S[II]{}\]:H$\alpha\!>$0.5 . Located next to this source is a SNR candidate reported in , which has a radio counterpart from the NVSS catalogue.  1481 also correlates with  source \[SPH97\] 284, which was identified as a SNR in SPH97 due to its spatial correlation with source 3-086. Figure \[Fig:src1481\_opt\] shows the  error circle over-plotted on LGGS images. From the  source position it looks more likely that the X-rays are emitted from the HII region rather than from the SNR candidate visible in the optical and radio wavelengths. Nevertheless the  source detected is point-like and its hardness ratios lie in the range expected for SNRs. If the X-ray emission originated from the -region, it should have been detected as spatially extended emission. Thus,  1481 is classified as SNR candidate. A puzzling fact, however, is the pronounced variability between  and  observations of $F_{{\mathrm}{var}}\!=\!9.82$ with a significance of $S_{{\mathrm}{var}}\!\approx\!4$ (see Table \[Tab:VarSNRs1\]), which is not consistent with the long term behaviour of SNRs. There is still the possibility that the detected X-ray emission does not belong to either the -region or a SNR at all. #### XMMM31 J004239.8+404318 ( 969) was already observed with  (SHP97, SHL2001) and  [@2004ApJ...609..735W s1-84]. No optical counterpart is visible on the LGGS images. The X-ray spectrum, which is shown in Fig.\[Fig:src969\_sp\], is well fitted by an absorbed non-equilibrium ionisation model with the following best fit values: an absorption of $N_{{\mathrm}{H}}=1.76^{+0.46}_{-0.60}$[$\times 10^{21}$ cm$^{-2}$]{}, a temperature of $k_{{\mathrm}{B}}T=219^{+32}_{-19}$eV, and an ionisation timescale of $\tau=1.75^{+0.82}_{-1.75}\times10^8$scm$^{-3}$. The unabsorbed 0.2–5keV luminosity is $L_{{\mathrm}{X}}\sim6.5$[$\times 10^{37}$ ]{}. The soft spectrum with the temperature of $\sim$200eV is in good agreement with spectra of old SNRs  in the SMC . Although the unabsorbed luminosity is rather high for an old SNR, it is still in the range found for other SNRs [  @2002ApJ...580L.125K; @2007ApJ...663..234G]. Hence, XMMM31 J004239.9+404318 is classified as a SNR candidate. ### Comparing SNRs and candidates in *XMM-Newton*, *Chandra* and *ROSAT* catalogues The second  PSPC catalogue (SHL2001) contains 16 sources classified as SNRs. The counterparts of 12 of these sources are also classified as SNRs or SNR candidates in the  catalogue. [rrccrcrrl]{}  & & & & & & fvar & svar & reason why indicated vaiability is not reliable\ & & & &\ 474 & 5.27 $\pm$ 0.56 & 21.18 $\pm$ 4.46 & & & & 4.01 & 3.54\ 668 & 7.94 $\pm$ 1.36 & 26.30 $\pm$ 6.69 & & & & 3.31 & 2.69\ 883 & 2.83 $\pm$ 0.33 & & & 3.33 $\pm$ 0.83 & & 1.18 & 0.56\ 1040 & 7.12 $\pm$ 0.47 & & & 12.49 $\pm$ 1.67 & & 1.75 & 3.11\ 1050 & 8.25 $\pm$ 0.70 & & & 2.50 $\pm$ 0.83 & & 3.30 & 5.28\ 1066 & 28.35 $\pm$ 1.16 & & 256.16$\pm$ 16.19 & 39.13 $\pm$ 3.33 & 25.29 $\pm$ 5.32 & 10.13 & 14.06 &  source is a blend of two  sources\ 1234 & 59.12 $\pm$ 1.10 & 152.91 $\pm$13.82 & 268.98 $\pm$17.09 & 54.11 $\pm$ 3.33 & 109.13 $\pm$ 11.31 & 4.97 & 12.34 & embedded in diffuse emission in central area of 31\ 1275 & 23.88 $\pm$ 1.08 & 53.50 $\pm$ 8.47 & 79.39 $\pm$ 9.90 & & & 3.32 & 5.58\ 1328 & 9.25 $\pm$ 0.74 & & 26.99 & & & 1.00 & 0.00\ 1351 & 4.96 $\pm$ 0.68 & 24.96 $\pm$ 8.92 & 17.77 & & & 1.00 & 0.00\ 1372 & 2.12 $\pm$ 0.84 & & 29.91 & & & 1.00 & 0.00\ 1410 & 7.40 $\pm$ 0.94 & 29.87 $\pm$ 7.13 & & & & 4.04 & 3.12\ 1481 & 3.43 $\pm$ 0.97 & 33.66 $\pm$ 7.36 & & & & 9.82 & 4.07 & see Sect.\[Sec:SNR\_Diss\]\ 1535 & 14.73 $\pm$ 1.31 & 53.94 $\pm$ 9.14 & 34.41 $\pm$ 7.20 & & & 3.66 & 4.25\ 1599 & 16.08 $\pm$ 0.92 & 54.39 $\pm$10.03 & 33.51 $\pm$ 6.97 & & & 3.38 & 3.80\ 1637 & 12.72 $\pm$ 1.33 & & 27.21 & & & 1.00 & 0.00\ \[Tab:VarSNRs1\] Notes:\ $^{ +~}$: KGP2002: @2002ApJ...577..738K,WGK2004: @2004ApJ...609..735W\  and  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, and ECF$_{{\mathrm}{KGP2002}}\!=\!8.325\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$. For WGK2004 the luminosity given in Table 2 of WGK2004 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!6.654\times$10$^{-15}\!\times\!L_{{\mathrm}{WGK2004}}$\[10$^{36}$ergs$^{-1}$\]. Table \[Tab:VarSNRs1\] lists the , , and  fluxes of all SNRs and SNR candidates from the  catalogue that have counterparts classified as SNRs in  or  source lists. In addition, the maximum flux variability and the maximum significance of the variability (following the variability calculation of Sect.\[Sec:DefVar\]) are given. Three SNRs that have  counterparts show variability changing in flux by more than a factor of five. The most variable source ( 1066) is discussed below, the second source was discussed in Sect.\[Sec:SNR\_Diss\] (XMMM31 J004434.8+412512,  1481), and the third source ( 1234) is embedded in the diffuse emission of the central area of 31. In this environment the larger PSF of  results in an overestimate of the source flux, since the contribution of the diffuse emission could not be totally separated from the emission of the point source. The remaining four  sources classified as SNRs and their  counterparts are discussed in the following paragraph. [lrccrcrrr]{}  & & & & & & fvar & svar & remark$^{\ddagger}$\ & & & &\ & & & & & & & &\ 294 & $18.50 \pm 0.85$ & & &$53.27 \pm 6.69$ & $46.78 \pm 7.87$ & 2.88 & 5.16 &\ 472 & $ 3.15 \pm 0.69$ & & & & $26.09 \pm 6.07$ & 8.28 & 3.76 & 468 brt\ 969 & $53.51 \pm 1.35$ & $84.51\pm 15.97^{+}$ & & $34.55 \pm 6.91$ & $89.06 \pm 11.92$ & 2.58 & 3.96 &\ 1079 & $ 4.19 \pm 0.59$ & & & $20.06 \pm 6.24 $& & 4.79 & 2.53 & brt\ 1291 & $14.55 \pm 0.75$ & $16.04^{*} $ & $>$24.0 & $35.22 \pm 8.47$ & $40.93 \pm 7.87$ & 2.81 & 3.33 &\ 1741 & $ 4.12 \pm 0.65$ & $4.17^{\dagger} $ & & & & 1.01 & — & brt\ 1793 & $ 3.70 \pm 0.52$ & & & $26.08 \pm 6.46$ & & 7.06 & 3.46 & 1799 brt\ \[Tab:VarSNRs\] Notes:\ $^{ \ddagger~}$: Source number (from  catalogue) of another (brighter)  source which correlate with the same  source as the  source given in Col. 1; brt:  flux is below the  detection threshold (5.3[$\times 10^{-15}$ ]{}).\  and  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{HRI}}\!=\!6.001\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, $^{ \dagger~}$: ECF$_{{\mathrm}{DKG2004}}\!=\!5.56\times$10$^{-12}$ergcm$^{-2}$cts$^{-1}$. $^{ +~}$: For WGK2004 the luminosity given in Table 2 of WGK2004 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!6.654\times$10$^{-15}\!\times\!L_{{\mathrm}{WGK2004}}$\[10$^{36}$ergs$^{-1}$\]. $^{ *~}$: For VG2007 the luminosity given in Table 2 of VG2007 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!9.433\times$10$^{-15}\!\times\!L_{{\mathrm}{VG2007}}$\[10$^{36}$ergs$^{-1}$\]. SHP97 report that \[SHP97\] 203 and \[SHP97\] 211 ($\hat{=}$\[SHL2001\] 206) correlate with the same SNR (\[DDB80\] 1.13), have the same spectral properties and have luminosities within the range of SNRs. A correlation with the  HRI catalogue (PFJ93) reveals that the true X-ray counterpart of \[DDB80\] 1.13 is located between the two  PSPC sources. Furthermore, PFJ93 report that this SNR is located ‘*within 19of a brighter X-ray source*’ which matches positionally with \[SHP97\] 211. These findings are confirmed by  and  observations. The X-ray counterpart of \[DDB80\] 1.13 is source  1066 in the  catalogue (or \[PFH2005\] 354 or r3-69 in [@2002ApJ...577..738K]). The second source, which correlates with \[SHP97\] 211, is the  source  1077, which has a “hard" spectrum and is $\sim\!6.7$ times brighter than  1066. Hence, \[SHP97\] 211 is a blend of the two  sources  1066 and  1077. This also explains the pronounced variability between \[SHL2001\] 206 and  1066 given in Table \[Tab:VarSNRs1\]. Comparing the  detections of the SNR counterpart with the  flux gives a variability factor of $F_{{\mathrm}{var}}\!\approx\!1.12$. The distance between \[SPH97\] 203 and \[DDB80\] 1.13 is $\ga\!20$. \[SPH97\] 203 was reported only in the first  PSPC catalogue. It was not detected in the observations of the second  PSPC catalogue or in any  or  observation of that region. Thus it seems very likely that \[SPH97\] 203 was either a transient source or a false detection. In both cases \[SPH97\] 203 cannot be a SNR. As the field of \[DDB80\] 1.13 was observed many times with , and as  has detected weak SNRs in the central part of 31 [@2003ApJ...590L..21K and  ],  should have detected X-ray emission corresponding to the  source \[SPH97\] 203, if it really belonged to a SNR. The remaining two  SNRs correlate with  sources, which were not classified as SNRs or SNR candidates. Source \[SHP97\] 258 correlates with source  1337 and has a 3$\sigma$ positional error of 30. From the improved spatial resolution of  the total positional error reduces to 23. Hence, we can see that the X-ray source belongs to a foreground star candidate ( Table 5) and not to the very nearby SNR. Source \[SHL2001\] 129 correlates with sources  743 and  761, which are classified as a GlC and a GlC candidate, respectively. The SNR candidate listed as the counterpart of \[SHL2001\] 129 is located between these two  sources. In addition PFH2005 gives a third source which lies within the error circle of \[SHL2001\] 129 and which is classified as an AGN candidate. Thus it is very likely that \[SHL2001\] 129 is a blend of these three  sources and that the correlation with the SNR candidate has to be considered as a chance coincidence. From the sources listed as SNRs in the different  studies many are re-detected. Nevertheless two SNRs from  were not detected in the  observations. Source n1-85 has been reported as spatially correlated with an optical SNR by @2004ApJ...609..735W, but has also been classified as a repeating transient source in the same paper. An  counterpart to n1-85 was detected neither in the study of PFH2005 nor in the  catalogue. The transient nature of this source is at odds with the SNR classification. Source CXOM31 J004247.8+411556 [@2003ApJ...590L..21K], which correlates with the radio source \[B90\] 95, is located in the vicinity of two bright sources and close to the centre of 31. Due to ’s larger point spread function this source cannot be resolved by  in this environment. The larger PSF of  is also the reason why source  1050 has a significant variability in Table \[Tab:VarSNRs1\], since this source is located within the central diffuse emission of 31. Finally, we wanted to determine whether any of the  SNRs and SNR candidates were previously observed, but not classified as SNRs. In total there are seven such sources. One of them ( 1741) is classified as a SNR candidate based on its  hardness ratios, and correlates with the  source n1-48 (DKG2004). The fluxes obtained with  and  are in good agreement (see Table \[Tab:VarSNRs\]), but below the  detection threshold (5.3 [$\times 10^{-15}$ ]{}). For a further four sources, the corresponding sources were only detected previously with ROSAT. One of them ( 1793$\hat{=}$\[SHP97\] 347) also correlates with a radio source [source 472 of @1990ApJS...72..761B] and is therefore identified as a SNR. The rather high flux variability between the  and  observations (see Table \[Tab:VarSNRs\]) can be attributed to source  1799, which is located within 199 of  1793. This suggests that \[SHP97\] 347 is a combination of both  sources, but as \[SHP97\] 347 was not detected in SHL2001, we cannot exclude a transient source or false detection as an explanation for the  source. Source  472 ($\hat{=}$\[SHL2001\] 84), source  294 ($\hat{=}$\[SHP97\] 53$\hat{=}$\[SHL2001\] 56), and source  1079 ($\hat{=}$\[SHP97\] 212) are SNR candidates based on their hardness ratios. The pronounced flux variability of source  472 is due to source  468, which is located within 185 of  472 and is $\sim$8.6 times brighter than  472. The observed flux for source  1079 was below the ROSAT threshold. Furthermore, the ROSAT source \[SHP97\] 212 was classified as a SNR, but did not appear in the SHL2001 catalogue. Hence ROSAT may have detected an unrelated transient instead. Sources corresponding to the remaining two  sources were detected with  and . Source  969 was detected in both  PSPC surveys (\[SHP97\] 185$\hat{=}$\[SHL2001\] 186) and correlates with  source s1-84 [@2006ApJ...643..356W]. We classify it as a SNR candidate due to its hardness ratios and X-ray spectrum (see XMMM31 J004239.8+404318). Counterparts for source  1291 were reported in the literature as \[PFJ93\] 84, \[SHP97\] 251, \[SHL2001\] 255, \[VG2007\] 261 and source 4 in Table 5 of @2006ApJ...643..844O. Based on the  hardness ratios and the correlation with radio source \[B90\] 166 [@1990ApJS...72..761B], we identified the source as a SNR. For sources  294,  969, and  1291 the variability between different observations may not be real because of systematic cross-calibration uncertainties. Therefore, we keep the $<$SNR$>$ and SNR classifications for these sources. ### The spatial distribution To examine the spatial distribution of the  SNRs and SNR candidates, we determined projected distances from the centre of M 31. The distribution of SNRs and SNR candidates (normalised per deg$^{2}$) is shown in Fig.\[Fig:SNRdepro\_dist\]. It shows an enhancement of sources around $\sim$3kpc, which corresponds to the SNR population in the ’inner spiral arms’ of M 31. In addition, a second enhancement of sources around $\sim$10kpc is detected; this corresponds to the well known dust ring or star formation ring in the disc of 31 [@2006Natur.443..832B]. Only a few sources are located beyond this ring. Figure \[Fig:SNR\_dudist\] shows the spatial distribution of the SNRs and SNR candidates from the  catalogue plotted over the IRAS 60$\mu$m image [@1994STIN...9522539W]. We see that most of the SNRs and SNR candidates are located on features that are visible in the IRAS image. This again demonstrates that SNRs and SNR candidates are coincident with the dust ring at $\sim\!10$kpc. In addition, the locations of star forming regions obtained from *GALEX* data [@2009ApJ...703..614K and private communication] are indicated in Fig.\[Fig:SNR\_dudist\]. We see that many of the SNRs and SNR candidates are located within or next to star forming regions in M 31. X-ray binaries {#SubSec:XRB} -------------- X-ray binaries consist of a compact object plus a companion star. The compact object can either be a white dwarf (these systems are a subclass of CVs), a neutron star (NS), or a black hole (BH). A common feature of all these systems is that a large amount of the emitted X-rays is produced due to the conversion of gravitational energy from the accreted matter into radiation by a mass-exchange from the companion star onto the compact object. X-ray binaries containing an NS or a BH are divided into two main classes, depending on the mass of the companion star: - Low mass X-ray binaries (LMXBs) contain companion stars of low mass ($M\la$ 1M) and late type (type A or later), and have a typical lifetime of $\sim$[$10^{8-9}$]{} yr . LMXBs can be located in globular clusters. Mass transfer from the companion star into an accretion disc around the compact object occurs via Roche-lobe overflow. - High mass X-ray binaries (HMXBs) contain a massive O or B star companion [$M_{{\mathrm}{star}}\ga10$M, @Verbunt1994] and are short-lived with lifetimes of $\sim$[$10^{6-7}$]{}yr . One has to distinguish between two main groups of HMXBs: super-giant and the Be/X-ray binaries. In these systems wind-driven accretion onto the compact object powers the X-ray emission. Mass-accretion via Roche-lobe overflow is less frequent in HMXBs, but is still known to occur in several bright systems ( LMCX-4, SMCX-1, CenX-3). HMXBs are expected to be located in areas of relatively recent star formation, between 25–60Myr ago [@2010ApJ...716L.140A]. We should expect about 45 LMXBs in 31, following a similar estimation as the one presented in Sect.\[Sec:SNR\_Diss\]. Here the number of LMXBs in the Galaxy was estimated from . In the  catalogue 88 sources are identified/classified as XRBs. This is not surprising as we may expect 31 to have a higher fraction of XRBs than the Galaxy since it is an earlier type galaxy composed of a higher fraction of old stars. XRBs are the main contribution to the population of “hard" X-ray sources in 31. Despite some more or less reliable candidates, not a single, definitely detected HMXB is known in 31. The results of a new search for HMXB candidates are presented in Sect.\[SubSec:XRB\_HMXB\]. The LMXBs can be separated into two sub-classes: the field LMXBs (discussed in this section) and those located in globular clusters. Sources belonging to the latter sub-class are discussed in Sect.\[SubSec:GlC\]. The sources presented here are classified as XRBs, because they have HRs indicating a $<$hard$>$ source and are either transient or show a variability factor larger than ten (see Sect.\[Sec:var\]). In total 10 sources are identified and 26 are classified as XRBs by us, according to the classification criteria given in Table \[Tab:class\]. Apart from source  57 (XMMM31 J003833.2+402133, see below), the identified XRBs had been reported as X-ray binaries in the literature (see comment column of Table 5). Figure \[Fig:XRB\_fldist\] (red histogram) shows the flux distribution of XRBs. We see that this class contains only rather bright sources. This is not surprising as the classification criterion for XRBs is based on their variability, which is more easily detected for brighter sources (  Sect.\[Sec:var\]). The XID fluxes range from 1.4[$\times 10^{-14}$ ]{} ( 378) to 3.75[$\times 10^{-12}$ ]{} ( 966), which correspond to luminosities from 1.0[$\times 10^{36}$ ]{} to 2.7[$\times 10^{38}$ ]{}. It is clear from Fig.\[Fig:XRB\_spdist\], which shows the spatial distribution of the XRBs, that nearly all sources classified or identified as XRBs (yellow dots) are located in fields that were observed more than once (centre and southern part of the disc). This is partly a selection effect, caused by the fact that these particular fields were observed several times, thus allowing the determination of source variability. For sources located outside these fields, especially the northern part of the disc, the transient nature must have been reported in the literature to mark them as an XRBs. The source density of LMXBs, which follows the overall stellar density, is higher in the centre than in the disc of 31. One would not expect HMXBs in the central region which is dominated by the bulge (old stellar population). From Fig.\[Fig:XRB\_spdist\_IRAS\], which shows the spatial distribution of the XRBs over-plotted on an IRAS 60$\mu$m image [@1994STIN...9522539W], we see that only a few sources, classified or identified as XRBs, are located in the vicinity of star forming regions. References for the sources, selected from their temporal variability, are given in Table \[Tab:varlist\]. TPC06 report on four bright X-ray transients, which they detected in the observations of July 2004 and suggested to be XRB candidates. We also found these sources and classified source  705 and identified sources  985,  1153,  1177 as XRBs. One of the identified XRBs ( 1177) shows a very soft spectrum. @2005ApJ...632.1086W observed source  1153 with  and *HST*. From the location and X-ray spectrum they suggest it to be an LMXB. They propose that the optical counterpart of the X-ray source is a star within the X-ray error box , which shows an optical brightness change (in B) by $\simeq$1 mag. Source  985 was first detected in January 1979 by TF91 with the  observatory. WGM06 rediscovered it in  observations from 2004. Their coordinated *HST* ACS imaging does not reveal any variable optical counterpart. From the X-ray spectrum and the lack of a bright star, WGM06 suggest that this source is an LMXB with a black hole primary. In the following subsections we discuss three transient XRBs in more detail. #### XMMM31 J003833.2+402133 ( 57) was first detected in the  observation from 02 January 2008 (s32) at an unabsorbed 0.2–10keV luminosity of $\sim\!2$[$\times 10^{38}$ ]{}. From two observations, taken about 0.5yr (s31) and 1.5yr (s3) earlier, we derived upper limits for the fluxes, which were more than a factor of 100 below the values obtained in January 2008. The combined EPIC spectrum from observation s32 (Fig.\[SubFig:spec\_1\]) is best fitted with an absorbed disc blackbody plus power-law model, with $N_{{\mathrm}{H}}\!=\!1.68^{+0.42}_{-0.48}$[$\times 10^{21}$ cm$^{-2}$]{}, temperature at the inner edge of the disc $k_{{\mathrm}{B}}T_{{\mathrm}{in}}\!=\!0.462\pm0.013$keV and power-law index of $2.55^{+0.33}_{-1.05}$. The contribution of the disc blackbody luminosity to the total luminosity is $\sim 59\,\%$. Formally acceptable fits are also obtained from an absorbed disc blackbody and an absorbed bremsstrahlung model (see Table \[Tab:specprop\]). We did not find any significant feature in a fast Fourier transformation (FFT) periodicity search. The combined EPIC light curve during observation s32 was consistent with a constant value. To identify possible optical counterparts we examined the LGGS images and the images taken with the  optical monitor during the X-ray observation (UVW1 and UVW2 filters). The absence of optical/UV counterparts and of variability on short timescales, as well as the spectral properties suggest that this source is a black hole LMXB in the steep power-law state [@2006csxs.book..157M]. #### CXOM31 J004059.2+411551: @2007ATel.1147....1G reported on the detection of a previously unseen X-ray source in a 5ks  ACIS-S observation from 05 July 2007. In an  ToO observation [sn11, @2007ATel.1191....1S] taken about 20 days after the  detection, the source ( 523) was still bright. The position agrees with that found by . We detected the source at an unabsorbed 0.2–10keV luminosity of $\sim\!1.1$[$\times 10^{38}$ ]{}. The combined EPIC spectrum (Fig.\[SubFig:spec\_2\]) can be well fitted with an absorbed disc blackbody model with $N_{{\mathrm}{H}}\!=\!{\left}(2.00\pm{0.16}{\right})$[$\times 10^{21}$ cm$^{-2}$]{} and with a temperature at the inner edge of the disc of $k_{{\mathrm}{B}}T_{{\mathrm}{in}}\!=\!0.538\pm0.017$keV (Table \[Tab:specprop\]). The spectral parameters and luminosity did not change significantly compared to the  values of @2007ATel.1147....1G. We did not find any significant feature in an FFT periodicity search. The combined EPIC light curve was consistent with a constant value. The examination of LGGS images and of images taken with the  optical monitor (UVW1 and UVW2 filters) during the X-ray observation did not reveal any possible optical/UV counterparts. The lack of bright optical counterparts and the X-ray parameters (X-ray spectrum, lack of periodicity, transient nature, luminosity) are consistent with this source being a black hole X-ray transient, as already mentioned in @2007ATel.1147....1G. #### XMMU J004144.7+411110 ( 705) was detected by @2006ApJ...645..277T in  observations b1–b4 (July 2004) at an unabsorbed luminosity of 3.1–4.4[$\times 10^{37}$ ]{} in the 0.3–7keV band, using a [DISKBB]{} model. We detected the source in observation sn11 (25 July 2007) with an unabsorbed 0.2–10keV luminosity of $\sim$1.8[$\times 10^{37}$ ]{}, using also a [DISKBB]{} model. In observation sn11, the source was bright enough to allow spectral analysis. The spectra can be well fitted with an absorbed power-law, disc blackbody or bremsstrahlung model (Table \[Tab:specprop\]). The obtained spectral shapes (absorption and temperature as well as photon index) are in agreement with the values of @2006ApJ...645..277T. An FFT periodicity search did not reveal any significant periodicities in the 0.3s to 2000s range. No optical counterparts were evident in the images taken with the  optical monitor UVW1 and UVW2 during the sn11 observation, nor in the LGGS images. The lack of a bright optical counterpart and the X-ray parameters support that this source is a black hole X-ray transient, as classified by @2006ApJ...645..277T. \ [ccccccccc]{} & & & & & & & &\ & & & & & & & &\ & & & & &\ s32 &PL+DISCBB&$1.68^{+0.42}_{-0.48}$&$0.462\pm0.013$&$106^{+9}_{-10}$ &$2.55^{+0.33}_{-1.05}$ &173.89(145)&2.04&PN+M1+M2\ s32 &DISCBB&$1.06\pm0.06$&$0.511\pm0.009$&$95\pm4$ & &270.01(147)&1.46&PN+M1+M2\ s32 &BREMSS&$1.91\pm0.07$&$1.082^{+0.029}_{-0.030}$& & &208.65(147)&2.12&PN+M1+M2\ & & & & &\ sn11 &DISCBB&$2.00\pm0.16$&$0.538\pm0.017$&$75\pm6$ & &97.70(79)&1.12&PN+M1+M2\ sn11 &BREMSS&$3.13\pm0.19$&$1.097^{+0.060}_{-0.056}$& & &93.17(79)&1.72&PN+M1+M2\ & & & & &\ sn11 &DISCBB&$2.32^{+1.03}_{-0.87}$&$0.586^{+0.100}_{-0.087}$&$26^{+13}_{-8}$ & &29.74(23)&0.18&PN+M1+M2\ sn11 &BREMSS&$3.72^{+1.14}_{-1.00}$&$1.216^{+0.373}_{-0.269}$& & &29.48(23)&0.29&PN+M1+M2\ sn11 &PL&$6.17^{+1.72}_{-1.47}$&& &$3.23^{+0.46}_{-0.40}$& 31.57(23)&1.12&PN+M1+M2\ \[Tab:specprop\] Notes:\ $^{ *~}$: effective inner disc radius, where $i$ is the inclination angle of the disc\ $^{ {\dagger}~}$: unabsorbed luminosity in the $0.2$–$10.0$keV energy range in units of [$10^{38}$ erg s$^{-1}$]{}\ ### Sources from the XMM-LP total catalogue that were not detected by *ROSAT* To search for additional XRB candidates, we selected all sources from the  catalogue, that were classified as $<$hard$>$ and which did not correlate with a source listed in the  catalogues (PFJ93, SHP97 and SHL2001). The flux distribution of the selected sources is shown in Fig.\[Fig:noROSdist\], and Table \[Tab:noROSdist\] gives the number of sources brighter than the indicated flux limit. [rr]{} &\ &\ 5.5E-15 & 541\ 1E-14 & 242\ 5E-14 & 7\ 1E-13 & 1\ \[Tab:noROSdist\] Possible, new XRB candidates are sources that have an XID flux that lies at least a factor of ten above the  detection threshold (5.3[$\times 10^{-15}$ ]{}). These sources fulfil the variability criterion used to classify XRBs ( Sect.\[Sec:var\]). The  catalogue lists five sources without  counterparts that have XID fluxes above 5.3[$\times 10^{-14}$ ]{}. These are:  239,  365,  910,  1164, and  1553. Between the  and  observations more than ten years have elapsed. On this time scale AGN can also show strong variability. To estimate the number of AGN among the five sources listed above, we investigated how many sources of the identified and classified background objects from the  catalogue with an XID flux larger than 5.3[$\times 10^{-14}$ ]{} were not detected by . The result is that  detected all background sources with an XID flux larger than 5.3[$\times 10^{-14}$ ]{} that are listed in the  catalogue. Thus, the probability that any of the five sources listed above is a background object is very small, in particular if the source is located within the D$_{25}$ ellipse of 31. Therefore, the two sources located within the D$_{25}$ ellipse are listed in the  catalogue as XRB candidates, while the remaining three sources, which are located outside the D$_{25}$ ellipse, are classified as $<$hard$>$. All five sources are marked in the comment column of Table 5 with ‘XRB cand. from  corr.’. ### Detection of high mass X-ray binaries {#SubSec:XRB_HMXB} As already mentioned, until now not a single secure HMXB in 31 has been confirmed. The reason for this is that the detection of HMXBs in 31 is difficult. @2004ApJ...602..231C showed that the hardness ratio method is very inefficient in selecting HMXBs in spiral galaxies. The selection process is complicated by the fact, that the spectral properties of BH HMXBs, which have power-law spectra with indices of $\sim$1– $\sim$2 are similar to LMXBs and AGN. Therefore the region in the HR diagrams where BH HMXB are located is contaminated by other hard sources (LMXBs, AGN, and Crab like SNRs). For the NS HMXBs, which have power-law indices of $\sim$1, and thus should be easier to select, the uncertainties in the hardness ratios lead at best to an overlap – in the worst case to a fusion – with the area occupied by other hard sources [@2004ApJ...602..231C]. Based on the spectral analysis of individual sources of 31, SBK2009 identified 18 HMXB candidates with power-law indices between 0.8 and 1.2. One of these sources (\[SBK2009\] 123) correlates with a globular cluster, and hence it is rather an LMXB in a very hard state rather than an HMXB [  @2004ApJ...616..821T]. Four of their sources (\[SBK2009\] 34, 106, 149, and 295) do not have counterparts in the  catalogue. @Peter developed a selection algorithm for HMXBs in the SMC, which also uses properties of the optical companion. X-ray sources were selected as HMXB candidates if they had HR2$+$EHR2$>$0.1 as well as an optical counterpart within 25 of the X-ray source, with $-0.5\!<$B$-$V$<\!0.5$mag, $-1.5\!<$U$-$B$<\!-0.2$mag and V$<$17mag. We tried to transfer this SMC selection algorithm to 31 sources. In doing so, we encountered two problems: The first problem is that the region of the U-B/B-V diagram is also populated by globular clusters (LMXB candidates) in 31. The second problem is that due to the much larger distance to 31, the range of detected V magnitudes of HMXBs in the SMC of $\sim$13$<$V$<$17mag translates to a $\sim$19$<$V$<$23mag criterion for 31. Thus the V magnitude of optical counterparts of possible HMXB candidates lies in the same range as the optical counterparts of AGN. Therefore the V mag criterion, which provided most of the discriminatory power in the case of the SMC, fails totally in the case of 31. A few of the sources selected from the optical colour-colour diagram and HR diagrams are bright enough to allow the creation of X-ray spectra. That way two additional ( not given in SBK2009) HMXB candidates were found. In addition, we determined the reddening free Q parameter: $$Q = (\rm{U}-\rm{B})-0.72(\rm{B}-\rm{V})$$ [for definition see   @cox2001allen] which allowed us to keep only the intrinsically bluest stars, using Q $\le\!-0.4$ [O-type stars typically have Q$<\!-0.9$, while -0.4 corresponds to a B5 dwarf or giant or an A0 supergiant, @2007AJ....134.2474M]. U$-$B and B$-$V were taken from the LGGS catalogue. #### XMMM31 J004557.0+414830 ( 1716) has an USNO-B1 (R2$=$18.72mag), a 2MASS and an LGGS (V$=$20.02mag; Q$=\!-0.44$) counterpart. The EPIC spectrum is best fitted ($\chi^2_{red}\!=\!0.93$) by an absorbed power-law with $=\!7.4^{+6.0}_{-3.9}$[$\times 10^{21}$ cm$^{-2}$]{} and photon index $\Gamma\!=\!1.2\pm0.4$. The absorption corrected X-ray luminosity in the 0.2–10keV band is $\sim$7.1[$\times 10^{36}$ ]{}. #### XMMM31 J004506.4+420615 ( 1579) has an USNO-B1 (B2$=$20.87mag), a 2MASS and an LGGS (V$=$20.77mag; Q$=\!-1.04$) counterpart. The EPIC PN spectrum is best fitted ($\chi^2_{red}\!=\!1.6$) by an absorbed power-law with $=\!0.48^{+2.4}_{-1.0}$[$\times 10^{21}$ cm$^{-2}$]{} and photon index $\Gamma\!=\!1.0^{+0.7}_{-0.5}$. The absorption corrected X-ray luminosity in the 0.2–10keV band is $\sim$8.6[$\times 10^{36}$ ]{}.\ To strengthen these classifications spectroscopic optical follow-up observations of the optical counterparts are needed. An FFT periodicity search did not reveal any significant periodicities for either of the two sources and the light curves do not show eclipses. From the sources reported as HMXB candidates in SBK2009, three sources (\[SBK2009\] 21, 236, and 256) are located in the region of the U-B/B-V diagram, that we used. Another three sources (\[SBK2009\] 123, 172, and 226) are located outside that region. The remaining sources of SBK2009 have either no counterparts with a U-B colour entry in the LGGS catalogue (\[SBK2009\] 99, 234, 294, and 302) or have no optical counterpart from the LGGS catalogue at all (\[SBK2009\] 9, 160, 197, and 305). The reddening free Q parameter for the SBK2009 sources that have counterparts in the LGGS catalogue are given in Table \[Tab:SBK\_Q\]. [rrlr]{} & & &\ 312 & 21 & J004001.50+403248.0 & -0.34\ 1668 & 236 & J004538.23+421236.0 & -0.77\ 1724 & 256 & J004558.98+420426.5 & -0.81\ 1109 & 123 & J004301.51+413017.5 & +1.77\ 1436 & 172 & J004420.98+413546.7 & -0.65\ & & J004421.01+413544.3$^{*}$ & -0.29\ 1630 & 226 & J004526.68+415631.5 & -0.92\ & & J004526.58+415633.1$^{*}$ & -0.72\ \[Tab:SBK\_Q\] Notes:\ $^{ *~}$: counterparts listed in SBK2009 Globular cluster sources {#SubSec:GlC} ------------------------ A significant number of the luminous X-ray sources in the Galaxy and in 31 are found in globular clusters. X-ray sources corresponding to globular clusters are identified by cross-correlating with globular cluster catalogues (see Sect.\[Sec:CrossCorr\_Tech\]). Therefore changes between the  catalogue and the catalogue of PFH2005 in the classification of sources related to globular clusters are based on the availability of and modifications in recent globular cluster catalogues. In total 52 sources of the  catalogue correlate with (possible) globular clusters. Of these sources 36 are identified as GlCs because their optical counterparts are listed as globular clusters in the catalogues given in Sect.\[Sec:CrossCorr\_Tech\], while the remaining 16 sources are only listed as globular cluster candidates. The range of source XID fluxes goes from 3.1[$\times 10^{-15}$ ]{} ( 924) to 2.7[$\times 10^{-12}$ ]{} ( 1057), or in luminosity from 2.3[$\times 10^{35}$ ]{} to 2.0[$\times 10^{38}$ ]{} (Fig.\[Fig:XRB\_fldist\]; green histogram). Compared to the fluxes found for the XRBs discussed in Sect.\[SubSec:XRB\], 14 sources that correlate with GlCs have fluxes below the lowest flux found for field XRBs. The reason for this finding is that the classification of field XRBs is based on the variable or transient nature of the sources, which can only be to detected for brighter sources ( Sect.\[Sec:var\]) and not just by positional coincidence that is also possible for faint sources. Figure \[Fig:GlC\_spdist\] shows the spatial distribution of the GlC sources. X-ray sources correlating with GlCs follow the distribution of the optical GlCs, which are also concentrated towards the central region of 31. The three brightest globular cluster sources, which are located in the northen disc of 31, are  1057 (XMMM31 J004252.0+413109),  694 (XMMM31 J004143.1+413420), and  1692 (XMMM31 J004545.8+413941). They are all brighter than 8.4[$\times 10^{37}$ ]{}. Source  694 was classified as a black hole candidate, due to its variability observed at such high luminosities. A detailed discussion of the three sources is given in @2008ApJ...689.1215B. XMMM31 J004303.2+412121 ( 1118) was identified as a foreground star in PFH2005, based on the classification in the “Revised Bologna Catalogue" . took the classification from , which is based on the velocity dispersion of that source. Recent ‘*HST images unambiguously reveal that this* \[B147\] *is a well resolved star cluster, as recently pointed out also by @2007AJ....133.2764B*’ . That is why source  1118 is now identified as an XRB located in globular cluster B147. ### Integrated optical properties of the globular clusters in which the X ray sources are located [lcclllc]{} & & & & & &\ B005 & confirmed & old & 15.69 & 1.29 & 1.15 & old\ SK055B & candidate & - & 18.991 & 0.388 & 0.248 & –\ B024 & confirmed & old & 16.8 & 1.15 & 1.01 & old\ SK100C & candidate & na & 18.218 & 1.181 & 1.041 & old\ B045 & confirmed & old & 15.78 & 1.27 & 1.13 & old\ B050 & confirmed & old & 16.84 & 1.18 & 1.04 & old\ B055 & confirmed & old & 16.67 & 1.68 & 1.54 & old\ B058 & confirmed & old:: & 14.97 & 1.1 & 0.96 & old-inter\ MITA140 & confirmed & old & 17 & 9999 & - & –\ B078 & confirmed & old & 17.42 & 1.62 & 1.48 & old\ B082 & confirmed & old & 15.54 & 1.91 & 1.77 & old\ B086 & confirmed & old & 15.18 & 1.26 & 1.12 & old\ SK050A & confirmed & - & 18.04 & 1.079 & 0.939 & old-inter\ B094 & confirmed & old & 15.55 & 1.26 & 1.12 & old\ B096 & confirmed & old & 16.61 & 1.48 & 1.34 & old\ B098 & confirmed & old & 16.21 & 1.13 & 0.99 & old-inter\ B107 & confirmed & old & 15.94 & 1.28 & 1.14 & old\ B110 & confirmed & old & 15.28 & 1.28 & 1.14 & old\ B117 & confirmed & old:: & 16.34 & 1 & 0.86 & inter\ B116 & confirmed & old & 16.79 & 1.86 & 1.72 & old\ B123 & confirmed & old & 17.416 & 1.29 & 1.15 & old\ B124 & confirmed & old & 14.777 & 1.147 & 1.007 & old\ B128 & confirmed & old:: & 16.88 & 1.12 & 0.98 & old-inter\ B135 & confirmed & old & 16.04 & 1.22 & 1.08 & old\ B143 & confirmed & old & 16 & 1.22 & 1.08 & old\ B144 & confirmed & old:: & 15.88 & 0.59 & 0.45 & young\ B091D & confirmed & old & 15.44 & 9999 & - & –\ B146 & confirmed & old:: & 16.95 & 1.09 & 0.95 & interm\ B147 & confirmed & old & 15.8 & 1.27 & 1.13 & old\ B148 & confirmed & old & 16.05 & 1.17 & 1.03 & old\ B150 & confirmed & old & 16.8 & 1.28 & 1.14 & old\ B153 & confirmed & old & 16.24 & 1.3 & 1.16 & old\ B158 & confirmed & old & 14.7 & 1.15 & 1.01 & old\ B159 & confirmed & old & 17.2 & 1.41 & 1.27 & old\ B161 & confirmed & old & 16.33 & 1.1 & 0.96 & old-inter\ B182 & confirmed & old & 15.43 & 1.29 & 1.15 & old\ B185 & confirmed & old & 15.54 & 1.18 & 1.04 & old\ B193 & confirmed & old & 15.33 & 1.28 & 1.14 & old\ SK132C & candidate & - & 18.342 & 1.84 & 1.7 & old\ B204 & confirmed & old & 15.75 & 1.17 & 1.03 & old\ B225 & confirmed & old & 14.15 & 1.39 & 1.25 & old\ B375 & confirmed & old & 17.61:: & 1.02 & 0.88 & interm\ B386 & confirmed & old & 15.547 & 1.154 & 1.014 & old\ \[Tab:GlC\_optprop\] Notes:\ $^{ *~}$: classification as confirmed or otherwise comes from the revised Bologna catalogue (December 2009, Version 4) <http://www.bo.astro.it/M31/RBC_Phot07_V4.tab>\ $^{ +~}$: age comes from @2009AJ....137...94C\ V, and V$-$I are integrated colours that come from the revised Bologna catalogue (December 2009, Version 4) <http://www.bo.astro.it/M31/RBC_Phot07_V4.tab>\ (V$-$I)$_{\rm{o}}$ is the dereddened V-I integrated colour, assuming E(B-V)=0.10+-0.03, which is the average of the reddenings of all 31 clusters in @2005AJ....129.2670R. (this E(B$-$V) corresponds to E(V$-$I)$=\!0.14$)\ $^{ \dagger~}$: This dereddened colour (V$-$I)$_{\rm{o}}$ is used to estimate the age on the basis of the plots (V$-$I)$_{\rm{o}}$ versus logAge from @2007AJ....133..290S. For each X-ray source which correlates with a globular cluster or globular cluster candidate in the optical, we investigated its integrated V-I colour and derived age estimates. Table \[Tab:GlC\_optprop\] lists the name of the optical counterpart, its classification according to RBC V.4 , the age classification of @2009AJ....137...94C, the V magnitude and V-I colour given in RBC V.4, the dereddened V-I colour, and the age estimate derived by ourselves. The integrated V$-$I colours of the clusters can be found in RBC V.4 and can be used to provide estimates of the ages of the clusters, in conjunction with reddening values. We have adopted a reddening of E(B$-$V)$=\!0.10\pm0.03$, which is the average of the reddenings of all 31 clusters in @2005AJ....129.2670R. Using these values, we have derived (V$-$I)$_{\rm{o}}$ for our clusters. In most cases (V$-$I)$_{\rm{o}}\!>\!1.0$ suggesting clusters older than $\simeq$2 Gyr according to @2007AJ....133..290S. The histogram in Fig.\[Fig:GlC\_agedist\] shows the distribution of (V$-$I)$_{\rm{o}}$ for our clusters, with the approximate age-ranges marked. In general there is good agreement between the @2009AJ....137...94C and our age estimates. This result indicates that the great majority of the objects are indeed old globular clusters. Figure \[Fig:optGlC\_agedist\] shows the distribution of (V$-$I)$_{\rm{o}}$ for all confirmed and candidate globular clusters, listed in the RBC V.4, which are located in the  field, and which have V as well as I magnitudes given. A comparison with Fig.\[Fig:GlC\_agedist\] again reveals that mainly counterparts of old globular clusters (age $\ga$2Gyr) are detected in X-rays. ### Comparing GlC and candidates in *XMM-Newton*, *Chandra* and *ROSAT* catalogues {#SubSub:comp_GlC} The combined  PSPC catalogue (SHP97 and SHL2001) contains 33 sources classified as globular cluster counterparts. Of these sources one is located outside the field observed with . Another two sources do not have counterparts in the  catalogue. The first one is \[SHL2001\] 232, which is not visible in any  observation taken before December 2006 as was already reported in @2004ApJ...616..821T. The second source (\[SHL2001\] 231) correlates with B164 which is identified as a globular cluster in RBC V3.5. In addition \[SHL2001\] 231 is listed in PFH2005 as the counterpart of the source \[PFH2005\] 423. Due to the improved positional accuracy of the X-ray source in the  observations, PFH2005 rejected the correlation with B164 and instead classified \[PFH2005\] 423 as a foreground star candidate. Three  GlC candidates have more than one counterpart in the  catalogue. \[SHL2001\] 249 correlates with sources  1262 and  1267, where the latter is the X-ray counterpart of the globular cluster B185. \[SHL2001\] 254 correlates with sources  1289 and  1293, where the former is the X-ray counterpart of the globular cluster candidate mita311 [@1993PhDT........41M]. \[SHL2001\] 258 has a 1$\sigma$ positional error of 48 and thus correlates with sources  1297,  1305 and  1357.[^26] The brightest of these three sources ( 1305), which is actually located closest to the  position, correlates with the globular cluster candidate SK132C (RBC V3.5). Table \[Tab:ROSAT\_GlC\_tvar\] gives the variability factors (Cols. 6, 8) and significance of variability (7, 9) for sources classified as GlC candidates in the  PSPC surveys. For most sources only low variability is detected. The two sources with the highest variability factors found ( 1262,  1293) belong to  sources with more than one  counterpart. In these cases the  sources that correlate with the same  source and the optical globular cluster source show much weaker variability. Interestingly, a few sources show low, but very significant variability. Among these sources is the Z-source identified in and two of the sources discussed in @2008ApJ...689.1215B [ 1057,  1692]. {width="12cm"} [rrrrrrrrrccc]{} & & & & & & & & & & &\ & & & & & & & & & & &\ 383 & 73 & 68 & 1.45E-12 & 1.05E-14 & 1.26 & 11.58 & 1.23 & 9.04 & GlC & \* & \*\ 403 & 79 & 74 & 2.55E-14 & 2.36E-15 & 2.61 & 2.09 & 7.57 & 4.92 & Gal & &\ 422 & & 76 & 2.01E-14 & 1.43E-15 & & & & & $<$hard$>$ & &\ 694 & 122 & 113 & 1.52E-12 & 1.04E-14 & 1.38 & 10.56 & 1.26 & 7.46 & GlC & \* & \*\ 793 & 138 & 136 & 4.76E-14 & 2.33E-15 & 1.10 & 0.55 & 1.02 & 0.14 & $<$Gal$>$ & \* &\ 841 & 150 & 147 & 1.40E-12 & 1.61E-14 & 1.77 & 19.81 & 2.11 & 26.37 & GlC & \* & –\ 855 & 158 & 154 & 4.21E-13 & 3.70E-15 & 1.01 & 0.14 & 3.54 & 48.75 & GlC & \* &\ 885 & 168 & 163 & 1.56E-14 & 1.60E-15 & 1.70 & 1.54 & & & GlC & &\ 923 & 175 & 175 & 1.47E-13 & 2.07E-15 & 1.87 & 7.68 & 1.30 & 3.44 & GlC & &\ 933 & 178 & & 3.67E-14 & 1.64E-15 & 3.03 & 7.03 & & & GlC & &\ 947 & 180 & 179 & 3.24E-13 & 7.46E-15 & 2.43 & 12.78 & 2.29 & 12.16 & GlC & \* & –\ 966 & 184 & 184 & 3.51E-12 & 9.21E-15 & 1.00 & 0.20 & 2.31 & 151.58 & XRB & &\ 1057 & 205 & 199 & 2.67E-12 & 2.05E-14 & 1.72 & 26.26 & 1.79 & 28.48 & GlC & \* & \*\ 1102 & 217 & 211 & 3.23E-13 & 2.93E-15 & 1.06 & 1.04 & 5.51 & 60.80 & GlC & \* &\ 1109 & 218 & 212 & 3.25E-13 & 9.08E-15 & 1.91 & 9.70 & 2.10 & 10.72 & GlC & \* & \*\ 1118 & 222 & 216 & 1.16E-13 & 2.03E-15 & 1.46 & 3.63 & 1.89 & 7.46 & GlC & &\ 1122 & 223 & 217 & 2.48E-13 & 2.72E-15 & 2.08 & 12.02 & 7.03 & 73.27 & GlC & &\ 1157 & 228 & 223 & 7.59E-13 & 4.48E-15 & 1.06 & 1.68 & 1.08 & 2.75 & GlC & \* & \*\ 1171 & 229 & 227 & 4.68E-13 & 4.93E-15 & 1.82 & 12.92 & 1.75 & 12.19 & GlC & \* & \*\ 1262 & 247 & 249 & 2.94E-14 & 3.10E-15 & 14.11 & 18.40 & 14.64 & 20.24 & & &\ 1267 & 247 & 249 & 4.80E-13 & 4.57E-15 & 1.16 & 3.04 & 1.11 & 2.44 & GlC & \* & \*\ 1289 & 250 & 254 & 2.88E-14 & 1.91E-15 & 1.16 & 0.57 & 1.98 & 3.15 & $<$GlC$>$ & \* &\ 1293 & 250 & 254 & 6.70E-15 & 9.42E-16 & 3.73 & 2.80 & 8.53 & 5.72 & $<$AGN$>$ & &\ 1296 & 253 & 257 & 3.89E-14 & 1.58E-15 & 4.03 & 9.38 & 1.25 & 1.17 & GlC & &\ 1297 & 252 & 258 & 5.59E-15 & 9.61E-16 & 4.46 & 2.69 & & & $<$hard$>$& &\ 1305 & & 258 & 1.69E-14 & 9.87E-16 & & & & & $<$GlC$>$ & &\ 1340 & 261 & 266 & 6.07E-14 & 3.01E-15 & 1.77 & 4.20 & 1.30 & 1.64 & GlC & & \*\ 1357 & & 258 & 7.04E-15 & 1.18E-15 & & & & & $<$hard$>$& &\ 1449 & 281 & 289 & 2.34E-14 & 1.00E-15 & 3.10 & 5.36 & 1.79 & 2.39 & fg Star & &\ 1463 & 282 & 290 & 7.51E-13 & 8.38E-15 & 1.13 & 3.33 & 1.33 & 7.64 & GlC & \* & \*\ 1634 & 302 & 316 & 7.70E-14 & 2.91E-15 & 3.14 & 5.60 & 1.89 & 4.33 & $<$hard$>$& \* & \*\ 1692 & 318 & 336 & 1.15E-12 & 2.00E-14 & 2.86 & 45.59 & 2.62 & 38.63 & GlC & \* &\ 1803 & 349 & 354 & 8.72E-13 & 9.17E-15 & 1.32 & 7.87 & 1.03 & 0.93 & GlC & \* & \*\ \[Tab:ROSAT\_GlC\_tvar\] Notes:\ $^{ *~}$: SI: SHP97, SII: SHL2001\ $^{ +~}$: XID Flux and error in ergcm$^{-2}$s$^{-1}$\ $^{ {\dagger}~}$: Variability factor and significance of variability, respectively, for comparisons of  XID fluxes to  fluxes listed in SPH97 and SHL2001, respectively.\ $^{ {\ddagger}~}$: An asterisk indicates that the XID flux is larger than the corresponding  flux.  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$ and ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$ The 18 X-ray sources correlating with globular clusters which were found in the  HRI observations (PFJ93) were all re-detected in the  data. From the numerous studies of X-ray globular cluster counterparts in 31 based on  observations , only eight sources are undetected in the present study. One of them (\[TP2004\] 1) is located far outside the field of 31 covered by the Deep  Survey. The transient nature of \[TP2004\] 35, and the fact that it is not observed in any  observation taken before December 2006 was mentioned in Sect.\[SubSec:Chcat\]. The six remaining sources (r2-15, r3-51, r3-71, \[VG2007\] 58, \[VG2007\] 65, \[VG2007\] 82) are located in the central area of 31 and are also not reported in PFH2005. Figure \[Fig:posGlCtrans\_pos\] shows the position of these six sources (in red) and the sources of the  catalogue (in yellow). If the brightness of the six sources had not changed between the  and  observations, they would be in principle bright enough to be detected by  in the merged observations of the central field, which have in total an exposure $\ga$ 100ks. Two sources (r2-15 and \[VG2007\] 65) are located next to sources detected by . Source r2-15 is located within 13 of  1012 and within 17 of  1017 and has – in the  observation – a similar luminosity to both  sources. The distance between  1012 and  1017 is 17 and within 20 of  1012,  detected source  1006, which is about a factor 4.6 fainter than  1012. Therefore, when in a bright state, source r2-15 should be detectable with . Source \[VG2007\] 65 is located within 17of  1100, which is at least 3.5 times brighter than \[VG2007\] 65. This may complicate the detection of \[VG2007\] 65 with . The variability of \[VG2007\] 58, \[VG2007\] 65, and \[VG2007\] 82 is supported by the fact that these three sources were not detected in any  study, prior to . Hence, these six sources are likely to be at least highly variable or even transient. Several sources identified with globular clusters in previous studies have counterparts in the  catalogue but are not classified as GlC sources by us. Source  403 (\[SHL2001\] 74) correlates with B007, which is now identified as a background galaxy [@2009AJ....137...94C; @2007AJ....134..706K RBC V3.5]. Sources  793 (\[SHL2001\] 136, s1-12) and  796 (s1-11) are the X-ray counterparts of B042D and B044D, respectively, which are also suggested as background objects by @2009AJ....137...94C. Source  948 (s1-83) correlates with B063D, which is listed as a globular cluster candidate in RBC V3.5, but might be a foreground star [@2009AJ....137...94C]. Due to this ambiguity in classification we classified the source as $<$hard$>$. Source  966 correlates with \[SHL2001\] 184, which was classified as the counterpart of the globular cluster NB21 (RBC V3.5) in the  PSPC survey (SHL2001). In addition, source  966 also correlates with the  source r2-26 [@2002ApJ...577..738K]. Due to the much better spatial resolution of  compared to , @2002ApJ...577..738K showed that source r2-26 does not correlate with the globular cluster NB21. identified this source as the first Z-source in 31. The nature of source  1078 is unclear as RBC V3.5 reported that source to be a foreground star, while @2009AJ....137...94C classified it as an old globular cluster. Due to this ambiguity in the classification and due to the fact that source  1078 is resolved into two  sources (r2-9, r2-10), we decided to classify the source as $<$hard$>$. Due to the transient nature [@2002ApJ...577..738K; @2006ApJ...643..356W] and the ambiguous classifications reported by RBC V3.5 (GlC) and @2009AJ....137...94C [H[II]{} region], we adopt the classification of PFH2005 ($<$XRB$>$) for source  1152. SBK2009 classified the source correlating with source  1293 as a globular cluster candidate. We are not able to confirm this classification, as none of the globular cluster catalogues used, contains an entry at the position of source  1293. Instead we found a radio counterpart in the catalogues of @2005ApJS..159..242G, @1990ApJS...72..761B and NVSS. We therefore classified the source as an AGN candidate, as was also done in PFH2005. For source  1449 (\[SHL2001\] 289) the situation is more complicated. SHL2001 report \[MA94a\] 380 as the globular cluster correlating with this X-ray source. Based on the same reference, @2005PASP..117.1236F included the optical source in their statistical study of globular cluster candidates. However, the paper with the acronym \[MA94a\] is not available. An intensive literature search of the papers by Magnier did not reveal any work relating to globular clusters in 31, apart from @1993PhDT........41M which is cited in @2005PASP..117.1236F as “MIT". In addition the source is not included in any other globular cluster catalogues listed in Sect.\[Sec:CrossCorr\_Tech\]. In the X-ray studies of @2004ApJ...609..735W and PFH2005 and in the source is classified as a foreground star (candidate). Hence, we also classified source  1449 as a foreground star candidate, but suggest optical follow-up observations of the source to clarify its true nature. A similar case is source  422 (\[SHL2001\] 76), which is classified as a globular cluster by SHL2001, based on a correlation with \[MA94a\] 16. Here again the source is not listed in any of the globular cluster catalogues used. We found one correlation of source  422 with an object in the USNO-B1 catalogue, which has no B2 and R2 magnitude. Two faint sources (V$>\!22.5$mag) of the LGGS catalogue are located within the X-ray positional error circle. Thus source  422 is classified as $<$hard$>$. While RBC V3.5 classified the optical counterpart of source  1634 (\[SHL2001\] 316) as a globular cluster candidate, @2009AJ....137...94C regard SK182C as being a source of unknown nature. Therefore we decided to classify source  1634 as $<$hard$>$. Conclusions {#Sec:Concl} =========== This paper presents the analysis of a large and deep  survey of the bright Local Group SA(s)b galaxy 31. The survey observations were taken between June 2006 and February 2008. Together with re-analysed archival observations, they provide for the first time full coverage of the M31 ${\mathrm}{D}_{25}$ ellipse down to a 0.2–4.5keV luminosity of $\sim$[$10^{35}$ erg s$^{-1}$]{}. The analysis of combined and individual observations allowed the study of faint persistent sources as well as brighter variable sources. The source catalogue of the Large  Survey of 31 contains 1897 sources in total, of which 914 sources were detected for the first time in X-rays. The XID source luminosities range from $\sim$4.4[$\times 10^{34}$ ]{} to 2.7[$\times 10^{38}$ ]{}. The previously found differences in the spatial distribution of bright ($\ga$[$10^{37}$ erg s$^{-1}$]{}) sources between the northern and southern disc could not be confirmed. The identification and classification of the sources was based on properties in the X-ray wavelength regime: hardness ratios, extent and temporal variability. In addition, information obtained from cross correlations with 31 catalogues in the radio, infra-red, optical and X-ray wavelength regimes were used. The source catalogue contains 12 sources with spatial extent between 62 and 230. From spectral investigation and comparison with optical images, five sources were classified as galaxy cluster candidates. 317 out of 1407 examined sources showed long term variability with a significance $>$3$\sigma$ between the  observations. These include 173 sources in the disc that were not covered in the study of the central field (SPH2008). Three sources located in the outskirts of the central field could not have been detected as variable in the study presented in SPH2008, as they only showed variability with a significance $>$3$\sigma$ between the archival and the “Large Project" observations. For 69 sources the flux varied by more than a factor of five between XMM-Newton observations; ten of these varied by a factor $>$100. Discrepancies in source detection between the Large  Survey catalogue and previous  catalogues could be explained by different search strategies, and differences in the processing of the data, in the parameter settings of the detection runs and in the software versions used. Correlations with previous  studies showed that those sources not detected in this study are strongly time variable, transient, or unresolved. This is particularly true for sources located close to the centre of 31, where ’s higher spatial resolution resolves more sources. Some of the undetected sources from previous  studies were located outside the field covered with . However, there were several sources detected by  that had a  detection likelihood larger than 15. If these sources were still in a bright state they should have been detected with . Thus, the fact that these sources are not detected with  implies that they are transient or at least highly variable sources. On the other hand 242 $<$hard$>$  sources were found with XID fluxes larger than [$10^{-14}$ erg cm$^{-2}$ s$^{-1}$]{}, which were not detected with . To study the properties of the different source populations of 31, it was necessary to separate foreground stars (40 plus 223 candidates) and background sources (11 AGN and 49 candidates, 4 galaxies and 19 candidates, 1 galaxy cluster and 5 candidates) from the sources of 31. 1247 sources could only be classified as $<$hard$>$, while 123 sources remained without identification or classification. The majority (about two-thirds, see Stiele et al. 2011 in preparation) of sources classified as $<$hard$>$ are expected to be background objects, especially AGN. The catalogue of the Large  survey of 31 contains 30 SSS candidates, with unabsorbed 0.2–1.0keV luminosities between 2.4[$\times 10^{35}$ ]{} and 2.8[$\times 10^{37}$ ]{}. SSSs are concentrated to the centre of 31, which can be explained by their correlation with optical novae, and by the overall spatial distribution of 31 late type stars ( enhanced density towards the centre). Of the 14 identifications made of optical novae, four were presented in more detail. The 25 identified and 31 classified SNRs had XID luminosities between 1.1[$\times 10^{35}$ ]{} and 4.3$\times$10$^{36}$ ergs$^{-1}$. Three of the 25 identified SNRs were detected for the first time in X-rays. For one SNR the  classification can be confirmed. Six of the SNR candidates were selected from correlations with sources in SNR catalogues from the literature. As these six sources had rather “hard" hardness ratios they are good candidates for “plerions". An investigation of the spatial distribution showed that most SNRs and candidates are located in regions of enhanced star formation, especially along the 10kpc dust ring in 31. This connection between SNRs and star forming regions, implies that most of the remnants are from type II supernovae. Most of the SNR classifications from previous studies have been confirmed. However, in five cases these classifications are doubtful. The population of “hard" 31 sources mainly consists of XRBs. These rather bright sources (XID luminosity range: 1.0[$\times 10^{36}$ ]{} to 2.7[$\times 10^{38}$ ]{}) were selected from their transient nature or strong long term variability (variability factor $>$10; 10 identified, 26 classified sources). The spectral properties of three transient sources were presented in more detail. A sub-class of LMXBs is located in globular clusters. They were selected from correlations with optical sources included in globular cluster catalogues (36 identified, 16 classified sources). The XID luminosity of GlCs ranges from 2.3[$\times 10^{35}$ ]{} to 1.0[$\times 10^{38}$ ]{}. The spatial distribution of this source class also showed an enhanced concentration to the centre of 31. From optical and X-ray colour-colour diagrams possible HMXB candidates were selected. If the sources were bright enough, an absorbed power-law model was fitted to the source spectra. Two of the candidates had a photon index consistent with the photon index range of NS HMXBs. Hence these two sources were suggested as new HMXB candidates. Follow-up studies in the optical as well as in radio are in progress or are planned. They will allow us to increase the number of identified sources and help us to classify or identify sources which can up to now only be classified as $<$hard$>$ or are without any classification. This work focused on the overall properties of the source population of individual classes and gave us deeper insights into the long-term variability, spatial and flux distribution of the sources in the field of 31 and thus helped us to improve our understanding of the X-ray source population of 31. This publication makes use of the USNOFS Image and Catalogue Archive operated by the United States Naval Observatory, Flagstaff Station (http://www.nofs.navy.mil/data/fchpix/), of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation, of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The XMM-Newton project is supported by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OX 0001) and the Max-Planck Society. HS acknowledges support by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OR 0405). \[App:XID\_Images\] [^1]: Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. [^2]: Tables 5 and 8 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ [^3]: <http://xmm.esac.esa.int/xsa/> [^4]: See also <http://xmm2.esac.esa.int/docs/documents/CAL-TN-0050-1-0.ps.gz> [^5]: This is a simplified description as [emldetect]{} transforms the derived likelihoods to equivalent likelihoods, corresponding to the case of two free parameters. This allows comparison between detection runs with different numbers of free parameters. [^6]: <http://heasarc.gsfc.gov/docs/xanadu/xspec> [^7]: <http://xmm2.esac.esa.int/external/xmm_sw_cal/calib/epic_files.shtml> [^8]: especially in the [emldetect]{} task [^9]: For the remainder of the subsection we will call all three catalogues “optical catalogues" for easier readability, although the 2MASS catalogue is an infrared catalogue. [^10]: From the LGGS catalogue only sources brighter than 21mag were used in order to be comparable to the brightness limit of the USNO-B1 catalogue. [^11]: the offset in declination is negligible [^12]: The combination of observations b1, b3 and b4 is called b. [^13]: <http://www.mpe.mpg.de/~m31novae/opt/m31/M31_table.html> [^14]: <http://www.cv.nrao.edu/nvss/NVSSlist.shtml> [^15]: <http://simbad.u-strasbg.fr/simbad> [^16]: <http://nedwww.ipac.caltech.edu> [^17]: in B2 magnitude [^18]: in B magnitude [^19]: The luminosity is based on XID Fluxes. Using the total 0.2–12keV band the result does not change (23 in the northern half and 24 in the southern half). [^20]: The source was observed with  on 11 January 2001. Obs. id.: 0065770101 [^21]: <http://www.mpe.mpg.de/~m31novae/opt/m31/M31_table.html> [^22]: <http://www.mpe.mpg.de/~m31novae/xray/index.php> [^23]: <http://www.supernovae.net/sn2005/novae.html> [^24]: <http://www.supernovae.net/sn2005/novae.html> [^25]: <http://cfa-www.harvard.edu/iau/CBAT_M31.html> [^26]: In addition \[SHL2001\] 258 correlates with  1275,  1289 and  1293. However these sources have each an additional  counterpart.

ArXiv

--- abstract: 'Gaussian mixture modeling is a fundamental tool in clustering, as well as discriminant analysis and semiparametric density estimation. However, estimating the optimal model for any given number of components is an NP-hard problem, and estimating the number of components is in some respects an even harder problem. In R, a popular package called `mclust` addresses both of these problems. However, Python has lacked such a package. We therefore introduce `AutoGMM`, a Python algorithm for automatic Gaussian mixture modeling. `AutoGMM` builds upon `scikit-learn`’s `AgglomerativeClustering` and `GaussianMixture` classes, with certain modifications to make the results more stable. Empirically, on several different applications, `AutoGMM` performs approximately as well as `mclust`. This algorithm is freely available and therefore further shrinks the gap between functionality of R and Python for data science.' author: - 'Thomas L. Athey' - 'Joshua T. Vogelstein$^,$' bibliography: - 'refs.bib' title: 'AutoGMM: Automatic Gaussian Mixture Modeling in Python' --- Introduction ============ Clustering is a fundamental problem in data analysis where a set of objects is partitioned into clusters according to similarities between the objects. Objects within a cluster are similar to each other, and objects across clusters are different, according to some criteria. Clustering has its roots in the 1960s [@cluster_og1; @cluster_og2], but is still researched heavily today [@cluster_review; @jain]. Clustering can be applied to many different problems such as separating potential customers into market segments [@cluster_market], segmenting satellite images to measure land cover [@cluster_satellite], or identifying when different images contain the same person [@cluster_face]. A popular technique for clustering is Gaussian mixture modeling. In this approach, a Gaussian mixture is fit to the observed data via maximum likelihood estimation. The flexibility of the Gaussian mixture model, however, comes at the cost hyperparameters that can be difficult to tune, and model assumptions that can be difficult to choose [@jain]. If users make assumptions about the model’s covariance matrices, they risk inappropriate model restriction. On the other hand, relaxing covariance assumptions leads to a large number of parameters to estimate. Users are also forced to choose the number of mixture components and how to initialize the estimation procedure. This paper presents `AutoGMM`, a Gaussian mixture model based algorithm implemented in python that automatically chooses the initialization, number of clusters and covariance constraints. Inspired by the `mclust` package in R [@mclust5], our algorithm iterates through different clustering options and cluster numbers and evaluates each according to the Bayesian Information Criterion. The algorithm starts with agglomerative clustering, then fits a Gaussian mixture model with a dynamic regularization scheme that discourages singleton clusters. We compared the algorithm to `mclust` on several datasets, and they perform similarly. Background ========== Gaussian Mixture Models ----------------------- The most popular statistical model of clustered data is the Gaussian mixture model (GMM). A Gaussian mixture is simply a composition of multiple normal distributions. Each component has a “weight”, $w_i$: the proportion of the overall data that belongs to that component. Therefore, the combined probability distribution, $f(x)$ is of the form: $$f(x) = \sum_{k=1}^{K} w_k f_k(x) = \sum_{k=1}^{K} \frac{w_k}{(2\pi)^{\frac{d}{2}}|\Sigma_k|^{-\frac{1}{2}}}\exp \left \{ {\frac{1}{2}(x-\mu_k)^T\Sigma_k^{-1}(x-\mu_k)} \right\}$$ where $k$ is the number of clusters, $d$ is the dimensionality of the data. The maximum likelihood estimate (MLE) of Gaussian mixture parameters cannot be directly computed, so the Expectation-Maximization (EM) algorithm is typically used to estimate model parameters [@mclachlan]. The EM algorithm is guaranteed to monotonically increase the likelihood with each iteration [@em]. A drawback of the EM algorithm, however, is that it can produce singular covariance matrices if not adequately constrained. The computational complexity of a single EM iteration with respect to the number of data points is $O(n)$. After running EM, the fitted GMM can be used to “hard cluster” data by calculating which mixture component was most likely to produce a data point. Soft clusterings of the data are also available upon running the EM algorithm, as each point is assigned a weight corresponding to all $k$ components. To initialize the EM algorithm, typically all points are assigned a cluster, which is then fed as input into the M-step. The key question in the initialization then becomes how to initially assign points to clusters. Initialization -------------- ### Random The simplest way to initialize the EM algorithm is by randomly choosing data points to serve as the initial mixture component means. This method is simple and fast, but different initializations can lead to drastically different results. In order to alleviate this issue, it is common to perform random initialization and subsequent EM several times, and choose the best result. However, there is no guarantee the random initializations will lead to satisfactory results, and running EM many times can be computationally costly. ### K-Means Another strategy is to use the k-means algorithm to initialize the mixture component means. K-means is perhaps the most popular clustering algorithm [@jain], and it seeks to minimize the squared distance within clusters. The k-means algorithm is usually fast, since the computational complexity of performing a fixed number iterations is $O(n)$ [@cluster_review]. K-means itself needs to be initialized, and k-means++ is a principled choice, since it bounds the k-means cost function [@kmeans++]. Since there is randomness in k-means++, running this algorithm on the same dataset may result in different clusterings. `GraSPy`, a Python package for graph statistics, performs EM initialization this way in its `GaussianCluster` class. ### Agglomerative Clustering Agglomerative clustering is a hierarchical technique that starts with every data point as its own cluster. Then, the two closest clusters are merged until the desired number of clusters is reached. In `scikit-learn`’s `AgglomerativeClustering` class, “closeness” between clusters can be quantified by L1 distance, L2 distance, or cosine similarity. Additionally, there are several linkage criteria that can be used to determine which clusters should be merged next. Complete linkage, which merges clusters according to the maximally distant data points within a pair of clusters, tends to find compact clusters of similar size. On the other hand, single linkage, which merges clusters according to the closest pairs of data points, is more likely to result in unbalanced clusters with more variable shape. Average linkage merges according to the average distance between points of different clusters, and Ward linkage merges clusters that cause the smallest increase in within-cluster variance. All four of these linkage criteria are implemented in `AgglomerativeClustering` and further comparisons between them can be found in @everitt. The computational complexity of agglomerative clustering can be prohibitive in large datasets [@xu]. Naively, agglomerative clustering has computational complexity of $\mc{O}(n^3)$. However, algorithmic improvements have improved this upper bound @hclust_eff. @scikit-learn uses minimum spanning tree and nearest neighbor chain methods to achieve $\mc{O}(n^2)$ complexity. Efforts to make faster agglomerative methods involve novel data structures [@birch], and cluster summary statistics [@cure], which approximate standard agglomeration methods. The algorithm in @mclust5 caps the number of data points on which it performs agglomeration by some number $N$. If the number of data points exceeds $N$, then it agglomerates a random subset of $N$ points, and uses those results to initialize the M step of the GMM initialization. So as $n$ increases beyond this cap, computational complexity of agglomeration remains constant with respect to $n$ per iteration. Covariance Constraints ---------------------- There are many possible constraints that can be made on the covariance matrices in Gaussian mixture modeling [@constraints; @mclust5]. Constraints lower the number of parameters in the model, which can reduce overfitting, but can introduce unnecessary bias. `scikit-learn`’s `GaussianMixture` class implements four covariance constraints (see Table \[tab:constraints\]). Constraint name Equivalent model in `mclust` Description ----------------- ------------------------------ ---------------------------------------------------------------- Full VVV Covariances are unconstrained and can vary between components. Tied EEE All components have the same, unconstrained, covariance. Diag VVI Covariances are diagonal and can vary between components Spherical VII Covariances are spherical and can vary between components Automatic Model Selection ------------------------- When clustering data, the user must decide how many clusters to use. In Gaussian mixture modeling, this cannot be done with the typical likelihood ratio test approach because mixture models do not satisfy regularity conditions [@mclachlan]. One approach to selecting the number of components is to use a Dirichlet process model [@rasmussen; @ferguson]. The Dirichlet process is an extension of the Dirichlet distribution which is the conjugate prior to the multinomial distribution. The Dirichlet process models the probability that a data point comes from the same mixture component as other data points, or a new component altogether. This approach requires approximating posterior probabilities of clusterings with a Markov Chain Monte Carlo method, which is rather computationally expensive. Another approach is to use metrics such as Bayesian information criterion (BIC) [@bic], or Akaike information criterion (AIC) [@aic]. BIC approximates the posterior probability of a model with a uniform prior, while AIC uses a prior that incorporates sample size and number of parameters [@aicbic]. From a practical perspective, BIC is more conservative because its penalty scales with $ln(n)$ and AIC does not directly depend on $n$. AIC and BIC can also be used to evaluate constraints on covariance matrices, unlike the Dirichlet process model. Our algorithm relies on BIC, as computed by: $$BIC = 2ln(\hat{L}) - p ln(n)$$ where $\hat{L}$ is the maximized data likelihood, $p$ is the number of parameters, and $n$ is the number of data points. We chose BIC as our evaluation criteria so we can make more direct comparisons with `mclust`, and because it performed empirically better than AIC on the datasets presented here (not shown). mclust ------ This work is directly inspired by `mclust`, a clustering package available only in R. The original `mclust` publication derived various agglomeration criteria from different covariance constraints [@mclust_original]. The different covariance constraints are denoted by three letter codes. For example, “EII” means that the covariance eigenvalues are *E*qual across mixture components, the covariance eigenvalues are *I*dentical to each other, and the orientation is given by the *I*dentity matrix (the eigenvectors are elements of the standard basis). In subsequent work, `mclust` was updated to include the fitting of GMMs, and the models were compared via BIC [@mclust_em]. Later, model selection was made according to a modified version of BIC that avoids singular covariance matrices [@mclust_regularize]. The most recent version of mclust was released in 2016 [@mclust5]. Comparing Clusterings --------------------- There are several ways to evaluate a given clustering, and they can be broadly divided into two categories. The first compares distances between points in the same cluster to distances between points in different clusters. The Silhouette Coefficient and the Davies-Bouldin Index are two examples of this type of metric. The second type of metric compares the estimated clustering to a ground truth clustering. Examples of this are Mutual Information, and Rand Index. The Rand Index is the fraction of times that two clusterings agree whether a pair of points are in the same cluster or different clusters. Adjusted Rand Index (ARI) corrects for chance and takes values in the interval $[-1,1]$. If the clusterings are identical, ARI is one, and if one of the clusterings is random, then the expected value of ARI is zero. Methods ======= Datasets {#sec:data} -------- We evaluate the performance of our algorithm as compared to `mclust` on three datasets. For each dataset, the algorithms search over all of their clustering options, and across all cluster numbers between 1 and 20. ### Synthetic Gaussian Mixture {#synthetic} For the synthetic Gaussian mixture dataset, we sampled 100 data points from a Gaussian mixture with three equally weighted components in three dimensions. All components have an identity covariance matrix and the means are: $$\mu_0 = \begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix},\; \mu_1 = \begin{bmatrix} 5 \\ 0 \\ 0\end{bmatrix},\; \mu_2 = \begin{bmatrix} 0 \\ 5 \\ 0\end{bmatrix},$$ We include this dataset to verify that the algorithms can cluster data with clear group structure. ### Wisconsin Breast Cancer Diagnostic Dataset {#bc} The Wisconsin Breast Cancer Diagnostic Dataset contains data from 569 breast masses that were biopsied with fine needle aspiration. Each data point includes 30 quantitative cytological features, and is labeled by the clinical diagnosis of benign or malignant. The dataset is available through the UCI Machine Learning Respository [@bc]. We include this dataset because it was used in one of the original `mclust` publications [@mclust_bc]. As in @mclust_bc, we only include the extreme area, extreme smoothness, and mean texture features. ### Spectral Embedding of Larval *Drosophila* Mushroom Body Connectome {#drosophila} @drosophila analyzes a *Drosophila* connectome that was obtained via electron microscopy [@drosophila_connectome]. As in @drosophila, we cluster the first six dimensions of the right hemisphere’s adjacency spectral embedding. The neuron types, Kenyon cells, input neurons, output neurons, and projection neurons, are considered the true clustering. `AutoGMM` --------- `AutoGMM` performs different combinations of clustering options and selects the method that results in the best BIC (see Appendix \[apdx:algs\] for details). 1. For each of the available 10 available agglomerative techniques (from\ `sklearn.cluster.AgglomerativeCluster`) perform initial clustering on up to $N$ points (user specified, default value is $2000$). We also perform 1 k-means clustering initialized with k-means++. 2. Compute cluster sample means and covariances in each clustering. These are the 11 initializations. 3. For each of the four available covariance constraints (from\ `sklearn.mixture.GaussianMixture`), initialize the M step of the EM algorithm with a result from Step 2. Run EM with no regularization. 1. If the EM algorithm diverges or any cluster contains only a single data point, restart the EM algorithm, this time adding $10^{-6}$ to the covariance matrices’ diagonals as regularization. 2. Increase the regularization by a factor of 10 if the EM algorithm diverges or any cluster contains only a single data point. If the regularization is increased beyond $10^0$, simply report that this GMM constraint has failed and proceed to Step 4. 3. If the EM algorithm successfully converges, save the resulting GMM and proceed to Step 4. 4. Repeat Step 3 for all of the 11 initializations from Step 2. 5. Repeat Steps 1-4 for all cluster numbers $k= 2\ldots 20$. 6. For each of the $11 \times 4 \times 20$ GMM’s that did not fail, compute BIC for the GMM. 7. Select the optimal clustering—the one with the largest BIC—as the triple of (i) initialization algorithm, (ii) number of clusters, and (iii) GMM covariance constraint. By default, `AutoGMM` iterates through all combinations of 11 agglomerative methods (Step 1), 4 EM methods (Step 3), and 20 cluster numbers (Step 5). However, users are allowed to restrict the set of options. `AutoGMM` limits the number of data points in the agglomerative step and limits the number of iterations of EM so the computational complexity with respect to number of data points is $O(n)$. The EM algorithm can run into singularities if clusters contain only a single element (“singleton clusters”), so the regularization scheme described above avoids such clusters. At first, EM is run with no regularization. However if this attempt fails, then EM is run again with regularization. As in @scikit-learn, regularization involves adding a regularization factor to the diagonal of the covariance matrices to ensure that they are positive. This method does not modify the eigenvectors of the covariance matrix, making it rotationally invariant [@ledoit]. `AutoGMM` has a `scikit-learn` compliant API and is freely available at <https://github.com/tathey1/graspy>. Reference Clustering Algorithms ------------------------------- We compare our algorithm to two other clustering algorithms. The first, `mclust v5.4.2`, is available on CRAN [@mclust5]. We use the package’s `Mclust` function. The second, which we call `GraSPyclust`, uses the `GaussianCluster` function in `GraSPy`. Since initialization is random in `GraSPyclust`, we perform clustering 50 times and select the result with the best BIC. Both of these algorithms limit the number EM iterations performed so their computational complexities are linear with respect to the number of data points. The data described in Section \[sec:data\] has underlying labels so we choose ARI to evaluate the clustering algorithms. Statistical Comparison ---------------------- In order to statistically compare the clustering methods, we evaluate their performances on random subsets of the data. For each dataset, we take ten independently generated, random subsamples, containing 80% of the total data points. We compile ARI and Runtime data from each clustering method on each subsample. Since the same subsamples are used by each clustering method, we can perform a Wilcoxon signed-rank test to statistically evaluate whether the methods perform differently on the datasets. We also cluster the complete data to analyze how each model was chosen, according to BIC. All algorithms were run on a single CPU core with 16GB RAM. Results {#sec:results} ======= Table \[tab:model\] shows the models that were chosen by each clustering algorithm on the complete datasets, and the corresponding BIC and ARI values. The actual clusterings are shown in Figures \[fig:synthetic\_cluster\]-\[fig:drosophila\_cluster\]. In the synthetic dataset, all three methods chose a spherical covariance constraint, which was the correct underlying covariance structure. The `GraSPyclust` algorithm, however, failed on this dataset, partitioning the data into 10 clusters. In the Wisconsin Breast Cancer dataset, the different algorithms chose component numbers of three or four, when there were only 2 underlying labels. All algorithms achieved similar BIC and ARI values. In the *Drosophila* dataset, all algorithms left the mixture model covariances completely unconstrained. Even though `AutoGMM` achieved the highest BIC, it had the lowest ARI. In both the Wisconsin Breast Cancer dataset and the *Drosophila* dataset, `AutoGMM` achieved ARI values between 0.5 and 0.6, which are not particularly impressive. In the *Drosophila* data, most of the disagreement between the `AutoGMM` clustering, and the neuron type classification arises from the subdivision of the Kenyon cell type into multiple subgroups (Figure \[fig:drosophila\_cluster\]). The authors of @drosophila, who used `mclust`, also note this result. --------------- ------------------ ----------- -------- ------------- --------- -- -- -- Dataset AutoGMM AutoGMM mclust GraSPyclust AutoGMM Synthetic L2, Ward Spherical EII Spherical 0 Breast Cancer None Diagonal VVI Diagonal 0 Drosophila Cosine, Complete Full VVV Full 0 --------------- ------------------ ----------- -------- ------------- --------- -- -- -- --------------- --------- -------- ------------- -- -- -- Dataset AutoGMM mclust GraSPyclust Synthetic 3 3 10 Breast Cancer 4 3 4 Drosophila 6 7 7 --------------- --------- -------- ------------- -- -- -- --------------- --------- -------- ------------- --------- -------- ------------- Dataset AutoGMM mclust GraSPyclust AutoGMM mclust GraSPyclust Synthetic -1120 -1111 -1100 1 1 0.64 Breast Cancer -8976 -8970 -8974 0.56 0.57 0.54 Drosophila 4608 4430 3299 0.5 0.57 0.55 --------------- --------- -------- ------------- --------- -------- ------------- ![Clustering results of different algorithms on the synthetic dataset (Section \[synthetic\]). **(b-c)** `AutoGMM` and `mclust` correctly clustered the data. **(c)** `GraSPyclust` erroneously subdivided the true clusters .[]{data-label="fig:synthetic_cluster"}](images/combined_synthetic_squarexcf.png){width="\textwidth"} ![Clustering results of different algorithms on the breast cancer dataset (Section \[bc\]). The original data was partitioned into two clusters (benign and malignant), but all algorithms here further subdivided the data into three or four clusters.[]{data-label="fig:bc_cluster"}](images/combined_bc_square.png){width="\textwidth"} ![Clustering results of different algorithms on the drosophila dataset (Section \[drosophila\]). There is considerable variation in the different algorithms’ results. One similarity, however, is that all algorithms subdivided the Kenyon cell cluster (red points in **(a)**) into several clusters.[]{data-label="fig:drosophila_cluster"}](images/combined_drosophila_square.png){width="\textwidth"} Figure \[fig:subset\] shows results from clustering random subsets of the data. The results were compared with the Wilcoxon signed-rank test at $\alpha=0.05$. On all three datasets, `AutoGMM` and `mclust` acheived similar ARI values. `GraSPyclust` resulted in lower ARI values on the synthetic dataset as compared to the `mclust`, but was not statistically different on the other datasets. Figure \[fig:subset\] shows that in all datasets, `mclust` was the fastest algorithm, and `GraSPyclust` was the second fastest algorithm. Figure \[fig:drosophila\_bicplot\] shows the BIC curves that demonstrate model selection of `AutoGMM` and `mclust` on the *Drosophila* dataset. We excluded the BIC curves from `GraSPyclust` for simplicity. The curves peak at the chosen models. ![By clustering random subsets of the data, ARI and Runtime values can be compared via the Wilcoxon signed-rank test ($\alpha=0.05$). **(a)** On the synthetic dataset, `GraSPyclust` had a significantly lower ARI than `mclust`. **(b), (c)** There were no statistically significant differences between the algorithms on the other datasets. **(d-f)** `mclust` was the fastest algorithm, and `AutoGMM` the slowest, on all three datasets. The p-value was the same (0.005) for all of the statistical tests in **(d-f)**[]{data-label="fig:subset"}](images/ari.png "fig:"){width="\textwidth"} ![By clustering random subsets of the data, ARI and Runtime values can be compared via the Wilcoxon signed-rank test ($\alpha=0.05$). **(a)** On the synthetic dataset, `GraSPyclust` had a significantly lower ARI than `mclust`. **(b), (c)** There were no statistically significant differences between the algorithms on the other datasets. **(d-f)** `mclust` was the fastest algorithm, and `AutoGMM` the slowest, on all three datasets. The p-value was the same (0.005) for all of the statistical tests in **(d-f)**[]{data-label="fig:subset"}](images/time.png "fig:"){width="\textwidth"} ![BIC values of all clustering options in `AutoGMM` and `mclust` on the *Drosophila* dataset. **(a)** There are 44 total clustering options in `AutoGMM`. Each curve corresponds to an agglomeration method, and each subplot corresponds to a covariance constraint (Table \[tab:constraints\]). **(b)** The 14 curves correspond to the 14 clustering options in `mclust`. The chosen models, from Table \[tab:model\], are marked with a vertical dashed line. Missing or truncated curves indicate that the algorithm did not converge to a satisfactory solution at those points.[]{data-label="fig:drosophila_bicplot"}](images/drosophila_v2_vert.png){width="80.00000%"} {width="60.00000%"} \[fig:runtimes\] We also investigated how algorithm runtimes scale with the number of data points. Figure \[fig:runtimes\] shows how the runtimes of all clustering options of the different algorithms scale with large datasets. We used linear regression on the log-log data to estimate computational complexity. The slopes for the `AutoGMM`, `mclust`, and `GraSPyclust` runtimes were 0.98, 0.86, and 1.09 respectively. This supports our calculations that the runtime of the three algorithms is linear with respect to $n$. Discussion ========== In this paper we present an algorithm, `AutoGMM`, that performs automatic model selection for Gaussian mixture modeling in Python. To the best of our knowledge, this algorithm is the first of its kind that is freely available in Python. `AutoGMM` iterates through 44 combinations of clustering options in Python’s `scikit-learn` package and chooses the GMM that achieves the best BIC. The algorithm avoids Gaussian mixtures whose likelihoods diverge, or have singleton clusters. `AutoGMM` was compared to `mclust`, a state of the art clustering package in R, and achieved similar BIC and ARI values on three datasets. Results from the synthetic Gaussian mixture (Table \[tab:model\], Figure \[fig:synthetic\_cluster\]) highlight the intuition behind `AutoGMM`’s regularization scheme. `GraSPyclust` did not perform well on the synthetic data, because it erroneously subdivided the 3 cluster data into 10 clusters. `AutoGMM` avoids this problem because its regularization does not allow singleton clusters. In all fairness, `GraSPyclust`’s performance on subsets of the synthetic data (Figure \[fig:subset\]) is much better than its performance on the complete data. However, its random initialization leaves it more susceptible to inconsistent results. Figure \[fig:subset\], shows that on our datasets, `mclust` is the fastest algorithm. However, computational complexity of all algorithms is linear with respect to the number of data points, and this is empirically validated by Figure \[fig:runtimes\]. Thus, `mclust` is faster by only a constant factor. Several features of the `mclust` algorithm contribute to this factor. One is that much of the computation in `mclust` is written in Fortran, a compiled programming language. `AutoGMM` and `GraSPyclust`, on the other hand, are written exclusively in Python, an interpreted programming language. Compiled programming languages are typically faster than interpreted ones. Another is that `mclust` evaluates fewer clustering methods (14) than `AutoGMM` (44). Indeed, using `AutoGMM` trades runtime for a larger space of GMMs. However, when users have an intuition into the structure of the data, they can restrict the modeling options and make the algorithm run faster. One opportunity to speed up the runtime of `AutoGMM` would involve a more principled approach to selecting the regularization factor. Currently, the algorithm iterates through 8 fixed regularization factors until it achieves a satisfactory solution. However, the algorithm could be improved to choose a regularization factor that is appropriate for the data at hand. The most obvious theoretical shortcoming of `AutoGMM` is that there is no explicit handling of outliers or singleton clusters. Since the algorithm does not allow for clusters with only one member, it may perform poorly with stray points of any sort. Future work to mitigate this problem could focus on the data or the model. Data-based approaches include preprocessing for outliers, or clustering subsets of the data. Alternatively, the model could be modified to allow singleton clusters, while still regularizing to avoid singularities. In the future, we are interested in high dimensional clustering using statistical models. The EM algorithm, a mainstay of GMM research, is even more likely to run into singularities in high dimensions, so we would need to modify `AutoGMM` accordingly. One possible approach would use random projections, as originally proposed by @dasgupta in the case where the mixture components have means that are adequately separated, and the same covariance. Another approach would involve computing spectral decompositions of the data, which can recover the true clustering and Gaussian mixture parameters under less restrictive conditions [@vempalaspectral],[@achspectral]. Algorithms {#apdx:algs} ========== - $X \in \R^{n \times d}$ - $n$ samples of $d$-dimensional data points. [Optional:]{} - $K_{min}$ - minimum number of clusters, default to 2. - $K_{max}$ - maximum number of clusters, default to 20. - affinities - affinities to use in agglomeration, subset of $[L2, L1, cosine, none]$, default to all. Note: $none$ indicates the k-means initialization option, so it is not paired with linkages. - linkages - linkages to use in agglomeration, subset of $[ward, complete, average, single]$, default to all. Note: $ward$ linkage is only compatible with $L2$ affinity. - cov\_constraints - covariance constraints to use in GMM, subset of $[full, tied, diag,spher$- $ical]$, default to all. <!-- --> - best\_clustering - label vector in $[1...K_{max}]^n$ indicating cluster assignment <!-- --> - best\_k - number of clusters - best\_init - initialization method - best\_cov\_constraint - covariance constraint - best\_reg\_covar - regularization factor continued on next page $ks$ = $[K_{min},...,K_{max}]$ best\_bic = $None$ $X\_subset$ = Subset($X$) init = sklearn.cluster.AgglomerativeCluster($k$,affinity,linkage).fit\_predict($X\_subset$) init = $None$ clustering, bic, reg\_covar = GaussianCluster($X$, $k$, cov\_constraint, init) best\_bic = bic best\_clustering = clustering best\_k = $k$ best\_init = \[affinity, linkage\] best\_cov\_constraint = cov\_constraint best\_reg\_covar = reg\_covar $X \in \R^{n \times d}$ - $n$ samples of $d$-dimensional data points. $X\_subset$ - random subset of the data. n\_samples = length($X$) $X\_subset = $ RandomSubset($X,2000$) $X\_subset = X$ - $X \in \R^{n \times d}$ - $n$ samples of $d$-dimensional data points. <!-- --> - $k$ - number of clusters. - cov\_constraint - covariance constraint. **Optional:** - init - initialization. <!-- --> - $y$ - cluster labels. <!-- --> - bic - Bayesian Information Criterion. - reg\_covar - regularization factor. weights\_init, means\_init, precisions\_init = EstimateGaussianParameters($X$, init) weights\_init, means\_init, precisions\_init = $None$,$None$,$None$ reg\_covar = $0$ bic = $None$ converged = $False$ gmm = sklearn.mixture.GaussianMixture(k,cov\_constraint,reg\_covar, weights\_init,means\_init,precisions\_init) $y$ = gmm.fit\_predict(X) reg\_covar = IncreaseReg(reg\_covar) bic = gmm.bic() converged = $True$ reg\_covar - current covariance regularization factor. reg\_covar\_new - incremented covariance regularization factor. reg\_covar\_new $=10^{-6}$ reg\_covar\_new $=$ reg\_covar $\times10$

ArXiv

--- abstract: 'The temporal characterization of ultrafast laser pulses has become a cornerstone capability of ultrafast optics laboratories and is routine both for optimizing laser pulse duration and designing custom fields. Beyond pure temporal characterization, spatio-temporal characterization provides a more complete measurement of the spatially-varying temporal properties of a laser pulse. These so-called spatio-temporal couplings (STCs) are generally nonseparable chromatic aberrations that can be induced by very common optical elements – for example diffraction gratings and thick lenses or prisms made from dispersive material. In this tutorial we introduce STCs and a detailed understanding of their behavior in order to have a background knowledge, but also to inform the design of characterization devices. We then overview a broad range of spatio-temporal characterization techniques with a view to mention most techniques, but also to provide greater details on a few chosen methods. The goal is to provide a reference and a comparison of various techniques for newcomers to the field. Lastly, we discuss nuances of analysis and visualization of spatio-temporal data, which is an often underappreciated and non-trivial part of ultrafast pulse characterization.' address: 'LIDYL, CEA, CNRS, Universit[é]{} Paris-Saclay, CEA Saclay, 91 191 Gif-sur-Yvette, France' author: - 'Spencer W. Jolly, Olivier Gobert, and Fabien Qu[é]{}r[é]{}' bibliography: - 'biblo\_tutorial.bib' --- originally from 2 March 2020, on arXiv 8 July 2020 Introduction {#sec:intro} ============ The frequency dependence of the spatial properties of a broadband light beam or of the optical response of a system is known as chromatism, and has been discussed for decades in many different fields of classical optics. In photography for example, chromatism of the imaging lens affects the ability to properly image an object illuminated by ambient white light, because slightly different images are produced for each color of the incident light. Due to the time-frequency uncertainty principle, ultrashort laser beams necessarily have significant spectral widths, and can therefore also be affected by chromatism. As for any other broadband light source, this impacts the spatial properties of the beam: if a chromatic ultrashort laser beam is focused by a perfect optic, its different frequency components are focused differently, resulting in a degradation of the spatial concentration of the laser light at focus. Yet, compared to incoherent broadband light, chromatism has further consequences for this peculiar type of light sources, now in the time domain: if the spectral properties (in amplitude and phase) of the laser beam are position-dependent, then by Fourier-transformation its temporal properties vary in space too. Such a dependence is known as a spatio-temporal coupling (STC), and implies that chromatism not only affects the concentration of light energy in space, but also its bunching in time, which is the key feature of ultrashort lasers. Properly assessing the impact of chromatism on ultrashort lasers therefore requires specific measurement methods, which give access to the full *spatio-temporal* structure of these beams. Developing such a spatio-temporal metrology, up to the point where it becomes part of the standard characterization routine of ultrashort lasers, is essential because STCs can have highly detrimental effects on the performance of these lasers. As is clear from previous qualitative analyses, they often have the effect of increasing the pulse duration and reducing intensity in focus [@bourassin-bouchet11], but can also have more complex yet very relevant effects, for example on pulse contrast [@li17; @li18-1]. On the other hand, STCs also provide extremely powerful ways of controlling the properties of light beams and therefore laser-matter interaction processes. Examples include optimization of non-colinear sum- or difference-frequency generation [@martinez89; @maznev98; @huangS-W12; @gobert14]), broadband THz generation [@stepanov03; @fulop14], isolated attosecond pulse generation by the attosecond lighthouse effect [@vincenti12; @wheeler12; @kim13; @quere14; @auguste16], improved non-linear microscopy using spatio-temporal focusing [@DURST20081796], and even laser machining [@sun18; @wangP18; @liQ19]. There is a broad collection of purely temporal laser diagnostics [@stibenz06; @walmsley09], which are meant to characterize the evolution of the electric field of a laser pulse in time. These measurements are generally either an average over a given aperture of the pulse, or essentially done at a single point (i.e. a small aperture), and therefore the result is only the local electric field resolved in time. These techniques include frequency-resolved optical gating (FROG) [@kane93; @trebino97; @oshea01; @bates10], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [@iaconis98; @gallmann99; @mairesse05; @radunsky07; @mahieu15], self-referenced spectral interferometry (SRSI, WIZZLER device) [@oksenhendler10; @moulet10; @trisorio12; @oksenhendler12], and D-Scan [@miranda11; @loriot13] among others. The devices and techniques to characterize a laser pulse spatio-temporally are often related to these purely temporal techniques, but also can employ completely separate schemes. Although not a pre-requisite, prior knowledge of temporal measurement techniques for ultrashort pulses will facilitate the reading of this tutorial. Extensive reviews, tutorials or even courses can be found in various past works [@Monmayrant_2010; @dorrer19]. This tutorial aims not to review the entire field of spatio-temporal metrology, especially since there has been an extremely comprehensive review done very recently [@dorrer19]. In contrast, it aims to introduce spatio-temporal couplings and a large range of techniques to diagnose them, in a manner to guide those without significant experience on this topic. We hope that scientists can use this tutorial to determine how to most simply and correctly diagnose or control spatio-temporal couplings in their specific situation. Section \[sec:concepts\] is mostly devoted to defining STCs in a pedagogical way, and introducing the characteristics of the most basic and common couplings. We finish this section by first touching upon techniques that require a minimal amount of specialized equipment, but may not be able to measure arbitrary STCs. In sections \[sec:spatial\] and \[sec:frequency\] we will then expand to more complete and advanced techniques, which are intended to determine the complete spatio-temporal structure of ultrashort laser beams. This ideally requires sampling a field in a three-dimensional space (two spatial coordinates, and time or frequency). This can be considered as one of the main difficulties of STC metrology, since the main light sensors available to date are cameras, which only have two dimensions. This problem has often been circumvented by resolving one spatial dimension only, obviously at the cost of a significant and potentially highly detrimental loss of information. Many present techniques are actually affected by this limitation, but will nonetheless be discussed in this tutorial due to their importance in the development of this field. Spatio-temporal or spatio-spectral metrology uses in general one of two methodologies: resolving a complete temporal or spectral characterization method in one (or more) spatial dimension(s) (’spatially-resolved spectral measurements’), or resolving the amplitude and phase of a spatial measurement at multiple frequencies (’frequency-resolved spatial measurements’). Although the separation based on these definitions can sometimes be difficult to distinguish, the two sections on ’complete’ techniques will be delineated according to our interpretation of these descriptions. The outcome of a complete measurement is a three-dimensional complex matrix describing the $E$-field of the laser beam in space-time or space-frequency. Interpreting and exploiting such a measurement result is far from straightforward, and the visualization and analysis of such datasets can therefore be considered as another significant difficulty of STC metrology. Specific tools have been developed over the last few years, and are summarized in the final section of this tutorial. Key concepts of spatio-temporal couplings and their metrology {#sec:concepts} ============================================================= Before discussing specific advanced methods to characterize the spatio-temporal properties of ultrashort laser pulses, it is necessary to understand exactly what STCs are, the implications on the beam properties in different parameter spaces, and the first very simple steps one might take to diagnose the presence of STCs, at least qualitatively. This is necessary to understand the capabilities of a given measurement device, i.e. it is crucial to understand what forms low- or high-order STCs may take at the measurement position. This is also helpful to finally analyze the result of any complete or incomplete measurement. The goal of any characterization device is to measure as completely as possible the 3-dimensional electric field of an ultrashort laser pulse $E$ in space and time $E(x,y,t)$, or in space and frequency $\hat{E}(x,y,\omega)$ (for the sake of simplicity, we will assume throughout this paper that the field is linearly-polarized, with the same polarization direction all across the beam). The quantities $E$ and $\hat{E}$ are related to each other by the one dimensional Fourier transform from time to frequency. We use $x$ and $y$ as the transverse dimensions, where the beam is propagating along $z$. Because a fully-characterized beam can be numerically propagated to any $z$, we are interested in the measurement of $E$ at only one $z$ that depends on the characterization device in use. In each case the field is composed of an amplitude term and a phase term, i.e.: $E(x,y,t)=\sqrt{I(x,y,t)}e^{i\phi(x,y,t)}$ and $\hat{E}(x,y,\omega)=\sqrt{\hat{I}(x,y,\omega)}e^{i\hat{\phi}(x,y,\omega)}$, where we will sometimes refer to the intensity $\hat{I}(x,y,\omega)$ or the amplitude $\hat{A}(x,y,\omega)=\sqrt{\hat{I}(x,y,\omega)}$. We will use these notations for the rest of the tutorial. The function $\hat{\phi}(x,y,\omega)$ is the ’spatio-spectral phase’, a crucial quantity for the properties of ultrashort laser beams. Much of the complexity of understanding and measuring STCs is actually concentrated in this function. It is closely related to the simple spectral phase $\hat{\phi}(\omega)$ provided by usual temporal measurement devices, in that $\hat{\phi}(\omega)$ is either the value of $\hat{\phi}(x,y,\omega)$ at a test position $(x_0, y_0)$, or a spatial average of this function over $x$ and $y$. Just as in the case of temporal metrology, it is a much simpler problem to measure only the spatio-spectral amplitude or intensity, but measuring both the amplitude and phase is more challenging and will be the topic of the two next sections of this tutorial. We feel that it is important to note finally that the term ’spatio-temporal’ and other similar versions of the term are often used to refer to the combination of measurements that are simply spatial and temporal. This has sometimes been the case in pulse characterization, but is much more often the case in fields that are more far-afield such as microscopy or spectroscopy, where it is less common to also have temporal information in the first place. Our definition is much stronger, i.e. in this tutorial a spatio-temporal measurement is not just the addition of a spatial measurement device and a temporal measurement device, but it is the measurement of the full spatio-temporal field (whether there are STCs present or not). The general concept of spatio-temporal coupling {#sec:concepts_general} ----------------------------------------------- For the purposes of this report we define the basic concept of what a spatio-temporal coupling actually is, in the most simple terms possible. That is: a spatio-temporal coupling is any property of an ultrashort laser pulse that results in the inability to describe the electric field of the laser pulse as a product of functions in space and time. Mathematically, if a beam has STCs, then the following statement is true: $$\label{eq:STC} E(x,y,t)\neq f(x,y)\times g(t) \quad \forall \quad f(x,y), g(t) .$$ In such a case, a similar inequality holds for $\hat{E}(x,y,\omega)$, since it is related to $E(x,y,t)$ by a simple Fourier transform with respect to time. In other words, as mentioned in the introduction, a beam with spatio-temporal couplings also has spatio-spectral couplings (i.e. chromatism), and we will often use these terms interchangeably. In fact the representation in frequency space is often the more convenient one to analyze the beam properties. An example of a nonseparable beam can be seen in a sketch in Fig. \[fig:STC\_concept\], where the beam in panel (a) has no STCs, and the beam in panel (b) does. The example with no STCs is perfectly described by separable functions $f(x,y)$ and $g(t)$ in space and time respectively. For the example in Fig. \[fig:STC\_concept\](b) there is both a varying arrival time of the pulse with the transverse dimension and some transverse variation in the temporal width. To account for the former effect, one may naively describe the field now in terms of $g(t-\tau_0(x))$ with $\gamma$ according to the magnitude of the tilt of the pulse. This is quite simple and potentially valid, but would still result in the full field $E(x,y,t)$ no longer being separable. ![Basic concept of STCs. Both panels show a sketch of the spatio-temporal electric field of an ultrashort laser beam. In (a), this a beam without STCs, where the full electric field can be expressed by separable functions, and the local pulse duration $\tau_0$ is valid globally. In (b), the beam has significant STCs, where the field is no longer separable and the local duration $\tau_0$ is different than the global duration $\tau_G$. The carrier wave here is a sketch and not meant to be to scale.[]{data-label="fig:STC_concept"}](STC_concept.pdf){width="83mm"} This distinction is simple to see when the mathematical descriptions of the fields are compared. We consider a Gaussian beam in space and time for convenience. If $r^2=x^2+y^2$ and the beam has a spatial width $w$, temporal width $\tau_0$, and central frequency $\omega_0$, then the case of Fig. \[fig:STC\_concept\](a) is written simply as $$\label{eq:GaussNoSTC} E_{1a}\propto e^{-r^2/w^2}e^{-t^2/\tau_0^2}e^{i\omega_0 t} .$$ This case is clearly separable. If the pulse has the properties shown in Fig. \[fig:STC\_concept\](b), then the field is written as $$\label{eq:GaussYesSTC} E_{1b}\propto e^{-r^2/w^2}e^{-(t-\tau_0(x))^2/\tau(r)^2}e^{i\omega_0 t} .$$ This is non-separable. As mentioned, this non-separability also has implications on the description of the electric field in frequency and space, but it then takes a different specific form, as will be further discussed in Section \[sec:concepts\_manifestations\]. Beyond having an impact on the mathematical description of the electric field, the presence of an STC will also affect measurable parameters. The most obvious is the temporal duration, which in the presence of some STCs could have spatial variation. It is not in the case of all STCs that the local duration will vary in space, but it is true that with any STC there will be a difference between the local pulse duration and the global pulse duration [@bourassin-bouchet11], referred to as $\tau_0$ and $\tau_G$ respectively in Fig. \[fig:STC\_concept\]. This generally results in a decrease in the peak intensity, and sometimes a varied spatial distribution of the different frequencies within the beam. The next few sections will discuss the nuances of the previous statements and classify a few of the well-known STCs. {width="171mm"} Introduction to low-order couplings {#sec:concepts_low-order} ----------------------------------- Here we introduce some common-place low-order STCs, which provide highly instructive examples. The term low-order refers to STCs where the field variations in space-time or space-frequency can be described by low-order polynomials of position coordinates and time/frequency, and that are therefore more likely to occur. For more complete analysis, we urge the reader to reference significant past work on describing and reviewing this topic [@akturk05; @gabolde07; @akturk10], and work that has gone over alternative matrix-based formalism specifically designed to describe dispersive optical systems [@kostenbauder90; @lin95; @marcus16]. The most prevalent and lowest-order STC is pulse-front tilt (PFT), where the duration of the beam is constant in space, but the arrival time varies linearly with one spatial dimension (Fig. \[fig:STC\_simple\](a)). With PFT, both the wavefront and the pulse-front (describing the location of the electromagnetic energy, i.e. the pulse envelope) of the beam are perfectly flat, but they are constantly at an angle to each other. In other words, the pulse front is tilted with respect to the propagation direction of the pulse. The next most common STC is pulse-front curvature (PFC), where the duration is still constant in space, but the arrival time now varies quadratically with the radial position (Fig. \[fig:STC\_simple\](b)). These two canonical STCs, PFT and PFC, can be caused by very simple and commonplace optical elements. For example, PFT can be induced via propagation through a wedged prism of any dispersive optical material (which includes glasses, the most ubiquitous optical materials), as seen in Fig. \[fig:STC\_simple\](a). The portion of the beam traveling though the thin part of the prism has traversed less material, so then the accumulated group-delay is less than that of the part of the beam passing through the thick portion of the prism. Because the thickness of the prism linearly depends on the transverse dimension in the plane of the page, then the accumulated group-delay will depend linearly on position as well, and this results in the rotation of the pulse front after propagation through the prism. For the same reason, the phase fronts also get tilted, but they do so according to the phase refractive index rather than the group refractive index. The output beam will have PFT if these two rotation angles are different, which occurs if the phase velocity $v_p$ is different than the group velocity $v_g$, i.e. the medium is dispersive. At higher orders, the output beam can also exhibit a spatially-dependent spectral chirp due to the different encountered thicknesses of glass. This generally has negligible impact compared to the PFT. PFC has a similar commonplace source, which is simple chromatic singlet lenses [@bor88; @bor89-1], as seen in Fig. \[fig:STC\_simple\](b). From the temporal point of view the portion of the beam at the outer edge of the lens will accrue less group delay than the center of the beam, resulting in a radially-varying arrival time. If the medium is dispersive ($v_p\neq v_g$) then the curvature of the pulse-front will be different than that for the phase-front after such a lens. Manifestations of couplings {#sec:concepts_manifestations} --------------------------- This section focuses on expanding the descriptions from the previous sections to be more quantitative and to describe couplings in different domains that are relevant for characterization devices and experiments. The main domains we will consider are the near-field (NF) where the beam is collimated (e.g. the output of a laser system), and the far-field (FF) where the beam is at a focus (e.g. where experiments are generally performed), both in time and frequency. The NF and FF of course have broader definitions in classical optics, but for simplicity we will refer to only these two planes as NF and FF throughout this tutorial. These NF and FF spaces defined in this way are related via the principles of Fourier optics, so that the NF is related to the FF by a two dimensional spatial Fourier transform and a coordinate change depending on the focal length ($x_\textrm{FF}=k_x \lambda f/2\pi$) [@doi:10.1002/0471213748.ch4]. Therefore the FF is technically equivalent to the $(k_x,k_y)$ reciprocal space at the NF plane (see Refs. [@akturk05; @akturk10]). However, from the authors’ point of view, considering different physical planes separated by propagation and focusing enables a simpler understanding and is physically more relevant than analyzing the field in different mathematical spaces. We must note that the previously mentioned Gaussian generalization [@akturk05; @akturk10] also provides important insight into the behavior of different couplings in different domains, and previous work has also gone into great detail specifically on the effect of couplings on the pulse duration in focus [@bourassin-bouchet11]. A beam that has no STCs can be described in a simple fashion in all four of the relevant domains (NF and FF, both in time and frequency). A beam with a Gaussian spatial distribution, flat wavefront, and a Gaussian temporal envelope in the NF will have the same temporal envelope in the FF and a Gaussian spatial distribution with a waist determined by the focusing conditions. The description in frequency will be similarly straightforward, regardless of if there is non-zero spectral phase. This is essentially the propagated and/or Fourier-transformed results of Eq. (\[eq:GaussNoSTC\]), where the Gaussian nature allows for analytical representations in all cases. We will take this example of a Gaussian STC-free beam as the baseline, which is pictured in all four domains, with amplitude and phase, in the top row (denoted with (a)) of Fig. \[fig:all\_couplings\]. The NF in frequency is in the pair of panels (i), the NF in time is in panels (ii), the FF in frequency is in (iii), and the FF in time is in (iv). We choose to describe a beam having a Fourier-limited duration $\tau_0$ and central wavelength $\omega_0$ such that $\omega_0\tau_0=10$ (i.e. the beam technically must be very broadband, but we do this to be able to visualize the carrier frequency in time). We plot in normalized units in order to ease the visualization, where for simplicity the NF and FF have a characteristic width noted as $w$ in both cases. In this section we will describe mostly the canonical couplings of PFT and PFC as well as one example beyond that, but we must stress that the final goal of any characterization device is to measure arbitrary couplings. Therefore the importance of this section is to develop the knowledge of how couplings manifest themselve in different spaces. The first very important intuition is for comparing the properties of a beam in time and in frequency domains, related by a 1D Fourier transform. To this end, in the cases of PFT (Fig. \[fig:all\_couplings\](b)(i)) and PFC (Fig. \[fig:all\_couplings\](c)(i)), we can first use a simple physical analysis of the optical systems that induce these couplings, before turning to a more formal description. As discussed in the previous section, PFT can be induced by a prism made of a dispersive glass. As is well-known, such a prism induces angular dispersion (AD), i.e. it results in different propagation directions at the prism output for the different frequency components. We can therefore expect PFT (time-domain description) to be equivalent to a frequency-dependent wavefront tilt (frequency-domain description). Similarly, PFC can be induced by a chromatic lens, which is known to induce a different wavefront curvature for the different frequency components (CC for chromatic curvature). We can therefore expect PFC (time-domain description) to be equivalent to a frequency-dependent wavefront curvature (frequency-domain description). We insist on the fact that PFT and AD (or similarly PFC and CC) correspond to the description of the very same beam, but considered in different spaces. Because of this equivalence, these canonical STCs will be referred to as AD/PFT and CC/PFC in the rest of this work. {width="171mm"} These correspondences between the time- and frequency-domain descriptions can of course be derived mathematically. A general derivation for arbitrary pulse-front distortions is provided in  \[sec:appendixA\] of this tutorial. This simple calculation shows that when considered in the frequency domain, a beam with AD/PFT is characterized by a spatio-spectral phase $\hat{\phi}(x,y,\omega)=\gamma x (\omega-\omega_0)$, where $\gamma$ represents the magnitude of the AD/PFT. This phase is plotted in Fig. \[fig:all\_couplings\](b)(i) and can be understood in two ways. One the one hand, this can be considered as a phase varying linearly in frequency, with a slope $\partial \hat{\phi}/\partial \omega$ (corresponding to a delay in the time domain) that varies linearly with position: this describes PFT. On the other hand, this can be considered as a phase varying linearly in position (i.e. a wavefront tilt), with a slope that varies linearly with frequency: this describes AD. Similarly, PFC is described by a spatio-spectral phase $\hat{\phi}(r,\omega)=\alpha r^2 (\omega-\omega_0)$, where $\alpha$ represents the magnitude of the CC/PFC. This is plotted in Fig. \[fig:all\_couplings\](c)(i), and can either be considered as a linear spectral phase with a slope varying quadratically with position (PFC), or as a quadratic spatial phase (wavefront curvature) varying linearly with frequency (CC). We have now emphasized multiple times that for a beam with STC, different frequencies have different spatial properties, and this has been nicely illustrated by the previous discussion on AD/PFT and CC/PFC. As result, when a beam affected by chromatism propagates, the different frequency components evolve differently. The beam’s spatio-spectral properties and spatio-temporal properties therefore change upon propagation. We now illustrate this point by considering the FF properties of beams that initially have AD/PFT and CC/PFC in the NF. To this end, we start from the frequency-domain description of these beams in the NF. For a beam with AD/PFT, the different frequencies have different wavefront tilt. Therefore, in the FF they must have a varying best-focus position in the transverse dimension. This is displayed in Fig. \[fig:all\_couplings\](b)(iii), and is known as ’transverse spatial chirp’. As a result of the transformation of the beam upon propagation, the temporal structure of the beam in the FF is also very different from that in the NF. In time, the pulse at focus no longer has any PFT, but has a longer local duration corresponding to the global duration in the NF. The focal spot is spatially larger than that of the perfect reference beam, since different frequencies are focused at different transverse positions. Finally, the spatio-temporal phase has a peculiar structure referred to as wavefront rotation (see the phase map of Fig. \[fig:all\_couplings\](b)(iv)). At negative times the spatial phase is tilted in one direction, and over time it changes to finally tilt in the opposite direction at positive times. This describes the fact that the propagation direction of light rotates in time on the scale of the pulse temporal envelope. This effect has interesting applications in high-intensity optics [@quere14], in particular for the generation of isolated attosecond pulses. For a beam with CC/PFC, the different frequencies have a different wavefront curvature in the NF. Therefore, in the FF they must have a varying best-focus position, now along the longitudinal dimension. At a single longitudinal position this manifests as a varying beam size according to frequency, and a spatio-spectral phase that represents the Guoy phase for each frequency. This is seen in panel (c)(iii) of Fig. \[fig:all\_couplings\]. The pulse in time at focus has a more complex amplitude profile, with a longer duration on-axis and a duration and arrival time that vary with the radial coordinate. Beyond the visualization just presented in Fig. \[fig:all\_couplings\], which had a flat spectral phase for at least one position in the beam, it could be such that a beam with either AD or CC were significantly chirped everywhere in space. This would not change much the spatio-spectral picture, since this simply corresponds to the addition of a spatially-homogeneous spectral phase, but would drastically change the picture in time. Because of this, pulse-front tilt and pulse-front curvature are only strictly proper names for these two couplings with no global chirp, and therefore angular dispersion or chromatic curvature are in a sense more general terms. To illustrate more complex cases, the fourth STC we look at (Fig. \[fig:all\_couplings\](d)) is a simple extension of the previous two couplings, where the spatio-spectral phase in the NF is now quadractic in frequency rather than linear (Fig. \[fig:all\_couplings\](d)(i)), i.e. $\hat{\phi}(x,y,\omega)=\zeta x (\omega-\omega_0)^2$. This can be understood as a transversely-varying linear temporal chirp, which in time corresponds to a transversely-varying pulse duration. This is also equivalent to the different colors having a wavefront tilt that varies quadratically with the frequency offset. In the focus this frequency-varying tilt manifests as a quadratically-varying best-focus position in the transverse dimension (Fig. \[fig:all\_couplings\](d)(iii)). In time at focus the pulse amplitude is quite complex, but the temporal phase no longer exhibits any chirp, because the chirps of different signs in the NF average-out at focus. The summaries above and the quantitative visualizations in Fig. \[fig:all\_couplings\] are on one hand relatively simple, and of low-order, but on the other hand can be quite difficult to digest in one sitting. However, understanding the difference between the spatio-spectral phases employed and the reasoning behind the relationships between time and frequency and also NF and FF is key to understanding STCs. This is true both of low-order STCs and those of arbitrary nature. All four of the cases in Fig. \[fig:all\_couplings\] have unique effects in focus and also in time on the collimated beam, but have identical spatio-spectral amplitudes in the NF. In the NF, it is *only* the spatio-spectral phases that differentiate them. From a practical point of view this makes sense, since we often imagine STCs being induced on the collimated beam in the form of chromatic phase aberrations, but in the general case we must also be open to more complex field configurations. In the next section, we briefly discuss some physically relevant cases which involve more complex couplings in the spectral domain. Examples of more complex couplings {#sec:concepts_complex} ---------------------------------- A simple example of pure amplitude coupling in the spectral domain and NF is the case of a beam that has been compressed by a single-pass grating compressor. In this case, the beam central frequency varies linearly with the transverse position, i.e. it has transverse chirp in the NF. If an overall temporal chirp is applied to such a beam, for instance by moving one of the gratings in the compressor, then the combination of these two effects obviously results in a tilt between the pulse-front and the wavefront [@akturk04] (Fig. \[fig:PFT\_akturk04\]). This is the same temporal intensity effect as in the ’standard’ AD/PFT, yet with a field configuration that is actually different both in the NF and FF. Hence this example is very instructive from the point of view of STC metrology, since a measurement of only the spatio-temporal intensity would not provide information on the full nature of the beam. ![A transversely varying central frequency and a spatially-homogeneous linear temporal chirp (quadratic spectral phase), shown in the top row, produce pulse-front tilt in time, shown in the bottom row. However, the field is different than the ’standard’ AD/PFT, despite the PFT in both cases. The color scale for phase goes from $-2\pi$ to $2\pi$.[]{data-label="fig:PFT_akturk04"}](PFT_akturk04.png){width="83mm"} It is important to stress as well that amplitude couplings can spontaneously arise when pulses are very broadband, even in the simple case of a freely propagating beam, due to the chromatic character of diffraction and propagation [@feng98; @porras02]. An example is the case of a broadband cavity operating with Kerr-lens mode-locking, where the Rayleigh range is fixed to be the same for all frequencies. This results in a beam-size that varies according to $\sqrt{\lambda}$ [@cundiff96]. This type of effect can become very significant when pulses approach the few-cycle limit, affecting even the Gouy phase and central frequency through a focus [@porras09; @hoff17]. Amplitude couplings can also easily occur in the misalignment of non-collinear OPAs [@harth18]. Beyond low-order couplings, high order couplings can have a myriad of sources, for example due to changes in laser gain medium [@tamer18] or temporal gain dynamics in highly-saturated Joule-level amplifiers [@jeandet19]. Simple or incomplete measurement techniques {#sec:concepts_measurements} ------------------------------------------- Before discussing advanced techniques, it is useful to discuss some experimentally simple techniques that can determine whether certain couplings are present, although not necessarily precisely their magnitudes. This is useful since these methods generally require very little specific or expensive devices and are therefore very accessible, and also apply to many of the real-world scenarios that scientists may encounter. The most well-known of these simple techniques is to diagnose the focus of an ultrashort laser beam with the full spectrum and compare to that with a narrow central part of the spectrum. In practice, this can be achieved using an appropriate band-pass spectral filter, placed in front of the sensor used to measure the focal spot profile. Referencing Fig. \[fig:all\_couplings\] can already hint that for both AD/PFT and CC/PFC in the NF, the effects of the couplings should be easily visible at the focus (FF). For AD/PFT the focus will be extended in one direction, i.e. elliptical, but will be round with only a narrow part of the original spectrum. Similarly a beam with CC/PFC will be larger than the expected diffraction-limited spot size in focus, but will get closer to this expectation with only a narrow part of the spectrum. In both cases an achromatic aberration may at first be suspected, for example astigmatism causing an elliptical focus, but the different nature of the focus with a narrower spectrum can make it possible to distinguish between chromatic and achromatic aberrations. ![An example of simple diagnostic of STCs, applied to two different cases. (a) and (b) are from the same focused beam with PFT, but (b) has a band-pass filter in front of the camera, while (a) is a measurement with the full beam spectrum. (c) and (d) compare the same types of measurements, now for a beam with PFC. In these two examples, the focus with the band-pass filter added shows the high quality of the focus, but the focus with the full spectrum reveals the coupling. Both sets of data are from different 800nm laser systems with large spectra, which have different focusing conditions. (c) and (d) are adapted from Ref. [@jolly20-1] The Optical Society.[]{data-label="fig:STC_focus_exp"}](STC_focus_exp.pdf){width="83mm"} Experimental examples of this for AD/PFT and CC/PFC in the NF (with different lasers and focusing conditions) can be seen in Fig. \[fig:STC\_focus\_exp\]. Due to transverse chirp, the elliptical focus obtained with the full spectrum in Fig. \[fig:STC\_focus\_exp\](a) is revealed to be round with only the central part of the spectrum in Fig. \[fig:STC\_focus\_exp\](b). Due to longitudinal chirp, the large beam in Fig. \[fig:STC\_focus\_exp\](c) is revealed to be smaller and more round with only the central part of the spectrum in Fig. \[fig:STC\_focus\_exp\](d). In this latter case of CC/PFC the beam with the entire spectrum (Fig. \[fig:STC\_focus\_exp\](c)) is more complex since the NF had a flat-top profile, so the frequencies not at best focus do not have a simple spatial distribution. Although this technique does show the presence of a coupling, only after a complex convolution of the spectrum and the measured profiles with and without the band-pass filter could one expect to quantify the coupling. Still, it can be a very useful technique due to the simplicity. This is why, when the source of the coupling is known and it is a simple step to tune the value, such a method can be very practical and useful. For example, minimizing the ellipticity of a beam with the full spectrum can be an indirect measure for minimizing AD/PFT when using a prism or grating compressor, where it is known that AD/PFT is very easily induced by misalignement. However, when the situation is more complex it cannot give much information, especially when the spectrum contains many features or there are a combination of multiple STCs and/or achromatic focusing aberrations. A further advancement of such a measurement was undertaken on a 100TW laser system and produced meaningful results, which was simply using an imaging spectrometer to spectrally resolve the focal spot along both spatial axes [@kahaly14]. Further extensions of this simple approach would consist in spatially scanning the beam in two dimensions with a fiber spectrometer, or scanning in one dimension with an imaging spectrometer, in order to reconstruct the full spatio-spectral amplitude. Although useful, the weakness of such measurements is that they do not provide any information on the spatio-spectral phase in the NF or FF. The measurement results of Ref. [@kahaly14], shown in Fig. \[fig:kahaly14\], provide an interesting illustration of this limitation. A curved spatio-spectral amplitude was observed in the FF, qualitatively similar to the case of the last coupling of Fig. \[fig:all\_couplings\](d). Yet, since the spatio-spectral phase remained unknown, there was no way to experimentally verify that the measured beam distortion was actually due to a spatially-varying temporal chirp in the NF. As a consequence, unambiguously identifying the nature and physical origin of this distortion in the laser system turned out to be impossible. ![Measurements of the spectrally-resolved focal spot profile along two slices in the FF of a 100TW laser beam (a). The slice shown in (b) has a quasi-parabolic dependence of central frequency on position, where the slice in (c) shows no significant STC. Results are taken from Ref. [@kahaly14] with permission.[]{data-label="fig:kahaly14"}](kahaly14.pdf){width="83mm"} There are many other examples of such ’incomplete’ measurements, of varying complexity, which produce results that are not complete representations of the pulse electric field. These include interferometric measurement of radial group-delay [@bor89-2; @netz00], extensions of single-shot autocorrelation to measure pulse-front tilt [@pretzler00; @sacks01; @akturk03; @figueira19] or pulse-front curvature [@wu16], multiple-slit spatio-temporal interferometry [@li18-2; @li19], There are more advanced diffractive methods that can do similar analysis, using a structured diffraction grating, referred to as “chromatic diversity” [@bahk18], or a measurement of angular chirp simultaneously in both spatial dimensions [@osvay05; @borzsonyi13]. There are also methods that are interested in only the temporal intensity profile (including the absolute intensity magnitude), for example the Temporally-Resolved Intensity Contouring (TRIC) technique [@haffa19]. Although of interest, such methods will not be discussed further in this tutorial. Spatially-resolved spectral measurement techniques {#sec:spatial} ================================================== We first address techniques that we deem are spatially-resolved spectral/temporal measurements, i.e. measurements that resolve the spectrum and spectral phase, extended to one or more spatial dimensions in order to resolve STCs. It is very important to emphasize that, maybe counter-intuitively, simply adding spatial resolution without caution to one of the usual techniques for purely temporal measurements, for instance by scanning the measurement device over space, actually does not provide a full spatio-temporal characterization of a laser beam. This is due to the fact that these techniques generally only measure the components of the spectral phase that affect the pulse duration and shape, but are blind to those that are constant or linear in frequency—which respectively correspond to the Carrier-Envelope relative Phase, and to the pulse arrival time. Therefore a device scanned across space would be able to detect the spatially varying envelope (due either to a change in nonlinear components of the spectral phase or to a change in the spectral width), but not something as simple as AD/PFT or CC/PFC, where the pulse shape does not vary at all spatially. ![Sketch explaining the necessity of spatial-temporal characterization. Since the temporal characterization device (“Device”) in this case is blind to spatial variations of the carrier phase and absolute arrival time of the true pulse in (a), the pulse-front tilt cannot be resolved in the reconstructed pulse shown in (b).[]{data-label="fig:STC_need"}](STC_need.pdf){width="83mm"} This important idea is illustrated via a sketch in Figure \[fig:STC\_need\], where the rastering of the device can resolve the more nuanced fluctuations in pulse length, but not the pulse-front tilt. Of course this may already be a useful amount of information, for example in pulse broadening in a plasma [@zair07; @beaurepaire16], but it is not a complete measurement. Furthermore, in practice the rastering process is itself limited due to the large number of measurements necessary to have a high resolution, especially if the measurement is performed on both transverse dimensions. We outline three different types of measurements that rely on spatially-resolving various methods of pulse characterization. Some of them make it possible to avoid the previous issue, while this can be a very tricky problem for others. The techniques are all based on forms of interferometry, so in every case the unknown beam needs to interfere either with a known reference, or with itself (so-called self-referenced interferometric techniques). One of the key challenges of this category of techniques is precisely to find ways to generate an appropriate reference beam. These techniques can be differentiated mainly by the type of reference used, and the method for resolving the measurement spatially and spectrally. These include: self-referenced techniques such as SPIDER or SRSI, resolved on an imaging spectrometer (section \[sec:SPIDER\], ’established techniques extended to spatial dimensions’); spectral interferometry raster-scanned over the spatial extent of a beam, which can be either externally-referenced or be referenced to a single point on the unknown beam (section \[sec:SEA-TADPOLE\], ’spatially-resolved spectral interferometry); and self-referenced Fourier-transform interferometry, where a spatially-extended reference is made from some central portion of the unknown beam and spectral resolution is obtained via Fourier-Transform spectroscopy (section \[sec:TERMITES\], ’spatially-resolved Fourier-transform interferometry’). Established techniques extended to spatial dimensions {#sec:SPIDER} ----------------------------------------------------- {width="171mm"} In this section we will discuss established techniques extended to spatial dimensions, including mainly the SPIDER and SRSI techniques. In each case the technique is expanded to one spatial dimension only, which already limits the information it provides. Regarding which STCs are accessible by this class of technique, there is some ambiguity in the literature, but we will provide our perspective here. SPIDER is a self-referenced interferometric technique for spectral/temporal measurements, where the reference beam consists of a spectrally-sheared version of the test pulse (TP) to be characterized. Implemented with a 1D spectrometer, this provides the local spectral amplitude and phase of the field in one shot [@iaconis98]. An obvious extension of this technique consists in rather using a 2D imaging spectrometer to obtain spatial resolution along one spatial axis. Historically, this has been one of the first approaches implemented for STC measurements  [@gallmann01], and this is why we discuss it first. We will however show that all spatially-resolved versions of SPIDER are affected by the limitation illustrated in Fig. \[fig:STC\_need\], and explain how only a more sophisticated measurement scheme can circumvent this limitation  [@dorrer02-3; @dorrer02-1; @dorrer02-2]. In order to understand the subtle issues involved in the generalization of SPIDER to STC measurements, it is useful to provide a more detailed description of the technique. To this end, we will focus on the particular implementation called SEA-F-SPIDER [@witting09-1], sketched in Fig. \[fig:SEA-SPIDER\], both because this is the latest and most advanced version, and because it has been used for the spatio-temporal characterization of few-cycle near-infrared and mid-infrared sources  [@witting12; @austin16; @witting16; @witting18]. For a review of the historical development of SPIDER and its different versions, and of the numerous practical advantages of SEA-F-SPIDER we refer the reader to Ref. [@Monmayrant_2010]. In any SPIDER device, the key operation is to create two replicas of the TP, which are sheared in frequency by a fraction of the TP’s spectral width. This is typically achieved by performing sum frequency generation of this test pulse in a second-order nonlinear crystal, with two quasi-monochromatic waves of slightly different frequencies. In SEA-F-SPIDER, these two waves are generated by producing two ancillary beams, obtained by passing two samples of the TP through separate narrowband spectral filters, placed at slightly different angles. For STC measurements, these ancillae can also be spatially filtered to avoid spatio-temporal distortions of the TP in the frequency conversion process  [@witting09-1; @wyatt11]. The technique then consists in comparing these two replicas, which is achieved by spectrally-resolved interferometry. As sketched in Fig. \[fig:SEA-SPIDER\], in SEA-SPIDER, the two replicas are recombined at an angle on the entrance slit on an imaging spectomerer, creating spatial fringes which encode the difference in spatio-spectral phase between the two beams. The mathematical expression of the interferogram measured in this scheme is: $$\begin{aligned} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}\left(x,\omega-\Omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}\left(x,\omega-\Omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)-k_x x\right] \label{eq:SEA-SPIDER}, \end{split}\end{aligned}$$ with $\Omega$ the induced spectral shear, and $k_x$ the transverse wave vector difference between the two beams. A straightforward processing, based on Fourier transformations and filtering and commonly used in interferometry, makes it possible to get the phase $\Delta \varphi(x,\omega)=\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)-k_x x$ from $S\left(x,\omega\right)$, provided $k_x$ is large enough. The next step, common to all implementations of SPIDER and called the calibration procedure, is to eliminate the phase term responsible for the interference fringes, which in the case of SEA-SPIDER is $-k_x x$. This phase term can be determined and then subtracted by performing a measurement of $S\left(x,\omega\right)$ without the spectral shear [@kosik05], leading to $\Delta \varphi(x,\omega)=-k_x x$. In SEA-F-SPIDER, this is achieved by simply rotating one of the spectral filters, so that both ancillae have the same frequency and no shear is induced between the two interfering replicas of the TP. We emphasize that this calibration step is crucial for SPIDER, and has to be performed with great care to make sure that no extra phase term is introduced in the procedure. This can become particularly tricky for STC measurements: in Ref. [@wyatt09], it was found that the SFG process of the TP with the spectrally-sheared ancillae in a non-collinear geometry introduces an extra phase term compared to the calibration configuration. This extra phase term corresponds to a spurious angular dispersion, and needs to be corrected for to get meaningful results. After this calibration step, and assuming no extra phase term has been introduced, one gets $\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)\approx \Omega \;\partial \hat{\phi}\left(x,\omega\right)/\partial \omega$. The spatio-spectral phase $\hat{\phi}\left(x,\omega\right)$ can then be obtained by an integration with respect to frequency, obviously up to an unknown frequency-independent phase term. In other words, this type of technique provides no information on the frequency-average wavefront of the beam. As we now explain, this unknown phase term prevents the determination of certain types of STC, such as pulse front distortions *in the measurement plane*. To demonstrate this point, we consider the simple case of a beam with AD/PFT in the measurement plane. As explained in section \[sec:concepts\_manifestations\] and demonstrated in  \[sec:appendixA\], such a beam is described in the spatio-spectral domain by a phase $\hat{\phi}\left(x,\omega\right)=\gamma (\omega-\omega_0) x$, with $\omega_0$ the central frequency of the beam. A SPIDER measurement will then provide $\partial \hat{\phi}\left(x,\omega\right)/\partial \omega= \gamma x$, whatever its specific implementation and assuming a perfect calibration procedure. It is then tempting to conclude that information about this coupling is indeed obtained, but this conclusion omits an essential point. As explained in  \[sec:appendixA\], a perfect STC-free beam that propagates at an angle (here with respect to optical axis of the SPIDER device) is described by a phase $\hat{\phi}\left(x,\omega\right)=\gamma \omega x$, which only differs from the phase of the beam with AD/PFT by a term independent of frequency. For such a beam, a SPIDER device also measures $\partial \hat{\phi}\left(x,\omega\right)/\partial \omega= \gamma x$. This shows that SPIDER cannot distinguish between a beam with AD/PFT and a simple tilted beam, due to the fact that it retrieves the phase up to an unknown frequency-independent term. Following the same reasoning, it is clear that it cannot either distinguish a beam with PFC from a converging or diverging beam. This is precisely the type of limitation illustrated in Fig. \[fig:STC\_need\]. Although this is a conceptual limitation of this technique, it is worth stressing that in most practical cases, this is probably not a significant shortcoming. Indeed, in SPIDER it is generally the laser field in the plane of the SFG crystal that is characterized. This corresponds to the far-field of the laser beam, while couplings such as PFT, PFC or other pulse front distortions typically occur in the NF and the nature of STCs change from NF to FF as explained in section \[sec:concepts\_manifestations\]. Techniques like SEA-F-SPIDER can therefore still provide very useful information on STC at focus. The measurement at focus of the peculiar PFT resulting from the combination of transverse spatial chirp and temporal chirp has for instance indeed been demonstrated [@witting16]. Overcoming this general limitation of the technique is possible, however, by combining a spatially-resolved SPIDER with a spectrally-resolved spatial-shearing interferometer [@dorrer02-3], in order to be able to measure not only $\partial\hat{\phi}/\partial\omega$ but also $\partial\hat{\phi}/\partial{x}$ [@dorrer02-1; @dorrer02-2]. This scheme is sometimes called 2D-SPIDER. From these two measurements it is possible to reconstruct the entire spatio-spectral phase along one spatial dimension. Considering again the comparison of a beam with AD/PFT and a tilted beam, these two cases can now be distinguished, since $\partial\hat{\phi}/\partial{x}=\gamma (\omega-\omega_0)$ in the first case, while $\partial\hat{\phi}/\partial{x}=\gamma \omega$ in the second. The technique is potentially single shot, but requires a rather complex optical set-up and a precise and very careful calibration, and has therefore not been in common use so far. SRSI [@oksenhendler10; @moulet10; @trisorio12] is another technique which is commonly used to determine the electric field of a pulse in time. In this technique, a nonlinear $\chi^{(3)}$ effect (cross-polarized wave generation, XPW [@minkovski04]) generates a temporally-filtered replica of the input TP, which is then used as a reference pulse for interferometric measurements. The key idea is that this reference pulse, which is spectrally broader than the TP, can be considered to have a nearly flat spectral phase. Imperfections of this reference pulse can be taken into account through the use of an iterative algorithm [@oksenhendler10]. In order to be able to obtain spatially-resolved measurement and potentially STCs, a tilt between the TP and the reference is implemented (SRSI-ETE) [@oksenhendler17], and the interferogram is measured with an imaging spectrometer. The interferogram obtained in this case at the sampled position $y_0$, with $k_x$ and $\tau$ representing the angle and temporal delay respectively, is given by: $$\begin{aligned} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}_{\textrm{XPW}}\left(x,\omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}_{\textrm{XPW}}\left(x,\omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}_{\textrm{XPW}}\left(x,\omega\right)-k_x x-\omega\tau\right] . \end{split}\end{aligned}$$ If the spectral phase of the reference is properly filtered by the XPW process, and with a proper calibration of $k_x x + \omega\tau$, this interferogram can be used to retrieve $\hat{\phi}\left(x,\omega\right)$. Yet, this technique should suffer from the same ambiguities as the SPIDER technique, again related to the issue emphasized in Fig. \[fig:STC\_need\]: the XPW process used to generate the reference does not filter the constant and linear terms of the spectral phase, corresponding in space-time to wave front and pulse front. As a result, the SRSI-ETE technique should not be able to resolve pulse-front distortions such as AD/PFT and CC/PFC. We note however that Ref. [@oksenhendler17] claimed a measurement of the former, which might have been possible in this specific implementation because the XPW process was carried out in the FF, and the SRSI-ETE measurement in the NF. The form of PFT due to the combination of spatial and temporal chirps would be resolved, since it is due to second-order spectral phase that would be filtered on the reference by the XPW process. There other examples of devices expanding upon the FROG technique, ImXFROG [@eilenberger13] and HcFROG [@mehta14]. However they have had very limited use and therefore fall outside of the scope of this tutorial. All the techniques described here have their own merits, but also tend to be experimentally complicated and require very precise calibration. Indeed, for various ambiguities it is not completely clear experimentally what the limits of the techniques are, and because they are self-referenced it is difficult to set a threshold for when the calibration has been done properly. So one should take great consideration when choosing if a technique in this section is suitable for one’s application. Spatially-resolved spectral interferometry {#sec:SEA-TADPOLE} ------------------------------------------ {width="171mm"} Spectral interferometry between an unknown pulse and a *known reference pulse* can provide the full spectral amplitude and phase information of the unknown pulse, and is a component of many standard temporal pulse characterization devices that have matured significantly to this point in time. We will outline devices that involve spatially-resolving spectral interferometry measurements in order to reconstruct the 3-D electric field, which are referred to as SEA-TADPOLE, STARFISH, and RED-SEA-TADPOLE. These are very different from the SEA-SPIDER and SRSI-ETE methods, in that they have no spectral or spatial shear, and therefore require an additional spectral phase measurement to reconstruct the complete spatio-temporal electric field, and spatial resolution is obtained by scanning the beam with an optical fiber. The original TADPOLE technique was developed to temporally characterize very weak pulses using a combination of a FROG measurement for a reference beam, and a single spectral interferogram between the reference and an unknown pulse [@fittinghoff96]. SEA-TADPOLE was then developed as a variant where the unknown and reference pulses are collected by monomode fibers, and then compared by spatio-spectral interferometry [@meshulach97] between the two separate fiber outputs [@bowlan06]. This was then naturally extended to spatio-temporal metrology by scanning one of the two fibers’ input across the spatial extent of the unknown beam, to measure its spatially-resolved spectral amplitude and phase [@bowlan07] through comparison with the fixed reference beam. The first measurements with this technique could achieve quite high spectral and spatial resolutions [@bowlan08]. The STARFISH technique [@alonso10; @alonso12; @alonso13] is essentially a simple variant of the same technique, which simplifies the experimental implementation by rather using standard fiber spectrometers and a fiber coupler, and relying on pure spectral interferometry, thus avoiding the need for a 2D spectrometer. In most practical cases, the reference beam is obtained from a part of the unknown beam itself, which is then separately characterized in time. We emphasize that this reference must cover the full spectral extent of the unknown beam, which can be an issue for the characterization of beams with strong inhomogeneities in spectral amplitude. A critical experimental issue is related to phase fluctuations in the monomode fibers, due e.g. to vibrations or temperature fluctuations. These are particularly difficult to avoid in these measurement schemes as at least one of the fibers needs to be scanned spatially—which necessarily implies a minimum amount of deformation. The main effect will generally be a randomly-varying overall phase term induced on the spectral phase of the test pulse as the beam is scanned. This implies that the wave front of the beam cannot be retrieved, and hence than STC such as PFT, PFC or any other pulse front distortions cannot be measured. A solution to this issue has been proposed and demonstrated in Ref. [@bowlan12], where SEA-TADPOLE measurements were done in multiple $z$ planes, and a standard phase retrieval algorithm (see section \[sec:phase\_retrieval\]) was then applied to the measurement results to retrieve the wavefront. This however requires scanning the beam transversely (in principle in 2D) with the fiber in *multiple planes*. The SEA-TADPOLE results cited here were generally applied on a focused beam or a beam near the focus, which is why the phase-retrieval technique proposed in Ref. [@bowlan12] was possible. However, if the beam to be measured has a pointing jitter such that fluctuations in the focus position become a significant fraction of the focal spot size, sampling this focus by scanning an optical fiber obviously becomes meaningless. Such a situation typically occurs on high-power ultrashort lasers. It is then necessary to rather scan the beam in the near-field, where the spatial extent is significantly larger. With a larger scanning range, the issues of phase fluctuations in the fiber, and of the stability and accuracy of the rastering stage and of the whole interferometer become much more critical. An evolution to the SEA-TADPOLE device, named RED-SEA-TADPOLE [@gallet14-2], was developed to solve these concerns and thus be able to apply this type of techniques to the collimated beams of e.g. high-power ultrashort lasers. The RED-SEA-TADPOLE device, shown in Fig. \[fig:SEA-TADPOLE\](a), uses a second reference beam which needs to fulfill stringent conditions (detailed in Ref. [@gallet14-2]), the main ones being that it must be in a spectral range different from that of the test beam, and that it has to be free of STCs. This reference beam is collected by the two fibers together with the unknown beam. The purpose of this spectrally distinct reference is to be able to independently measure imperfections in the spatial scanning and fluctuations in the fibers and to subtract them from the final measurement of the unknown beam. Essentially, since the spectrally distinct reference is homogeneous and free of STCs, any distortion of this beam retrieved by the measurement must be due to the stage wobbling or the fiber fluctuations. The production of a suitable spatially-extended reference beam with sufficient photon flux is however far from trivial. This was done in Ref. [@gallet14-2] with an expanded photonic-crystal fiber-based supercontinuum source and a band-pass filter outside of the band of the unknown laser pulse spectrum. So with the RED-SEA-TADPOLE device a large collimated beam can now be measured with all scanning imperfections accounted for. In Ref. [@gallet14-2] this was successfully utilized to measure both AD/PFT and CC/PFC due to a detuned grating compressor and a chromatic lens respectively. However, there remains one important issue that is relevant to SEA-TADPOLE, STARFISH, and RED-SEA-TADPOLE (despite the improvements in the latter). That is, due to ambiguities in phase-unwrapping, the spatial scanning resolution necessary to truly resolve an unknown pulse in the nearfield are very stringent. This phase unwrapping ambiguity is shown in Fig. \[fig:SEA-TADPOLE\](b-c). If the measured pulse is simply tilted with respect to the plane of the rastering, then the true phase across one spatial cut will be linear with a slope depending on the tilt (shown in the dashed lines). This is relevant because in practice it is extremely difficult to align the plane of a 2D stage exactly with the plane perpendicular with the laser propagation. The raw measured phase, which is necessarily wrapped, will require unwrapping to resolve this tilt. When the beam is scanned with high resolution as in Fig. \[fig:SEA-TADPOLE\](b), there is no ambiguity in the phase unwrapping, so the result is correct. However, when the scanning resolution is much lower as in Fig. \[fig:SEA-TADPOLE\](c), the wrapped phase is no longer unambiguous, because it is not known a priori how many $2\pi$ phase jumps occur between each measurement point. When the phase is unwrapped there is a high likelyhood of producing an incorrect unwrapped phase. This is a general issue in phase unwrapping, but in the context of RED-SEA-TADPOLE combined with a large beam in the NF, this implies that rastering in both tranverse dimension requires millions of sample points. This issue means that for practical purposed SEA-TADPOLE on a collimated beam is often just too inconvenient to implement. However, for beams in-focus it is still a reasonable solution, because in such a configuration having a very high spatial sampling is practically achievable [@bowlan08]. We however note that in all measurements performed with these techniques so far, the beam was in practice sampled along one spatial direction only, due to the burden of finely scanning the fiber tip in 2D. As a last point, it is theoretically possible to create a multiplexed version of SEA-TADPOLE for measurements in the near field, to reduce or eliminate the need for scanning in 2D, referred to as MUFFIN in Ref. [@gallet14-1]. But in this case the same difficulties regarding the phase-jumps in space exist, and the experimental complexity of adding fibers and the lack of high spatial resolution in the final data make it not very attractive in the end, and in fact it was never implemented for the characterization of STCs. Spatially-resolved Fourier-transform spectroscopy {#sec:TERMITES} ------------------------------------------------- A major drawback of the previous techniques is that the unknown beam needs to be scanned spatially in 2D to measure the full 3D spatio-temporal or spatio-spectral field. It would obviously be more straightforward to directly resolve the two transverse spatial coordinates of the beam on a camera. But one then needs to find a way to resolve the third coordinate, i.e. to get spectral resolution. We now explain how this can be achieved by exploiting Fourier-transform spectroscopy (FTS), leading to different techniques where only one physical parameter—a temporal delay- now needs to be scanned, described in this section and later in section \[sec:phase\_retrieval\]. FTS is a powerful method to resolve the spectrum of an unknown beam by measuring the evolving signal on a photodiode as the unknown beam temporally interferes with a delayed copy of itself. The resulting interferometic trace, composed of the signal measured at all of the scanned delays $\tau$, contains the first-order autocorrelation function of the field, and can be Fourier-transformed to frequency in order to directly produce the spectral intensity $\hat{I}(\omega)$, when selecting only the spectral information at the positive frequency peak. More explicitly: $$\hat{I}(\omega)=\left\lbrace \mathcal{F}_{\tau\rightarrow\omega}\left[\int \left|E(t)+E(t-\tau)\right|^2 dt\right]\right\rbrace_{+\omega} ,\label{eq:FTS}$$ which is essentially the Wiener-Khinchin theorem [@doi:10.1002/0471213748.ch11]. Since the beams interfering are copies of each other, the resulting beam has a zero spectral phase regardless of the phase on the input beam. This has the benefit that the spectral phase of the beam to be measured does not matter, but of course the downside is that FTS cannot resolve the spectral phase. Since FTS is a scanning method there are significant implications of shot-to-shot fluctuations and noise on the resulting spectrum [@dorrer00]. This makes FTS generally less preferred to simple fiber-coupled grating spectrometers (when available) to measure the 1-D spectrum, but in the case of full spatio-temporal characterization it has found a new application. FTS can indeed be spatially-resolved quite easily, just by resolving the interfering beams on a camera. When the interfering beams are exact copies of each other, this provides a straightforward way to measure the spatio-spectral intensity. However, there is no phase information just as in the 1-D case. A self-referenced version of spatially-resolved FTS that can resolve the spatio-spectral phase was developed simultaneously and independently in  [@miranda14] and  [@gallet-thesis], and later named TERMITES [@pariente16], which will be the focus of the rest of this section. TERMITES has been used successfully with varying specific experimental layouts for measurements on a Terawatt laser [@pariente16], a Petawatt laser [@jeandet19], ultrafast optical vortices [@miranda17], and a femtosecond OPCPA used for high repetition-rate high-order harmonic generation [@harth18]. In each case it could resolve both standard STCs and more complex couplings in both amplitude and phase. The TERMITES technique involves interfering the unknown beam with a spatial portion of itself, which has been expanded to overlap across the whole spatial extent of the beam. The key idea is that this expanded portion of the original beam can be used as a reference, because it comes from a small enough portion of the original beam to be free from STCs. Since the reference is then not strictly an exact copy of the unknown beam, the spatially-resolved FTS will have both amplitude and phase information, as will be explained more precisely below. Comparing TERMITES to SEA-TADPOLE, the reference in TERMITES has been expanded spatially so that on a single 2D image there is interference at all points, eliminating the need to scan in 2D that was present in SEA-TADPOLE. {width="171mm"} An example of experimental implementation of TERMITES using a modified Michelson interferometer with one curved end-mirror is shown in Fig. \[fig:TERMITES\_1\](a). The overlapping beam and reference are either viewed directly on a camera chip, or put on to a scattering screen which is then imaged using a viewing objective and camera. The FTS is performed by stepping through the delay $\tau$ with steps small enough to beat the Shannon limit. Therefore significantly sub-cycle accuracy on the delay stage is essential: for characterizing 800nm wavelength beams we have employed steps of 150nm using a piezo-electric stage with fluctuations on the 10nm level [@pariente16]. In the end a TERMITES measurement produces a 3-D matrix of interferometric signals in $x$, $y$, and $\tau$. This data represents the spatially-resolved interferences of the unknown beam and the reference, with the addition of a large curvature value due to the fact that the reference is diverging. The first analysis steps involve Fourier-transforming from $\tau$ to $\omega$, selecting only relevant positive frequencies, and subtracting the known curvature. These steps are shown in Fig. \[fig:TERMITES\_1\](b-d). After these steps there remains the “cross-spectral density” $\hat{s}(x,y,\omega)$, which corresponds to the product of the complex spectral amplitude of the unknown beam, and the conjugate of the complex spectral amplitude of the reference beam. Algorithmically this corresponds to $$\begin{aligned} \label{termites-equation} \hat{s}(x,y,\omega)&=\hat{E}(x,y,\omega)\hat{E}_R^*(x,y,\omega) \\ &=A(x,y,\omega)A_R(x,y,\omega)e^{\hat{\phi}(x,y,\omega)-\hat{\phi}_R(x,y,\omega)} ,\end{aligned}$$ where the curvature of the reference beam has already been removed. At each point of the beam, one can thus get the difference in spectral phase $\hat{\phi}(x,y,\omega)-\hat{\phi}_R(x,y,\omega)$ between the unknown and reference beams. This is why we include TERMITES in the category of spatially-resolved spectral measurement techniques. The final step of the data processing leading to the reconstruction of the unknown beam depends on the way TERMITES is implemented. In the first version utilized in Lund [@miranda14], the reference came from a very small portion of the original beam, which was moreover spatially filtered before interfering with the unknown beam. The reference beam can then be considered as originating from a point source, such that it can be assumed to be free from STC, i.e. $\hat{\phi}_R(x,y,\omega)=\hat{\phi}_R^0(\omega)$ (and similarly for the spatio-spectral amplitude). Since TERMITES is self-referenced, it is blind to any spatially homogeneous spectral phase of the unknown beam. Therefore, to reconstruct the field in the spatio-temporal domain, a single temporal measurement is still necessary, either on the reference beam or at any single point of the unknown beam (see section \[sec:phase-stitching\], phase stitching). Note however that even without this final measurement, all pure spatio-spectral effects are resolved. In the implementation presented in Fig. \[fig:TERMITES\_1\](a), the reference rather comes from some finite central portion of the unknown beam, due to the practical constraints imposed by the application to a laser beam of large diameter. For instance, in the version used in Ref. [@pariente16], the reference came from the central half of the beam. This scheme is then similar to radial-shearing interferometry [@doi:10.1002/9780470135976.ch5], and the reference may itself still contain STCs. Based on the fact that it originates from a sub-pupil of the unknown beam, an iterative algorithm can be applied to the spectral amplitude and phase provided by Eq. (\[termites-equation\]), to eliminate the contribution of the reference and reconstruct the complex spatio-spectral field of the unknown beam. As before, a temporal measurement at a single point of the beam is still required to determine the field in space-time. There are strict requirements on the camera properties in a TERMITES device. Due to the varying angle between the reference and unknown beam, the interference fringes increase in spatial frequency towards the outer part of the beam (see example images in Fig. \[fig:TERMITES\_1\](b)). The input beam diameter of the collimated beam $D$, the focal length $f$ of the convex mirror for the reference, and the fraction $\beta$ of the unknown beam diameter used to produce the reference fix the pixel size required to resolve the fringes at the outer part of the beam. If we say at the edge of the beam $p$ pixels are required for each fringe, then the linear number of pixels needed across the beam is $N=pD^2\beta/2\lambda_0|f|=pD^2(1-\beta)/2\lambda_0 L$, which leads to a total image size of at least $N^2$ pixels (see supplementary material of Ref. [@pariente16], and note the geometric constraint of $\beta=|f|/(|f|+L)$, where L is related to the total path length of the device). This can lead to requirements of tens of megapixels for beam diameters of a few centimeters, having a big impact on the data size of the final measurement (routinely many Gigabytes for a single measurement). A nuanced distinction regarding the required pixels, which is not discussed in previous work, is that the true constraint is on the size of the pixels such that the camera signal is not averaging over a significant portion of a spatial interference fringe. If a camera chip was decimated so that fewer total pixels were chosen, but the pixel size was still small enough, then the proper signal would still be measured (although with lower resolution), now with a reduced data transfer time and smaller final data size. Additionally, the pixel size at the center of the beam could theoretically be much larger, since the constraint on pixel size is only an important constraint near the edge of the beam. But in reality both of these cases would require more advanced analysis or expensive hardware, and have not been presently investigated. There is also a geometry of TERMITES where rather than the unknown beam staying collimated and the reference being diverging, the curved mirror can have a positive focal length and is used to reflect the unknown beam, such that the reference now remains collimated while the unknown beam converges. In this case the reference is still larger than the unknown beam on the camera, but the overall size of the relevant image is smaller. This is shown in Figure \[fig:TERMITES\_1\](e). The advantage of such a setup is that the size of the camera chip can be much smaller. If $\beta$ is still defined as the ratio of the unknown beam size to the reference beam size on the camera, a simple calculation shows that the requirement for the linear number of pixels across the beam in this configuration is $N=pD^2\beta/2\lambda_0|f|=pD^2\beta(1-\beta)/2\lambda_0 L$. So essentially, in this converging setup with the same device size determined by $L$, the number of pixels across the beam can be reduced by a factor of $\beta$ compared to the diverging setup, resulting in a total image with $\beta^2$ fewer total pixels. Due to the fixed geometrical relationship between $\beta$, $f$, and $L$ the focal length of the converging mirror must be longer to have a device with the same $\beta$ and $L$ ($\beta=(f-L)/f$ in the converging case). The most important parameter when designing a TERMITES device is the fraction $\beta$ of the unknown beam diameter used to produce the reference. For example, a smaller $\beta$ will make the iterative algorithm more likely to converge to the true physical result since the reference would be from a smaller portion of the unknown beam, and therefore more likely to be free of STCs. But according to the previous discussion, a small $\beta$ will also result in more stringent geometric constraints, potentially making the setup untenable in size or price. There are also limits on $\beta$ since the best signal-to-noise ratio in the computed complex spectrum is when there is perfect fringe contrast. However, since the reference is diverging this requires the beamsplitter in the interferometer to be different than 50:50. As $\beta$ become smaller this becomes a bigger issue and one must eventually compromise with non-ideal fringe contrast. We now emphasize certain advantages and limitations of the TERMITES technique. Because of the strict requirements on pixel size the device will generally have a very good spatial resolution. This totally eliminates the phase unwrapping ambiguity previously encountered with SEA-TADPOLE and its variants, and will also provide TERMITES with remarkable ability to resolve very fine spatial features, as demonstrated in Ref.[@jeandet19]. On the other hand, it also causes the initial data files to be in the tens of Gigabytes. From a physical point of view, the spectrum of the whole beam can only be resolved properly if the reference is as spectrally broad or broader than the unknown beam. So, if the center of the unknown beam where the reference is taken from has a narrower spectrum than the outer portions, then the resulting spectrum after all analysis steps will be narrower than in reality. The TERMITES technique (and FTS in general) requires taking camera images based on many successive laser shots. Because of this any fluctuations of parameters can have an effect on the final result: intensity fluctuations can be accounted for using a portion of the camera image that is not interfering or by using the integrated signal on the detector, but fluctuations in spectrum, pointing, wavefront, or the STCs on the pulse could have a significant effect. Additionally, since delay steps are not perfect, fluctuations in the delay above a certain level can cause a degradation of the retrieved spectrum [@dorrer00]. Despite this inherent shortcoming of being multi-shot, the few measurements on very large laser systems that tend to have non-trivial fluctuations have been successful [@jeandet19; @pariente16]. ![The SEA-TERMITES technique is a single-shot version of TERMITES whereby an imaging spectrometer resolves the cross-spectral density $\hat{s}$ in only one spatial dimension. The idea was proposed in Ref. [@gallet-thesis], with the visualization style in this figure adopted from Ref. [@pariente16].[]{data-label="fig:TERMITES_2"}](SEA-TERMITES.pdf){width="83mm"} Category Technique(s) Spatial dim. Spectral information obtained by Single-shot? Complete? Type of reference beam -------------------------------------------- --------------------------------- ---------------- ---------------------------------- -------------- -------------------- -------------------------------------- Extension of established techniques 2D-SPIDER, SEA-SPIDER, SRSI-ETE 1D Spectrometer Yes No Shearing of unknown beam, XPW effect Spatially-resolved spectral-interferometry SEA-TADPOLE, STARFISH 2D, usually 1D Spectrometer No Yes, but difficult Independent or part of unknown beam Spatially-resolved FTS TERMITES 2D FTS No Yes Part of unknown beam As a last discussion related to the TERMITES principle, there exists a version of TERMITES that is single-shot, but only resolved in one spatial dimension. We refer to this as SEA-TERMITES [@gallet-thesis], which is pictured in Fig. \[fig:TERMITES\_2\]. The concept is simple, rather than scanning over $\tau$ the device can be set up at a single delay and an imaging spectrometer can be installed at the location of the screen or camera. Then a single-shot measurement will produce the TERMITES data at one spatial slice via spectral interferometry. Following this measurement a similar analysis procedure must be followed (except that the data is already resolved in frequency) with vastly decreased data size and therefore speed, with the price of loosing information in one spatial dimension. In the case of TERMITES the success of the iterative algorithm requires that the measured slice is precisely that going through the mutual center of the reference and the unknown beam, and cylindrical symmetry must be assumed. This same equivalence between spectral interferometry and FTS will be seen again briefly in the case of the INSIGHT device, with the same trade-off of being single-shot, but losing one dimension of information and having restrictions on the symmetry of the beam. As a conclusion of this section, the main properties of the different techniques discussed in this section are summarized in Table \[tab:table1\]. Frequency-resolved spatial measurement techniques {#sec:frequency} ================================================= The second major classification of characterization methods corresponds to the techniques that can be referred to as frequency-resolved spatial measurements. This mostly takes the form of techniques that frequency resolve a measurement that is generally used to determine the spatial amplitude and phase profiles of a beam, but will also include more advanced examples that are either similar in philosophy or in methods. This sometimes results in loss of one spatial coordinate, or is made at the cost of resolution. The measurement of the laser wavefront is important for many experiments involving focused laser beams, especially those operating at high-intensity. Measurement of a laser wavefront generally uses one of the following three approaches: 1- measuring the local slope of the wavefront, e.g. using an array of micro-lenses to create an array of foci whose position depends on the local wavefront (Shack-Hartmann method), 2- interferometry, either externally-referenced or self-referenced (as in the common four-wave lateral shearing interferometry [@Primot:93; @chanteloup04]), 3- phase-retrieval algorithms applied to amplitude-only measurements in different $z$ planes  [@10.1117/12.472377]. Essentially, the techniques described in this section are derived from one of these approaches. Extensions of established wavefront sensing techniques {#sec:wavefront} ------------------------------------------------------ What we call here “direct” spatial phase measurements correspond to industry-standard wavefront measurements, which in this section are expanded to include the frequency dimension. We include here both Shack-Hartmann and Four-wave shearing interferometry, although the latter is also related to a following subsection that discusses interferometric methods. Wavefront autocorrelation was already attempted early on [@grunwald03], where autocorrelation was performed on the sub-foci of an all-reflective Shack-Hartmann device. However, this result was not expanded upon in the literature, so we rather focus on other methods that have been detailed more recently. A first measurement involving a wavefront measurement at a small number of frequencies and propagation calculations [@hauri05] has confirmed that resolving the wavefront spectrally is indeed a valid method for reconstructing or approximating the entire electric field, providing a good foundation for these techniques. A further result resolved in frequency, combined with spectral phase stitching (see section \[sec:phase-stitching\]), was the Shackled-FROG technique [@rubino09]. This technique, visualized in Fig. \[fig:HAMSTER\](a) is the combination of an imaging spectrometer with a Shack-Hartmann wavefront sensor, as well as a single FROG measurement. Essentially the Shack-Hartmann wavefront sensor is placed at the image plane of an imaging spectrometer, producing an array of sub-foci that are resolved in frequency along the dispersive axis of the grating, and resolved in one spatial dimension. This measurement results in the spatio-spectral phase $\hat{\phi}_S(x,\omega)$ at one sampled $y_0$ position, up to an unknown spatially homogeneous spectral phase. When combined with a FROG measurement, in this case at one point on the same spatial slice $y_0$, the total spatio-spectral phase can be reconstructed. The intensity of the sub-foci can lead to a proxy measurement for the spectral intensity, which means that the spatio-spectral electric field is fully known along one spatial axis. ![(a) Shackled-FROG technique, based on the schematic from Ref. [@rubino09]. (b) HAMSTER technique, based on the schematic from Ref. [@cousin12].[]{data-label="fig:HAMSTER"}](HAMSTER.pdf){width="83mm"} The HAMSTER technique [@cousin12], pictured in Fig. \[fig:HAMSTER\](b) also uses a Shack-Hartman device, but keeps both spatial dimensions. This is accomplished by using an acouto-optic dispersive filter (AOPDF) before the wavefront sensor to select narrow regions of the original spectrum. After making multiple such measurements the spatio-spectral phase $\hat{\phi}_S(x,\omega)$ is known, again without pure spectral phase knowledge. A local FROG measurement can lead to the reconstruction of the full spatio-spectral phase, and again the intensity of the sub-foci on the Shack-Hartman device leads to knowledge of the spatio-spectral amplitude. This device then is capable of measuring the full spatio-spectral electric field, but has a few restrictions. In particular, in order to select the narrow spectral regions without adverse effects, the AOPDF must be behind any amplifiers. Because the AOPDF generally has a very small aperture, this limits the type of beams that can be measured without prior demagnification. Taking a more direct approach, a Shack-Hartman device has been used in combination with various band-pass and long-pass filters (shown in Fig. \[fig:wavefront\](a)) to assess the wavefront of broadband laser sources, such as a white-light continuum [@hauri05; @kueny18] . This method is conceptually similar to the HAMSTER technique, but potentially cheaper to implement and is only limited spatially by the aperture of the sensor and the filters. However, it is difficult to design filters that have a narrow transmission bandwidth, so the wavefront must be constructed via a very small number of frequencies as in Ref. [@hauri05] or a set of measurements that had overlapping spectra as in Ref.[@kueny18]. This could lead to errors if the transmission of the filters is not well-known or the spectrum is highly modulated, and will regardless lead to a relatively low spectral resolution. It must also be taken into account that the spectral filters may themselves impart wavefront imperfections. From this point of view the problem of spatio-spectral measurement is somewhat shifted to filter calibration in this technique. There is also a device in development that uses a single filter rotated or translated to shift its transmission [@ranc19], mitigating some of the mentioned limitations. Rather than filtering the incoming beam and then measuring with a wavefront sensor, it has also been shown that one can use a multi-spectral camera, combined with either a Hartmann mask or a checkerboard mask (for four-wave lateral shearing interferometry) [@dorrer18] (shown in Fig. \[fig:wavefront\](b)). This approach is elegant and simple, but suffers from a very low spatial resolution, low wavelength resolution, and requires a multi-spectral camera that is generally quite expensive. For example, two common pixel patterns are shown in the inset of Fig. \[fig:wavefront\](b) that only have two or three colors, and reduce the resolution of the CCD by a factor of 4. However, the rapid industrial progress in so-called “snapshot” multi-spectral imaging techniques [@hagen13] could be adapted for the spatio-temporal characterization of ultrashort beams. This application of multi-spectral or hyper-spectral cameras could be transformative. {width="171mm"} An example of a snapshot multi-spectral imaging technique was deployed to rapidly characterize a scattering medium [@boniface19] (shown Fig. \[fig:wavefront\](c)) and could be adapted for pulse characterization. This technique used the combination of a lens array and a tilted grating to create an array of sub-foci on a 2D sensor, intimately related to the field of integral field spectroscopy common to astronomy [@allington-smith06]. Because the grating was tilted relative to the axis of the lens array, the dispersed sub-foci do not overlap. With a proper patterning of the lens array, the ability to pack a given spectrum on the 2D sensor can be maximized, although there is still a limitation on the bandwidth of the pulse to be measured. It could be that other methods applied to study scattering media [@liX19], or the multi-spectral properties of scattering media or multi-mode fibers themselves [@xiong20; @xiong19-2; @ziv20] could be used to characterize ultrashort pulses, although for now it is highly speculative. Frequency-resolved detection of the wavefront of the trains of attosecond pulses produced via high-harmonics generation has been done in various implementations [@frumker09; @austin11; @freisem18; @dacasa19], but we emphasize that the discrete nature of the high-harmonic spectra makes it a significantly different problem than for a single ultrafast pulse with a continuous spectrum. Iterative phase retrieval with frequency resolution {#sec:phase_retrieval} --------------------------------------------------- In this section, we look at measurement methods that rely on iterative phase retrieval algorithms. This type of approach is well known for wavefront measurements of monochromatic beams. The principle is to measure the spatial intensity profile of a beam –which is straightforward to do using a camera– at multiple $z$ planes separated by known distances. The evolution of this profile as the beam propagates obviously depends on the phase properties of the beam. Iterative algorithms such as the Gerchberg-Saxton one have thus been developed to extract this information from a few measured profiles [@gonsalves79; @fienup82; @matsuoka00]. Such phase-retrieval methods are already used in concert with deformable mirrors in order to optimize the *on-target* focal spot of high-power lasers [@pharao; @oasys; @beamtuner]. However, directly applying this type of approach to a broadband laser beam is doomed to fail if this beam is affected by significant chromatic effects. Indeed, in such a case, the measured intensity profile is the incoherent sum of multiple and potentially different intensity profiles associated to each frequency, $I(x,y)=\sum_{\omega}{I(x,y,\omega)}$, which each evolve in their own way along the propagation axis. Such effects are not taken into account in the iterative phase retrieval algorithm, which will then either poorly converge, or converge to a wrong solution. However, if the intensity profile of the beam is known at each frequency, then this approach can be safely applied independently to all frequencies of the beam. This is the approach discussed in this section, which can be implemented in different ways. ![CROAK technique, based on the schematic from Ref. [@bragheri08], but improved to have less complicated steps.[]{data-label="fig:CROAK"}](CROAK.pdf){width="83mm"} The first implementation of such a phase-retrieval technique for spatio-temporal measurements was termed the CROAK technique, standing for Complete Retrieval of the Optical Amplitude and phase using the $(k,\omega)$ spectrum [@bragheri08]. This method is detailed in Figure \[fig:CROAK\] with a simplified geometry compared to that in the original reference. The technique requires three steps. The first step, in Fig. \[fig:CROAK\](a) is to measure the spectral phase at a well-known position via any method (FROG in this example). The second step, also in Fig. \[fig:CROAK\](a) involves measuring the spatially resolved spectrum of the fundamental beam to be measured in the near-field along one axis using an imaging spectrometer (including the point where the FROG measurement was done). And lastly, the third step requires measuring the spatially resolved spectrum directly in the focus of a lens with a known focal length (and without aberrations) with the same imaging spectrometer, shown in Fig. \[fig:CROAK\](b). The combination of the latter two steps allows for reconstruction of the one-dimensional spatial phase at each frequency using phase retrieval algorithms, within certain limits and requiring certain assumptions. When combined with the spectral phase measurements of the first step this could produce the complete E-field in one spatial dimension and in either frequency or time. There are many issues with this method, however. Firstly, the measurement of the spatio-spectral amplitude in the near-field and at the far-field must be done on exactly the same slice of the beam, and any aberrations on the beam which lack cylindrical symmetry about the axis of the spectrometer slit could cause significant errors. And secondly, since the spatio-spectral amplitude is measured in the near-field and then at the far-field, there are very tight restrictions on either the size of the near-field beam, the focal length, or the number and size of the pixels in the imaging spectrometer. {width="171mm"} A significant improvement in spatio-temporal measurement using phase retrieval is the INSIGHT technique [@borot18]. Rather than using an imaging spectrometer, the INSIGHT technique resolves the spectral amplitude at each point of the measured beam via spatially-resolved Fourier-transform interferometry around the focus. Knowing the spectrum at each point of the beam, one can obviously determine the spatial profile at each frequency, which is the information required for proper phase-retrieval on broadband beams. The approach is implemented by splitting the beam near its focus into two copies as shown in Fig. \[fig:INSIGHT\](a), and resolving the interference of these two copies on a standard CCD camera. This has the advantage of resolving the spatial properties in two dimensions. The FTS is performed by taking camera images as the delay $\tau$ is stepped through with sub-cycle accuracy (Fig. \[fig:INSIGHT\](b)), and the spatially-resolved spectrum is calculated via taking the Fourier transform with respect to $\tau$ and selecting only the positive frequencies (Fig. \[fig:INSIGHT\](c)). This procedure removes the large and expensive imaging spectrometer device when compared to CROAK and resolves the second spatial dimension, but requires a delay stage capable of delay steps of a fraction of a wavelength as was the case for the TERMITES technique. Performing this temporal scan for one $z$ plane already provides the beam spatio-spectral amplitude. In order to obtain the spatio-spectral phase, the INSIGHT technique requires this spatio-spectral amplitude at multiple planes, just as with the CROAK technique. In order to allow for a better convergence of the phase-retrieval algorithm the FTS is repeated at two additional planes out-of-focus (at $\pm\delta{z}$). Once the spatio-spectral intensity is found around the focus (Fig. \[fig:INSIGHT\](d)) and out-of-focus (Fig. \[fig:INSIGHT\](e)) the phase-retrieval algorithm is done at each frequency to compute the spatio-spectral phase (Fig. \[fig:INSIGHT\](f)). Finally with a single measurement of the spectral phase a one position ($x_0$,$y_0$) the spatio-temporal electric field can be computed (Fig. \[fig:INSIGHT\](g)). Since INSIGHT is done in-focus it allows for the optics of the measurement device to be very small and lightweight, and also allows the measurement to be done in exactly the location of eventual experiments. The out-of-focus planes are generally measured at $\delta{z}\approx3-10 z_R$, so the camera properties can be optimized to have high-resolution. However, the camera chip should be significantly larger than the beam focus so that high spatial frequencies in the near-field can still be resolved. The INSIGHT device was used successfully for measurements on Terawatt [@borot18] and Petawatt lasers [@jeandet19]. With the addition of a second camera looking at the leak-through of one interferometer arm (shown in Fig. \[fig:INSIGHT\](a)) pointing fluctuations can be measured for each laser shot and numerically corrected, which significantly increases the fidelity of the computed spatially-resolved spectrum [@borot18]. This is especially important for measurement on high-power and low rep-rate systems, where pointing fluctuations can be significant. A birefringent delay line [@harvey04; @brida12] has recently been used for hyperspectral imaging [@perri19] (spatially-resolved Fourier-spectroscopy without phase information), and we have recently implemented this scheme in an INSIGHT device as well [@jolly_prisms]. Finally, using an imaging spectrometer near the focus of the INSIGHT device could essentially be a single-shot version that is resolved in only one spatial dimension (similar to the single-shot version of TERMITES, SEA-TERMITES), and would be very similar to steps 2 and 3 of the CROAK technique (but done around the focus). However, when compared to SEA-TERMITES, operating very close to the focus would still allow for the optics to be small and lightweight. But we consider the loss of one spatial dimension and the demand for cylindrical symmetry and measurement of the exact same spatial slice to be severe downsides of such a modified version of INSIGHT. If suitable hyperspectral cameras eventually become available, it might be possible to implement a single-shot version of INSIGHT that does not suffer from these limitations, by fitting multiple replicas of the beam into the camera’s chip, to measure the spectrally-resolved spatial intensity profile of the beam in multiple $z$ planes in a single-shot. Implementation of this idea, however, is far from straightforward. Interferometric Techniques {#sec:interferometric} -------------------------- Techniques of this class rely on the interference of the unknown beam with a second beam, considered as a reference, which can either be obtained from the unknown beam itself (self-referenced interferometry), or be an independent perfectly characterized beam. A self-referenced interferometric technique commonly used for the spatial characterization of laser beams is spatial shearing interferometry. This is the spatial analogous of the SPIDER technique, where the unknown beam is interfered with a spatially-sheared replica of itself. The resulting interference pattern can be used to determined the spatial derivative of the spatial phase. This technique has been extended for the spatio-spectral characterization of ultrashort laser beams, leading to a technique called “spectrally-resolved spatial-shearing interferometry” [@dorrer02-3]. In this technique, a Michelson interferometer is used to generate, on the slit of an imaging spectrometer (oriented along the $x$ axis at the sampling point $y_0$), two beams separated by a delay $\tau$ , with an angle offset represented by $k_x$ and with a spatial shear $X$. The interferogram signal $S$ generated can then be written as follows: $$\begin{aligned} \label{eq:dorrer} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}\left(x-X,\omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}\left(x-X,\omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x-X,\omega\right)-k_x x-\omega\tau\right] . \end{split}\end{aligned}$$ By nulling the shear $X$, it is possible to calibrate for the term $-k_x x-\omega\tau$. After adding the shear again, one has access to the phase difference $\hat{\phi}\left(x,\omega \right)-\hat{\phi}\left(x-X,\omega\right)$, proportional directly to $\partial\hat{\phi}\left(x,\omega\right)/\partial{x}$. From this data, we can reconstruct the phase $\hat{\phi}_S\left(x,\omega\right)=\hat{\phi}\left(x,\omega\right)+\alpha(\omega)$ with $\alpha$ being an arbitrary function of $\omega$. This gives the spatio-spectral phase up to an unknown overall spectral phase, and does not directly give the spectral amplitude. Essentially it is a measurement of the one-dimensional wavefront (i.e. along $x$) resolved in frequency. This combined with a spatially-resolved SPIDER is the 2D-SPIDER technique referenced earlier [@dorrer02-1]. We note that in principle, by sweeping the position $y$ of the beam on the slit of the spectrometer and then by rotating the beam by 90${}^{\circ}$ around the $z$ axis and by measuring the interferogram at a position $x_0$, the full STC phase $\hat{\phi}_S(x,y,\omega)$ could be obtained (with still an ambiguity of an arbitrary function of $\omega$). However, this is very difficult to do in practice and in fact has not been demonstrated. As we have seen already, diffraction, including of course the standard linear diffraction grating, is useful for separating the frequencies for doing spatio-spectral measurements. This has so far been to essentially use one dimension of a 2D detector to resolve the frequency while the other dimension remains for one spatial axis (as in SEA-SPIDER [@witting16], Shackled-FROG [@rubino09], CROAK [@bragheri08]). However, we saw already briefly in Fig. \[fig:wavefront\](c) [@boniface19] that a cleverly oriented grating can orient the frequency information in such a way that the 2D detector has continuous 2D spatial information along with discrete frequency information. Indeed, this shifts the difficulty from the sensor to the analysis. The STRIPED-FISH technique, which is complete and single-shot, uses a tilted 2D diffraction grating and a frequency filter to create spatially-separated profiles at discrete frequencies on a standard 2D CCD chip [@gabolde06; @gabolde08; @guang14; @guang15]. The interference of these dispersed intensity profiles with a *spatio-temporally perfect* reference, or a reference that has been *perfectly characterized in space-time*, in principle allows for measurement of the full spatio-spectral amplitude and phase, and hence for the reconstruction of the full spatio-temporal field. This technique is summarized in Fig. \[fig:STRIPED-FISH\]. ![STRIPED-FISH technique. The setup in (a) requires both the unknown beam and a characterized reference, a 2D diffraction grating, an interferometric bandpass filter (IBPF), and a 2D CCD detector. The analysis in (b) involves the standard spatial FFT, but also requires segmenting and overlaying the discrete frequency components. Images modified with permission from Refs. [@gabolde06; @gabolde08] The Optical Society.[]{data-label="fig:STRIPED-FISH"}](STRIPED-FISH.pdf){width="84mm"} Category Technique(s) Spatial dim. Spectral information obtained by Single-shot? Complete? ------------------------------------------------------- --------------------------------------------------------- ------------------- -------------------------------------------------------- -------------- ----------- -- Extension of established wavefront sensing techniques HAMSTER, Shackled FROG, hyperspectral wavefront sensors 1D or 2D Spectrometer/grating or filter or multispectral camera Yes Yes Spectrally-resolved phase retrieval CROAK, INSIGHT 2D (INSIGHT only) Spectrometer or FTS Not yet Yes Interferometry STRIPED FISH 2D Filter Yes Yes As shown in Fig. \[fig:STRIPED-FISH\](a) the reference and unknown beams are incident with an angle $\alpha$ on a 2D diffraction grating that is tilted by an angle $\varphi$ in a plane perpendicular to the propagation direction of the unknown beam. The grating produces many diffraction orders that are of course diffracted at larger angles from that of the incident beams. An interferometric bandpass filter (IBPF) is placed after the grating, tilted at an angle $\beta$, yet relative to a different plane (see Fig. \[fig:STRIPED-FISH\](a)). Because it is an interferometric filter the transmitted spectrum varies with incidence angle on the filter. Since different diffraction orders of the grating have a different angle of incidence, this causes each order to be filtered to a different bandpass wavelength. The result is a mosaic of spots on the 2D CCD sensor which correspond to the different frequency components of the incident beams. Since the unknown beam and reference beam have a relative angle $\alpha$, this produces spatial interference fringes on each spot, which allows one to extract phase information as well. The analysis steps involve the standard spatial FFT to calculate the amplitude and phase, segmenting the acquired image so that each frequency component can be analyzed independently, and finally stacking the information properly to produce the 3D amplitude and phase (see Fig. \[fig:STRIPED-FISH\](b)). The strength of STRIPED-FISH is that it is both a complete 3D technique, and single-shot—a significant advantage over techniques like INSIGHT and TERMITES for instance, especially for lasers that have low repetition-rates or fluctuate from shot to shot. Yet, it has significant drawbacks and limitations. The main limitation is one of principle: it requires a reference beam that covers the full spectrum of the unknown beam, and either has no STCs, or has been fully characterized in space-time. In most real-life cases, this reference needs to be produced from the unknown beam itself, which actually makes STRIPED FISH a self-referenced technique. In practice, this means that this technique is mostly suited to the measurement of the spatio-temporal effects induced by an optical system on a laser beam. A pick-off of this beam prior to this optical system can then be used as a reference, provided this system does not increase or shift the beam spectral content—i.e. this system should be linear. In terms of performance, in order to pack all of the interferograms for all frequencies on the 2D detector, there are strong limits on either the size of the detector, the size of the beam, the number of frequencies, or the spatial resolution. Finally, accurately calibrating the wavelength of each diffraction order, properly stacking the different diffraction orders, and finally producing the correct spatio-spectral phase all require a very careful calibration for each unique device and a very robust analysis algorithm. The STRIPED-FISH technique has for instance been used to measure the ultrafast lighthouse effect [@guang16] and beams from a multi-mode fiber [@guang17], where a pick-off on the input beam prior to the optical system under investigation was used as a reference. But due to the previous limitations, this technique has not been in common use so far, despite its conceptual elegance and its complete and single-shot character. In particular, because of the difficulty of producing a reliable reference, it has not yet been used to directly characterize spatio-temporally the output of a complex laser system, to the best of our knowledge. As a conclusion of this section, the properties of the main techniques discussed in this section are summarized in Table \[tab:table2\]. Analysis and visualization {#sec:analysis_visualization} ========================== The results of a 2-D or 3-D spatio-temporal or spatio-spectral measurement in general consist of a large matrix of complex numbers describing the laser field. For example, in the case of TERMITES or INSIGHT measurements this could be a 3-D complex-valued matrix of a size above 300$\times$300$\times$50 pixels (${x}\times{y}\times\omega$, 4.5 million points). With such a large set of complex data in more than two dimensions, it is not always straightforward to analyze, to extract meaningful physical information, or to visualize the results of a successful measurement. We will outline some methods to analyze the data produced from these measurements with various goals in mind, and in each case will also provide examples of how to effectively visualize the data and the individual analysis steps. Phase-stitching {#sec:phase-stitching} --------------- We start with a post-processing treatment of the measurement data that in practice is often the very first step of the analysis. For many of the techniques described in this tutorial, we have seen that the spatio-spectral phase is actually measured up to an unknown overall spectral phase, that equally applies to all points of the beam. This includes TERMITES, INSIGHT, and direct wavefront measurements with filters or multi-spectral cameras. Therefore these techniques by themselves should be more rigorously called spatio-spectral—rather than spatio-temporal—characterization devices. For other techniques such as SEA-TADPOLE and STRIPED-FISH, only when the technique is done with a suitably characterized reference can the measured spatio-spectral phase be considered complete. All spatio-spectral couplings are still resolved well in every case, and as we will see below, a lot can be said about the beam properties even with this remaining indeterminacy. But without knowledge of the overall spectral phase, the actual spatio-temporal field cannot be calculated. ![For some measurement techniques, phase stitching is necessary to have the proper phase relationship between frequencies, and be able to calculate the field in the spatio-temporal domain by a Fourier transformation. A measurement technique may be blind to spectral phase as in (a), but produce spatial phase results at many frequencies. A single measurement of spectral phase (b), in this case at $x=2$mm, will fix the phase relation and produce the correct spatio-spectral phase at all positions. This figure is courtesy of A. Jeandet.[]{data-label="fig:phase-stitching"}](phase-stitching-tutorial.pdf){width="83mm"} In all cases, a measurement of the spectral phase at a known single point in space can resolve this issue. This single measurement gives the relationship between the retrieved spatial phase maps at different frequencies. Using this measurement one can do “phase-stitching” to transform the data from the spatio-spectral device, with the data at different frequencies being essentially independent of each other, to data having the complete *physical* spatio-spectral phase. This phase-stitching procedure is illustrated in Fig. \[fig:phase-stitching\] for 2D data (one spatial coordinate only), but the concept applies equally well for 3D data. Mathematically, this phase stitching operation consists in applying the following transformation to the measured spatio-spectral phase $\hat{\phi}_{meas}(x,y,\omega)$ (displayed in Fig. \[fig:phase-stitching\](a)), to obtain the physical spatio-spectral phase $\hat{\phi}(x,y,\omega)$ (displayed in Fig. \[fig:phase-stitching\](b)): $$\hat{\phi}(x,y,\omega)=\hat{\phi}_{meas}(x,y,\omega)-\hat{\phi}_{meas}(x_0,y_0,\omega)+\varphi(\omega),$$ where $\varphi(\omega)$ is the spectral phase measured at a given point $(x_0,y_0)$ of the unknown beam (green line in Fig. \[fig:phase-stitching\](b)), using a temporal measurement device such as a SPIDER, FROG, or D-scan for instance. Because performing a local measurement of the spectral phase is much easier on an unfocused beam, this procedure is generally applied to the spatio-spectral phase in the NF. In such a case, when the spatio-spectral measurement is performed in the FF (like in INSIGHT), the measured field needs to be numerically propagated from the FF to the NF before applying phase stitching. Calculating the magnitude of low-order couplings {#sec:analysis_low} ------------------------------------------------ One of the key steps when analyzing measured data from a spatio-temporal characterization device is generally estimating the magnitude of the lowest-order couplings, or that of the couplings expected to be present based on the nature of the source. This is an essential step because the most common couplings are also typically those of lowest-orders. Returning to the canonical couplings of AD/PFT and CC/PFC, we can find a straightforward way to calculate the magnitude of these couplings using the phase data of the 3-D matrix. If we consider the NF, these couplings are only concerning the phase, so we reference the reconstructed spatio-spectral phase $\hat{\phi}(x,y,\omega)$ or $\hat{\phi}(r,\omega)$. We can find the AD/PFT via the following relation $$\gamma_x=\frac{\partial}{\partial\omega}\frac{\partial \hat{\phi}(x,y,\omega)}{\partial x} ,$$ and we can find the CC/PFC via the following relation $$\alpha=\frac{1}{2}\frac{\partial}{\partial\omega}\frac{\partial^2 \hat{\phi}(r,\omega)}{\partial r^2} .$$ In order to be insensitive to the spectral phase of the measured pulse, the spatial derivate must be performed first. This is important especially for measurements that do not have a pure spectral phase measurement included (i.e. without spectral phase stitching). ![Examples of low-order couplings analysis for (a) AD/PFT and (b) CC/PFC. A slice of the spatio-spectral phase at $y=0$ is shown in the left column. lineouts of this phase at discrete frequencies $\omega_i$ are shown in the central column, which have varying linear or quadratic coefficients for $x$, $c(\omega)$, if the coupling is AD/PFT or CC/PFC respectively. Fitting a linear curve to $c(\omega)$ in each case, as shown in the right column, can result in the magnitude of the coupling. In the case of CC/PFC the behavior is the same in terms of the radial coordinate $r$, but a slice of $x$ at $y=0$ is shown here for simplicity.[]{data-label="fig:analysis_1"}](Low_analysis.png){width="83mm"} These are simple relationships that make it easy to calculate these STCs when the spatio-spectral phase is a known function, but of course when the phase is represented not as a continuous function, but rather as a discrete data set, derivatives cannot be taken as such. In practice, to find the magnitude of these couplings, the spatial phase at each frequency should rather be fit to a polynomial in space. Then the relevant coefficients at each frequency (either linear in position for AD/PFT or quadratic in radius for CC/PFC) should be fit to a polynomial in frequency. The linear component of this polynomial in frequency is the magnitude of the coupling. More explicitly for AD/PFT: $$\hat{\phi}(x,y,\omega)=c_1(\omega)\times x ,\quad c_1(\omega)=\gamma_x\times (\omega-\omega_0) ,$$ and for CC/PFC: $$\hat{\phi}(r,\omega)=c_2(\omega)\times r^2 ,\quad c_2(\omega)=\alpha\times (\omega-\omega_0) ,$$ where in each case $c_i(\omega)$ and the subsequent coupling (either $\gamma$ or $\alpha$) are found via a least-squares regression. Figure \[fig:analysis\_1\] shows this procedure for both AD/PFT and CC/PFC. It is very important to do these analysis steps on phase data, only within the spectral region where there is significant intensity. Most measurement devices will produce random or highly irregular phase data outside of the real spectral region of the measured beam, which must be ignored because it would negatively influence any fitting. Addressing the magnitude of arbitrary phase couplings {#sec:analysis_high} ----------------------------------------------------- Beyond low-order couplings addressed in the previous section, it may be that higher-order phase couplings are expected, or that it is clear there are some effects not of low-order. And besides expectations, in general it can be tedious to fit individual polynomials to 3-D data. Furthermore, when looking at the 3-D data resulting from a spatio-temporal or spatio-spectral measurement, it cannot always be clear whether the high-order aberrations that are present are chromatic or not. Therefore it is necessary to have some type of general way to address this, especially if there is no particular expectation or prediction (which is often the case in the real world). We borrow a standard technique from monochromatic wavefront analysis, and propose to utilize frequency-resolved Zernike polynomials to describe the general phase aberrations present. This was introduced and implemented with great utility in recent work [@borot18; @jeandet19]. The Zernike polynomials are a way to represent an arbitrary function over the unit disk via a set of orthogonal polynomials [@zernike34]. These polynomials can be used to represent the spatial phase of a laser beam over a defined pupil [@born99], which corresponds to the area where there is significant intensity. Without going into detail, we simply remind that the Zernike polynomials $Z_n^m$ generally have two indices $m$ and $n$ (with $\left|m\right|\le n$) that correspond to the azimuthal and radial degrees of freedom respectively, where $m$ can be negative, but $n$ is limited to the natural numbers. When the phase map is decomposed onto this basis of Zernike polynomials, the result is a list of constants $C_n^m$ corresponding to the amplitude of each polynomial. Algorithmically this is much simpler than fitting arbitrary polynomials to the spatial data, since it amounts to decomposing a known function on a complete orthonormal basis set. We emphasize that these coefficients can be divided by $k=2 \pi /\lambda$, with $\lambda$ the wavelength of the beam under consideration, such that they describe the actual physical distance of displacement/deformation of the wavefront across the beam. Although this is not necessarily the standard practice in wavefront sensing, it is preferable to use this normalization of the coefficients for the analysis of chromatic effects discussed below, and this is what we will assume in the rest of this section. The extension of the Zernike polynomials to include frequency is quite straightforward. At each frequency $\omega$, the spatial phase map is decomposed on the basis of Zernike polynomials, leading to coefficients $C_n^m(\omega)$ that can now depend on frequency. When a given term does depend on frequency, it can then be concluded that there is a chromatic effect on that Zernike component. For instance, for a beam with PFT, at least one of the coefficients $C_1^{\pm1}$ will vary with frequency - while they would be exactly constant for a tilted beam (provided the normalization mentioned above is used). Such a picture is quite powerful, since it intuitively can show the chromatic nature of different phase aberrations. The intuition of the various aberrations (defocus, astigmatism, etc.) can be utilized to attempt to understand the data that now has the additional dimension of frequency. An example of this data for a beam having mostly CC/PFC in shown in Fig. \[fig:Zernike\](a). {width="171mm"} We now explain how to relate these coefficients to different coupling parameters introduced earlier in this tutorial for the canonical couplings AD/PFT and CC/PFT (see section \[sec:concepts\_manifestations\]). For calculating the magnitude of the AD/PFT coefficient $\gamma$, the tilt Zernike terms $C_1^{\pm 1}(\omega)$ first need to be related to the frequency-varying wavefront tilt $\theta$ (in direction $x$ or $y$), through the following relationship: $$\theta_{x,y}(\omega)\approx\frac{d}{R_p}=\frac{2 C_1^{\pm 1}(\omega)}{ R_p} \label{eq:ZernikeTilt} \\$$ where $R_p$ is the pupil radius used for the Zernike computation, and $d$ is the displacement of the wavefront at a given frequency at the edge of the unit disc defined by the pupil. Note that this relationship assumes that the Zernike modes are normalized to have a modulus of $\pi$ over the unit disc **(really?)**. The AD/PFT coefficient $\gamma$ is then directly related to the linear slope of $\theta$ via: $$\gamma_{x,y}=\frac{\omega_0}{c}\frac{\partial\theta_{x,y}}{\partial\omega}\Big|_{\omega_0} \label{eq:ZernikePFT},$$ with $\omega_0$ the central frequency of the pulse. Following the same reasoning, the CC/PFC coefficient $\alpha$ can be deduced from the frequency-varying Zernike terms for defocus $C_2^0(\omega)$, by relating both quantities to the frequency-resolved wavefront curvature $1/R(\omega)$: $$\begin{aligned} \frac{1}{R(\omega)}&=\frac{2d}{d^2 + R_p^2}\approx\frac{2d}{R_p^2}=\frac{4\sqrt{3} C_2^0}{ R_p^2} \label{eq:ZernikeDefocus} \\ \alpha&=\frac{\omega_0}{2c}\frac{\partial(1/R(\omega))}{\partial\omega}\Big|_{\omega_0} \label{eq:ZernikePFC},\end{aligned}$$ where $d$ is the same as before. A schematic of this is shown in Fig. \[fig:Zernike\](b)–(c). The phase maps at three frequencies in Fig. \[fig:Zernike\](b) show qualitatively the varying curvature, but the linear fit to the frequency-dependence of $1/R(\omega)$ in Fig. \[fig:Zernike\](c) produces the clear quantitative value of the CC/PFC based on Eq. (\[eq:ZernikePFC\]). Both of these examples are essentially identical to the straightforward approach outlined in the previous section for low-order couplings, but require a different set of steps. Depending on one’s priorities and capabilities either method should result in the same quantitative result. The obvious advantages of the Zernike polynomial method are that the higher-order aberrations are generated for free, a similar analysis can be done for the chromatic nature of these higher-order aberrations, and all of the technical considerations of fitting at low or high orders are not relevant. However, it is important to realize that choosing an appropriate pupil for the Zernike calculations is very important for calculating the correct result. Assessing the total effect of couplings {#sec:analysis_total} --------------------------------------- As a complementary step to quantifying the magnitude of specific couplings, which have identified causes or effects, it is useful to quantify the overall impact of all couplings present. There are methods to quantify the various effects of phase and amplitude couplings separately or together, and again they depend on the application. In some optical setups, for example a NOPA [@harth18] or a multi-pass cell for pulse compression [@weitenberg17; @lavenu18], there may be a significant effect on the homogeneity of the spectral amplitude when the system is not properly aligned. So in these cases a quantity can be used to assess this level of homogeneity when optimizing. This assessment of spectral homogeneity over the spectrum can be defined for example by the spectral overlap integral $V$ [@weitenberg17; @lavenu18] $$V(r)=\frac{\left[\int\sqrt{I(\lambda,r)\times I(\lambda,r=0)}d\lambda\right]^2}{\left(\int I(\lambda,r)d\lambda\right)\times\left(\int I(\lambda,r=0)d\lambda\right)}.$$ This integral is essentially comparing the spectrum at off-axis positions to the spectrum on-axis. As the spectrum becomes more homogeneous this integral will approach 1 at every transverse position. An example of this calculation is shown in Fig. \[fig:analysis\_3\]. ![A perfect pulse (a), with the spectral amplitude shown on the left and the calculated (perfect) overlap integral shown on the right. An example pulse with spectral amplitude on the left in (b) having a decreasing spectral width with increasing $y$, with the spectral overlap integral calculated the right, showing a decrease away from the axis.[]{data-label="fig:analysis_3"}](total_analysis1.png){width="83mm"} Of course in many applications the phase is important, and in general the phase on the NF beam (which is more often analyzed) will have a larger effect than the amplitude on desired parameters, such as in-focus pulse duration or peak intensity. So in addition to looking at the spectral amplitude, various spatio-temporal Strehl ratios can quantify the impact of spatio-temporal phase distortions [@pariente16; @jeandet19]. The beam measured has a spatio-temporal or spatio-spectral amplitude $A(x,y,\omega)$ and phase $\hat{\phi}(x,y,\omega)$. The commonly known Strehl ratio ($\textrm{SR}_\textrm{WFS}$, associated with standard wavefront sensors) quantifies the effect of the frequency-averaged wavefront on the focusing of the frequency-averaged beam profile. This is usually performed on data that is already averaged (via measurement on a CCD camera), i.e. $\textrm{SR}_\textrm{WFS}=I\left[\overline{A}(x,y)e^{i\overline{\phi}(x,y)}\right]/I\left[\overline{A}(x,y)\right]$, where the upper bar symbol indicates an average over frequency. We use $I[ ]$ to denote the calculation of the focused intensity of a given beam. With knowledge of the full 3D intensity and phase, more nuanced versions of this quantity can be calculated as we now show. Although there are many possible definitions of a spatio-temporal Strehl ratio, we will focus on only a few versions to demonstrate the concept, which have been used in the previous works [@pariente16; @jeandet19]. The Strehl ratio assessing the impact of all phase distortions both chromatic and not, termed $\textrm{SR}_\textrm{Full}$, compares the fully measured beam with a beam having zero phase at every frequency $$\textrm{SR}_\textrm{Full}=\frac{I\left[\hat{A}(x,y,\omega)e^{i\hat{\phi}(x,y,\omega)}\right]}{I\left[A(x,y,\omega)\right]}.$$ When $\textrm{SR}_\textrm{Full}$ is less than one, it represents the departure in focused intensity from the perfect case of the fully measured beam. It should be the representation of the physically existing pulse intensity. Note that with this definition, the value of $\textrm{SR}_\textrm{Full}$ also depends on the spectral phase of the beam -which not the case with the usual, spatial-only, definition of the Strehl ratio of laser beams: indeed, a chirped laser pulse, even without any spatio-temporal coupling (i.e. the chirp is spatially homogeneous), necessarily has $\textrm{SR}_\textrm{Full}<1$. The Strehl ratio assessing the impact of only the chromatic phase distortions, which we call $\textrm{SR}_\textrm{STC}$, compares the measured beam with the frequency-averaged wavefront subtracted to a beam having zero phase at each frequency $$\textrm{SR}_\textrm{STC}=\frac{I\left[\hat{A}(x,y,\omega)e^{i\left(\hat{\phi}(x,y,\omega)-\overline{\phi}(x,y)\right)}\right]}{I\left[A(x,y,\omega)\right]}.$$ The physical case that $\textrm{SR}_\textrm{STC}$ describes is that where a deformable mirror was implemented perfectly so as to remove all non-chromatic wavefront distortions, but all chromatic aberrations still remain. This is useful if it is known that achromatic aberrations exist (and are either impossible to remove or not necessary to remove at that moment), and one wants to assess the impact of STCs only. A simple example is shown in Fig. \[fig:analysis\_4\]. ![Visualization of the Strehl ratio calculations for one simple case. In (a) a perfect beam, having flat phase, is defined to have an intensity of 1 in the far-field. In (b) a beam with AD/PFT in the nearfield produces a beam with a lower intensity in the far-field, which would correspond to a calculation of $\textrm{SR}_\textrm{Full}=\textrm{SR}_\textrm{STC}=0.35$.[]{data-label="fig:analysis_4"}](total_analysis2.png){width="83mm"} Note that if there are no chromatic phase distortions then $\textrm{SR}_\textrm{Full}=\textrm{SR}_\textrm{STC}$. These definitions of $\textrm{SR}_\textrm{Full}$ and $\textrm{SR}_\textrm{STC}$ were used on measurements of the BELLA PW system in recent work [@jeandet19]. Lastly, If one wants to quantify the effect of all spatio-temporal distortions, both in phase and amplitude, then a mixed ratio can be calculated as was done in [@pariente16]. In that case it was calculated by comparing the measured beam to a beam with zero phase in space and frequency, and also with the amplitude replaced by the average in space and the average in frequency, i.e. $\textrm{SR}_\textrm{mixed}=I\left[\hat{A}(x,y,\omega)e^{i\left(\hat{\phi}(x,y,\omega)\right)}\right]/I\left[\overline{A}(x,y)\overline{A}(\omega)\right]$. However, we believe that looking at the impact of phase and amplitude effects separately generally provides more insight. Visualization {#sec:visualization} ------------- Due to the complexity and multi-dimensional nature of spatio-temporal couplings, visualization is an important issue [@rhodes17; @li18-1]. Even after a successful measurement using one of the devices described, it is not trivial to properly discern the couplings present, nor is it simple to properly communicate the magnitude of the couplings. Therefore visualization is crucial to both assess initial measurements to guide the analysis priorities, but also to communicate the impact after analysis has taken place. We showcase a few examples of visualization options in Figure \[fig:visualization\] (taken from Ref. [@borot18]). These methods are: spatial properties visualized at discrete frequencies (Fig. \[fig:visualization\](a)), spectral and/or temporal properties visualized at discrete spatial coordinates (Fig. \[fig:visualization\](b)) and, as already discussed, the frequency-resolved Zernike coefficients visualized in a 3-D format (Fig. \[fig:Zernike\](a)). {width="150mm"} The method used for visualization depends strongly on the desired knowledge. If one is applying a certain spatio-temporal coupling in order to induce a given mechanism at the experimental focus, then visualizations exactly as in Fig. \[fig:visualization\](a) or (b) are likely most helpful. This is because they give a direct visualization of certain properties where they are important. However, if one is a laser physicist looking to remove undesired spatio-temporal couplings, then the same views as in Fig. \[fig:visualization\](a) or (b) may be desired, but on the collimated beam rather than at the focus. This is because the bulk of most laser amplifiers and the optics that may induce undesired couplings act on the collimated beam (although likely at increasing diameter throughout a laser chain). Because of this the views in the near-field may provide more direct input into the source of undesired couplings. The last important view was already shown in Fig. \[fig:Zernike\](a), the 3D view of the frequency-resolved Zernike polynomials, is likely useful to all scientists. This is because, although the Zernike coefficients are calculated based on the near-field beam, their nature also provides direct input in to the manifestation of any given chromatic effect in the focus. For this reason the frequency-resolved Zernike coefficients may be the most universal and helpful view of all. Since they show no amplitude information, other views will always be necessary as a complement. The data shown in Fig. \[fig:Zernike\](a) is for a beam with CC/PFC (i.e. a linear slope in frequency of the focus) to contrast with that already shown in Ref. [@borot18]. There are more compact methods to visualize spatio-temporal couplings. One example is where rather than a color scale representing the amplitude or intensity, it corresponds to the local instantaneous frequency. This is especially relevant for couplings where the spatio-spectral amplitude changes with propagation. A systematic review of visualization using this method along with many examples was presented in Ref. [@rhodes17]. Additionally, three dimensional stationary views of pulse intensity can be made with quadrants cut out (see many examples in Ref. [@li18-1]) or with constant intensity contours (see for example Fig. 7 of Ref. [@borot18]). Beyond the stationary views discussed, it is often useful and instructive to use movies to achieve multiple objectives. This includes: 1) panning or rotating a fixed 3-D plot in order to have a more immediate sense of the 3-D presentation (as in Supplementary movies 1 and 3 of Ref. [@pariente16], or Visualization 2 of Ref. [@borot18]), 2) showing a 2-D map and stepping through a third parameter (time or frequency) as the movie progresses (as in Supplementary movie 2 of Ref. [@pariente16]), or movies 1 and 2 of Ref. [@jeandet19]), or 3) visualizing a 2-D or 3-D property as the beam is numerically propagated through space where the movie steps through time or propagation distance (as in Visualization 1 of Ref. [@borot18], or movie 3 of Ref. [@jeandet19]). Although movies are not necessarily as useful as stationary plots within scientific journal articles, they are becoming better integrated in certain journals and their use is becoming more prevalent. Moreover, movies are an extremely useful tool for analysis for a scientist when interpreting results, so for the reader of this tutorial they could be important. Conclusion {#sec:conclusion} ========== In this tutorial we have introduced spatio-temporal couplings in a detailed fashion and reviewed techniques ranging from the simple to the complex for characterizing ultrashort laser pulses completely. This included very simple qualitative techniques, established temporal characterization methods extended to include one or all spatial dimensions, and advanced methods using a variety of techniques. The fact that this work is a tutorial was especially stressed in the order of introducing techniques and the level of detail included for a small number of them. From this point of view, it should not be treated as a full review of STC characterization (of which there is a good recent example [@dorrer19]). In addition to some past results or techniques that have not been discussed, there are many up-and-coming techniques which may prove to be integral to making spatio-temporal characterization more widespread in the community. For example, the STRIPED-FISH technique is the only technique employed for ultrashort pulses that is single-shot, although it requires a reference. A reference-free single-shot method that is more simple to implement experimentally is the grand challenge of this field. Indeed, techniques such as TERMITES and INSIGHT are functioning well and on the road to becoming available products for the community, but they are still methods that require scanning over many independent pulses. It may be that intuition from the mature but separate world of hyperspectral imaging [@hagen13] may provide innovation for pulse characterization if they are improved to handle the broad spectrum of ultrashort pulses, or even via the field of imaging through scattering media [@boniface19; @liX19]. The methods exposed in this paper for visible and near-infrared can be used as inspiration for characterization of sources in other wavelength ranges. This includes the much shorter wavelengths in attosecond pulse (see Ref. [@dacasa19]) and the longer wavelengths of a growing number of mid- and far-infrared ultrafast sources. This is important because, for example, attosecond pulses generated from gases are considered to often have extreme levels of spatio-temporal couplings depending on the precise generation parameters [@wikmark19]. The chief difficulty in developing devices for these exotic wavelengths is generally the components: optics such as mirrors, beamsplitters, and filters are commonplace for near-infrared sources, but can be quite bulky, expensive, or perform worse for extreme wavelengths. Even beyond sources of different wavelengths, ultrafast vector beams — beams with a spatially-varying polarization — add a completely new challenge to characterization. There are some solutions in development [@alonso19], and this will surely become a very active area. Most of the examples of either simulated or real STCs in this tutorial were simple in nature, mostly in order to clearly demonstrate the concept. These simple STCs have many applications as discussed in the introduction. However, there are many exotic STCs or exotic scenarios where STCs can be an avenue for fine control of physical mechanisms. These mechanisms include: Simultaneous space-and-time focusing caused by focusing a beam with spatial chirp in the nearfield [@zhu05; @durfee12; @heF14]; Spatio-temporal light springs [@pariente15], relevant potentially for laser-plasma acceleration [@vieira18], and extended to the attosecond regime [@porras19-3]; A “Flying Focus” in the focus of a beam with longitudinal chromatism and temporal chirp [@sainte-marie17; @froula18; @jolly20-1] for Raman amplification [@turnbull18-1], ionization waves of arbitrary velocity [@palastro18; @turnbull18-2], or photon acceleration [@howard19]; spatial chirp or chromatic focusing for high harmonic generation in gases [@hernandez-garcia16; @holgado17]; steering of beams in laser-plasma acceleration due to pulse-front tilt [@popp10; @thevenet19-2; @mittelberger19] and the effect on the polarization of betatron radiation [@schnell13]; circumventing intrinsic limits of laser-plasma acceleration [@debus19]; pulse-front tilt for dielectric laser acceleration [@plettner08; @wei17]; THz beams with tilted pulse-front for traveling-wave electron acceleration [@walsh17]; In-band noise filtering of high-power lasers [@wangJ18]; Diffraction-free space-time wave packets [@kondakci16; @kondakci17; @kondakci19-1; @bhaduri19-1; @bhaduri19-2; @kondakci19-2; @yessenov19-1; @yessenov19-2], among many others. The recent activity in designing new spatio-temporal characterization devices and the wealth of applications in ultrafast physics underscores the importance of the field. With many Terawatt and Petawatt lasers coming online across the world [@danson19], and pulses with few-cycle duration becoming ever more commonplace, the increase in familiarity with the concepts in this tutorial is paramount for the community to successfully utilize these sources and to characterize and troublehoot their spatio-temporal performance. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge Antoine Jeandet for general discussions and for supplying the phase-stiching figure. calculation of the spatio-spectral phase of beams with pulse-front distortions {#sec:appendixA} ============================================================================== We consider a beam of central frequency $\omega_0$ whose spatio-temporal field is described by a function of the following form: $$\nonumber E(x,t)=f(x) g[t-t_0(x)] e^{i \omega_0 t}$$ This corresponds to a beam whose temporal profile $g(t)$ is invariant in space, but whose arrival time $t_0(x)$ depends on the position $x$ in the beam—while the wave front at the carrier frequency, described by the last term of the equation, is assumed to be flat and normal to the $z$ axis. The term $f(x)$ describes the spatial envelope of the beam, and will be omitted in all following calculations, as it appears as an overall factor in all equations. For simplicity, we restrict the analysis to one transverse spatial dimension only, but it can easily be generalized to two transverse coordinates. The spatio-spectral description of this beam is obtained by performing a Fourier-transformation with respect to $t$. To carry out this transformation, we first rewrite the previous equation as: $$\nonumber E(x,t)=[g(t) \otimes \delta[t-t_0(x)] ] \times e^{i \omega_0 t},$$ where $\otimes$ is the symbol for the convolution product. The Fourier-transform of this field is then given by: $$\begin{aligned} \nonumber \hat{E}(x,\omega)&=[\hat{g}(\omega) \times FT\left\{\delta[t-t_0(x)]\right\} ] \otimes FT\left\{e^{i \omega_0 t}\right\} \\ &=[\hat{g}(\omega) \times e^{i\omega t_0(x)}] \otimes \delta(\omega-\omega_0) \nonumber\\ &=e^{i\delta\omega \: t_0(x)}\hat{g}(\delta \omega), \nonumber\end{aligned}$$ where $\hat{g}(\omega)$ is the Fourier-transform of $g(t)$, and $\delta \omega=\omega-\omega_0$ is the frequency offset from the central frequency $\omega_0$. This equation shows that a pure pulse front distortion $t_0(x)$ in the time domain is entirely described in the spectral domain by a spatio-spectral phase $\hat{\phi}(x,\omega)=\delta \omega \: t_0(x)$. In the case of pulse front tilt, we have $t_0(x)=\gamma x$ leading to $\hat{\phi}(x,\omega)=\gamma \: \delta \omega \: x$. In the case of pulse front curvature, $t_0(x)=\alpha x^2$ leading to $\hat{\phi}(x,\omega)=\alpha \:\delta \omega \: x^2$. These expressions of the spatio-spectral phase are the ones discussed in section \[sec:concepts\_manifestations\] of the main text. We now provide a more detailed discussion of these two cases, which is actually very useful to understand some of the subtleties of STCs and their metrology. We first analyze the mathematical differences between a beam with PFT, and a perfect STC-free beam propagating at an angle with the $z$ axis. In the later case, the field writes: $$\begin{aligned} \nonumber E(x,t)&=g(t-\gamma x) e^{i \omega_0 (t-\gamma x)} \\ &=[g(t)e^{i \omega_0 t}]\otimes \delta(t-\gamma x). \nonumber\end{aligned}$$ When going to the spectral domain, a calculation similar to the previous one leads to: $$\hat{E}(x,\omega)= e^{i \gamma x \omega}\hat{g}(\delta \omega). \nonumber$$ This shows that in the spatio-spectral domain, the only difference between a beam with PFT, and a perfect tilted beam, lies in a subtle difference in the spatio spectral phase: in the former case, $\hat{\phi}(x,\omega)=\gamma \delta \omega x$, while in the later, $\hat{\phi}(x,\omega)=\gamma \omega x$. This has important consequences for STC metrology: a measurement method can only distinguish these two types of beams, and hence detect PFT, if it can differentiate these two types of spatio-spectral phases. A similar analysis shows that in the spatio-spectral domain, a perfect STC-free curved beam (e.g. a perfect beam just after a perfect focusing optic) is described by the spatio-spectral phase $\hat{\phi}(x,\omega)=\alpha \omega x^2$. This again only slightly differs from the case of a beam with PFC, where $\hat{\phi}(x,\omega)=\alpha \delta \omega x^2$. This last case actually leads to an interesting question. Let us consider again a perfect beam without STC, which goes through a perfect focusing optic. Just after this optic, the field has the form: $$E(x,t)=g(t-\alpha x^2) e^{i \omega_0 (t-\alpha x^2)}. \nonumber$$ The beam wave front and pulse front are both curved by the same amount. This form of field is not separable as a product of a function of time and a function of space. According to the definition of section \[sec:concepts\_general\] (Eq. (\[eq:STC\])), this would imply that such a beam presents STC. Yet, intuitively, one would not consider this beam as suffering from STC—or equivalently chromatic aberrations. The point of view of the authors is that this apparent contradiction is a weakness or flaw of the present commonly-used definition of STC, which will have to be clarified by further theoretical work. One potential solution would be to consider that a beam has no STC when there exists a transformation $t'=h(t,x)$ such that $E(x,t')$ can be decomposed as $E(x,t')=f(x)g(t')$. With such a definition, a perfect curved beam would then be free of STC, since the transformation $t'=t-\alpha x^2$ makes it separable. Whether this definition makes sense in a more general case remains on open question. References {#references .unnumbered} ==========

ArXiv

--- abstract: 'Collisionless plasmas, mostly present in astrophysical and space environments, often require a kinetic treatment as given by the Vlasov equation. Unfortunately, the six-dimensional Vlasov equation can only be solved on very small parts of the considered spatial domain. However, in some cases, e.g. magnetic reconnection, it is sufficient to solve the Vlasov equation in a localized domain and solve the remaining domain by appropriate fluid models. In this paper, we describe a hierarchical treatment of collisionless plasmas in the following way. On the finest level of description, the Vlasov equation is solved both for ions and electrons. The next courser description treats electrons with a 10-moment fluid model incorporating a simplified treatment of Landau damping. At the boundary between the electron kinetic and fluid region, the central question is how the fluid moments influence the electron distribution function. On the next coarser level of description the ions are treated by an 10-moment fluid model as well. It may turn out that in some spatial regions far away from the reconnection zone the temperature tensor in the 10-moment description is nearly isotopic. In this case it is even possible to switch to a 5-moment description. This change can be done separately for ions and electrons. To test this multiphysics approach, we apply this full physics-adaptive simulations to the Geospace Environmental Modeling (GEM) challenge of magnetic reconnection.' author: - bibliography: - 'lit.bib' title: Multiphysics simulations of collisionless plasmas --- Introduction ============ One of the most important challenges in astrophysical, space and fusion plasmas is the treatment of different spatial and temporal scales and the correct physical description on each of these different scales. In order to give a rough estimate for different plasma systems, let us first consider the warm ionized phase (diffuse ionized hydrogen) in the interstellar medium. Here, the smallest relevant kinetic scales are in the order of magnitude of kilometres, while the global scale of the system is about $10^{13}$km. In the heliosphere the scales are altogether smaller (kinetic scales about $2$km, system scale about $10^8$km), but the ratio of global to kinetic scales is still astronomical in the truest sense. The situation is similar in fusion plasmas: the electron skin depth is about $5\cdot 10^{-4}$m and the vessel measures about $10$meters. In all these cases, it is not possible to carry out simulations which represent all scales with the finest level (kinetic equations) of the physical description. Most of these plasmas can be considered as collisionless, since collision times are orders of magnitude larger than time scales relevant for the dynamical evolution of the plasma. Such plasmas can be modelled with the kinetic Vlasov equation. Nevertheless, kinetic models are inherently computationally expensive, so that large–scale simulations of typical phenomena, as for example magnetic reconnection or collisionless shocks, are hardly feasible and only possible in localized regions of interest. As an alternative, much cheaper fluid models can be considered, but they lack the expressiveness and some physics of full kinetic models, even though some of the effects may be included. Simple treatments and modelling of Landau damping in the same context were proposed and analyzed in [@wang-et-al:2015; @ng-hakim-etal:2017; @allmann-rahn-trost-grauer:2018]. These studies were based on the closure introduced by @hammett-perkins:1990 and successive work in this direction [@hammett-dorland-perkins:1992; @passot-sulem:2003]. An extension providing heat fluxes in the parallel and perpendicular directions (with respect to the magnetic field) was presented in [@sharma-hammett-etal:2003]. An excellent overview is given in @chust-belmont:2006. Fortunately, many relevant problems like magnetic reconnection or collisionless shocks exhibit a rather clear separation of scales and regimes such that an adaptive approach is promising and might combine the best of the two worlds: cheap models where they are sufficient and detailed models where they are necessary and interesting. The idea of coupling different physical models is not new and has been applied in different physical contexts. Schulze et al. [@Schulze2003] couple kinetic Monte-Carlo and continuum models in the context of epitaxial growth. Considerable efforts have been made to couple kinetic Boltzmann descriptions with fluid models (see e.g. [@deg2010; @del2003; @gou2013; @tiwari-klar:1998; @Tal1997]). In the context of plasma physics Sugiyama and Kusano [@Sug2007], Markidis et al. [@Mar2014] and Daldorf et al. [@daldorff-et-al:2014] show ways to combine PIC and MHD fluid models, and Kolobov and Arslanbekov [@Kol2012] describe the transition from neutral gas models to models of weakly ionized plasmas. We take a slightly different route in solving the Vlasov equation on the finest relevant scales and then adaptively use less and less detailed fluid models outside the kinetic region. In this way we have some control where to use which kind of physical model at the expense of dealing with a substantially more complicated computational infrastructure. Our group has developed and is continuously developing and improving methods and codes that are capable of combining kinetic and fluid models during runtime [@rieke-et-al:2015], making it possible to consider problems of the type mentioned above at much lower expenses than before. A sketch of this hierarchy is depicted in figure \[fig:sketch\]. In the inner zone, both ions and electrons are treated kinetically and solved with the Vlasov equation. Adjacent to this zone, ions are still modelled with the Vlasov equation but electrons are described with a 10-moment fluid model. On the next coarser level of description, the ions are also described by a 10-moment fluid model. To ease the transition from the kinetic to the 10-moment fluid description we apply the Landau closure developed in [@wang-et-al:2015] in the fluid description. ![Oversimplified sketch of a multiphysics approach for tail reconnection[]{data-label="fig:sketch"}](earthfieldlines){width="90.00000%"} It may turn out that in some spatial regions outside the reconnection zone the temperature tensor in the 10-moment description is nearly isotopic. In this case it is even possible to switch to a 5-moment description. This change can be done separately for ions and electron. In future studies we will also try to include the coupling of the 5-moment model to magnetohydrodynamic (MHD) models (with generalised Ohms law) which would represent the last step in this hierarchy. With this multiphysics strategy, these codes can be applied to problem sizes that are otherwise impossible to reach with kinetic simulations and the understanding of the impact of small scale phenomena on the dynamics on global scales is in reach. The outline of the paper is the following: first we briefly describe all the plasma models and the necessary numerical schemes (Vlasov equation, 10- and 5-moment fluid equations, Maxwell’s equations, the coupling procedure, the Landau fluid closure). We will then study the Geospace Environmental Modeling (GEM) reconnection setup [@birn2001] and perform comparisons to pure kinetic and pure fluid simulations. Plasma Models ============= The plasma models that we have to consider are: i) the Vlasov equation, ii) Maxwell’s equations and iii) the 10- and 5-moment fluid equations. We will briefly summarise these sets of equations. Vlasov equation --------------- Collisionless plasmas on the finest level of description are governed by the Vlasov equation $$\label{eq:vlasov-eq} \partial_t f_s({\textbf{x}},{\textbf{v}},t) + {\textbf{v}}\cdot \nabla_{{\textbf{x}}}f_s({\textbf{x}},{\textbf{v}},t) + \frac{q_s}{m_s}\big({\textbf{E}} + {\textbf{v}} \times {\textbf{B}} \big)\cdot \nabla_{{\textbf{v}}}f_s({\textbf{x}},{\textbf{v}},t) = 0\;,$$ where $f_s({\textbf{x}},{\textbf{v}},t)$ denotes the phase-space density, $q_s$ and $m_s$ the particle charge and mass for species $s \in \{e,i\}$ (electrons and ions). The electric and magnetic fields ${\textbf{E}}$ and ${\textbf{B}}$ are given by Maxwell’s equations: \[eq:maxwell-eq\] $$\begin{aligned} \nabla \cdot {\textbf{E}} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot {\textbf{B}} &= 0 \\ \partial_t {\textbf{B}} &= - \nabla \times {\textbf{E}} \label{eq:faradays-law} \\ \partial_t {\textbf{E}} &= c^2\left(\nabla\times{\textbf{B}} - \mu_0 {\textbf{j}}\right) \label{eq:amperes-law} \end{aligned}$$ with speed of light $c$ and electric constant $\varepsilon_0$. Maxwell’s equations depend on charge and current densities $\rho$ and ${\textbf{j}}$, which are obtained from the phase-space densities $f_s({\textbf{x}},{\textbf{v}},t)$: $$\begin{aligned} \rho &:= \sum_s q_s \int\! f_s({\textbf{x}},{\textbf{v}},t)\,{\mathrm{d}^}3 v, \label{eq:charge-density} \\ {\textbf{j}} &:= \sum_s q_s \int\! {\textbf{v}} f_s({\textbf{x}},{\textbf{v}},t) \,{\mathrm{d}^}3 v\;. \label{eq:current-density} \end{aligned}$$ Vlasov equation and Maxwell’s equations form a closed set of equations and constitute the most fundamental description of a collisionless plasma. Two-species fluid equations --------------------------- Fluid descriptions can be obtained from the Vlasov equation by taking moments of the phase-space density $f_s$, $$\label{eq:def-nth-moment} \mu_{n,s} := \int\! {\textbf{v}}^n f_s({\textbf{x}},{\textbf{v}},t) \,{\mathrm{d}^}3 v \, .$$ Here, ${\textbf{v}}^n$ denotes the n-fold tensor product of ${\textbf{v}}$ with itself, ${\textbf{v}}^0 := 1$. Typically, only the first few moments are considered since a Gaussian distribution $f_s({\textbf{x}},{\textbf{v}},t)$ is exactly represented by the moments $\mu_{0,s}$, $\mu_{1,s}$, $\mu_{2,s}$ (and all other moments equaling zero). We will subsequently the describe the 10- and 5-moment equations. Consider the lowest moments up to $\mu_{3,s}$ : \[eq:mass-density\] & & n\_s &:= \_[0,s]{} = f\_s([**x**]{},[**v**]{},t) \^3 v &\ \[eq:momentum-density\] & & \_s &:= = v f\_s([**x**]{},[**v**]{},t) \^3 v &\ \[eq:energy-tensor\] & & \_s &:= m\_s \_[2,s]{} = m\_s v\^2 f\_s([**x**]{},[**v**]{},t) \^3 v &\ \[eq:heat-flux\] & & \_s &:= m\_s \_[3,s]{} = m\_s v\^3 f\_s([**x**]{},[**v**]{},t) \^3 v & $n_s$, $\hat{{\textbf{u}}}_s$, ${\mathbbm{E}}_s$ are evolved by the following equations, obtained from the Vlasov equation : \[eq:10mom-eq\] $$\begin{aligned} \partial_t n_s =& - \nabla \cdot (n_s\hat{{\textbf{u}}}_s) \label{eq:continuity-3d} \\ \partial_t (m_sn_s\hat{{\textbf{u}}}_s) =& - \nabla\cdot{\mathbbm{E}}_s + q_s \big( n_s{\textbf{E}} + n_s\hat{{\textbf{u}}}_s \times {\textbf{B}} \big) \label{eq:momentum-3d} \\ \partial_t {\mathbbm{E}}_s =& -\nabla\cdot{\mathbbm{Q}}_s + 2 q_s {\operatorname{sym}}\!\left(n_s\hat{{\textbf{u}}}_s {\textbf{E}} + \frac 1 {m_s}{\mathbbm{E}}_s\times {\textbf{B}}\right) \label{eq:energy-3d} \end{aligned}$$ Naturally, these equations are not closed. Designing appropriate fluid closures have a long history. An excellent overview is given in @chust-belmont:2006. In order to mimic kinetic Landau damping effects, several closures have been developed (see [@hammett-dorland-perkins:1992; @passot-sulem:2003]), all based on the early @hammett-perkins:1990 model. Wang et al. [@wang-et-al:2015] suggested a heat flux closure which approximates a spectrum of wave numbers by one single wave number $k_0$. Following this idea, Allmann-Rahn et al. [@allmann-rahn-trost-grauer:2018] developed an improved model that is able to correctly describe the kinetic scaling of average reconnection rate $(\lambda/d_{i})^{-0.73}$ as a function of the distance between the islands’ $O$-points $\lambda$ and where $d_i$ denotes the ion skin depth (see figure 9 in [@allmann-rahn-trost-grauer:2018]). In the simulations used in this paper the original $k_0$-closure from @wang-et-al:2015 is used. It approximates the divergence of the heat flux with an expression that forces an anisotropic pressure tensor to a more isotropic one. More precisely, the expression reads $$\label{eq:closure-ten-moment} \nabla \cdot {\mathbbm{Q}}_s = v_{\mathrm{th},s}|k_0|({\mathbbm{P}}_s - p_s{\mathbbm{1}})\;,$$ with the pressure tensor ${\mathbbm{P}}_s = {\mathbbm{E}}_s - m_s n_s \hat{{\textbf{u}}}_s\hat{{\textbf{u}}}_s$, the scalar pressure $p_s = \frac 1 3 {\operatorname{tr}}{\mathbbm{P}}_s$ and thermal velocity $v_{\mathrm{th}}$. The parameter $k_0$ is choosen on the order of the inverse Debye length. Together with , this constitutes a closed set of ten fluid equations. In future simulations it planed to switch to the improved model introduced in @allmann-rahn-trost-grauer:2018. As an alternative to the 10-moment description, an even simpler 5-moment description can be introduced where the energy density tensor and heatflux tensor are replaced by the scalar energy density $\mathcal{E}_s = \frac 1 2 {\operatorname{tr}}{\mathbbm{E}}_s$ and vector heat flux ${\textbf{Q}}_s$. The scalar energy density evolves in time according to: $$\label{eq:energy-1d} \partial_t\mathcal{E}_s = -\nabla\cdot{\textbf{Q}}_s - \nabla\cdot\left(\frac{5} 2 p_s{\textbf{u}}_s - \frac 1 2 m_sn_s({\textbf{u}}_s\cdot{\textbf{u}}_s){\textbf{u}}_s \right) + q_sn_s{\textbf{u}}_s\cdot {\textbf{E}}$$ Together with the assumption of adiabaticity, $\nabla\cdot{\textbf{Q}}_s \equiv 0$, (\[eq:continuity-3d\], \[eq:momentum-3d\], \[eq:energy-1d\]) form the set of five moment equations. Completely analogous to the case of the Vlasov equation, the source terms $\rho$ and $\mathbf{j}$ in Maxwell’s equations  are formed from the particle densities $n_s$ (see equation ). Actually, as will be described in section \[sec:maxwell\], only the current density $\mathbf{j}$ is needed to propagate Maxwell’s equations. Numerical Methods ================= Vlasov equation --------------- In order to circumvent the complexity that could arise from the high dimensionality of the phase space, the Vlasov equation is split into five one-dimensional problems using Strang splitting [@strang:1968]. These one-dimensional advection problems are solved with a third order semi-Lagrangian flux-conservative scheme introduced by @filbet2001. In order to minimize the error due to the Strang splitting when calculating the backward characteristics needed in the semi-Lagrangian method, the cascade interpolation [@leslie-purser:1995] is combined with the Boris step [@boris:1970] to form the backsubstitution method introduced in @schmitz-grauer:2006a. Details of this procedure can be found in [@schmitz-grauer:2006b]. The code is fully parallelized using the message passing interface (MPI) [@MPI:1994] where the Vlasov part is solved in parallel on distributed graphics cards using CUDA programming tools [@programming-guide]. Two-species fluid equations --------------------------- Both the 10-moment and the 5-moment two-species fluid models are all discretized with the same numerical methods. For the discretization in space, we use the CWENO scheme introduced by @kurganov-levy:2000, an easy to implement third order finite-volume scheme which is a perfect compromise between sharp shock resolution and high-order approximation in spatially smooth regions. A third order strong-stability-preserving Runge-Kutta scheme [@shu-osher:1988] is employed for the time integration. Maxwell’s equation {#sec:maxwell} ------------------ The electromagnetic fields are positioned on a staggered Yee grid [@yee:1966] in order to maintain the divergence free condition for the magnetic field: $\nabla \cdot \mathbf{B} = 0$. Equations (\[eq:faradays-law\], \[eq:amperes-law\]) are evolved through the FDTD method presented in Taflove and Brodwin [@taflove1995]. Here, only the current density $\mathbf{j}$ enters as a source term. Since the speed of light exceeds all other speeds found in the plasma by far, subcycling is used in order to resolve lightwaves while keeping the global timestep as large as possible. In addition, the speed of light is artificially reduced to $20$ times the Alfvén speed. Adaptive Coupling ----------------- The coupling strategy is the most important and at the same time the most critical part of the multiphysics simulations. The coupling strategy involves two separate problems: first, providing the correct boundary conditions at interfaces between different physical models and second, designing criteria to decide which model can be used in which part of the computational domain in an adaptive way. We start with discussing the strategy for obtaining boundary conditions at the interfaces. Providing boundary conditions for the fluid part at the kinetic/fluid interface is rather straightforward: the fluid boundary conditions are obtained by taking the necessary moments of the phase-space densities $f_s$ at the interface. Providing boundary conditions for the phase-space densities $f_s$ from the fluid description is far less trivial and described in detail in @rieke-et-al:2015. In short the procedure can be summarised as follows: we first extrapolate the phase-space density $f_s$ to the boundary region. Next, we adjust the extrapolated phase-space density $f_s$ such that the moments equal the moments from the 10-moment fluid description. In this way, we only manipulate the phase-space density $f_s$ rather “smoothly” with minimal changes and do not force $f_s$ to a Gaussian shape. The coupling between the 10-moment and 5-moment fluid regions is done in a very natural way. The boundary conditions for the 5-moment description is simply obtained by calculating the energy scalar from the trace of the energy density tensor. In the other direction, the energy tensor has to be constructed from the energy scalar by assuming a diagonal shape at the 10-/5-moment interface. The criteria to decide which of the available models shall be used in which subregion of the domain is a highly non-trivial issue. Presently, our strategy is still in a phase of proof of concept and further work has to be invested. For the case of magnetic reconnection, we implemented heuristic criteria based on the current density $j_z$ since it is a good indicator for regions of high reconnection. In order to allow for a finer detachment of electrons and ions, we actually use the velocity $u_{z,s} = n_s \hat u_{z,s}$ for electrons and ions separately. This is reasonable as the current density is given by $j_z = \sum_s q_s u_{z,s}$ (compare ). It should be stated clearly that a criterion based on the current density is rather heuristic and far from universally applicable approach and further work has to be invested on robust criteria. In addition, once a criterion is considered satisfactory for the context, thresholds have to be defined that mirror a good trade-off between the need to use a higher-information model for a correct representation and the opportunity to save computational resources with a lower-information model. Up to know we can only state that this is based on educated guesses. $2^0$ $2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ $2^7$ ---------- ------- ------- ------- ------- ------- ------- ------- ------- $2^{26}$ $2^{28}$ $2^{30}$ $2^{32}$ $2^{34}$ : \[tab\_scaling\] Scaling behavior of [*muphy*]{} on JURECA across different problem sizes (absolut number of grid cells) and number of GPUs. Given are the times needed for 2000 steps (in hours) and the relative speedup normalized to $2^{30}$ cells on one GPU. \[tab:performance\] ![Scaling behavior of [*muphy*]{} on JURECA. Same data as in table \[tab:performance\]. \[fig:performance\] ](scaling_complete){width="\textwidth"} Code performance ---------------- The described numerical codes, the adaptive coupling procedures and the parallelisation framework based on space-filling curves [@dreher-grauer:2005] is build in our framework called *muphy*. This framework has been developed over the last 10 years. It is written in C++/CUDA, runs partly on GPUs and partly on CPUs and employs MPI for parallelization. Scaling runs have been performed on the JURECA supercomputer at the FZ Jülich, Germany [@jureca], on a fully kinetic Whistler-wave setting [@daldorff-et-al:2014]. Scaling results are excellent as shown in table \[tab:performance\] and figure \[fig:performance\]. Note that the number of GPUs was only restricted by the actual configuration of JURECA. Results ======= We apply the described models and the multiphysics coupling strategy to the Geospace Environmental Modeling (GEM) challenge [@birn2001]. The domain size is chosen as $4\pi\, d_i$ in $x$- and as $2\pi\, d_i$ in $y$-direction, where $d_i$ denotes ion skin depth. The symmetry properties of the GEM problem make it sufficient to calculate only one quarter of the spatial domain. We use a uniform cell-width of d$x =\; $d$y = \frac{\pi}{128}\, d_i$ in 2d physical space and a uniform resolution of $32$ cells in each direction in 3d velocity space. To reduce the computational costs, we apply the common values for the reduced mass ratio $\frac{m_i}{m_e} = 25$ and reduced speed of light of $20$ times the Alfvén speed. The numerical setup is depicted in table \[tab:gem-setup\]. ----------------------------------------------- -------------------------------------------------- dimensions of physical domain $\{x,y\} \in \{[-2\pi..2\pi],[-\pi..\pi]\}\,d_i$ cell-width of physical space d$x = $d$y = \frac{\pi}{128}\,d_i$ size of subregions in physical space $\tilde N_x = \tilde N_y = 32$cells resolution of velocity space (kinetic region) $N_{v_x} = N_{v_y} = N_{v_z} = 32$cells mass ratio $\frac{m_i}{m_e} = 25$ speed of light $c = 20\,v_{\mathrm A}$ ----------------------------------------------- -------------------------------------------------- : \[tab:gem-setup\] Numerical setup of GEM In the simulation we use the $u_{z,s}$-based criterion with a thresholds depicted in table \[tab:thresholds-uz\]. The criterion is reassessed every $0.1\,\omega_{c,i}^{-1}$ (inverse ion-gyrofrequencies) for every subregion. kinetic iff … ten-moment iff not kinetic and … -- ----------- ------------------------------------------------------ ------------------------------------------------------- electrons $\max {\left|u_{z,e}\right|} \geq 0.3 v_{\mathrm A}$ $\max {\left|u_{z,e}\right|} \geq 0.1\,v_{\mathrm A}$ ions $\max {\left|u_{z,i}\right|} \geq 0.6 v_{\mathrm A}$ $\max {\left|u_{z,i}\right|} \geq 0.2\,v_{\mathrm A}$ : \[tab:thresholds-uz\] Thresholds for the $u_z$-criterion in units of Alfvén-speed $v_{\mathrm A}$. They are evaluated for every subregion separately. The five-moment model is used iff neither the kinetic nor the ten-moment thresholds are met In figure \[fig:time-series-uz1\] the fields $j_z$, $u_{z,e}$ and $u_{z,i}$ are shown together with the areas depicting the different physical models for different times of the simulation. From these figures one can deduce that substantial saving in simulation time can be achieved since the performance gain is approximately proportional to the ratio of the computational domain to the area where the Vlasov equation is solved. ![ \[fig:time-series-uz1\] ${\left|u_{z,e}\right|}$, $j_z$ and ${\left|u_{z,i}\right|}$ for different times in units of inverse ion-gyrofrequencies $\omega_{c,i}^{-1}$. On the outsides, the used models are depicted based on the values of $u_{z,s}$ and threshold as given in table \[tab:thresholds-uz\]. Red areas are solved with the kinetic solver, blue areas with the ten-moment and yellow areas with the five-moment fluid solver. Note that the initial models at time $t=0$ have been prescribed.](time-series-uz-c1) In figure \[fig:model-comparison\], a comparison of different simulations are shown and compared to the multiphysics run. Depicted is the current density $j_z$ at the time of the highest reconnection rate. The fully kinetic Vlasov simulation agrees rather well with the multiphysics simulation. The overall agreement is substantially better than the results obtained from the 10- and 5-moment simulations. However, differences especially in the precise values of the absolute maxima of the current density, are visible. Whether this is an effect of the model selection criteria has to be tested in further investigations. ![ \[fig:model-comparison\] $j_z$ for the coupled run and some uncoupled comparison runs with a single solver](comparison) As a more quantitative comparison, the reconnecting flux for the multiphysics simulation is plotted in figure \[fig:reconnection-flux\] together with purely kinetic and fluid runs. There are a number of things to observe from the plot: While the reconnecting flux of the fluid runs does not saturate within the simulation time, the kinetic and the multiphysics runs saturate. In addition, they both saturate at the same level and thus capture essentially the same small scale physics which is not possible with the fluid models. ![ \[fig:reconnection-flux\] Reconnection fluxes for the coupled run and some uncoupled comparison runs with a single solver. Crosses mark the point of highest reconnection rate throughout the respective run](reconnection-flux) Alone the presence of electron and ion kinetic regions in the very center of reconnection zone and 10-moment fluid regions around it seems to ensure the characteristic behaviour of the full kinetic reconnection scenario. Discussion ========== We showed that the proposed multiphysics coupling hierarchy can give excellent results even when only a small part of computational domain near the reconnection zone is captured with a kinetic model. However, still many questions and challenges remain and it is clear that the present simulations are only on the level of a proof of concept. Most important is the issue of designing robust physics refinement criteria and their thresholds. First attempts based on the heat flux are under investigation. In addition, the multiphysics coupling strategy should be formulated as an asymptotic preserving scheme [@degond-deluzet:2017; @hu-jin-li:2017]. The coupling of the 10- and 5-moment models is already in this state when incorporating the effect of Landau damping [@wang-et-al:2015; @allmann-rahn-trost-grauer:2018]. Presently, we are also reformulating the coupling between the Vlasov and the 10-moment model. For this, we formulate the kinetic description as an adaptive (in time and space) $\delta f$ method and ease the transition to the fluid description as an asymptotic preserving scheme. Finally, the multiphysics hierarchy should not stop at the level of the 5-moment fluid description. Work to couple the 5-moment model to MHD is in progress. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge interesting discussions with F. Allmann-Rahn, J. Dreher and T. Trost. Computations were conducted on the Davinci cluster at TP1 Plasma Research Department and on the JURECA cluster at Jülich Supercomputing Center under the project number HBO43. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC).

ArXiv

--- abstract: 'We predict a non-monotonous temperature dependence of the persistent currents in a ballistic ring coupled strongly to a stub in the grand canonical as well as in the canonical case. We also show that such a non-monotonous temperature dependence can naturally lead to a $\phi_0/2$ periodicity of the persistent currents, where $\phi_0$=h/e. There is a crossover temperature $T^*$, below which persistent currents increase in amplitude with temperature while they decrease above this temperature. This is in contrast to persistent currents in rings being monotonously affected by temperature. $T^*$ is parameter-dependent but of the order of $\Delta_u/\pi^2k_B$, where $\Delta_u$ is the level spacing of the isolated ring. For the grand-canonical case $T^*$ is half of that for the canonical case.' address: | $^a$Fl.48, 93-a prospect Il’icha, 310020 Khar’kov, Ukraine\ $^b$S.N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake City, Calcutta 98, India. author: - 'M. V. Moskalets$^a$ and P. Singha Deo$^{b,}$[@eml]' title: '**[Temperature enhanced persistent currents and “$\phi_0/2$ periodicity”]{}**' --- Introduction ============ Although the magnitude of persistent current amplitudes in metallic and semiconductor mesoscopic rings [@but] has received experimental attention [@exp], much attention has not been given to qualitative features of the persistent current. Qualitative features reflect the underlying phenomena, and are more important than the order of magnitude. Incidentally, the order of magnitude and sign of the persistent currents in metallic rings is still not understood. With this background in mind, we study the temperature dependence of persistent currents in a ring strongly coupled to a stub [@buet]. We predict a non-monotonous temperature dependence of the amplitude of persistent currents in this geometry both for the grand-canonical as well as for the canonical case. We show that there is a crossover temperature ($T^*$) above which it decreases with temperature and below which it increases with temperature, and energy scales determining this crossover temperature are quantified. This is in contrast to the fact that in the ring, temperature monotonously affects the amplitude of persistent currents. However, so does dephasing and impurity scattering, which are again directly or indirectly temperature dependent [@but; @Cheung], except perhaps in very restrictive parameter regimes where it is possible to realize a Luttinger liquid in the ring in the presence of a potential barrier [@krive]. Recent study, however, shows that in the framework of a Luttinger liquid, a single potential barrier leads to a monotonous temperature dependence of the persistent currents for non-interacting as well as for interacting electrons [@mos99]. We also show a temperature-induced switch over from $\phi_0$ periodicity to $\phi_0/2$ periodicity. This is a very non-trivial temperature dependence of the fundamental periodicity that cannot be obtained in the ring geometry. There is also another motivation behind studying the temperature dependence of persistent currents in this ring-stub system. In the ring, the monotonous behavior of the persistent current amplitude with temperature stems from the fact that the states in a ring pierced by a magnetic flux exhibit a strong parity effect [@Cheung]. There are two ways of defining this parity effect in the single channel ring (multichannel rings can be generalized using the same concepts and mentioned briefly at the end of this paragraph). In the single-particle picture (possible only in the absence of electron-electron interaction), it can be defined as follows: states with an even number of nodes in the wave function carry diamagnetic currents (positive slope of the eigenenergy versus flux) while states with an odd number of nodes in the wave function carry paramagnetic currents (negative slope of the eigenenergy versus flux) [@Cheung]. In the many-body picture (without any electron-electron interaction), it can be defined as follows: if $N$ is the number of electrons (spinless) in the ring, the persistent current carried by the $N$-body state is diamagnetic if $N$ is odd and paramagnetic if $N$ is even [@Cheung]. Leggett conjectured [@leg] that this parity effect remains unchanged in the presence of electron-electron interaction and impurity scattering of any form. His arguments can be simplified to say that when electrons move in the ring, they pick up three different kinds of phases: 1) the Aharonov-Bohm phase due to the flux through the ring, 2) the statistical phase due to electrons being Fermions and 3) the phase due to the wave-like motion of electrons depending on their wave vector. The parity effect is due to competition between these three phases along with the constraint that the many-body wave function satisfy the periodic boundary condition (which means if one electron is taken around the ring with the other electrons fixed, the many-body wave function should pick up a phase of 2$\pi$ in all). Electron-electron interaction or simple potential scattering cannot introduce any additional phase, although it can change the kinetic energy or the wave vector and hence modify the third phase. Simple variational calculations showed that the parity effect still holds [@leg]. Multichannel rings can be understood by treating impurities as perturbations to decoupled multiple channels, which means small impurities just open up small gaps at level crossings within the Brillouin zone and keep all qualitative features of the parity effect unchanged. Strong impurity scattering in the multichannel ring can, however, introduce strong level correlations, which is an additional phenomenon. Whether and how the parity effect gets modified by these correlations is an interesting problem. In a one-dimensional (1D) system where we have a stub of length $v$ strongly coupled to a ring of length $u$ (see the left bottom corner in Fig. 1), we can have a bunching of levels with the same sign of persistent currents, [@Deo95] i.e., many consecutive levels carry persistent currents of the same sign. This is essentially a breakdown of the parity effect. The parity effect breaks down in this single channel system because there is a new phase that does not belong to any of the three phases discussed by Leggett and mentioned in the preceding paragraph. This new phase cancels the statistical phase and so the N-body state and the (N+1)-body state behave in similar ways or carry persistent currents of the same sign [@deo96; @sre]. When the Fermi energy is above the value where we have a node at the foot of the stub (that results in a transmission zero in transport across the stub), there is an additional phase of $\pi$ arising due to a slip in the Bloch phase [@deo96] (the Bloch phase is the third kind of phase discussed above, but the extra phase $\pi$ due to slips in the Bloch phase is completely different from any of the three phases discussed above because this phase change of the wave function is not associated with a change in the group velocity or kinetic energy or the wave vector of the electron [@deo96; @sre]). The origin of this phase slip can be understood by studying the scattering properties of the stub structure. One can map the stub into a $\delta$-function potential of the form $k \cot (kv) \delta (x-x_0)$ [@deo96]. So one can see that the strength of the effective potential is $k \cot (kv)$ and is energy dependent. Also the strength of the effective potential is discontinuous at $kv=n \pi$. Infinitesimally above $\pi$ an electron faces a positive potential while infinitesimally below it faces a negative potential. As the effective potential is discontinuous as a function of energy, the scattering phase, which is otherwise a continuous function of energy, in this case turns out to be discontinuous as the Fermi energy sweeps across the point $kv=\pi$. As the scattering phase of the stub is discontinuous, the Bloch phase of the electron in the ring-stub system is also discontinuous. This is pictorially demonstrated in Figs. 2 and 3 of Ref. [@deo96]. In an energy scale $\Delta_u\propto 1/u$ (typical level spacing for the isolated ring of length $u$) if there are $n_b\sim\Delta_u/\Delta_v$ (where $\Delta_v\propto 1/v$, the typical level spacing of the isolated stub) such phase slips, then each phase slip gives rise to an additional state with the same slope and there are $n_b$ states of the same slope or ithe same parity bunching together with a phase slip of $\pi$ between each of them [@deo96]. The fact that there is a phase slip of $\pi$ between two states of the same parity was generalized later, arguing from the oscillation theorem, which is equivalent to Leggett’s conjecture for the parity effect [@lee]. Transmission zeros are an inherent property of Fano resonance generically occurring in mesoscopic systems and this phase slip is believed to be observed [@the] in a transport measurement [@sch]. For an elaborate discussion on this, see Ref. [@tan]. A similar case was studied in Ref. [@wu], where they show the transmission zeros and abrupt phase changes arise due to degeneracy of “dot states” with states of the “complementary part” and hence these are also Fano-type resonances. The purpose of this work is to show a very non-trivial temperature dependence of persistent currents due to the breakdown of the parity effect. The temperature effects predicted here, if observed experimentally, will further confirm the existence of parity-violating states, which is a consequence of this new phase. To be precise, the new phase is the key source of the results discussed in this work. Theoretical treatment ===================== We concentrate on the single channel system to bring out the essential physics. The multichannel ring also shows a very strong bunching of levels even though the rotational symmetry is completely broken by the strongly coupled stub and wide gaps open up at the level crossings [@sre] within the Brillouin zone. Hence let us consider a one-dimensional loop of circumference $u$ with a one-dimensional stub of length $v$, which contain noninteracting spinless electrons. The quantum-mechanical potential is zero everywhere. A magnetic flux $\phi$ penetrates the ring (see the left bottom corner in Fig. 1). In this paper we consider both the grand-canonical case (when the particle exchange with a reservoir at temperature $T$ is present and the reservoir fixes the chemical potential $\mu$; in this case we will denote the persistent current as $I_\mu$) and the canonical case (when the number $N$ of particles in the ring-stub system is conserved; in this case we will denote the persistent current as $I_N$). For the grand canonical case we suppose that the coupling to a reservoir is weak enough and the eigenvalues of electron wave number $k$ are not affected by the reservoir [@Cheung]. They are defined by the following equation [@Deo95]. $$\cos(\alpha)=0.5\sin(ku)\cot(kv)+\cos(ku), \label{Eq1}$$ \ where $\alpha=2\pi\phi/\phi_0$, with $\phi_0=h/e$ being the flux quantum. Note that Eq. (\[Eq1\]) is obtained under the Griffith boundary conditions, [@Griffith] which take into account both the continuity of an electron wave function and the conservation of current at the junction of the ring and the stub; and the hard-wall boundary condition at the dead end of the stub. Each of the roots $k_n$ of Eq.(\[Eq1\]) determines the one-electron eigenstate with an energy $\epsilon_n=\hbar^2k_n^2/(2m)$ as a function of the magnetic flux $\phi$. Further we calculate the persistent current $I_{N/\mu}=-\partial F_{N/\mu}/\partial \phi$ [@Byers], where $F_N$ is the free energy for the regime $N=const$ and $F_\mu$ is the thermodynamic potential for the regime $\mu=const$. In the latter case for the system of noninteracting electrons the problem is greatly simplified as we can use the Fermi distribution function $f_0(\epsilon)=(1+\exp[(\epsilon-\mu)/T] )^{-1}$ when we fill up the energy levels in the ring-stub system and we can write the persistent current as follows [@Cheung]. $$I_\mu=\sum_n I_n f_0(\epsilon_n), \label{Eq2}$$ \ where $I_n$ is a quantum-mechanical current carried by the $n$th level and is given by [@Deo95] $$\frac{\hbar I_n}{e}=\frac{2k_n\sin(\alpha)}{\frac{u}{2}\cos(k_nu) \cot(k_nv)-[\frac{v}{2}{\rm cosec}^2(k_nv)+u]\sin(k_nu)}. \label{Eq3}$$ \ For the case of $N=const$ we must calculate the partition function $Z$, which determines the free energy $F_N=-T\ln(Z)$ [@Landau], $$Z=\sum_m \exp\left( -\frac{E_m}{T} \right), \label{Eq4}$$ \ where $E_m$ is the energy of a many-electron level. For the system of $N$ spinless noninteracting electrons $E_m$ is a sum over $N$ different (pursuant to the Pauli principle) one-electron energies $E_m=\sum_{i=1}^{N} \epsilon_{n_i}$, where the index $m$ numbers the different series $\{\epsilon_{n_1},...,\epsilon_{n_N}\}_m$. For instance, the ground-state energy is $E_0=\sum_{n=1}^{N}\epsilon_n$. Results and discussions ======================= First we consider the peculiarities of the persistent current $I_\mu$, $i.e.,$ for the regime $\mu=const$. In this case the persistent current is determined by Eqs.(\[Eq1\])-(\[Eq3\]). Our calculations show that the character of the temperature dependence of the persistent currents is essentially dependent on the position of the Fermi level $\mu$ relative to the groups of levels with similar currents. If the Fermi level lies symmetrically between two groups (which occurs if $u/\lambda_F=n$ or $n+0.5$, where $n$ is an integer and $\lambda_F$ is the Fermi wavelength), then the current changes monotonously with the temperature that is depicted in Fig. 1 (the dashed curve). In this case the low-lying excited levels carry a current which is opposite to that of the ground-state; the line shape of the curve is similar to that of the ring [@Cheung]. On the other hand, if the Fermi level lies within a group ($u/\lambda_F\sim n+0.25$) then low-lying excited states carry persistent currents with the same sign. In that case there is an increase of a current at low temperatures as shown in Fig. 1 (the dotted curve). At low temperatures the currents carried by the low-lying excited states add up with the ground-state current. However, these excited states are only populated at the cost of the ground state population. Although in the clean ring higher levels carry larger persistent currents, this is not true for the ring-stub system. This is because the scattering properties of the stub are energy-dependent and at a higher energy the stub can scatter more strongly. Hence a lot of energy scales such as temperature, Fermi energy and number of levels populated compete with each other to determine the temperature dependence. A considerable amount of enhancement in persistent current amplitudes as obtained in our calculations appears for all choices of parameters whenever the Fermi energy is approximately at the middle of a group of levels that have the same slope. At higher temperatures when a large number of states get populated, the current decreases exponentially. So in this case the current amplitude has a maximum as a function of the temperature and we can define the temperature corresponding to the maximum as the crossover temperature $T^*$. It is worth mentioning that in the ring system, although there is no enhancement of persistent currents due to temperature, one can define a crossover temperature below which persistent currents decrease less rapidly with temperature. Essentially this is because at low temperatures thermal excitations are not possible because of the large single-particle level spacings. Hence this crossover temperature is the same as the energy scale that separates two single-particle levels, $i.e.,$ the crossover temperature is proportional to the level spacing $\Delta=hv_F/L$ in the ideal ring at the Fermi surface, where $v_F$ is the Fermi velocity and $L$ is the length of the ring. The crossover temperature obtained by us in the ring-stub system is of the same order of magnitude, $i.e.,$ $\Delta_u=hv_F/u$, although different in meaning. In the case of $u/\lambda_F=n+0.25$ at low temperatures we show the possibility of obtaining $\phi_0/2$ periodicity, although the parity effect is absent in this system. This is shown in Fig. 2, where we plot $I_{\mu}/I_0$ versus $\phi/\phi_0$ at a temperature $k_BT/\Delta_u$=0.01 in solid lines, which clearly show a $\phi_0/2$ periodicity. Previously two mechanisms are known that can give rise to $\phi_0/2$ periodicity of persistent currents. The first is due to the parity effect [@los], which does not exist in our system, and the second is due to the destructive interference of the first harmonic that can only appear in a system coupled to a reservoir so that the Fermi energy is an externally adjustable parameter. The later mechanism can be understood by putting $k_FL$=$(2n\pi+\pi /2)$ in eq. 2.11 in Ref. [@Cheung]. If this later case is the case in our situation, then the periodicity should remain unaffected by temperature and for fixed $N$ we should only get $\phi_0$ periodicity [@Cheung], because then the Fermi energy is not an externally adjustable parameter but is determined by $N$. We show in Fig. 2 (dashed curve) that the periodicity changes with temperature and in the next two paragraphs we will also show that one can obtain $\phi_0/2$ periodicity for fixed $N$. The dashed curve in Fig. 2 is obtained at a temperature $k_BT/\Delta_u$=0.15 and it shows a $\phi_0$ periodicity. As it is known, the crossover temperature depends on the harmonic number $m$: $T^*_m=T^*/m$ [@Cheung], in this case a particular harmonic can actually increase with temperature initially and decrease later, different harmonics reaching their peaks at different temperatures. Therefore, the second harmonic that peaks at a lower temperature than the first harmonic can exceed the first harmonic in certain temperature regimes. At higher temperatures it decreases with the temperature faster than the first harmonic and so at higher temperature $\phi_0$ periodicity is recovered. In view of a strong dependence of the considered features on the chemical potential, we consider further the persistent current $I_N$ in the ring-stub system with a fixed number of particles $N=const$. In this case we calculate the persistent current using the partition function (Eq.(\[Eq4\])). The numerical calculations show that in this case there is also a non-monotonous temperature dependence of the persistent current amplitude in the canonical case as in the grand-canonical case. This is shown in Fig. 1 by the solid curve. The maximum of $I_N(T)$ is more pronounced if $v/u$ is large and the number of electrons ($N$) is small. Besides, if the number of electrons is more than $n_b/2$, then the maximum does not exist. The crossover temperature is higher by a factor 2 as compared to that in $I_\mu$. This was also found for the 1D ring [@Cheung; @Loss92], where, as mentioned before, the crossover temperature has a different meaning. To show that one can have $\phi_0/2$ periodicity for fixed $N$, we plot in the inset of Fig. 2 the first harmonic $I_1/I_0$ (solid curve) and the second harmonic $I_2/I_0$ (dotted curve) of $I_N$ for $N$=5, $v$=7$k_F$ and $u$=2.5$k_F$. At low temperature the second harmonic exceeds the first harmonic because the stub reduces the level spacing and in a sense can adjust the Fermi energy in the ring to create partial but not exact destruction of the first harmonic. There are distinct temperature regimes where $I_1$ exceeds $I_2$ and vice versa and the two curves peak in completely different temperatures. $I_2$ also exhibits more than one maxima. Experimentally different harmonics can be measured separately and the first harmonic as shown in Fig. 2 can show tremendous enhancement with temperature. An important conclusion that can be made from Fig. 2 is that observation of $\phi_0/2$ periodicity as well as $\phi_0$ is possible even in the absence of the parity effect quite naturally because the absence of the parity effect also means one can obtain an enhancement of the persistent current amplitude with temperature, and as a result an enhancement of a particular harmonic with temperature, resulting in different harmonics peaking at different temperatures. Conclusions =========== In summary, we would like to state that the temperature dependence of persistent currents in a ring strongly coupled to a stub exhibits very nontrivial features. Namely, at small temperatures it can show an enhancement of the amplitude of persistent currents in the grand-canonical as well as in the canonical case. The fundamental periodicity of the persistent currents can change with temperature. If detected experimentally, these can lead to a better understanding of the qualitative features of persistent currents. It will also confirm the existence of parity-violating states that is only possible if there is a new phase apart from the three phases considered by Leggett [@leg] while generalizing the parity effect. This new phase is the sole cause of the nontrivial temperature dependence. There is a crossover temperature $T^*$ above which the amplitude of persistent currents decreases with temperature. How the crossover temperature is affected by electron correlation effects and dephasing should lead to interesting theoretical and experimental explorations in the future. Finally, with the large discrepancies between theory and experiments for the persistent currents in disordered rings, one cannot completely rule out the possibility of parity violation in the ring system as well. The stub is not the only way to produce this new phase that leads to a violation of the parity effect. There can be more general ways of getting transmission zeros [@lee] that may also be parity violation. In that case, the ring-stub system may prove useful as a theoretical model to understand the consequences of parity violation. Its consequences on the temperature dependence shown here may motivate future works in this direction. [99]{} email: [email protected] M. B[ü]{}ttiker, Y. Imry, and R. Landauer, Phys. Lett. [**96A**]{}, 365 (1983). L. P. Levy et al, Phys. Rev. Lett. [**64**]{}, 2074 (1990); V. Chandrasekhar et al, Phys. Rev. Lett. [**67**]{}, 3578 (1991); D. Mailly et al, Phys. Rev. Lett. [**70**]{}, 2020 (1993). The weak coupling limit was earlier studied by M. B[ü]{}ttiker, Phys. Scripta T [**54**]{}, 104 (1994). Coupled rings was studied by T.P. Pareek and A.M. Jayannavar, Phys. Rev. B [**54**]{}, 6376 (1996). H.F.Cheung, Y.Gefen, E.K.Riedel, W.H.Shih, Phys. Rev. B [**37**]{}, 6050 (1988). I.V. Krive et al, Phys. Rev. B [**52**]{}, 16451 (1995); A.S.Rozhavsky, J. Phys.: Condens. Matter [**9**]{}, 1521 (1997); I. V. Krive et al, cond-mat/9704151. M.V. Moskalets, Physica E [**5**]{}, 124 (1999). A.J. Leggett in: Granular nano-electronics, eds. D. K. Ferry, J.R. Barker and C. Jacobony, NATO ASI Ser. B [**251**]{} (Plenum, New York, 1991) p. 297. P.Singha Deo, Phys. Rev. B [**51**]{}, 5441 (1995). P. Singha Deo, Phys. Rev. B [**53**]{}, 15447 (1996). P. A. Sreeram and P. Singha Deo, Physica B [**228**]{}, 345 (1996). H.-W.Lee, Phys. Rev. Lett., [**82**]{}, 2358 (1999). P.Singha Deo and A.M.Jayannavar, Mod. Phys. Lett. B [**10**]{}, 787 (1996); P.Singha Deo, Solid St. Communication [**107**]{}, 69 (1998); C.M.Ryu et al, Phys. Rev. B [**58**]{}, 3572 (1998); Hongki Xu et al, Phys. Rev. B, [**57**]{}, 11903 (1998). R. Schuster et al, Nature [**385**]{}, 417 (1997). T. Taniguchi and M. Büttiker, Phys. Rev. B [**60**]{}, 13814 (1999). J.Wu et al, Phys. Rev. Lett. [**80**]{}, 1952 (1998). S. Griffith, Trans. Faraday. Soc. [**49**]{}, 650 (1953). N. Byers, C.N. Yang, Phys. Rev. Lett. [**7**]{}, 46 (1961); F. Bloch, Phys. Rev. B [**2**]{}, 109 (1970). L.D. Landau, E.M. Lifschitz, (1959) Statistical Physics (Pergamon, London). D. Loss and P. Goldbart, Phys. Rev. B [**43**]{}, 13762 (1991). D. Loss, Phys. Rev. Lett. [**69**]{}, 343 (1992).  \ Fig. 1. The ring of length $u$ with a stub (resonant cavity) of length $v$ threaded by a magnetic flux $\phi$ (left bottom corner). The dependence of the current amplitude $I_\mu$ in units of $I_0=ev_F/u$ on the temperature $T$ in units of $\Delta_u/2\pi^2k_B$ for the regime $\mu=const$ with $v=15\lambda_F$ and $u=(5+x)\lambda_F$ at $x=0$ (dashed curve) and $x=0.25$ (dotted curve); and $I_N/I_0$ for the isolated ring-stub system with $v/u=10$, and $N=3$ (solid curve). For the appropriate scale the curves 2 and 3 are multiplied by factors of 3 and 15, respectively. Fig. 2. The dependence of the persistent current $I_\mu$ in units of $I_0=ev_F/u$ on the magnetic flux $\phi$ in units of $\phi_0$ for the regime $\mu=const$ with $v=15\lambda_F$ and $u=5.25\lambda_F$ for $T/\Delta_u=0.01$ (dashed curve) and $T/\Delta_u=0.15$ (solid curve). The curve 2 is multiplied by a factor of 5 for the appropriate scale. The inset shows the first harmonic $I_1$ (solid curve) and second harmonic $I_2$ (dotted curve) of $I_N$ in units of $I_0$ for N fixed at 5, $v$=7$k_F$ and $u$=2.5$k_F$ versus temperature in units of $\Delta_u/2\pi^2k_B$.

ArXiv

--- abstract: 'In atomic and molecular phase measurements using laser-induced fluorescence detection, optical cycling can enhance the effective photon detection efficiency and hence improve sensitivity. We show that detecting many photons per atom or molecule, while necessary, is not a sufficient condition to approach the quantum projection limit for detection of the phase in a two-level system. In particular, detecting the maximum number of photons from an imperfectly closed optical cycle reduces the signal-to-noise ratio (SNR) by a factor of $\sqrt{2}$, compared to the ideal case in which leakage from the optical cycle is sufficiently small. We derive a general result for the SNR in a system in terms of the photon detection efficiency, probability for leakage out of the optical cycle per scattered photon, and the product of the average photon scattering rate and total scattering time per atom or molecule.' author: - Zack Lasner - 'D. DeMille' bibliography: - 'cyclingPaperBibFinal.bib' title: 'Statistical sensitivity of phase measurements via laser-induced fluorescence with optical cycling detection' --- Atoms and molecules are powerful platforms to probe phenomena at quantum-projection-limited precision. In many atomic and molecular experiments, a quantum state is read out by laser-induced fluorescence (LIF), in which population is driven to a short-lived state and the resulting fluorescence photons are detected. Due to geometric constraints on optical collection and technological limitations of photodetectors, the majority of emitted photons are typically undetected, reducing the experimental signal. Optical cycling transitions can be exploited to overcome these limitations, by scattering many photons per particle. In the limit that many photons from each particle are detected, the signal-to-noise ratio (SNR) may be limited by the quantum projection (QP) noise (often referred to as atom or molecule shot noise). LIF detection with photon cycling is commonly used in ultra-precise atomic clock [@Wynands2005; @Zelevinsky2008] and atom interferometer [@Cronin2009] experiments to approach the QP limit. Molecules possess additional features, beyond those in atoms, that make them favorable probes of fundamental symmetry violation [@ACMECollaboration2014; @Collaboration2018; @Hudson2011; @Devlin2015; @Hunter2012; @Kozyryev2017] and fundamental constant variation [@Borkowski2018; @Beloy2011; @DeMille2008; @Zelevinsky2008; @Shelkovnikov2008; @Kozyryev2018], as well as promising platforms for quantum information and simulation [@DeMille2002; @Liu2018; @Micheli2006; @Sundar2018; @Wall2015]. Many molecular experiments that have been proposed, or which are now being actively pursued, will rely on optical cycling to enhance measurement sensitivity while using LIF detection [@Collaboration2018; @Hunter2012; @Kozyryev2018; @Kozyryev2017; @ACMECollaboration2014; @Devlin2015]. Due to the absence of selection rules governing vibrational decays, fully closed molecular optical cycling transitions cannot be obtained: each photon emission is associated with a non-zero probability of decaying to a “dark” state that is no longer driven to an excited state by any lasers. However, for some molecules many photons can be scattered using a single excitation laser, and up to $\sim10^{6}$ photons have been scattered using multiple repumping lasers to return population from vibrationally excited states into the optical cycle [@DiRosa2004; @Shuman2009]. This has enabled, for example, laser cooling and magneto-optical trapping of molecules [@Shuman2010; @Barry2014; @Hummon2013; @Collopy2018; @Zhelyazkova2014; @Truppe2017; @Chae2017; @Anderegg2017]. Furthermore, some precision measurements rely on atoms in which no simply closed optical cycle exists [@Regan2002; @Parker2015]; our discussion here will be equally applicable to such species. These considerations motivate a careful study of LIF detection for precision measurement under the constraint of imperfectly closed optical cycling. Some consequences of loss during the cycling process have been considered in [@Rocco2014]. However, the effect of the statistical nature of the cycling process on the optimal noise performance has not been previously explored. In particular, the number of photons scattered before a particle (an atom or molecule) decays to an unaddressed dark state, and therefore ceases to fluoresce, is governed by a statistical distribution rather than a fixed finite number. We show that due to the width of this distribution, a naive cycling scheme reduces the SNR to below the QP limit. In particular, we find that in addition to the intuitive requirement that many photons from every particle are detected, to approach the QP limit it is also necessary that the probability of each particle exiting the cycling transition (via decay to a dark state outside the cycle) is negligible during detection. If this second condition is not satisfied, so that each particle scatters enough photons that it is very likely to have been optically pumped into a dark state, then the SNR is decreased by a factor of $\sqrt{2}$ below the QP limit. Consider an ensemble of $N$ particles in an effective two-level system, in a state of the form $$|\psi\rangle=(e^{-i\phi}|\uparrow\rangle+e^{i\phi}|\downarrow\rangle)/\sqrt{2}.$$ The relative phase $\phi$ is the quantity of interest in this discussion. It can be measured, for example, by projecting the wavefunction onto an orthonormal basis $\{|X\rangle\propto|\uparrow\rangle+|\downarrow\rangle,\,|Y\rangle\propto|\uparrow\rangle-|\downarrow\rangle\}$ such that $|\langle X|\psi\rangle|^{2}=\cos^{2}(\phi)$ and $|\langle Y|\psi\rangle|^{2}=\sin^{2}(\phi)$. In the LIF technique, this can be achieved by driving state-selective transitions, each addressing either $|X\rangle$ or $|Y\rangle$, through an excited state that subsequently decays to a ground state and emits a fluorescence photon. This light is detected, and the resulting total signals, $S_{X}$ and $S_{Y}$, are associated with each state. (This protocol is equivalent to the more standard Ramsey method, in which each spin is reoriented for detection by a spin-flip pulse and the population of spin-up and spin-down particles is measured [@Ramsey1950].) The measured value of the phase, $\tilde{\phi},$ is computed from the observed values of $S_{X}$ and $S_{Y}$. In the absence of optical cycling, the statistical uncertainty of the phase measurement is $\sigma_{\tilde{\phi}}=\frac{1}{2\sqrt{N\epsilon}}$, where $\epsilon$ is the photon detection efficiency and $0<\epsilon\leq1$. Note that $N\epsilon$ is the average number of detected photons; hence, this result is often referred to as the “photon shot noise limit.” In the ideal case of $\epsilon=1$, the QP limit (a.k.a. the atom or molecule shot noise limit) limit $\sigma_{\tilde{\phi}}=\frac{1}{2\sqrt{N}}$ is obtained. This scaling is derived as a limiting case of our general treatment below, where the effects of optical cycling are also considered. We suppose that the phase is projected onto the $\{|X\rangle,\,|Y\rangle\}$ basis independently for each particle. Repeated over the ensemble of particles, the total number of particles $N_{X}$ projected along $|X\rangle$ is drawn from a binomial distribution, $N_{X}\sim B(N,\,\cos^{2}\phi)$, where $x\sim f(\alpha_{1},\cdots,\alpha_{k})$ denotes that the random variable $x$ is drawn from the probability distribution $f$ parametrized by $\alpha_{1},\cdots,\alpha_{k}$, and $B(\nu,\,\rho)$ is the binomial distribution for the total number of successes in a sequence of $\nu$ independent trials that each have a probability $\rho$ of success. Therefore, $\overline{N_{X}}=N\,\cos^{2}\phi$ and $\sigma_{N_{X}}^{2}=N\,\cos^{2}\phi\sin^{2}\phi$, where $\bar{x}$ is the expectation value of a random variable $x$ and $\sigma_{x}$ is its standard deviation over many repetitions of an experiment. We define the number of photons scattered from the $i$-th particle to be $n_{i}$, where a “photon scatter” denotes laser excitation followed by emission of one spontaneous decay photon, and define $\overline{n_{i}}=\bar{n}$ (the average number of photons scattered per particle) and $\sigma_{n_{i}}=\sigma_{n}$. Note that these quantities are assumed to be the same for all particles (i.e., independent of $i$). The probability of detecting any given photon (including both imperfect optical collection and detector quantum efficiency) is $\epsilon$, such that each photon is randomly either detected or not detected. We define $d_{ij}$ to be a binary variable indexing whether the $j$-th photon scattered from the $i$-th particle is detected. Therefore, $d_{ij}\sim B(1,\,\epsilon)$, and it follows that $\overline{d_{ij}}=\epsilon$ and $\sigma_{d_{ij}}^{2}=\epsilon(1-\epsilon)$. We define the signal of the measurement of a particular quadrature $|X\rangle$ or $|Y\rangle$ from the ensemble, when projecting onto that quadrature, to be the total number of photons detected. For example, the signal $S_{X}$ from particles projected along $|X\rangle$ is $$S_{X}=\sum_{i=1}^{N_{X}}\sum_{j=1}^{n_{i}}d_{ij}.\label{eq:Sx definition}$$ $\noindent$Explicitly, among $N$ total particles, $N_{X}$ are projected by the excitation light onto the $|X\rangle$ state and the rest are projected onto $|Y\rangle$. The $i$-th particle projected onto $|X\rangle$ scatters a total of $n_{i}$ photons, and we count each photon that is detected (in which case $d_{ij}=1)$. The right-hand side of Eq. \[eq:Sx definition\] depends on $\phi$ implicitly through $N_{X}$, and we use this dependence to compute $\tilde{\phi}$, the measured value of $\phi$. Because $N_{X},\,n_{i},$ and $d_{ij}$ are all statistical quantities, the extracted value $\tilde{\phi}$ has a statistical uncertainty. The QP limit is achieved when the only contribution to uncertainty arises from $N_{X}$ due to projection onto the $\{|X\rangle,|Y\rangle\}$ basis. We can compute $\overline{S_{X}}$ by repeated application of Wald’s lemma ([@Bruss1991; @Wald2013]), $\overline{\sum_{i=1}^{m}x}=\bar{m}\bar{x}$. This results in $$\overline{S_{X}}=N\cos^{2}\phi\,\bar{n}\epsilon.\label{eq:ESx}$$ $\noindent$That is, the expected signal from projecting onto the $|X\rangle$ state is (as could be anticipated) simply the product of the average number of particles in $|X\rangle$, $N\cos^{2}\phi$, the number of photons scattered per particle, $\bar{n}$, and the probability of detecting each photon, $\epsilon$. We compute the variance in $S_{X}$ by repeated use of the law of total variance [@Blitzstein], $\sigma_{a}^{2}=\overline{\sigma_{a|b}^{2}}+\sigma_{\overline{a|b}}^{2}$, where $\overline{a|b}$ denotes the mean of $a$ conditional on a fixed value of $b$ and, analogously, $\sigma_{a|b}^{2}$ denotes the variance of $a$ conditional on a fixed value of $b$. This gives $$\sigma_{S_{X}}^{2}=N\cos^{2}\phi\,\bar{n}\epsilon^{2}\left(\frac{1}{\epsilon}+\frac{\sigma_{n}^{2}}{\bar{n}}-1+\bar{n}\sin^{2}\phi\right).$$ $\noindent$The results for $S_{Y}$ are identical, with the substitution $\cos^{2}\phi\leftrightarrow\sin^{2}\phi$. Many atomic clocks [@Weyers2001; @Jefferts2002; @Kurosu2004; @Levi2004; @Szymaniec2005a] and some molecular precision measurement experiments [@ACMECollaboration2014; @Devlin2015] measure both $S_{X}$ and $S_{Y}$, while others detect only a single state [@Collaboration2018; @Hudson2011; @Regan2002; @Parker2015]. In what follows, we assume that both states are probed. The case of detecting only one state, with some means of normalizing for variations in $N\bar{n}\epsilon$, can be worked out using similar considerations. In the regime $\phi=\pm\frac{\pi}{4}+\delta\phi$, where $\delta\phi\ll1$, sensitivity to small changes in phase, $\delta\phi$, is maximized. In this case, we define the measured phase deviation $\delta\tilde{\phi}$ by $\tilde{\phi}=\pm\frac{\pi}{4}+\delta\tilde{\phi}$. This is related to measured quantities via the asymmetry $\mathcal{A}=\frac{S_{X}-S_{Y}}{S_{X}+S_{Y}}=\mp\sin(2\delta\tilde{\phi})\approx\mp2\delta\tilde{\phi}$. When $N\gg1$, the average value of $\tilde{\phi}$ computed in this way is equal to the phase $\phi$ of the two-level system. The uncertainty in the asymmetry, $\sigma_{\mathcal{A}}\approx\frac{1}{N}\sqrt{\sigma_{S_{X}}^{2}+\sigma_{S_{Y}}^{2}-2\sigma_{S_{X},S_{Y}}^{2}}$, can be computed to leading order in $\delta\phi$ from $\sigma_{S_{X}}$, $\sigma_{S_{Y}}$, and the covariance $\sigma_{S_{X},S_{Y}}^{2}=\overline{S_{X}S_{Y}}-\overline{S_{X}}\,\overline{S_{Y}}$ using standard error propagation [@Bevington1969]. We relate $\sigma_{\mathcal{A}}$ to the uncertainty in the measured phase by $\sigma_{\mathcal{A}}=2\sigma_{\tilde{\phi}}$. This relationship defines the statistical uncertainty in $\tilde{\phi}$, the measured value of $\phi$, for the protocol described here. The covariance, $\sigma_{S_{X},S_{Y}}^{2}=-\frac{N}{4}\bar{n}^{2}\epsilon^{2}$, can be calculated directly using the same methods already described. This result can be understood as follows: the photon scattering and detection processes for particles projected onto $|X\rangle$ and $|Y\rangle$ are independent, so the covariance between signals $S_{X}$ and $S_{Y}$ only arises from quantum projection. In the simplest case of perfectly efficient, noise-free detection and photon scattering, e.g., $\epsilon=1$, $\bar{n}=1$, and $\sigma_{n}=0$, the quantum projection noise leads to signal variances $\sigma_{S_{X}}^{2}=\sigma_{S_{Y}}^{2}=\frac{N}{4}$. The covariance is negative because a larger number of particles projected onto $|X\rangle$ is associated with a smaller number of particles projected onto $|Y\rangle$. The additional factor of $\bar{n}^{2}\epsilon^{2}$ for the general case accounts for the fact that both signals $S_{X}$ and $S_{Y}$ are scaled by $\bar{n}\epsilon$ when $\bar{n}$ photons are scattered per particle and a proportion $\epsilon$ of those photons are detected on average. The uncertainty in the measured phase, computed using the procedure just described, has the form $\sigma_{\tilde{\phi}}=\frac{1}{2\sqrt{N}}\sqrt{F}$, where we have defined the “excess noise factor” $F$ given in this phase regime by $$F=1+\frac{1}{\bar{n}}\left(\frac{1}{\epsilon}-1\right)+\frac{\sigma_{n}^{2}}{\bar{n}^{2}}.$$ It is instructive to evaluate this expression in some simple limiting cases. For example, consider the case when exactly one photon is scattered per particle so that $\bar{n}=1$ and $\sigma_{n}=0$. (This is typical for experiments with molecules, where optical excitation essentially always leads to decay into a dark state.) In this case, $F=\frac{1}{\epsilon}$ and the uncertainty in the phase measurement is $\sigma_{\tilde{\phi}}=\frac{1}{2\sqrt{N\epsilon}}$, as stated previously. Alternatively, as $\bar{n}\rightarrow\infty$, $F\rightarrow1+\left(\frac{\sigma_{n}}{\bar{n}}\right)^{2}$. This is in exact analogy with the excess noise of a photodetector whose average gain is $\bar{n}$ and whose variance in gain is $\sigma_{n}^{2}$ [@Knoll2010]. By inspection, the ideal result of $F\rightarrow1$ can be achieved only if $\frac{\sigma_{n}}{\bar{n}}\rightarrow0$, and either $\epsilon\rightarrow1$ or $\bar{n}\rightarrow\infty$. We now compute $\bar{n}$ and $\sigma_{n}^{2}$ for a realistic optical cycling process. We define the branching fraction to dark states, which are lost from the optical cycle, to be $b_{\ell}$. We assume that each particle interacts with the excitation laser light for a time $T$, during which the scattering rate of a particle in the optical cycle is $r$. Therefore, an average of $rT$ photons would be scattered in the absence of decay to dark states, i.e. when $b_{\ell}=0$. (All of our results hold for a time-dependent scattering rate $r(t)$, with the substitution $rT\rightarrow\int r(t)dt$.) Note that in the limit $rT\rightarrow\infty$, $1/b_{\ell}$ photons are scattered per particle on average. Recall that the number of photons scattered from the $i$-th particle, when projected to a given state, is $n_{i}$. We define the probability that a particle emits exactly $n_{i}$ photons to be $P(n_{i};\,rT,b_{\ell})$. This probability distribution can be computed by first ignoring the decay to dark states. For the case where $b_{\ell}=0$, the number of photons emitted in time $T$ follows a Poisson distribution with average number of scattered photons $rT$. For the more general case where $b_{\ell}>0$, we assign a binary label to each photon depending on whether it is associated with a decay to a dark state. Each decay is characterized by a Bernoulli process, and we use the conventional labels of “successful” (corresponding to decay to an optical cycling state) and “unsuccessful” (corresponding to decay to a dark state) for each outcome. Then $P(n_{i};\,rT,b_{\ell})$ is the probability that there are exactly $n_{i}$ events in the Poisson process, all of which are successful, or there are at least $n_{i}$ events such that the first $n_{i}-1$ are successful and the $n_{i}$-th is unsuccessful. (For concreteness, we have assumed that unsuccessful decays, i.e., those that populate dark states, emit photons with the same detection probability as all successful decays. The opposite case, in which decays to dark states are always undetected, can be worked out with the same approach and leads to similar conclusions.) Direct calculation gives $$\bar{n}=\frac{1-e^{-b_{\ell}rT}}{b_{\ell}}{\rm \,and}$$ $$\sigma_{n}^{2}=\frac{1-b_{\ell}+e^{-b_{\ell}rT}b_{\ell}(2b_{\ell}rT-2rT+1)-e^{-2b_{\ell}rT}}{b_{\ell}^{2}}.$$ Therefore, $$F=1+\frac{1}{1-e^{-b_{\ell}rT}}\left(\frac{b_{\ell}}{\epsilon}+\frac{1-2b_{\ell}+2b_{\ell}e^{-b_{\ell}rT}(1-rT(1-b_{\ell}))-e^{-2b_{\ell}rT}}{1-e^{-b_{\ell}rT}}\right).\label{eq:sigmaPhi}$$ $\noindent$The behavior of the SNR (proportional to $1/\sqrt{F}$) arising from Eq. \[eq:sigmaPhi\] is illustrated in Fig. \[fig:snr\]. ![$1/\sqrt{F}$, the SNR resulting from Eq. \[eq:sigmaPhi\], normalized to the ideal case of the QP limit ($F=1$). This plot assumes $\epsilon=0.1$. When few photons per particle can be detected, i.e., when $\epsilon/b_{\ell}\ll1$ (far left of plot), cycling to very deep completion $(b_{\ell}rT\gg1$) does not significantly affect the SNR. Even when one photon per particle can be detected on average, i.e., when $\epsilon/b_{\ell}=1$ (dashed red line), the SNR never exceeds roughly half its ideal value. By further closing the optical cycle, i.e. such that $\epsilon/b_{\ell}\gg1$ (right of dashed red line), the SNR can be improved to near the optimal value given by the QP limit. However, to reach this optimal regime, the number of photons that would be scattered in the absence of dark states, $rT$, must be small compared to the average number that can be scattered before a particle exits the optical cycle, $1/b_{\ell}$. For example, with $1/b_{\ell}=1,000$ (green dashed line) and $rT=100$ so that $b_{\ell}rT=0.1$ (lower circle), the SNR is more than 30% larger than in the case when $rT=10,000$ and $b_{\ell}rT=10$ (upper circle). \[fig:snr\]](\string"snr_plot_3\string".pdf){width="8cm"} To understand the implications of this result, we consider several special cases, summarized in Table \[tab:special-cases\]. We first consider the simple case when cycling is allowed to proceed until all particles decay to dark states, i.e., $b_{\ell}rT\rightarrow\infty$. We refer to this as the case of “cycling to completion.” In this case, for the generically applicable regime $\epsilon\leq\frac{1}{2}$ we find $F\geq2$, even as the transition becomes perfectly closed ($b_{\ell}\rightarrow0$). We can understand this result intuitively as follows. As the optical cycling proceeds, the number of particles that will still be in the optical cycle after each photon scatter is proportional to the number of particles that are currently in the optical cycle, $\frac{dP}{dn_{i}}\propto P$. Hence, we expect $P(n_{i};\,rT\rightarrow\infty,b_{\ell})\propto e^{-\alpha n_{i}}$ for some characteristic constant $\alpha$. In fact, one can show that for $rT\rightarrow\infty$, this result holds with $\alpha\approx b_{\ell}$. The width $\sigma_{n}$ of this exponential distribution is given by the mean $\bar{n}$; that is, $\sigma_{n}\approx\bar{n}$. Therefore, we should expect that cycling to completion reduces the SNR by a factor of $\sqrt{F}=\sqrt{1+(\sigma_{n}/\bar{n})^{2}}\rightarrow\sqrt{2}$ compared to the ideal case of $F=1$, which requires $\frac{\sigma_{n}}{\bar{n}}=0$. Surprisingly, this reduction in SNR can be partially recovered for an imperfectly closed optical cycle, by choosing a finite cycling time, $rT<\infty$, to minimize $\sigma_{\tilde{\phi}}$. The best limiting case, as found from Eq. \[eq:sigmaPhi\], preserves the condition that many photons are detected per particle, $rT\epsilon\gg1$, but additionally requires that the probability of decaying to a dark state remains small, $rTb_{\ell}\ll1$. In this case, photon emission is approximately a Poisson process for which $\left(\frac{\sigma_{n}}{\bar{n}}\right)^{2}\approx\frac{1}{rT}\ll1$, and the excess noise factor, $F$, does not have a significant contribution from the variation in scattered photon number. The optimal value of $rT$ for a finite proportion of decays to dark states, $b_{\ell}$, and detection efficiency, $\epsilon$, lies in the intermediate regime and can be computed numerically. A special case of “cycling to completion,” which must be considered separately, occurs when every particle scatters exactly one photon, corresponding to parameter values $b_{\ell}=1$ and $rT\gg1$ so that $\bar{n}=1$ and $\sigma_{n}=0$. As we have already seen, in this case there is no contribution to the excess noise arising from variation in the scattered photon number, and hence the SNR is limited only by photon shot noise: $F=\frac{1}{\epsilon}$. In atomic physics experiments with essentially completely closed optical cycles, $b_{\ell}\approx0$, the limit $b_{\ell}rT\rightarrow\infty$ is not obtained even for very long cycling times where $rT\gg1$. Instead, in this case $b_{\ell}rT\rightarrow0$ and hence $F\rightarrow1+\frac{1}{rT\epsilon}$, which approaches unity as the probability to detect a photon from each particle becomes large, $rT\epsilon\gg1$. Therefore, the reduction in the SNR associated with the distribution of scattered photons does not occur in this limit of a completely closed optical cycle. Condition Sub-conditon $F$ ---- ------------------------------- -------------------------------- -------------------------------------------------------------------- 1a $b_{\ell}rT\rightarrow\infty$ $2+b_{\ell}(\frac{1}{\epsilon}-2)$ 1b $b_{\ell}rT\rightarrow\infty$ $\epsilon\leq0.5$ $\geq2$ 2a $b_{\ell}rT\rightarrow0$ $1+\frac{1}{rT\epsilon}+\frac{1}{2}b_{\ell}(\frac{1}{\epsilon}-2)$ 2b $b_{\ell}rT\rightarrow0$ $\epsilon rT\rightarrow\infty$ 1 3a **$b_{\ell}\rightarrow1$** $\frac{1}{\epsilon}\frac{1}{1-e^{-rT}}$ 3b $b_{\ell}\rightarrow1$ $rT\rightarrow\infty$ $\frac{1}{\epsilon}$ : The excess noise factor $F$ in some special cases. (1a) All particles are lost to dark states during cycling. (1b) With all particles lost and realistic detection efficiency, $\epsilon\leq0.5$, $F\geq2$. (2a) No particles are lost to dark states. (2b) No particles are lost, but many photons per particle are detected. The QP limit is reached. (3a) Up to one photon can be scattered per particle. (3b) Exactly one photon is scattered per particle and the photon shot noise limit is reached.\[tab:special-cases\] We have also considered how the additional noise due to optical cycling combines with other noise sources in the detection process. For example, consider intrinsic noise in the photodetector itself. Commonly, a photodetector (such as a photomultiplier or avalanche photodiode) has average intrinsic gain $\bar{G}$ and variance in the gain $\sigma_{G}^{2}$, with resulting excess noise factor $f=1+\frac{\sigma_{G}^{2}}{\bar{G}^{2}}$. Including this imperfection in the model considered here leaves Eq. \[eq:sigmaPhi\] unchanged up to the substitution $\epsilon\rightarrow\epsilon/f$. Similar derivations can be performed assuming a statistical distribution of $N$ or $\phi$ to obtain qualitatively similar but more cumbersome results. In conclusion, we have shown that a quantum phase measurement, with detection via laser-induced fluorescence using optical cycling on an open transition, when driven to completion, incurs a reduction in the SNR by a factor of $\sqrt{2}$ compared to the QP limit when the optical cycle is driven to completion. This effect has been understood as due to the distribution of the number of scattered photons for this particular case. This reduction of the SNR does not occur for typical atomic systems, where decay out of the optical cycle and into dark states is negligible over the timescale of the measurement. An expression for the SNR has been derived for the general case, in which the cycling time is finite and the probability of decay to dark states is non-zero. For a given decay rate to dark states, an optimal combination of cycling rate and time can be computed numerically to obtain a SNR that most closely approaches the QP limit. This ideal limit can be obtained only when the photon cycling proceeds long enough for many photons from each atom or molecule to be detected, but not long enough for most atoms or molecules to exit the optical cycle by decaying to an unaddressed dark state. This work was supported by the NSF.

ArXiv

--- abstract: | The Schr" odinger picture of the Dirac quantum mechanics is defined in charts with spatially flat Robertson-Walker metrics and Cartesian coordinates. The main observables of this picture are identified, including the interacting part of the Hamiltonian operator produced by the minimal coupling with the gravitational field. It is shown that in this approach new Dirac quantum modes on de Sitter spacetimes may be found analytically solving the Dirac equation. Pacs: 04.62.+v author: - | Ion I. Cotăescu [^1]\ [*West University of Timişoara,*]{}\ [*V. P\^ arvan Ave. 4, RO-300223 Timişoara, Romania*]{} title: 'The Schr" odinger picture of the Dirac quantum mechanics on spatially flat Robertson-Walker backgrounds' --- The relativistic quantum mechanics of the spin-half particle on a given background can be constructed as the one-particle restriction of the quantum theory of the free Dirac field on this background, considered as a perturbation that does not affect the geometry. The central piece is the Dirac equation whose form depends on the local chart (or natural frame) and the tetrad fields defining the local frames and co-frames. This type of quantum mechanics has two virtues. First of all the charge conjugation of the Dirac field is point-independent indicating that the vacuum of the original field theory is stable in any geometry [@co; @cot]. Therefore, the resulted one-particle Dirac quantum mechanics can be seen as a coherent theory similar to that of special relativity. The second virtue is just the spin which generates specific terms helping us to correctly interpret the physical meaning of principal operators. In the non-relativistic quantum mechanics the time evolution can be studied in different pictures (e. g., Schr" odinger, Heisenberg, Interaction) which transform among themselves through specific time-dependent unitary transformations. It is known that the form of the Hamiltonian operator and the time dependence of other operators strongly depend on the picture choice. In special and general relativity, despite of its importance, the problem of time-evolution pictures is less studied because of the difficulties in finding suitable Hamiltonian operators for scalar or vector fields. However, the Dirac quantum mechanics is a convenient framework for studying this problem since the Dirac equation can be put in Hamiltonian form at any time. In this paper we should like to show that at least two different pictures of the Dirac quantum mechanics can be identified in the case of backgrounds with spatially flat Robertson-Walker (RW) metrics. We start with the simple conjecture of the Dirac equation in diagonal gauge and Cartesian coordinates considering that this constitutes the [*natural*]{} picture. Furthermore, we define the Schr" odinger picture such that the kinetic part of the Dirac equation should take the standard form known from special relativity. In this picture we identify the momentum and the Hamiltonian operators pointing out that they represent a generalization of the similar operators we obtained previously on de Sitter spacetimes [@cot]. Let us start denoting by $\{t,\vec{x}\}$ the Cartesian coordinates $x^{\mu}$ ($\mu,\nu,...=0,1,2,3 $) of a chart with the RW line element $$ds^2=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}=dt^2-\alpha(t)^2 (d\vec{x}\cdot d\vec{x})$$ where $\alpha$ is an arbitrary time dependent function. In this chart we introduce the tetrad fields $e_{\hat\mu}(x)$ that define the local frames and those defining the corresponding coframes, $\hat e^{\hat\mu}(x)$ [@SW]. These fields are labeled by the local indices ($\hat\mu,\hat\nu,...=0,1,2,3$) of the Minkowski metric $\eta=$diag$(1,-1,-1,-1)$, satisfy $e_{\hat\mu}(x)\hat e^{\hat\mu}(x)=1_{4\times4}$ and give the metric tensor as $g_{\mu \nu}=\eta_{\hat\alpha\hat\beta}\hat e^{\hat\alpha}_{\mu}\hat e^{\hat\beta}_{\nu}$. Here we consider the tetrad fields of the diagonal gauge that have non-vanishing components [@BD; @SHI], $$\label{tt} e^{0}_{0}=1\,, \quad e^{i}_{j}=\frac{1}{\alpha(t)}\delta^{i}_{j}\,,\quad \hat e^{0}_{0}=1\,, \quad \hat e^{i}_{j}=\alpha(t)\delta^{i}_{j}\,,\quad i,j,...=1,2,3\,,$$ determining the form of the Dirac equation [@BD], $$\label{ED1} \left(i\gamma^0\partial_{t}+i\frac{1}{\alpha(t)}\gamma^i\partial_i +\frac{3i}{2}\frac{\dot{\alpha}(t)}{\alpha(t)}\gamma^{0}-m\right)\psi(x)=0\,.$$ This is expressed in terms of Dirac $\gamma$-matrices [@TH] and the fermion mass $m$, with the notation $\dot{\alpha}(t)=\partial_t\alpha(t)$. Thus we obtain the natural picture in which the time evolution is governed by the Dirac equation (\[ED1\]). The principal operators of this picture, the energy $\hat H$, momentum $\vec{\hat P}$ and coordinate $\vec{\hat X}$, can be defined as in special relativity, $$\label{ON} (\hat H \psi)(x)=i\partial_t\psi(x)\,,\quad (\hat P^i \psi)(x)=-i\partial_i\psi_S(x)\,,\quad (\hat X^i \psi)(x)=x^i\psi(x)\,.$$ The operators $\hat X^i$ and $\hat P^i$ are time-independent and satisfy the well-known canonical commutation relations $$\label{com} \left[\hat X^i, \hat P^j\right]=i\delta_{ij}I\,,\quad \left[\hat H, \hat X^i\right]=\left[\hat H,\hat P^i\right]=0\,,$$ where $I$ is the identity operator. Other operators are formed by orbital parts and suitable spin parts that can be point-dependent too. In general, the orbital terms are freely generated by the basic orbital operators $\hat X^i$ and $\hat P^i$. An example is the total angular momentum $\vec{J}=\vec{L}+\vec{S}$ where $\vec{L}=\vec{\hat X}\times\vec{\hat P}$ and $\vec{S}$ is the spin operator. We specify that the operators $\hat P^i$ and $J^i$ are generators of the spinor representation of the isometry group $E(3)$ of the spatially flat RW manifolds [@cot]. Therefore, these operators are [*conserved*]{} in the sense that they commute with the Dirac operator [@CML; @ES]. The natural picture can be changed using point-dependent operators which could be even non-unitary operators since the relativistic scalar product does not have a direct physical meaning as that of the non-relativistic quantum mechanics. We exploit this opportunity for defining the Schr" odinger picture as the picture in which the kinetic part of the Dirac operator takes the standard form $i\gamma^0\partial_t+i\gamma^i\partial_i$. The transformation $\psi(x)\to \psi_S(x)=U_S(x)\psi(x)$ leading to the Schr" odinger picture is produced by the operator of time dependent [*dilatations*]{} $$\label{U} U_S(x)=\exp\left[-\ln(\alpha(t))(\vec{x}\cdot\vec{\partial})\right]\,,$$ which has the following suitable action $$U_S(x)F(\vec{x})U_S(x)^{-1}=F\left(\frac{1}{\alpha(t)}\vec{x}\right)\,,\quad U_S(x)G(\vec{\partial})U_S(x)^{-1}=G\left(\alpha(t)\vec{\partial}\right)\,,$$ upon any analytical functions $F$ and $G$. Performing this transformation we obtain the Dirac equation of the Schr" odinger picture $$\label{ED2} \left[i\gamma^0\partial_{t}+i\vec{\gamma}\cdot\vec{\partial} -m +i\gamma^{0}\frac{\dot{\alpha}(t)}{\alpha(t)} \left(\vec{x}\cdot\vec{\partial}+\frac{3}{2}\right)\right]\psi_S(x)=0\,.$$ Hereby we have to identify the specific operators of this picture, the energy $H_S$ and the operators $P^i_S$ and $X^i_S$ that must be time-independent, as in the non-relativistic case. We assume that these operators are defined as $$\label{OS} (H_S \psi_S)(x)=i\partial_t\psi_S(x)\,,\quad (P^i_S \psi_S)(x)=-i\partial_i\psi_S(x)\,,\quad (X^i_S \psi_S)(x)=x^i\psi_S(x)\,,$$ obeying commutation relations similar to Eqs. (\[com\]). The Dirac equation (\[ED2\]) can be put in Hamiltonian form, $H_S\psi_S={\cal H}_S\psi_S$, where the Hamiltonian operator ${\cal H}_S={\cal H}_0 + {\cal H}_{int}$ has the standard kinetic term ${\cal H}_0=\gamma^0\vec{\gamma}\cdot \vec{P}_S+\gamma^0 m$ and the interaction term with the gravitational field, $$\label{Hint} {\cal H}_{int}=\frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{X}_S\cdot \vec{P}_S-\frac{3i}{2}I\right)=\frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{\hat X}\cdot \vec{\hat P}-\frac{3i}{2}I\right)\,,$$ which vanishes in the absence of gravitation when $\alpha$ reduces to a constant. The sets of operators (\[OS\]) and (\[ON\]) are defined in different manners such that they have similar expressions but in different pictures. For analyzing the relations among these operators it is convenient to turn back to the natural picture. Performing the inverse transformation we find that in this picture the operators (\[OS\]) become new interesting time-dependent operators, $$\begin{aligned} H(t)&=&U_S(x)^{-1}H_SU_S(x)=\hat H+\frac{\dot{\alpha}(t)}{\alpha(t)} \vec{\hat X}\cdot\vec{\hat P}\,,\label{Ht}\\ X^i(t)&=&U_S(x)^{-1}X_S^iU_S(x)=\alpha(t) \hat X^i\,,\\ P^i(t)&=&U_S(x)^{-1}P_S^iU_S(x)=\frac{1}{\alpha(t)}\hat P^i\,,\label{Pt}\end{aligned}$$ satisfying the usual commutation relations (\[com\]). Notice that the total angular momentum and the operator (\[Hint\]) have the same expressions in both these pictures since they commute with $U_S(x)$. Now the problem is to select the set of operators with a good physical meaning. We observe that in the moving charts with RW metrics of the de Sitter spacetime the operator (\[Ht\]) is time-independent (since $\dot{\alpha}/\alpha =$const.) and [*conserved*]{}, corresponding to the unique time-like Killing vector of the $SO(4,1)$ isometries [@cot]. This is an argument indicating that the correct physical observables are the operators (\[Ht\])-(\[Pt\]) while the operators (\[ON\]) may be considered as auxiliary ones. In fact these are just the usual operators of the relativistic quantum mechanics on Minkowski spacetime where is no gravitation. The examples we worked out [@cot; @C; @C1] convinced us that this is the most plausible interpretation even though this is not in accordance with other attempts [@Guo]. The Schr" odinger picture we defined above may offer one some technical advantages in solving problems of quantum systems interacting with the gravitational field. For example, in this picture we can derive the non-relativistic limit (in the sense of special relativity) replacing ${\cal H}_0$ directly by the Schr" odinger kinetic term $\frac{1}{2m}{\vec{P}_S}^2$. Thus we obtain the Schr" odinger equation $$\label{Sc} \left[-\frac{1}{2m}\Delta -i\, \frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{x}\cdot \vec{\partial }+\frac{3}{2}\right)\right]\phi(x)=i\partial_t \phi(x)\,,$$ for the wave-function $\phi$ of a spinless particle of mass $m$. Moreover, using standard methods one can derive the next approximations in $1/c$ producing characteristic spin terms. It is remarkable that the non-relativistic Hamiltonian operators obtained in this way are Hermitian with respect to the usual non-relativistic scalar product. In the particular case of the de Sitter spacetime, the Schr" odinger picture will lead to important new results for the Dirac and Schr" odinger equations in moving charts with RW metrics and spherical coordinates. In these charts where $H=H(t)$ is conserved both the mentioned equations, namely Eqs. (\[ED2\]) and (\[Sc\]), are analytically solvable in terms of Gauss hypergeometric functions and, respectively, Whittaker ones [@coco]. Therefore, in this picture it appears the opportunity of deriving new Dirac quantum modes determined by the set of commuting operators $\{H, {\vec{J}\,}^2, J_3, K\}$ where $K=\gamma^0(2\vec{L}\cdot\vec{S}+1)$ is the Dirac angular operator. We specify that common eigenspinors of this set of operators were written down in [*static*]{} central charts with spherical coordinates [@C] but never in moving charts. We remind the reader that in moving charts with spherical coordinates one knows only the Shishkin’s solutions of the Dirac equation [@SHI] derived in the natural picture. We have shown [@C1] that there are suitable linear combinations of these solutions representing common eigenspinors of the set $\{{\vec{\hat P}\,}^2,{\vec{J}\,}^2, J_3, K\}$. Taking into account that the operators $H$ and ${\vec{\hat P}\,}^2$ do not commute with each other we understand the importance of the new quantum modes that could be showed off grace to our Schr" odinger picture. Finally we note that the quantum mechanics developed here is a specific approach working only in spatially flat RW geometries. This may be completed with new pictures (as the Heisenberg one) after we shall find the general mechanisms of time evolution in the relativistic quantum mechanics. However, now it is premature to look for general principles before to carefully analyze other significant particular examples. Acknowledgments {#acknowledgments .unnumbered} --------------- We are grateful to Erhardt Papp for interesting and useful discussions on closely related subjects. [20]{} I. I. Cotăescu, [*Int. J. Mod. Phys. A*]{} [**19**]{}, 2117 (2004). I. I. Cotăescu, [*Phys. Rev. D*]{} [**65**]{}, 084008 (2002). S. Weinberg, [*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*]{} (Wiley, New York, 1972). A. O. Barut and I. H. Duru, [*Phys. Rev. D*]{} [**36**]{}, 3705 (1987). G. V. Shishkin, [*Class. Quantum Grav.*]{} [**8**]{}, 175 (1991). B. Thaller, [*The Dirac Equation*]{}, Springer Verlag, Berlin Heidelberg, 1992 B. Carter and R. G. McLenaghan, [*Phys. Rev. D*]{} [**19**]{}, 1093 (1979). I. I. Cotăescu, [*J. Phys. A: Math. Gen.*]{} [**33**]{}, 1977 (2000). I. I. Cotăescu, [*Mod. Phys. Lett. A*]{} [**13**]{}, 2991 (1998). I. I. Cotăescu, Radu Racoceanu and Cosmin Crucean, Mod. Phys. Lett. A [**21**]{}, 1313 (2006). H.-Y. Guo, C.-G. Huang, Z. Xu and B. Zhou, [*Mod. Phys. Lett. A*]{} [**19**]{}, 1701 (2004); id. [hep-th/0311156]{}; E. A. Tigrov, [gr-qc/0011011]{}. I. I. Cotăescu, in preparation. [^1]: E-mail:   [email protected]

ArXiv

--- abstract: 'This report reviews the recent experimental results from the CLAS collaboration (Hall B of Jefferson Lab, or JLab) on Deeply Virtual Compton Scattering (DVCS) and Deeply Virtual Meson Production (DVMP) and discusses their interpretation in the framework of Generalized Parton Distributions (GPDs). The impact of the experimental data on the applicability of the GPD mechanism to these exclusive reactions is discussed. Initial results obtained from JLab 6 GeV data indicate that DVCS might already be interpretable in this framework while GPD models fail to describe the exclusive meson production (DVMP) data with the GPD parameterizations presently used. An exception is the $\phi$ meson production for which the GPD mechanism appears to apply. The recent global analyses aiming to extract GPDs from fitting DVCS CLAS and world data are discussed. The GPD experimental program at CLAS12, planned with the upcoming 12 GeV upgrade of JLab, is briefly presented.' author: - | Hyon-Suk Jo\ \ Institut de Physique Nucléaire d’Orsay, 91406 Orsay, France title: '[**Deeply Virtual Compton Scattering and Meson Production at JLab/CLAS**]{}' --- Introduction {#introduction .unnumbered} ============ Generalized Parton Distributions take the description of the complex internal structure of the nucleon to a new level by providing access to, among other things, the correlations between the (transverse) position and (longitudinal) momentum distributions of the partons in the nucleon. They also give access to the orbital momentum contribution of partons to the spin of the nucleon. GPDs can be accessed via Deeply Virtual Compton Scattering and exclusive meson electroproduction, processes where an electron interacts with a parton from the nucleon by the exchange of a virtual photon and that parton radiates a real photon (in the case of DVCS) or hadronizes into a meson (in the case of DVMP). The amplitude of the studied process can be factorized into a hard-scattering part, exactly calculable in pQCD or QED, and a non-perturbative part, representing the soft structure of the nucleon, parametrized by the GPDs. At leading twist and leading order approximation, there are four independent quark helicity conserving GPDs for the nucleon: $H$, $E$, $\tilde{H}$ and $\tilde{E}$. These GPDs are functions depending on three variables $x$, $\xi$ and $t$, among which only $\xi$ and $t$ are experimentally accessible. The quantities $x+\xi$ and $x-\xi$ represent respectively the longitudinal momentum fractions carried by the initial and final parton. The variable $\xi$ is linked to the Bjorken variable $x_{B}$ through the asymptotic formula: $\xi=\frac{x_{B}}{2-x_{B}}$. The variable $t$ is the squared momentum transfer between the initial and final nucleon. Since the variable $x$ is not experimentally accessible, only Compton Form Factors, or CFFs (${\cal H}$, ${\cal E}$, $\tilde{{\cal H}}$ and $\tilde{{\cal E}}$), which real parts are weighted integrals of GPDs over $x$ and imaginary parts are combinations of GPDs at the lines $x=\pm\xi$, can be extracted. The reader is referred to Refs. [@gpd1; @gpd2; @gpd3; @gpd4; @gpd5; @gpd6; @gpd7; @gpd8; @vgg1; @vgg2; @bmk] for detailed reviews on the GPDs and the theoretical formalism. Deeply Virtual Compton Scattering {#deeply-virtual-compton-scattering .unnumbered} ================================= ![Handbag diagram for DVCS (left) and diagrams for Bethe-Heitler (right), the two processes contributing to the amplitude of the $eN \to eN\gamma$ reaction.[]{data-label="fig:diagrams"}](dvcs_bh.png){height="0.11\textheight"} Among the exclusive reactions allowing access to GPDs, Deeply Virtual Compton Scattering (DVCS), which corresponds to the electroproduction of a real photon off a nucleon $eN \to eN\gamma$, is the key reaction since it offers the simplest, most straighforward theoretical interpretation in terms of GPDs. The DVCS amplitude interferes with the amplitude of the Bethe-Heitler (BH) process which leads to the exact same final state. In the BH process, the real photon is emitted by either the incoming or the scattered electron while in the case of DVCS, it is emitted by the target nucleon (see Figure \[fig:diagrams\]). Although these two processes are experimentally indistinguishable, the BH is well known and exactly calculable in QED. At current JLab energies (6 GeV), the BH process is highly dominant (in most of the phase space) but the DVCS process can be accessed via the interference term rising from the two processes. With a polarized beam or/and a polarized target, different types of asymmetries can be extracted: beam-spin asymmetries ($A_{LU}$), longitudinally polarized target-spin asymmetries ($A_{UL}$), transversely polarized target-spin asymmetries ($A_{UT}$), double-spin asymmetries ($A_{LL}$, $A_{LT}$). Each type of asymmetry gives access to a different combination of Compton Form Factors. ![DVCS beam-spin asymmetries as a function of $-t$, for different values of $Q^{2}$ and $x_{B}$. The (black) circles represent the latest CLAS results [@bsa3], the (red) squares and the (green) triangles are the results, respectively, from Ref. [@bsa1] and Ref. [@halla]. The black dashed curves represent Regge calculations [@jml]. The blue curves correspond to the GPD calculations of Ref. [@vgg1] (VGG) at twist-2 (solid) and twist-3 (dashed) levels, with the contribution of the GPD $H$ only.[]{data-label="fig:bsa"}](bsa.png){height="0.34\textheight"} The first results on DVCS beam-spin asymmetries published by the CLAS collaboration were extracted using data from non-dedicated experiments [@bsa1; @bsa2]. Also using non-dedicated data, CLAS published DVCS longitudinally polarized target-spin asymmetries in 2006 [@tsa]. In 2005, the first part of the e1-DVCS experiment was carried out in the Hall B of JLab using the CLAS spectrometer [@clas] and an additional electromagnetic calorimeter, made of 424 lead-tungstate scintillating crystals read out via avalanche photodiodes, specially designed and built for the experiment. This additional calorimeter was located at the forward angles, where the DVCS/BH photons are mostly emitted, as the standard CLAS configuration does not allow detection at those forward angles. This first CLAS experiment dedicated to DVCS measurements, with this upgraded setup allowing a fully exclusive measurement, ran using a 5.766 GeV polarized electron beam and a liquid-hydrogen target. From this experiment data, CLAS published in 2008 the largest set of DVCS beam-spin asymmetries ever extracted in the valence quark region [@bsa3]. Figure \[fig:bsa\] shows the corresponding results as a function of $-t$ for different bins in ($Q^{2}$, $x_{B}$). The predictions using the GPD model from VGG (Vanderhaeghen, Guichon, Guidal) [@vgg1; @vgg2] overestimate the asymmetries at low $|-t|$, especially for small values of $Q^{2}$ which can be expected since the GPD mechanism is supposed to be valid at high $Q^{2}$. Regge calculations [@jml] are in fair agreement with the results at low $Q^{2}$ but fail to describe them at high $Q^{2}$ as expected. We are currently working on extracting DVCS unpolarized and polarized absolute cross sections from the e1-DVCS data [@hsj]. Having both the beam-spin asymmetries and the longitudinally polarized target-spin asymmetries, a largely model-independent GPD analysis in leading twist was performed, fitting simultaneously the values for $A_{LU}$ and $A_{UL}$ obtained with CLAS at three values of $t$ and fixed $x_{B}$, to extract numerical constraints on the imaginary parts of the Compton Form Factors (CFFs) ${\cal H}$ and $\tilde{{\cal H}}$, with average uncertainties of the order of 30% [@guidal_clas]. Before that, the same analysis was performed fitting the DVCS unpolarized and polarized cross sections published by the JLab Hall A collaboration [@halla] to extract numerical constraints on the real and imaginary parts of the CFF ${\cal H}$ [@guidal_fitter_code]. Another GPD analysis in leading twist, assuming the dominance of the GPD $H$ (the contributions of $\tilde{H}$, $E$ and $\tilde{E}$ being neglicted) and using the CLAS $A_{LU}$ data as well as the DVCS JLab Hall A data, was performed to extract constraints on the real and imaginary parts of the CFF ${\cal H}$ [@moutarde]. Similar analyses were performed using results published by the HERMES collaboration [@guidal_moutarde; @guidal_hermes]. A third approach was developped, using a model-based global fit on the available world data to calculate the real and imaginary parts of the CFF ${\cal H}$ [@km]. When we compare the different results of those analyses for the imaginary part of ${\cal H}$, they appear to be relatively compatible (such a comparison plot can be found in Ref. [@km2]). Deeply Virtual Meson Production {#deeply-virtual-meson-production .unnumbered} =============================== The CLAS collaboration published several results on pseudoscalar meson electroproduction ($\pi^{0}$, $\pi^{+}$) [@ps1; @ps2]. However, those are not reviewed in this paper, limiting itself to vector mesons. CLAS published cross-section measurements for the following vector mesons: $\rho^{0}$ [@rho01; @rho02], $\omega$ [@omega] and $\phi$ [@phi1; @phi2], contributing significantly to the world data on vector mesons with measurements in the valence quark region, corresponding to low $W$ ($W<5$ GeV). First measurements of $\rho^{+}$ electroproduction are being extracted from the e1-DVCS data mentionned above [@rho+]. ![Longitudinal cross sections for $\rho^{0}$ as a function of $W$ at fixed $Q^{2}$ (CLAS and world data). The results from CLAS are shown as full circles. The blue curves are VGG GPD-based predictions. The red curves represent GK GPD-based predictions: total (solid), valence quarks (dashed), sea quarks and gluons (dot-dashed).[]{data-label="fig:rho"}](rho_vgg_gk.png){height="0.37\textheight"} As the leading-twist handbag diagram is only valid for the longitudinal part of the cross section of those vector mesons, it is required to separate the longitudinal and transverse parts of the cross sections extracted from the experimental data by analyzing the decay angular distribution of the meson. Figure \[fig:rho\] shows the longitudinal cross sections of the $\rho^{0}$ meson production $\sigma_{L}(\gamma^{*} p \to p \rho^{0})$ as a function of $W$ at fixed $Q^{2}$, for different bins in $Q^{2}$. As a function of increasing $W$, those cross sections first drops at low $W$ ($W<5$ GeV, corresponding to the valence quark region) and then slightly rise at higher $W$. The longitudinal cross sections of the $\omega$ meson production seems to show the same behavior as a function of $W$ as the one observed for the $\rho^{0}$ meson. The GPD-based predictions from VGG and from GK (Goloskokov, Kroll) [@gk] describe quite well those results at high $W$ but both GPD models fail by large to reproduce the behavior at low $W$ (see the curves on Figure \[fig:rho\]). The $\phi$ meson production, which is mostly sensitive to gluon GPDs, is a different case as its longitudinal cross sections as a function of $W$ show a different behavior by continuously rising with increasing $W$ all the way from the lowest $W$ region; these cross sections are very well described by the GPD model predictions [@gk_phi]. The reason why the GPD models fail to describe the data for the $\rho^{0}$ and $\omega$ mesons at low $W$ (valence quark region) is unsure at this point. The handbag mechanism might not be dominant in the low $W$ valence region as the minimum value of $|-t|$ increases with decreasing $W$ and higher-twist effects grow with $t$. Another possibility is that the handbag mechanism might actually be dominant in the low $W$ valence region but there is an important contribution missing in the GPD models. DVCS and DVMP at CLAS12 {#dvcs-and-dvmp-at-clas12 .unnumbered} ======================= With the upcoming 12 GeV upgrade of JLab’s CEBAF accelerator, the instrumentation in the experimental halls will be upgraded as well. In Hall B, the CLAS detector will be replaced by the new CLAS12 spectrometer, under construction, with the study of Generalized Parton Distributions as one of the highest priorities of its future experimental program. The experiments currently proposed have the following goals: - DVCS beam-spin asymmetries on the proton, - DVCS longitudinal target-spin asymmetries on the proton, - DVCS transverse target-spin asymmetries on the proton, - DVCS on the neutron, - DVCS unpolarized and polarized cross sections, - DVMP: pseudoscalar meson electroproduction, - DVMP: vector meson electroproduction. To study DVCS on the neutron, a central neutron detector was designed to be added to the base equipment of the CLAS12 spectrometer. A combined analysis of DVCS on the proton and on the neutron allows flavor separation of GPDs. ![The CLAS12 detector currently under construction.[]{data-label="fig:clas12"}](clas12.pdf){height="0.3\textheight"} JLab 12 GeV will provide high luminosity (L$\sim10^{35}$cm$^{-2}$s$^{-1}$) for high accuracy measurements to study GPDs in the valence quark region and test the models on a large $x_{B}$ scale. The new CLAS12 spectrometer, with its large acceptance allowing measurements on a large kinematic range, will be perfectly fitted for a rich GPD experimental program. Conclusions {#conclusions .unnumbered} =========== The CLAS collaboration produced the largest set of data for DVCS and exclusive vector meson production ever extracted in the valence quark region. The VGG GPD model fairly agrees with the DVCS asymmetry data at high $Q^{2}$ but fails to reproduce it at lower $Q^{2}$. As for the exclusive vector meson data, GPD models describe well the longitudinal cross sections at high $W$ (region corresponding to sea quarks and/or gluons) which seem to be interpretable in terms of leading-twist handbag diagram (quark/gluon GPDs) but fail by large for $W<5$ GeV (corresponding to the valence quark region) except for the $\phi$ meson for which the GPD formalism seems to apply. We need experimental data of higher $Q^{2}$ while staying in the valence quark region to extend the DVCS data on a larger kinematic domain and provide more constraints for the GPD models, and to test the GPD mechanism validity regime for DVMP. JLab 12 GeV will provide high luminosity for high accuracy measurements to test models on a large $x_{B}$ scale and thus will be a great facility to study GPDs in the valence quark region. The new CLAS12 spectrometer, with its large acceptance, will be well suited for a rich and exciting GPD experimental program. Acknowledgments {#acknowledgments .unnumbered} =============== Thanks to P. Stoler and R. Ent for the opportunity to give this presentation. Thanks to M. Guidal and S. Niccolai for useful discussions and for providing slides used for the preparation of this talk. [99]{} D. Müller, D. Robaschik, B. Geyer, F.-M. Dittes, and J. Horejsi, Fortschr. Phys. [**4**2]{}, 101 (1994). X. Ji, Phys. Rev. Lett. 78, 610 (1997); Phys. Rev. D [**5**5]{}, 7114 (1997). A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; Phys. Rev. D [**5**6]{}, 5524 (1997). J.C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. D [**5**6]{}, 2982 (1997). K. Goeke, M.V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. [**4**7]{}, 401 (2001). M. Diehl, Phys. Rept. [**3**88]{}, 41 (2003). A.V. Belitsky, A.V. Radyushkin, Phys. Rept. [**4**18]{}, 1 (2005). M. Guidal, Prog. Part. Nucl. Phys. [**6**1]{}, 89 (2008). M. Vanderhaeghen, P.A.M. Guichon, and M. Guidal, Phys. Rev. D [**6**0]{}, 094017 (1999). M. Guidal, M.V. Polyakov, A.V. Radyushkin and M. Vanderhaeghen, Phys. Rev. D [**7**2]{}, 054013 (2005). A. Belitsky, D. Müller and A. Kirchner, Nucl. Phys. B [**6**29]{}, 323 (2002). S. Stepanyan [*e*t al.]{} (CLAS Collaboration), Phys. Rev. Lett. [**8**7]{}, 182002 (2001). G. Gavalian [*e*t al.]{} (CLAS Collaboration), Phys. Rev. C [**8**0]{}, 035206 (2009). S. Chen [*e*t al.]{} (CLAS Collaboration), Phys. Rev. Lett. [**9**7]{}, 072002 (2006). B. Mecking [*e*t al.]{}, Nucl. Instrum. Meth. A [**5**03]{}, 513 (2003). F.X. Girod [*e*t al.]{} (CLAS Collaboration), Phys. Rev. Lett. [**1**00]{}, 162002 (2008). J.M. Laget, Phys. Rev. C [**7**6]{}, 052201(R) (2007). H.S. Jo, Ph.D. thesis, Université Paris-Sud, Orsay, France (2007). M. Guidal, Phys. Lett. B [**6**89]{}, 156 (2010). C. Munoz Camacho [*e*t al.]{} (JLab Hall A Collaboration), Phys. Rev. Lett. [**9**7]{}, 262002 (2006). M. Guidal, Eur. Phys. J. A [**3**7]{}, 319 (2008) \[Erratum-ibid. A [**4**0]{}, 119 (2009)\]. H. Moutarde, Phys. Rev. D [**7**9]{}, 094021 (2009). M. Guidal and H. Moutarde, Eur. Phys. J. A [**4**2]{}, 71 (2009). M. Guidal, Phys. Lett. B [**6**93]{}, 17 (2010). K. Kumeri[č]{}ki and D. Müller, Nucl. Phys. B [**8**41]{}, 1 (2010). K. Kumeri[č]{}ki and D. Müller, arXiv:1008.2762 \[hep-ph\]. R. De Masi [*e*t al.]{} (CLAS Collaboration), Phys. Rev. C [**7**7]{}, 042201(R) (2008). K. Park [*e*t al.]{} (CLAS Collaboration), Phys. Rev. C [**7**7]{}, 015208 (2008). C. Hadjidakis [*e*t al.]{} (CLAS Collaboration), Phys. Lett. B [**6**05]{}, 256-264 (2005). S. Morrow [*e*t al.]{} (CLAS Collaboration), Eur. Phys. J. A [**3**9]{}, 5-31 (2009). L. Morand [*e*t al.]{} (CLAS Collaboration), Eur. Phys. J. A [**2**4]{}, 445-458 (2005). K. Lukashin [*e*t al.]{} (CLAS Collaboration), Phys. Rev. C [**6**3]{}, 065205 (2001). J. Santoro [*e*t al.]{} (CLAS Collaboration), Phys. Rev. C [**7**8]{}, 025210 (2008). A. Fradi, Ph.D. thesis, Université Paris-Sud, Orsay, France (2009). S.V. Goloskokov and P. Kroll, Eur. Phys. J. C [**4**2]{}, 281 (2005); Eur. Phys. J. C [**5**0]{}, 829 (2007). S.V. Goloskokov, arXiv:0910.4308 \[hep-ph\].

ArXiv

--- abstract: 'A real Bott manifold is the total space of a sequence of $\R P^1$ bundles starting with a point, where each $\R P^1$ bundle is the projectivization of a Whitney sum of two real line bundles. A real Bott manifold is a real toric manifold which admits a flat riemannian metric. An upper triangular $(0,1)$ matrix with zero diagonal entries uniquely determines such a sequence of $\R P^1$ bundles but different matrices may produce diffeomorphic real Bott manifolds. In this paper we determine when two such matrices produce diffeomorphic real Bott manifolds. The argument also proves that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. In particular, this implies the main theorem of [@ka-ma08] which asserts that two real Bott manifolds are diffeomorphic if and only if their cohomology rings with $\Z/2$ coefficients are isomorphic as graded rings. We also prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors.' address: 'Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan.' author: - Mikiya Masuda title: Classification of real Bott manifolds --- [^1] Introduction ============ A [*real Bott tower*]{} of height $n$, which is a real analogue of a Bott tower introduced in [@gr-ka94], is a sequence of $\R P^1$ bundles $$\label{tower} M_n\stackrel{\R P^1}\longrightarrow M_{n-1}\stackrel{\R P^1}\longrightarrow \cdots\stackrel{\R P^1}\longrightarrow M_1 \stackrel{\R P^1}\longrightarrow M_0=\{\textrm{a point}\}$$ such that $M_j\to M_{j-1}$ for $j=1,\dots,n$ is the projective bundle of the Whitney sum of a real line bundle $L_{j-1}$ and the trivial real line bundle over $M_{j-1}$, and we call $M_n$ a [*real Bott manifold*]{}. A real Bott manifold naturally supports an action of an elementary abelian 2-group and provides an example of a real toric manifold which admits a flat riemannian metric invariant under the action. Conversely, it is shown in [@ka-ma08] that a real toric manifold which admits a flat riemannian metric invariant under an action of an elementary abelian 2-group is a real Bott manifold. Real line bundles are classified by their first Stiefel-Whitney classes as is well-known and $H^1(M_{j-1};\Z/2)$, where $\Z/2=\{0,1\}$, is isomorphic to $(\Z/2)^{j-1}$ through a canonical basis, so the line bundle $L_{j-1}$ is determined by a vector $A_j$ in $(\Z/2)^{j-1}$. We regard $A_j$ as a column vector in $(\Z/2)^n$ by adding zero’s and form an $n\times n$ matrix $A$ by putting $A_j$ as the $j$-th column. This gives a bijective correspondence between the set of real Bott towers of height $n$ and the set $\T(n)$ of $n\times n$ upper triangular $(0,1)$ matrices with zero diagonal entries. Because of this reason, we may denote the real Bott manifold $M_n$ by $M(A)$. Although $M(A)$ is determined by the matrix $A$, it happens that two different matrices in $\T(n)$ produce (affinely) diffeomorphic real Bott manifolds. In this paper we introduce three operations on $\T(n)$ and say that two elements in $\T(n)$ are [*Bott equivalent*]{} if one is transformed to the other through a sequence of the three operations. Our first main result is the following. \[main\] The following are equivalent for $A,B$ in $\T(n)$: 1. $A$ and $B$ are Bott equivalent. 2. $M(A)$ and $M(B)$ are affinely diffeomorphic. 3. $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$ are isomorphic as graded rings. Moreover, any graded ring isomorphism from $H^*(M(A);\Z/2)$ to $H^*(M(B);\Z/2))$ is induced by an affine diffeomorphism from $M(B)$ to $M(A)$. In particular, we obtain the following main theorem of [@ka-ma08]. \[maincoro\] Two real Bott manifolds are diffeomorphic if and only if their cohomology rings with $\Z/2$ coefficients are isomorphic as graded rings. It is asked in [@ka-ma08] whether Corollary \[maincoro\] holds for any real toric manifolds but a counterexample is given in [@masu08]. We say that a real Bott manifold is *indecomposable* if it is not diffeomorphic to a product of more than one real Bott manifolds. Using Corollary \[maincoro\] together with an idea used to prove Theorem \[main\], we are able to prove our second main result. \[main1\] The decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. In particular, we have \[main1coro\] Let $M$ and $M'$ be real Bott manifolds. If $S^1\times M$ and $S^1\times M'$ are diffeomorphic, then $M$ and $M'$ are diffeomorphic. It would be interesting to ask whether Theorem \[main1\] and Corollary \[main1coro\] hold for any real toric manifolds. The author learned from Y. Kamishima that Corollary \[main1coro\] can also be obtained from the method developed in [@ka-na08] and [@nazr08] and that the cancellation property above fails to hold for general compact flat riemannian manifolds, see [@char65-1]. This paper is organized as follows. In Section \[sect:rbott\] we describe $M(A)$ and its cohomology rings explicitly in terms of the matrix $A$. In Section \[sect:matrix\] we introduce the three operations on $\T(n)$. To each operation we associate an affine diffeomorphism between real Bott manifolds in Section \[sect:affine\], which implies the implication (1) $\Rightarrow$ (2) in Theorem \[main\]. The implication (2) $\Rightarrow$ (3) is trivial. In Section \[sect:cohom\] we prove the latter statement in Theorem \[main\]. The argument also establishes the implication (3) $\Rightarrow$ (1). In the proof we introduce a notion of eigen-element and eigen-space in the first cohomology group of a real Bott manifold using the multiplicative structure of the cohomology ring and they play an important role on the analysis of isomorphisms between cohomology rings. Using this notion, we prove Theorem \[main1\] in Section \[sect:decom\]. Real Bott manifolds and their cohomology rings {#sect:rbott} ============================================== As mentioned in the Introduction, a real Bott manifold $M(A)$ of dimension $n$ is associated to a matrix $A\in\T(n)$. In this section we give an explicit description of $M(A)$ and its cohomology ring. We set up some notation. Let $S^1$ denote the unit circle consisting of complex numbers with unit length. For elements $z\in S^1$ and $a\in \Z/2$ we use the following notation $$z(a):=\begin{cases} z \quad&\text{if $a=0$}\\ \bar z\quad&\text{if $a=1$}. \end{cases}$$ For a matrix $A$ we denote by $A^i_j$ the $(i,j)$ entry of $A$ and by $A^i$ (resp. $A_j$) the $i$-th row (resp. $j$-th column) of $A$. Now we take $A$ from $\T(n)$ and define involutions $a_i$’s on $T^n:=(S^1)^n$ by $$\label{ai} a_i(z_1,\dots,z_n):=(z_1,\dots,z_{i-1},-z_i,z_{i+1}(A^i_{i+1}),\dots, z_n(A^i_n))$$ for $i=1,\dots,n$. These involutions $a_i$’s commute with each other and generate an elementary abelian 2-group of rank $n$, denoted by $G(A)$. The action of $G(A)$ on $T^n$ is free and the orbit space is the desired real Bott manifold $M(A)$. $M(A)$ is a flat riemannian manifold. In fact, Euclidean motions $s_i$’s $(i=1,\dots,n)$ on $\R^n$ defined by $$s_i(u_1,\dots,u_n):=(u_1,\dots,u_{i-1}, u_i+\frac{1}{2}, (-1)^{A_{i+1}^i}u_{i+1},\dots, (-1)^{A_{n}^i}u_n)$$ generate a crystallographic group $\Gamma(A)$, where the subgroup generated by $s_1^2,\dots,s_n^2$ consists of all translations by $\Z^n$, and the action of $\G(A)$ on $\R^n$ is free and the orbit space $\R^n/\G(A)$ agrees with $M(A)$ through an identification $\R/\Z$ with $S^1$ via an exponential map $u\to \exp(2\pi\sqrt{-1}u)$. $M(A)$ admits an action of an elementary abelian 2-group defined by $(u_1,\dots,u_n)\to (\pm u_1,\dots,\pm u_n)$ and this action preserves the flat riemannian metric on $M(A)$. Let $G_k$ $(k=1,\dots,n)$ be a subgroup of $G(A)$ generated by $a_1,\dots,a_k$. Needless to say $G_n=G(A)$. Let $T^k:=(S^1)^k$ be a product of first $k$-factors in $T^n=(S^1)^n$. Then $G_k$ acts on $T^k$ by restricting the action of $G_k$ on $T^n$ to $T^k$ and the orbit space $T^k/G_k$ is a real Bott manifold of dimension $k$. Natural projections $T^k\to T^{k-1}$ for $k=1,\dots,n$ produce a real Bott tower $$M(A)=T^n/G_n\to T^{n-1}/G_{n-1} \to \dots\to T^1/G_1\to \text{\{a point\}}.$$ The graded ring structure of $H^*(M(A);\Z/2)$ can be described explicitly in terms of the matrix $A$. We shall recall it. For a homomorphism $\lambda\colon G(A)\to \Z_2=\{\pm 1\}$ we denote by $\R(\lambda)$ the real one-dimensional $G(A)$-module associated with $\lambda$. Then the orbit space of $T^n\times \R(\lambda)$ by the diagonal action of $G(A)$, denoted by $L(\lambda)$, defines a real line bundle over $M(A)$ with the first projection. Let $\lambda_j\colon G(A)\to \Z_2$ $(j=1,\dots,n)$ be a homomorphism sending $a_i$ to $-1$ for $i=j$ and $1$ for $i\not=j$, and we set $$x_j=w_1(L(\lambda_j))$$ where $w_1$ denotes the first Stiefel-Whitney class. \[cohoA\] As a graded ring $$H^*(M(A);\Z/2)=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n).$$ Let $B$ be another element of $\T(n)$. Since $M(A)=T^n/G(A)$ and $M(B)=T^n/G(B)$, an affine automorphism $\f$ of $T^n$ together with a group isomorphism $\phi\colon G(B)\to G(A)$ induces an affine diffeomorphism $f\colon M(B)\to M(A)$ if $\f$ is $\phi$-equivariant, i.e., $\f(gz)=\phi(g)\f(z)$ for $g\in G(B)$ and $z\in T^n$. Since the actions of $G(A)$ and $G(B)$ on $T^n$ are free, the isomorphism $\phi$ will be uniquely determined by $\f$ if it exists. We shall use $b_i$ and $y_j$ for $M(B)$ in place of $a_i$ and $x_j$ for $M(A)$. \[f\*\] If $\phi(b_i)=\prod_{j=1}^na_j^{F^i_j}$ with $F^i_j\in \Z/2$, then $f^*(x_j)=\sum_{i=1}^nF^i_jy_i$. A map $T^n\times \R(\lambda\circ\phi)\to T^n\times \R(\lambda)$ sending $(z,u)$ to $(\f(z),u)$ induces a bundle map $L(\lambda\circ\phi) \to L(\lambda)$ covering $f\colon M(B)\to M(A)$. Since $(\lambda_j\circ\phi)(b_i)=F^i_j$, this implies the lemma. Three matrix operations {#sect:matrix} ======================= In this section we introduce three operations on matrices used in later sections to analyze when $M(A)$ and $M(B)$ (resp. $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$) are diffeomorphic (resp. isomorphic) for $A,B\in \T(n)$. In the following $A$ will denote an element of $\T(n)$. [**1st operation (Op1).**]{} For a permutation matrix $S$ of size $n$ we define $$\Phi_S(A):=SAS^{-1}.$$ To be more precise, there is a permutation $\sigma$ on a set $\{1,\dots,n\}$ such that $S^i_j=1$ if $i=\sigma(j)$ and $S^i_j=0$ otherwise. We note that if we set $B=\Phi_S(A)$, then $SA=BS$ and $$\label{SA=BA} A^i_j=(SA)^{\sigma(i)}_j=(BS)^{\sigma(i)}_j=B^{\sigma(i)}_{\sigma(j)}.$$ $\Phi_S(A)$ may not be in $\T(n)$ but we will perform the operation $\Phi_S$ on $A$ only when $\Phi_S(A)$ stays in $\T(n)$. [**2nd operation (Op2).**]{} For $k\in \{1,\dots,n\}$ we define a square matrix $\Phi^k(A)$ of size $n$ by $$\label{2nd} \Phi^k(A)_j:=A_j+A^k_jA_k\quad\text{for $j=1,\dots,n$}.$$ $\Phi^k(A)$ stays in $\T(n)$ and since the diagonal entries of $A$ are all zero and we are working over $\Z/2$, the composition $\Phi^k\circ \Phi^k$ is the identity; so $\Phi^k$ is bijective on $\T(n)$. [**3rd operation (Op3).**]{} Let $I$ be a subset of $\{1,\dots,n\}$ such that $A_i=A_j$ for $i,j\in I$ and $A_i\not=A_j$ for $i\in I$ and $j\notin I$. Since the diagonal entries of $A$ are all zero, the condition $A_i=A_j$ for $i,j\in I$ implies that $A^i_j=0$ for $i,j\in I$. Let $C=(C^i_k)_{i,k\in I}$ with $C^i_k\in \Z/2$ be an invertible matrix of size $|I|$. Then we define a square matrix $\Phi^I_C(A)$ of size $n$ by $$\label{3rd} \Phi^I_C(A)^i_j:=\begin{cases} \sum_{k\in I}C^i_kA^k_j\quad&\text{$(i\in I)$}\\ A^i_j\quad&\text{$(i\notin I)$}. \end{cases}$$ $\Phi^I_C(A)$ stays in $\T(n)$ and since $C$ is invertible, $\Phi^I_C$ is bijective on $\T(n)$. We say that two elements in $\T(n)$ are [*Bott equivalent*]{} if one is transformed to the other through a sequence of the three operations (Op1), (Op2) and (Op3). $\T(2)$ has two elements and they are not Bott equivalent. $\T(3)$ has $2^3=8$ elements and they are classified into four Bott equivalence classes as follows: 1. The zero matrix of size $3$ 2. ${\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$ 3. ${\tiny \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix} }$ 4. ${\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$ $\T(4)$ has $2^6=64$ elements and one can check that it has twelve Bott equivalence classes, see [@ka-ma08] and [@nazr08]. Furthermore, $\T(5)$ has $2^{10}=1024$ elements and one can check that it has $54$ Bott equivalence classes. The author learned from Admi Nazra that he classified real Bott manifolds of dimension 5 up to diffeomorhism from a different viewpoint (see [@ka-na08], [@nazr08]) and found the 54 Bott equivalence classes in $\T(5)$. The author does not know the number of Bott equivalence classes in $\T(n)$ for $n\ge 6$ although it is in between $2^{(n-2)(n-3)/2}$ and $2^{n(n-1)/2}$ (see Example \[Deltan\] below). Let $\T_k(n)$ $(1\le k\le n-1)$ be a subset of $\T(n)$ such that $A\in \T(n)$ is in $\T_k(n)$ if and only if $A$ has exactly $k$ non-zero columns. There is only one Bott equivalence class in $\T_1(n)$ and the corresponding real Bott manifold is the product of a Klein bottle and $(\R P^1)^{n-2}$. $\T_2(3)$ has two Bott equivalence classes represented by $${\tiny \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}}\quad \text{}\quad {\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$$ But $\T_2(n)$ for $n\ge 4$ has four Bott equivalence classes; two of them are represented by $n\times n$ matrices with the above $3\times 3$ matrices at the right-low corner and $0$ in others, and the other two are represented by $n\times n$ matrices with the following $4\times 4$ matrices at the right-low corner and $0$ in others $${\tiny \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\end{pmatrix}}\quad \text{}\quad {\tiny \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix} }$$ \[Deltan\] Let $\Delta(n)$ be a subset of $\T(n)$ such that $A\in \T(n)$ is in $\Delta(n)$ if and only if $A^{i}_{i+1}=1$ for $i=1,\dots,n-1$. Only the operation (Op2) is available on $\Delta(n)$ and one can change $(i,i+2)$ entry into $0$ for $i=1,\dots,n-2$ using the operation, so that $A$ is Bott equivalent to a matrix $\bA$ of this form [$$\label{reduced} \bA=\begin{pmatrix} 0&1&0&\bA^1_4&\bA^1_5&\dots&\bA^1_{n-1}&\bA^1_n\\ 0&0&1&0&\bA^2_5&\dots&\bA^2_{n-1}&\bA^2_n\\ \vdots&\vdots& \ddots &\ddots &\ddots&\ddots &\vdots & \vdots\\ 0& 0& \dots&0&1 &0 & \bA^{n-4}_{n-1} & \bA^{n-4}_n\\ 0& 0& \dots&0&0 &1 & 0 & \bA^{n-3}_n\\ 0& 0& \dots&0&0 &0 & 1 & 0\\ 0& 0& \dots&0&0 &0 & 0 & 1\\ 0& 0& \dots&0&0 &0 & 0 & 0 \end{pmatrix}$$]{} $\bA$ is uniquely determined by $A$ and two elements $A,B\in \Delta(n)$ are Bott equivalent if and only if $\bA=\bB$. Therefore there are exactly $2^{(n-2)(n-3)/2}$ Bott equivalent classes in $\Delta(n)$ for $n\ge 2$. As remarked above $\Phi_S(A)$ may not stay in $\T(n)$. This awkwardness can be resolved if we consider the union of $\Phi_S(\T(n))$ over all permutation matrices $S$. The three operations above preserve the union and are bijective on it. This union is a natural object. In fact, it is shown in [@ma-pa08 Lemma 3.3] that a square matrix $A$ of size $n$ with entries in $\Z/2$ lies in the union if and only if all principal minors of $A+E$ (even the determinant of $A+E$ itself) are one in $\Z/2$ where $E$ denotes the identity matrix of size $n$. Affine diffeomorphisms {#sect:affine} ====================== In this section we associate an affine diffeomorphism between real Bott manifolds to each operation introduced in the previous section, and prove the implication $(1)\Rightarrow (2)$ in Theorem \[main\], that is If $A, B\in \T(n)$ are Bott equivalent, then the associated real Bott manifolds $M(A)$ and $M(B)$ are affinely diffeomorphic. We set $B=\Phi_S(A), \Phi^k(A), \Phi^I_C(A)$ respectively for the three operations introduced in the previous section. In order to prove the proposition above, it suffices to find a group isomorphism $\phi\colon G(B)\to G(A)$ and a $\phi$-equivariant affine automorphism $\f$ of $T^n$ which induces an affine diffeomorphism from $M(B)$ to $M(A)$. [*The case of the operation (Op1).*]{} Let $S$ and $\sigma$ be as before. We define a group isomorphism $\phi_S\colon G(B)\to G(A)$ by $$\label{phiS} \phi_S(b_{\sigma(i)}):=a_{i}$$ and an affine automorphism $\f_S$ of $T^n$ by $$\f_S(z_1,\dots,z_n):=(z_{\sigma(1)},\dots,z_{\sigma(n)}).$$ Then it follows from (applied to $b_{\sigma(i)}$) that the $j$-th component of $\f_S(b_{\sigma(i)}(z))$ is $z_{\sigma(j)}(B^{\sigma(i)}_{\sigma(j)})$ for $j\not=i$ and $-z_{\sigma(i)}$ for $j=i$ while that of $a_i(\f_S(z))$ is $z_{\sigma(j)}(A^i_j)$ for $j\not=i$ and $-z_{\sigma(i)}$ for $j=i$. Since $A^i_j=B^{\sigma(i)}_{\sigma(j)}$ by , this shows that $\f_S$ is $\phi_S$-equivariant. It follows from Lemma \[f\*\] and that the affine diffeomorphism $f_S\colon M(B)\to M(A)$ induced from $\f_S$ satisfies $$\label{op1coho} f_S^*(x_j)=y_{\sigma(j)}\quad \text{for $j=1,\dots,n$.}$$ [*The case of the operation (Op2).*]{} We define a group isomorphism $\phi^k\colon G(B)\to G(A)$ by $$\label{phik} \phi^k(b_i):=a_ia_k^{A^i_k} $$ and an affine automorphism $\f^k$ of $T^n$ by $$\f^k(z_1,\dots,z_n):=(z_1,\dots,z_{k-1},\sqrt{-1}z_k,z_{k+1},\dots,z_n).$$ We shall check that $\f^k$ is $\phi^k$-equivariant, i.e., $$\label{fkb} \f^k(b_i(z))= a_ia_k^{A^i_k}(\f^k(z)).$$ The identity is obvious when $i=k$ because $A^k_k=0$ and $B^k_j=A^k_j$ for any $j$ by . Suppose $i\not=k$. Then the $j$-th component of the left hand side of is given by $$\begin{cases}z_j(B^i_j)\quad&\text{for $j\not=i,k$},\\ -z_i \quad&\text{for $j=i$},\\ \sqrt{-1}(z_k(B^i_k)) \quad&\text{for $j=k$}, \end{cases}$$ while that of the right hand side of is given by $$\begin{cases}z_j(A^i_j+A^k_jA^i_k)\quad&\text{for $j\not=i,k$},\\ -z_i(A^k_iA^i_k) \quad&\text{for $j=i$},\\ (-1)^{A^i_k}(\sqrt{-1}z_k)(A^i_k) \quad&\text{for $j=k$}. \end{cases}$$ Since $B^i_j=A^i_j+A^k_jA^i_k$ by , the $j$-th components above agree for $j\not=i,k$. They also agree for $j=i$ because either $A^k_i$ or $A^i_k$ is zero. We note that $B^i_k=A^i_k$ by , and the $k$-th components above are both $\sqrt{-1}z_k$ when $B^i_k=A^i_k=0$ and $\sqrt{-1}\bar{z_k}$ when $B^i_k=A^i_k=1$. Thus the $j$-th components above agree for any $j$. Since $A^i_k=B^i_k$ for any $i$, it follows from Lemma \[f\*\] and that the affine diffeomorphism $f^k\colon M(B)\to M(A)$ induced from $\f^k$ satisfies $$\label{op2coho} (f^k)^*(x_j)=y_j\quad\text{for $j\not=k$}, \quad \quad (f^k)^*(x_k)=y_k+\sum_{i=1}^nB^i_ky_i.$$ [*The case of the operation (Op3).*]{} The homomorphism $\operatorname{GL}(m;\Z)\to \operatorname{GL}(m;\Z/2)$ induced from the surjective homomorphism $\Z\to \Z/2$ is known (and easily proved) to be surjective. We take a lift of the matrix $C=(C^i_k)_{i,k\in I}$ to $\operatorname{GL}(|I|,\Z)$ and denote the lift by $\tC$. Then we define a group isomorphism $\phi^I_C\colon G(B)\to G(A)$ by $$\label{phiIC} \phi^I_C(b_i):=\begin{cases} \prod_{k\in I}a_k^{C^i_k}\quad& \text{for $i\in I$},\\ a_i\quad&\text{for $i\notin I$,} \end{cases}$$ and the $j$-th component of an affine automorphism $\f^I_{\tC}$ of $T^n$ by $$\label{fIC} \f^I_{\tC}(z)_j:=\begin{cases} \prod_{\ell\in I}z_\ell^{\tC^\ell_j} \quad&\text{for $j\in I$},\\ z_j\quad&\text{for $j\notin I$.} \end{cases}$$ We shall check that $\f^I_{\tC}$ is $\phi^I_C$-equivariant. To simplify notation we abbreviate $\f^I_{\tC}$ and $\phi^I_C$ as $\f$ and $\phi$ respectively. What we prove is the identity $$\label{fb=bf} \f(b_i(z))_j=\phi(b_i)\f(z)_j.$$ We distinguish four cases. [*Case 1.*]{} The case where $i,j\in I$. As remarked in the definition of (Op3), $A^k_\ell=0$ whenever $k,\ell\in I$, so $B^i_\ell=0$ for any $\ell\in I$ by . It follows from and that $$\f(b_i(z))_j=(-z_i)^{\tC^i_j} \prod_{\ell\in I,\ell\not=i}z_\ell(B^i_\ell)^{\tC^\ell_j} =(-1)^{C^i_j}\prod_{\ell\in I}z_\ell^{\tC^\ell_j}$$ while $$\phi(b_i)\f(z)_j=(\prod_{k\in I}a_k^{C^i_k})\f(z)_j =(-1)^{C^i_j}\prod_{\ell\in I}z_\ell^{\tC^\ell_j}.$$ [*Case 2.*]{} The case where $i\in I$ but $j\notin I$. In this case we have $$\f(b_i(z))_j=z_j(B^i_j)$$ while $$\phi(b_i)\f(z)_j =z_j(\sum_{k\in I}C^i_kA^k_j)=z_j(B^i_j)$$ where the last identity follows from . [*Case 3.*]{} The case where $i\notin I$ but $j\in I$. In this case we have $$\f(b_i(z))_j=\prod_{\ell\in I}z_\ell(B^i_\ell)^{\tC^\ell_j}$$ while $$\phi(b_i)\f(z)_j =(\prod_{\ell\in I}z_\ell^{\tC^\ell_j})(A^i_j) =\prod_{\ell\in I}z_\ell(A^i_j)^{\tC^\ell_j}.$$ Since $B^i_\ell=A^i_\ell$ for $i\notin I$ by , the above verifies . [*Case 4.*]{} The case where $i,j\notin I$. In this case $$\f(b_i(z))_j=z_j(B^i_j)$$ while $$\phi(b_i)\f(z)_j=z_j(A^i_j).$$ Since $B^i_j=A^i_j$ for $i\notin I$ by , the above verifies . It follows from Lemma \[f\*\] and that the affine diffeomorphism $f^I_C\colon M(B)\to M(A)$ induced from $\f^I_C$ satisfies $$\label{op3coho} (f^I_C)^*(x_j)=\begin{cases}\sum_{i\in I}C^i_jy_i \quad&\text{for $j\in I$,}\\ y_j\quad&\text{for $j\notin I$.} \end{cases}$$ Cohomology isomorphisms {#sect:cohom} ======================= In this section we prove the latter statement in Theorem \[main\] and the implication (3) $\Rightarrow$ (1) at the same time, i.e. the purpose of this section is to prove the following. \[MAMBcoho\] Any isomorphism $H^*(M(A);\Z/2)\to H^*(M(B);\Z/2)$ is induced from a composition of affine diffeomorphisms corresponding to the three operations (Op1), (Op2) and (Op3), and if $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$ are isomorphic as graded rings, then $A$ and $B$ are Bott equivalent. We introduce a notion and prepare a lemma. Remember that $$\label{HMA} H^*(M(A);\Z/2)=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n).$$ One easily sees that products $x_{i_1}\dots x_{i_q}$ $(1\le i_1<\dots<i_q\le n)$ form a basis of $H^q(M(A);\Z/2)$ as a vector space over $\Z/2$ so that the dimension of $H^q(M(A);\Z/2)$ is $\binom{n}{q}$ (see [@ma-pa08 Lemma 5.3]). We set $$\label{alphaj} \alpha_j=\sum_{i=1}^nA^i_jx_i\quad\text{for $j=1,\dots,n$}$$ where $\alpha_1=0$ since $A$ is an upper triangular matrix with zero diagonal entries. Then the relations in are written as $$\label{alpha} x_j^2=\alpha_jx_j \quad\text{for $j=1,\dots,n$.}$$ Motivated by this identity we introduce the following notion. We call an element $\alpha\in H^1(M(A);\Z/2)$ an [*eigen-element*]{} of $H^*(M(A);\Z/2)$ if there exists $x\in H^1(M(A);\Z/2)$ such that $x^2=\alpha x$, $x\not=0$ and $x\not=\alpha$. The set of all elements $x\in H^1(M(A);\Z/2)$ satisfying the equation $x^2=\alpha x$ is a vector subspace of $H^1(M(A);\Z/2)$ which we call the [*eigen-space*]{} of $\alpha$ and denote by $\EA(\alpha)$. We also introduce a notation $\tEA(\alpha)$ which is the quotient of $\EA(\alpha)$ by the subspace spanned by $\alpha$, and call it the [*reduced eigen-space*]{} of $\alpha$. Eigen-elements and (reduced) eigen-spaces are invariants preserved under graded ring isomorphisms. By $\alpha_j$’s are eigen-elements of $H^*(M(A);\Z/2)$ and the following lemma shows that these are the only eigen-elements. \[EAa\] If $\alpha$ is an eigen-element of $H^*(M(A);\Z/2)$, then $\alpha=\alpha_j$ for some $j$ and the eigen-space $\EA(\alpha)$ of $\alpha$ is generated by $\alpha$ and $x_i$’s with $\alpha_i=\alpha$. By the definition of eigen-element there exists a non-zero element $x\in H^1(M(A);\Z/2)$ different from $\alpha$ such that $x^2=\alpha x$. Since both $x$ and $x+\alpha$ are non-zero, there exist $i$ and $j$ such that $x=x_i+p_i$ and $x+\alpha=x_j+q_j$ where $p_i$ is a linear combination of $x_1,\dots,x_{i-1}$ and $q_j$ is a linear combination of $x_1,\dots,x_{j-1}$. Then $$x_ix_j+x_iq_j+x_jp_i+p_iq_j=0$$ because $x(x+\alpha)=0$. As remarked above, products $x_{i_1}x_{i_2}$ $(1\le i_1<i_2\le n)$ form a basis of $H^2(M(A);\Z/2)$, so $i$ must be equal to $j$ for the identity above to hold. Then as $x_j^2=x_j\alpha_j$, it follows from the identity above that $\alpha_j=q_j+p_i$ (and $p_iq_j=0$). This implies that $\alpha=\alpha_j$, proving the former statement of the lemma. We express a non-zero element $x\in \EA(\alpha)$ as $\sum_{i=1}^nc_ix_i$ $(c_i\in \Z/2)$ and let $m$ be the maximum number among $i$’s with $c_i\not=0$. *Case 1.* The case where $x_m$ appears when we express $\alpha$ as a linear combination of $x_1,\dots,x_n$. We express $x(x+\alpha)$ as a linear combination of the basis elements $x_{i_1}x_{i_2}$ $(1\le i_1<i_2\le n)$. Since $x_m$ appears in both $x$ and $\alpha$, it does not appear in $x+\alpha$. Therefore the term in $x(x+\alpha)$ which contains $x_m$ is $x_m(x+\alpha)$ and it must vanish because $x(x+\alpha)=0$. Therefore $x=\alpha$. *Case 2.* The case where $x_m$ does not appear in the linear expression of $\alpha$. In this case, the term in $x(x+\alpha)$ which contains $x_m$ is $x_m(x_m+\alpha)=x_m(\alpha_m+\alpha)$ since $x_m^2=\alpha_m x_m$, and it must vanish because $x(x+\alpha)=0$. Therefore $\alpha_m=\alpha$. The sum $x+x_m$ is again an element of $\EA(\alpha)$. If $x\not=x_m$ (equivalently $x+x_m$ is non-zero), then the same argument applied to $x+x_m$ shows that there exists $m_1(\not=m)$ such that $\alpha_{m_1}=\alpha$ and $x+x_m+x_{m_1}$ is again an element of $\EA(\alpha)$. Repeating this argument, $x$ ends up with a linear combination of $x_i$’s with $\alpha_i=\alpha$. With this preparation we shall prove Proposition \[MAMBcoho\]. Let $B$ be another element of $\T(n)$. We denote the canonical basis of $H^*(M(B);\Z/2)$ by $y_1,\dots,y_n$ and the elements in $H^1(M(B);\Z/2)$ corresponding to $\alpha_j$’s by $\beta_j$’s, i.e., $\beta_j=\sum_{i=1}B^i_jy_i$ for $j=1,\dots,n$. Let $\varphi\colon H^*(M(A);\Z/2)\to H^*(M(B);\Z/2)$ be a graded ring isomorphism. It preserves the eigen-elements and (reduced) eigen-spaces. In the following we shall show that we can change $\varphi$ into the identity map by composing isomorphisms induced from affine diffeomorphisms corresponding to the three operations (Op1), (Op2) and (Op3). Through the operation (Op1) we may assume that $\varphi(\alpha_j)=\beta_j$ for any $j$ because of . Then $\varphi$ restricts to an isomorphism $\EA(\alpha_j) \to \EB(\beta_j)$ between eigen-spaces and induces an isomorphism $\tEA(\alpha_j)\to \tEB(\beta_j)$ between reduced eigen-spaces. Let $\alpha$ (resp. $\beta$) stand for $\alpha_j$ (resp. $\beta_j$) and suppose that $\varphi(\alpha)=\beta$. Let $I$ be a subset of $\{1,\dots,n\}$ such that $\alpha_i=\alpha$ if and only if $i\in I$. We denote the image of $x_i$ (resp. $y_i$) in $\tEA(\alpha)$ (resp. $\tEB(\beta)$) by $\bar x_i$ (resp. $\bar y_i$). The $\bar x_i$’s (resp. $\bar y_i$’s) for $i\in I$ form a basis of $\tEA(\alpha)$ (resp. $\tEB(\beta)$) by Lemma \[EAa\], so if we express $\varphi(\bar x_j)= \sum_{i\in I}C^i_j\bar y_i$ with $C^i_j\in \Z/2$, then the matrix $C=(C^i_j)_{i,j\in I}$ is invertible. Therefore, through the operation (Op3), we may assume that $C$ is the identity matrix because of . This means that we may assume that $\varphi(x_j)=y_j$ or $y_j+\beta_j$ for each $j=1,\dots,n$. Finally through the operation (Op2), we may assume that $\varphi(x_j)=y_j$ for any $j$ because of and hence $A=B$ (and $\varphi$ is the identity) because $\varphi(\alpha_j) =\beta_j$, $\alpha_j=\sum_{i=1}^nA^i_jx_i$ and $\beta_j=\sum_{i=1}^nB^i_jy_i$ for any $j$, proving the proposition. Unique decomposition of real Bott manifolds {#sect:decom} =========================================== We say that a real Bott manifold is *indecomposable* if it is not diffeomorphic to a product of more than one real Bott manifolds. The purpose of this section is to prove Theorem \[main1\] in the Introduction, that is \[bdeco\] The decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Namely, if $\prod_{i=1}^k M_i$ is diffeomorphic to $\prod_{j=1}^\ell N_j$ where $M_i$ and $N_j$ are indecomposable real Bott manifolds, then $k=\ell$ and there is a permutation $\sigma$ on $\{1,\dots,k=\ell\}$ such that $M_i$ is diffeomorphic to $N_{\sigma(i)}$ for $i=1,\dots,k$. $H^*(\prod_{i=1}^kM_i;\Z/2)=\bigotimes_{i=1}^kH^*(M_i;\Z/2)$ by Künneth formula and the diffeomorphism types of real Bott manifolds are detected by cohomology rings with $\Z/2$ coefficient by Corollary \[maincoro\], so the theorem above reduces to a problem on the decomposition of a cohomology ring into tensor products over $\Z/2$. We call a graded ring over $\Z/2$ a [*Bott ring*]{} of rank $n$ if it is isomorphic to the cohomology ring with $\Z/2$ coefficient of a real Bott manifold of dimension $n$. Let $\MH$ be a Bott ring of rank $n$, so it has an expression $$\label{MH} \MH=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n)$$ with $A\in\T(n)$. The eigen-elements of $\MH$ are $$\label{eigenj} \text{$\alpha_j=\sum_{i=1}^nA^i_jx_i$\quad $(j=1,\dots,n)$.}$$ We denote by $\MH^q$ the degree $q$ part of $\MH$ and define $$\begin{split} N(\MH)&:=\{ x\in \MH^1\mid x^2=0\},\text{ and}\\ S(\MH)&:=\{x\in \MH^1\backslash\{0\}\mid \exists \bx\in\MH^1\backslash\{0\} \ \text{with $x\bx=0$ and $\bx\not=x$}\}. \end{split}$$ In terms of eigen-elements and eigen-spaces, $N(\MH)$ is the eigen-space of the zero eigen-element. Also, if we write $\bx=x+\alpha$ with $\alpha\in\MH^1$, then $x\bx=0$ means that $x^2=\alpha x$; so $S(\MH)$ with the zero element added is the union of eigen-spaces of all non-zero eigen-elements in $\MH$. The latter statement in Lemma \[EAa\] shows that the eigen-element $\alpha$ is uniquely determined by $x$, hence so is $\bx$. $N(\MH)=\MH^1$ if and only if $A$ in is the zero matrix. Unless $N(\MH)=\MH^1$, $S(\MH)\not=\emptyset$. \[MHS\] The graded subring $\MH_S$ of a Bott ring $\MH$ generated by $S(\MH)$ is a Bott ring. The isomorphism class of $\MH$ does not change through the three operations (Op1), (Op2) and (Op3). Through (Op1) we may assume that the first $\ell$ columns of the matrix $A$ in are all zero but none of the remaining columns is zero. If the maximum number of linearly independent vectors in the first $\ell$ rows of $A$ is $m$, then we may assume that the first $\ell-m$ rows are zero by applying the operation (Op3) to the first $\ell$ columns. Then $\MH_S$ is the Bott ring associated with the $(n-\ell+m)\times(n-\ell+m)$ submatrix of $A$ at the right-low corner of $A$. Let $\MH_S$ be as in Lemma \[MHS\] and let $V$ be a subspace of $N(\MH)$ complementary to $N(\MH)\cap \MH_S^1$. The dimension of $V$ is $\ell-m$ in the proof of Lemma \[MHS\]. The graded subalgebra of $\MH$ generated by $V$ is an exterior algebra $\Lambda(V)$, so $$\label{MH0} \MH=\Lambda(V)\otimes \MH_S.$$ We say that a Bott ring $\MH$ is *semisimple* if $\MH$ is generated by $S(\MH)$. Clearly $\MH_S$ is semisimple and $\MH$ is semisimple if and only if $\MH=\MH_S$. \[YMA\] Let $\MH$ be a Bott ring. If $\MH=\bigotimes_{i=1}^r\MH_i$ with Bott subrings $\MH_i$’s of $\MH$, then $S(\MH)=\coprod_{i=1}^rS(\MH_i)$. Therefore $\MH$ is semisimple if and only if all $\MH_i$’s are semisimple. Let $x\in S(\MH)$ and write $x=\sum_{i=1}^r y_i$ and $\bx= \sum_{i=1}^r z_i$ with $y_i,z_i\in \MH_i$. Since $x\bx=0$, we have $$\text{$y_iz_j+y_jz_i=0$ for all $i\not=j$. }$$ Suppose that $y_i\not=0$ and $z_j\not=0$ for some $i\not=j$. Then $y_i=z_i$ and $y_j=z_j$ to satisfy the equations above. This shows that $x=\bx$, which contradicts the fact that $x\in S(\MH)$. Therefore $x=y_i$ and $\bx=z_i$ for some $i$, proving the lemma. Recall that a Bott ring $\MH$ has a decomposition $\Lambda(V)\otimes \MH_S$ in . \[semis\] If $\MH$ has another decomposition $\Lambda(U)\otimes \MS$ where $U$ is a subspace of $N(\MH)$ and $\MS$ is a semisimple subring of $\MH$, then $\dim U=\dim V$ and $\MS=\MH_S$. Since both $S(\Lambda(U))$ abd $S(\Lambda(V))$ are empty, $S(\MH)=S(\MS)$ by Lemma \[YMA\] and this implies the corollary. \[factor\] Let $\MH=\bigotimes_{i=1}^r\MH_i$ be as in Lemma \[YMA\] and $\pi_i\colon \MH\to \MH_i$ be the projection. Let $\ML$ be a semisimple Bott ring and let $\psi\colon \ML\to \MH$ be a graded ring monomorphism. If the composition $\pi_i\circ \psi\colon \ML\to \MH_i$ is an isomorphism for some $i$, then $\psi(\ML)=\MH_i$. Let $y\in S(\ML)$. Then $\psi(y)\in S(\MH)$ because $\psi$ is a graded ring monomorphism, and it is actually in $S(\MH_i)$ by Lemma \[YMA\] since $(\pi_i\circ \psi)(y)\not=0$. This shows that $\psi(S(\ML))\subset S(\MH_i)$ but since $\pi_i\circ\psi$ is an isomorphism, the inclusion should be the equality. Therefore $\psi(\ML)=\MH_i$ because $\ML$ and $\MH_i$ are both semisimple. We say that a semisimple Bott ring is *simple* if it is not isomorphic to the tensor product (over $\Z/2)$ of more than one semisimple Bott rings, in other words, a simple Bott ring is a Bott ring isomorphic to the cohomology ring (with $\Z/2$ coefficient) of an indecomposable real Bott manifold different from $S^1$. A Bott ring isomorphic to the cohomology ring of the Klein bottle with $\Z/2$ coefficient is simple and we call it especially a *Klein ring*. If an element $x\in S(\MH)$ satisfies $(x+\bx)^2=0$, then the subring generated by $x$ and $\bx$ is a Klein ring and we call such a pair $\{x,\bx\}$ a *Klein pair*. We note that $x$ and $\bx$ have the same eigen-element and $\{x,\bx\}$ is a Klein pair if and only if the eigen-element of $x$ and $\bx$, that is $x+\bx$, lies in $N(\MH)$. \[klein\] If $S(\MH)\not=\emptyset$, then a Klein pair exists in $\MH$ and the quotient of $\MH$ by the ideal generated by a Klein pair is again a Bott ring. Let $\MH$ be of the form . The assumption $S(\MH)\not=\emptyset$ is equivalent to $A$ being non-zero as remarked before. As in the proof of Lemm \[MHS\], we may assume through the operation (Op1) that the first $\ell$ columns of $A$ are zero and none of the remaining columns is zero. Then $x_1,\dots,x_{\ell}$ are elements of $N(\MH)$ and the eigen-element $\alpha_{\ell+1}$ of $x_{\ell+1}$ is a linear combination of $x_1,\dots,x_{\ell}$, so $\alpha_{\ell+1}$ lies in $N(\MH)$ which means that $\{x_{\ell+1},\bx_{\ell+1}\}$ is a Klein pair. If $\{x,\bx\}$ is a Klein pair, then the eigen-element of $x$ is non-zero and belongs to $N(\MH)$, so through the operation (Op1) we may assume that it is $\alpha_{\ell+1}$. Then, applying the operation (Op3) to the eigen-space of $\alpha_{\ell+1}$, we may assume $x=x_{\ell+1}$. We further may assume $\alpha_{\ell+1}=x_\ell$ by applying the operation (Op3) to $N(\MH)$. The quotient ring of $\MH$ by the ideal generated by the Klein pair $\{x,\bx\}$ is then nothing but to take $x_\ell=x_{\ell+1}=0$ in $\MH$, so it is a Bott ring associated with a $(n-2)\times(n-2)$ matrix obtained from $A$ by deleting $\ell$-th and $\ell+1$-st columns and rows. Now we are in a position to prove the unique decomposition of a semisimple Bott ring into a tensor product of simple Bott rings. \[simpl\] Let ${\MC}_i$ $(i=1,\dots,p)$ and ${\MD}_j$ $(j=1,\dots,q)$ be simple Bott rings. If there exists a graded ring isomorphism $$\label{MA} \vf\colon \bigotimes_{i=1}^p\MC_i\to \bigotimes_{j=1}^q \MD_j,$$ then $p=q$ and $\vf$ preserves the factors, i.e. there is a permutation $\rho$ on $\{1,\dots,p=q\}$ such that $\vf(\MC_i)=\MD_{\rho(i)}$ for $i=1,\dots,p$. We set $\MC=\bigotimes_{i=1}^p\MC_i$ and $\MD=\bigotimes_{j=1}^q\MD_j$. If either $\MC$ or $\MD$ is simple (i.e. $p=1$ or $q=1$), then both of them must be simple and the proposition is trivial. In the sequel we will assume that both $\MC$ and $\MD$ are not simple (so that $p\ge 2$ and $q\ge 2$), and prove the proposition by induction on the rank of $\MC$, that is, $\dim \MC^1$. If $\vf(\MC_i)=\MD_j$ for some $i$ and $j$, say $\vf(\MC_p)=\MD_q$, then we factorize them so that $\vf$ induces an isomorphism $\bar \vf\colon \bigotimes_{i=1}^{p-1}\MC_i\to \bigotimes_{j=1}^{q-1} \MD_j$. By the induction assumption, we conclude $p=q$ and may assume that $\bar \vf(\MC_i)=\MD_i$ for $i=1,\dots,p-1$ if necessary by permuting the suffixes of $\MD_j$’s. Then it follows from Lemma \[factor\] that $\vf(\MC_i)=\MD_{i}$ for $i=1,\dots,p-1$. This together with $\vf(\MC_p)=\MD_q$ where $p=q$ proves the statement in the lemma. In the sequel, it suffices to show that $\vf(\MC_i)=\MD_j$ for some $i$ and $j$ when we have an isomorphism $\vf$ in the proposition. *Case 1.* The case where some $\MC_i$ or $\MD_j$ is a Klein ring. We may assume that $\MC_p$ is a Klein ring without loss of generality. Let $\{x,\bx\}$ be a Klein pair in $\MC_p$. Its image by $\vf$ sits in some $\MD_j$ by Lemma \[YMA\] and we may assume that it sits in $\MD_q$. If $\MD_q$ is also a Klein ring, then $\vf(\MC_p)=\MD_q$. Therefore we may assume that $\MD_q$ is not a Klein ring in the following. Our isomorphism $\vf$ induces an isomorphism $$\bar \vf \colon \MC/(x,\bx)=\bigotimes_{i=1}^{p-1}\MC_i\cong \bigotimes_{j=1}^{q-1}\MD_j\otimes (\MD_q/(\vf(x),\vf(\bx)))$$ where $(u,v)$ denotes the ideal generated by the elements $u$ and $v$ and $\MD_q/(\vf(x),\vf(\bx))$ is a Bott ring by Lemma \[klein\]. Since $\operatorname{rank}(\MC/(x,\bx))=\operatorname{rank}\MC-2$, it follows from the induction assumption that $p-1\ge q$ and we may assume that $\bar \vf(\MC_i)=\MD_i$ for $i=1,\dots,q-1$ and $\bar \vf(\otimes_{i=q}^{p-1}\MC_i)=\MD_q/(\vf(x),\vf(\bx))$, in particular, $\bar \vf(\MC_1)=\MD_1$ as $q\ge 2$. Then, it follows from Lemma \[factor\] that $\vf(\MC_1)=\MD_1$. *Case 2.* The case where none of $\MC_i$’s and $\MD_j$’s is a Klein ring. Let $\{x,\bx\}$ be a Klein pair of $\MC_p$ and we may assume that its image by $\vf$ sits in $\MD_q$ as before. Then $\vf$ induces an isomorphism $$\bar \vf\colon \MC/(x,\bx)=\bigotimes_{i=1}^{p-1}\MC_i\otimes (\MC_p/(x,\bx)) \to \bigotimes_{j=1}^{q-1}\MD_j\otimes (\MD_q/(\vf(x),\vf(\bx))),$$ where the quotients $\MC_p/(x,\bx)$ and $\MD_q/(\vf(x),\vf(\bx))$ are both Bott rings by Lemma \[klein\]. The induction assumption can be applied to this situation as before. If $\bar \vf(\MC_i)=\MD_j$ for some $1\le i\le p-1$ and $1\le j\le q-1$, then $\vf(\MC_i)=\MD_j$ by Lemma \[factor\]. If $\bar \vf(\bigotimes_{i=1}^{p-1}\MC_i)=\MD_q/(\vf(x),\vf(\bx))$ and $\bar \vf(\MC_p/(x,\bx))=\bigotimes_{j=1}^{q-1}\MD_j$, then $\vf$ restricts to an isomorphism $$(\bigotimes_{i=1}^{p-1}\MC_i)\otimes\langle x,\bx\rangle\to \MD_q$$ where $\langle x,\bx\rangle$ denotes the Klein ring generated by $x$ and $\bx$, and this contradicts the fact that $\MD_q$ is simple as $p\ge 2$. Now Theorem \[bdeco\] follows from Corollaries \[maincoro\], \[semis\] and Proposition \[simpl\]. [**Acknowledgment**]{}. I would like to thank Y. Kamishima for communications which stimulated this research and A. Nazra for informing me of his classification of real Bott manifolds of dimension $\le 5$ up to diffeomorphism. [19]{} L. S. Charlap, *Compact flat Riemannian manifolds: I*, Ann. of Math. 81 (1965), 15–30. M. Grossberg and Y. Karshon, *Bott towers, complete integrability, and the extended character of representations*, Duke Math. J 76 (1994), 23–58. Y. Kamishima and M. Masuda, *Cohomological rigidity of real Bott manifolds*, preprint, arXiv:0807.4263. Y. Kamishima and A. Nazra, *Seifert fibered structure and rigidity on real Bott towers*, in preparation. M. Masuda, *Cohomological non-rigidity of generalized real Bott manifolds of height 2*, preprint, arXiv:0809.2215. M. Masuda and T. Panov, *Semifree circle actions, Bott towers, and quasitoric manifolds*, Sbornik Math. (to appear), arXiv:math.AT/0607094. A. Nazra, *Real Bott tower*, Tokyo Metropolitan University, Master Thesis 2008. [^1]: The author was partially supported by Grant-in-Aid for Scientific Research 19204007

ArXiv

--- address: | Department of Physics, University of California\ Riverside, CA 92521, USA author: - ERNEST MA title: | MODELS OF NEUTRINO MASS AND INTERACTIONS\ FOR NEUTRINO OSCILLATIONS --- \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} Neutrino Masses =============== In the minimal standard model, under the gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$, the leptons transform as: $$\left[ \begin{array} {c} \nu_e \\ e \end{array} \right]_L, \left[ \begin{array} {c} \nu_\mu \\ \mu \end{array} \right]_L, \left[ \begin{array} {c} \nu_\tau \\ \tau \end{array} \right]_L \sim (1, 2, -1/2); ~~~ e_R, ~ \mu_R, ~ \tau_R \sim (1, 1, -1).$$ There is also the Higgs scalar doublet $(\phi^+, \phi^0) \sim (1, 2, 1/2)$ whose nonzero vacuum expectation value $\langle \phi^0 \rangle = v$ breaks $SU(2)_L \times U(1)_Y$ to $U(1)_Q$. Whereas charged leptons acquire masses proportional to $v$, the absence of $\nu_R$ implies that $m_{\nu_i} = 0$. If nonzero neutrino masses are desired (which are of course necessary for neutrino oscillations), then we must ask “What is the nature of this mass?" and “What new physics goes with it?" If $\nu_R$ does not exist, one way to have $m_\nu \neq 0$ is to add a Higgs triplet $(\xi^{++}, \xi^+, \xi^0)$. Each $\nu_L$ then gets a Majorana mass. However, $\langle \xi^0 \rangle$ must be very small, and if the lepton number being carried by $\xi$ is spontaneously violated [@1], the decay of $Z$ to the associated massless Goldstone boson (the triplet Majoron) and its partner would count as two extra neutrinos. Since the effective number of light neutrinos in $Z$ decay is now measured [@2] to be $2.989 \pm 0.012$, the triplet Majoron model is clearly ruled out. If one $\nu_R \sim (1, 1, 0)$ exists for each $\nu_L$, the most general $2 \times 2$ neutrino mass matrix linking $(\bar \nu_L, \bar \nu_R^c)$ to $(\nu_L^c, \nu_R)$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c} m_L & m_D \\ m_D & m_R \end{array} \right].$$ If $m_L = 0$ and $m_D << m_R$, we get the famous seesaw mechanism [@3] $$m_\nu \sim {m_D^2 \over m_R}.$$ Here, $\nu_L - \nu_R^c$ mixing is $m_D/m_R$ and $m_R$ is the scale of new physics. In this minimal scenario, new physics enters only through $m_R$, hence there is no other observable effect except for a nonzero $m_\nu$. Actually, $m_D/m_R$ is in principle observable but it is in practice far too small. In general, the mass matrix of Eq. (2) yields two nondegenerate interacting Majorana neutrinos (unless $m_L = m_R = 0$ is maintained exactly). If both eigenvalues are small, the effective number of neutrinos counted in Big Bang Nucleosynthesis may be as high as six, instead of the usual three, depending on the mass splitting and mixing in each case [@4]. The smallness of neutrino masses may be indicative of their radiative origin. Many papers have been written on the subject. For a brief review, see Ref. 5. There are three one-loop mechanisms: the exchange of two scalar bosons with one fermion mass insertion; the exchange of one scalar boson with three fermion mass insertions; and the coupling to a scalar boson which gets a radiative vacuum expectation value through a fermion loop with five mass insertions. A prime example of the first mechanism is the Zee model [@6]. Here the minimal standard model is extended to include a charged scalar singlet $\chi^+$ and a second scalar doublet $(\eta^+, \eta^0)$. We then have the coupling $$f_{ij} \chi^+ (\nu_i l_j - l_i \nu_j),$$ which by itself would require $\chi^+$ to have lepton number $-2$. However, this model also allows the cubic scalar coupling $$\chi^- (\phi^+ \eta^0 - \phi^0 \eta^+),$$ hence lepton number is broken explicitly. A radiative Majorana mass matrix is thus obtained through the exchange and mixing of $\chi^+$ and the physical linear combination of $\phi^+$ and $\eta^+$. Since $f_{ij}$ of Eq. (4) is zero for $i = j$ and $\phi^+$ couples $\nu_i$ to $l_i$ with strength proportional to $m_{l_i}$ which is also the one fermion mass insertion required, the $3 \times 3$ neutrino mass matrix for $\nu_e$, $\nu_\mu$ and $\nu_\tau$ is of the form $${\cal M}_\nu \propto \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & 0 & f_{e \tau} m_\tau^2 \\ 0 & 0 & f_{\mu \tau} m_\tau^2 \\ f_{e \tau} m_\tau^2 & f_{\mu \tau} m_\tau^2 & 0 \end{array} \right] + {\cal O} (m_\mu^2).$$ This means that $\nu_\tau$ is almost degenerate with a linear combination of $\nu_\mu$ and $\nu_e$ in this model. This may have a practical application in present neutrino-oscillation phenomenology [@7]. There are also three two-loop mechanisms: the exchange of three scalar bosons which are tied together by a cubic coupling; the exchange of two $W$ bosons; and the exchange of $W_L$ and $W_R$ which mix at the one-loop level. The second mechanism [@8] is unique in that it requires only one additional $\nu_R$ beyond the standard model. In this specific case, one $\nu_L$ gets a seesaw mass and the other two get two-loop masses proportional to this mass and as functions of the charged-lepton masses with double GIM suppression [@9]. A detailed analytical and numerical study of this mechanism has been made [@10]. Finally let me return to the triplet-Higgs mechanism. If lepton number is violated explicitly by the coupling of $\xi$ to the scalar doublet $\phi$, then one may let $\xi$ be very heavy and integrate it out to obtain the following effective nonrenormalizable interaction: $${1 \over M} [\phi^0 \phi^0 \nu_i \nu_j + \phi^+ \phi^0 (\nu_i l_j + l_i \nu_j) + \phi^+ \phi^+ l_i l_j] + h.c.$$ For $M \sim 10^{13}$ GeV, one gets $m_\nu \sim$ few eV. This is the most economical solution and could also be a realistic model of leptogenesis [@11] in the early universe which gets converted at the electroweak phase transition into the present observed baryon asymmetry. Neutrino Oscillations ===================== Present experimental evidence for neutrino oscillations [@12] includes the solar $\nu_e$ deficit which requires $\Delta m^2$ of around $10^{-5}$ eV$^2$ for the MSW explanation or $10^{-10}$ eV$^2$ for the vacuum-oscillation solution, the atmospheric neutrino deficit in the ratio $\nu_\mu + \bar \nu_\mu / \nu_e + \bar \nu_e$ which implies a $\Delta m^2$ of around $10^{-2}$ eV$^2$, and the LSND experiment which indicates a $\Delta m^2$ of around 1 eV$^2$. Three different $\Delta m^2$ necessitate four neutrinos, but the invisible width of the $Z$ boson as well as Big Bang Nucleosynthesis allow only three. If all of the above-mentioned experiments are interpreted correctly as due to neutrino oscillations, we are faced with a theoretical challenge in trying to understand how three can equal four. I will focus on addressing this issue rather than trying to review the many theoretical models for the three known neutrinos. Three Neutrinos and One Light Singlet ===================================== One possibility is that there is a light singlet neutrino $\nu_S$ in addition to the three known doublet neutrinos $\nu_e$, $\nu_\mu$, and $\nu_\tau$. This is necessary so that it is not counted in the effective number of neutrinos in $Z$ decay [@2]. On the other hand, it has to mix with the doublet neutrinos for it to be relevant to oscillation experiments. Hence it is also contrained [@4] by Big Bang Nucleosynthesis. Using all available data, a model-independent analysis [@13] shows that the $4 \times 4$ neutrino mass matrix must separate approximately into two blocks: one for $\nu_e - \nu_S$ and the other for $\nu_\mu - \nu_\tau$, the latter with large mixing. An example of a specific model of this kind already exists [@14]. The neutrino interaction eigenstates are related to the mass eigenstates by $$\left[ \begin{array} {c} \nu_S \\ \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} \right] = \left[ \begin{array} {c@{\quad}c@{\quad}c@{\quad}c} -s & c & s''/\sqrt 2 & s''/\sqrt 2 \\ c & s & -s'/\sqrt 2 & s'/\sqrt 2 \\ -s' & 0 & -1/\sqrt 2 & 1/\sqrt 2 \\ 0 & -s'' & 1/\sqrt 2 & 1/\sqrt 2 \end{array} \right] \left[ \begin{array} {c} \nu_1 \\ \nu_2 \\ \nu_3 \\ \nu_4 \end{array} \right],$$ where $m_1 = 0$, $m_2 \sim 2.5 \times 10^{-3}$ eV, $m_3 \sim m_4 \sim 2.5$ eV, with $\Delta m_{34}^2 \sim 1.8 \times 10^{-2}$ eV$^2$; $s \sim s' \sim 0.04$, but $s''$ is undetermined. Note that $m_{\nu_e} < m_{\nu_S}$ is necessary for the MSW solution [@15] of the solar neutrino deficit. Note also that $\nu_\mu$ and $\nu_\tau$ are pseudo-Dirac partners, hence the mixing angle for atmospheric neutrino oscillations is 45 degrees. What is the nature of this light singlet? and how does it mix with the usual neutrinos? There have been some discussions on these questions in the past two or three years. One idea [@16] is that it is the fermion partner of the massless Goldstone boson of a sponatneously broken global symmetry, such as lepton number (hence a Majorino) in supersymmetry. Another [@17] is that it is the fermion partner of a scalar field corresponding to a flat direction (hence a modulino) in the supersymmetric Higgs potential. If the standard model is extended to include a mirror $[SU(2) \times U(1)]'$ sector, then $\nu_S$ may be identified as a mirror neutrino, either in a theory where the mirror symmetry is broken [@18] or one where it is exact [@19]. In the latter case, maximal mixing between the three known neutrinos and their mirror counterparts would occur and Big Bang Nucleosynthesis would count six neutrinos under normal conditions. Both questions can be answered naturally in a model [@20] based on $E_6$ inspired by superstring theory. In the fundamental [**27**]{} representation of $E_6$, outside the 15 fields belonging to the minimal standard model, there are 2 neutral singlets. One ($N$) is identifiable with the right-handed neutrino because it is a member of the [**16**]{} representation of $SO(10)$; the other ($S$) is a singlet also under $SO(10)$. In the reduction of $E_6$ to $SU(3)_C \times SU(2)_L \times U(1)_Y$, an extra U(1) gauge factor may survive down to the TeV energy scale. It could be chosen such that $N$ is trivial under it, but $S$ is not. This means that $N$ is allowed to have a large Majorana mass so that the usual seesaw mechanism works for the three doublet neutrinos. At the same time, $S$ is protected from having a mass by the extra U(1) gauge symmetry, which I call $U(1)_N$. However, it does acquire a small mass from an analog of the usual seesaw mechanism because it can couple to doublet neutral fermions which are present in the [**27**]{} of $E_6$ outside the [**16**]{} of $SO(10)$. Renaming $S$ as $\nu_S$, the $3 \times 3$ mass matrix spanning $\nu_S$, $\nu_E$, and $N_E^c$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & m_1 & m_2 \\ m_1 & 0 & m_E \\ m_2 & m_E & 0 \end{array} \right].$$ Hence $m_{\nu_S} \sim 2 m_1 m_2/m_E$, which is a singlet-doublet seesaw rather than the usual doublet-singlet seesaw. Furthermore, the mixing of $\nu_S$ with the doublet neutrinos is also possible through these extra doublet neutral fermions. The spontaneous breaking of $U(1)_N$ is not possible without also breaking the supersymmetry, hence both are assumed to occur at the TeV energy scale, resulting in a rich $Z'$ and Higgs phenomenology [@21]. Three Neutrinos and One Anomalous Interaction ============================================= If one insists on keeping only the usual three neutrinos and yet try to accommodate all present data, how far can one go? It has been pointed out by many authors [@22] that both solar and LSND data can be explained, as well as most of the atmospheric data except for the zenith-angle dependence. It is thus worthwhile to consider the following scenario [@23] whereby a possible anomalously large $\nu_\tau$-quark interaction may mimic the observed zenith-angle dependence of the atmospheric data. Consider first the following approximate mass eigenstates: $$\begin{aligned} \nu_1 &\sim& \nu_e ~~~ {\rm with} ~ m_1 \sim 0, \\ \nu_2 &\sim& c_0 \nu_\mu + s_0 \nu_\tau ~~~ {\rm with} ~ m_2 \sim 10^{-2} ~{\rm eV}, \\ \nu_3 &\sim& -s_0 \nu_\mu + c_0 \nu_\tau ~~~ {\rm with} ~ m_3 \sim 0.5 ~{\rm eV},\end{aligned}$$ where $c_0 \equiv \cos \theta_0$, $s_0 \equiv \sin \theta_0$, and $\theta_0$ is not small. Allow $\nu_1$ to mix with $\nu_3$ with a small angle $\theta '$ and the new $\nu_1$ to mix with $\nu_2$ with a small angle $\theta$, then the LSND data can be explained with $\Delta m^2 \sim 0.25$ eV$^2$ and $2 s_0 s' c' \sim 0.19$ and the solar data can be understood as follows. Consider the basis $\nu_e$ and $\nu_\alpha \equiv c_0 \nu_\mu + s_0 \nu_\tau$. Then $$-i {d \over {dt}} |\nu \rangle_{e,\alpha} = \left( p + {{\cal M}^2 \over {2p}} \right) |\nu \rangle_{e,\alpha},$$ where $${\cal M}^2 = {\cal U} \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_2^2 \end{array} \right] {\cal U}^\dagger + \left[ \begin{array} {c@{\quad}c} A+B & 0 \\ 0 & B+C \end{array} \right].$$ In the above, $A$ comes from the charged-current interaction of $\nu_e$ with $e$, $B$ from the neutral-cuurent interaction of $\nu_e$ and $\nu_\alpha$ with the quarks and electrons, and $C$ from the assumed anomalous $\nu_\tau$-quark interaction. Let $${\cal U} = \left( \begin{array} {c@{\quad}c} c & s \\ -s & c \end{array} \right),$$ then the resonance condition is $$m_2^2 \cos 2 \theta - A + C = 0,$$ where [@24] $$A - C = 2 \sqrt 2 G_F (N_e - s_0^2 \epsilon'_q N_q) p.$$ In order to have a large $\epsilon'_q$ and yet satisfy the resonance condition for solar-neutrino flavor conversion, $m_2$ should be larger than its canonical value of $2.5 \times 10^{-3}$ eV, and $\epsilon'_q$ should be negative. \[If $\epsilon'_q$ comes from $R$-parity violating squark exchange, then it must be positive, in which case an inverted mass hierarchy, [*i.e.*]{} $m_2 < m_1$ would be needed. If it comes from vector exchange, it may be of either sign.\] Assuming as a crude approximation that $N_q \simeq 4 N_e$ in the sun, the usual MSW solution with $\Delta m^2 = 6 \times 10^{-6}$ eV$^2$ is reproduced here with $$s_0^2 \epsilon'_q \simeq -3.92 = -4.17 (m_2^2/10^{-4}{\rm eV}^2) + 0.25.$$ The seemingly arbitrary choice of $\Delta m_{21}^2 \sim 10^{-4}$ eV$^2$ is now sen as a reasonable value so that $\epsilon'_q$ can be large enough to be relevant for the following discussion on the atmospheric neutrino data. Atmospheric neutrino oscillations occur between $\nu_\mu$ and $\nu_\tau$ in this model with $\Delta m^2_{32} \sim 0.25$ eV$^2$, the same as for the LSND data, but now it is large relative to the $E/L$ ratio of the experiment, hence the factor $\cos \Delta m^2 (L/2E)$ washes out and $$P_0 (\nu_\mu \rightarrow \nu_\mu) = 1 - {1 \over 2} \sin^2 2 \theta_0 \simeq 0.66 ~~ {\rm for} ~~ s_0 \simeq 0.47.$$ In the standard model, this would hold for all zenith angles. Hence it cannot explain the present experimental evidence that the depletion is more severe for neutrinos coming upward to the detector through the earth than for neutrinos coming downward through only the atmosphere. This zenith-angle dependence appears also mostly in the multi-GeV data and not in the sub-GeV data. It is this trend which determines $\Delta m^2$ to be around $10^{-2}$ eV$^2$ in this case. As shown below, the hypothesis that $\nu_\tau$ has anomalously large interactions with quarks will mimic this zenith-angle dependence even though $\Delta m^2$ is chosen to be much larger, [*i.e.*]{} 0.25 eV$^2$. Consider the basis $\nu_\mu$ and $\nu_\tau$. Then the analog of Eq. (13) holds with Eq. (14) replaced by $${\cal M}^2 = {\cal U}_0 \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_3^2 \end{array} \right] {\cal U}_0^\dagger + \left[ \begin{array} {c@{\quad}c} B & 0 \\ 0 & B + C \end{array} \right].$$ The resonance condition is then $$m_3^2 \cos 2 \theta_0 + C = 0,$$ where $N_q$ in $C$ now refers to the quark number density inside the earth and the factor $s_0^2$ in Eq. (17) is not there. If $C$ is large enough, the probability $P_0$ would not be the same as the one in matter. Using the estimate $N_q \sim 9 \times 10^{30}$ m$^{-3}$ and defining $$X \equiv \epsilon'_q E_\nu/(10 ~{\rm GeV}),$$ the effective mixing angles in matter are given by $$\begin{aligned} \tan 2 \theta_m^E &=& {{\sin 2 \theta_0} \over {\cos 2 \theta_0 + 0.091 X}} ~~ {\rm for} ~ \nu, \\ \tan 2 \bar \theta_m^E &=& {{\sin 2 \theta_0} \over {\cos 2 \theta_0 - 0.091 X}} ~~ {\rm for} ~ \bar \nu.\end{aligned}$$ For sub-GeV neutrinos, $X$ is small so matter effects are not very important. For multi-GeV neutrinos, $X$ may be large enough to satisfy the resonance condition of Eq. (21). Assuming adiabaticity, the neutrino and antineutrino survival probabilities are given by $$\begin{aligned} P(\nu_\mu \rightarrow \nu_\mu) &=& {1 \over 2} (1 + \cos 2 \theta_0 \cos 2 \theta_m^E), \\ \bar P(\bar \nu_\mu \rightarrow \bar \nu_\mu) &=& {1 \over 2} (1 + \cos 2 \theta_0 \cos 2 \bar \theta_m^E).\end{aligned}$$ Since $\sigma_\nu \simeq 3 \sigma_{\bar \nu}$, the observed ratio of $\nu + \bar \nu$ events is then $$P_m \simeq {{3 r P + \bar P} \over {3 r + 1}},$$ where $r$ is the ratio of the $\nu_\mu$ to $\bar \nu_\mu$ flux in the upper atmosphere. The atmospheric data are then interpreted as follows. For neutrinos coming down through only the atmosphere, $P_0 = 0.66$ applies. For neutrinos coming up through the earth, $P_m \simeq P_0 \simeq 0.66$ as well for the sub-GeV data. However, for the multi-GeV data, if $X = -15$, then $P = 0.31$ and $\bar P = 0.76$, hence $P_m$ is lowered to 0.39 if $r = 1.5$ or 0.42 if $r = 1.0$. The apparent zenith-angle dependence of the data may be explained. Conclusion and Outlook ====================== If all present experimental indications of neutrino oscillations turn out to be correct, then either there is at least one sterile neutrino beyond the usual $\nu_e$, $\nu_\mu$, and $\nu_\tau$, or there is an anomalously large $\nu_\tau$-quark interaction. The latter can be tested at the forthcoming Sudbury Neutrino Observatory (SNO) which has the capability of neutral-current detection. The predicted $\Delta m^2$ of 0.25 eV$^2$ in $\nu_\mu$ to $\nu_e$ and $\nu_\tau$ oscillations will also be tested at the long-baseline neutrino experiments such as Fermilab to Soudan 2 (MINOS), KEK to Super-Kamiokande (K2K), and CERN to Gran Sasso. More immediately, the new data from Super-Kamiokande, Soudan 2, and MACRO on $\nu_\mu + \bar \nu_\mu$ events through the earth should be analyzed for such an effect. For a zenith angle near zero, the $\Delta m^2 \sim 10^{-2}$ eV$^2$ oscillation scenario should have $R \sim 1$, whereas the $\Delta m^2 \sim 0.25$ eV$^2$ oscillation scenario (with anomalous interaction) would have $R = P_0 \sim 0.66$. Furthermore, if $\nu$ and $\bar \nu$ can be distinguished (as proposed in the HANUL experiment), then to the extent that $CP$ is conserved, matter effects can be isolated. Neutrino physics is on the verge of major breakthroughs. New experiments in the next several years will be decisive in leading us forward in its theoretical understanding, and may even discover radically new physics beyond the standard model. Acknowledgments {#acknowledgments .unnumbered} =============== I thank Biswarup Mukhopadhyaya and all the other organizers for their great hospitality and a stimulating workshop. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. References {#references .unnumbered} ========== [99]{} G. B. Gelmini and M. Roncadelli, Phys. Lett. [**99B**]{}, 411 (1981). P. B. Renton, Int. J. Mod. Phys. [**A12**]{}, 4109 (1997). M. Gell-Mann, P. Ramond, and R. Slansky, in [*Supergravity, Proc. of the Workshop*]{}, Stony Brook, New York, 1979, eds. P. Van Nieuwenhuizen and D. Z. Freedman (North-Holland, 1979), p. 315; T. Yanagida, in [*Proc. of the Workshop on the Unified Theory and the Baryon Number in the Universe*]{}, Tsukuba, Japan, 1979, eds. O. Sawada and A. Sugamoto (KEK Report No. 79-18, Tsukuba, Japan, 1979). K. Enqvist [*et al.*]{}, Nucl. Phys. [**B373**]{}, 498 (1992); X. Shi [*et al.*]{}, Phys. Rev. [**D48**]{}, 2563 (1993). K. S. Babu and E. Ma, Mod. Phys. Lett. [**A4**]{}, 1975 (1989). A. Zee, Phys. Lett. [**93B**]{}, 389 (1980). A. Yu. Smirnov and M. Tanimoto, Phys. Rev. [**D55**]{}, 1665 (1997). K. S. Babu and E. Ma, Phys. Rev. Lett. [**61**]{}, 674 (1988); Phys. Lett. [**B228**]{}, 508 (1989). See also S. T. Petcov and S. T. Toshev, Phys. Lett. [**B143**]{}, 175 (1984). S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. [**D2**]{}, 1285 (1970). D. Choudhury, R. Gandhi, J. A. Gracey, and B. Mukhopadhyaya, Phys. Rev. [**D50**]{}, 3468 (1994). E. Ma and U. Sarkar, UCRHEP-T204 (in preparation). For an updated review, see the talks by S. Pakvasa and by P. Pal in these Proceedings. N. Okada and O. Yasuda, Int. J. Mod. Phys. [**A12**]{}, 3669 (1997); S. M. Bilenky, C. Giunti, and W. Grimus, Eur. Phys. J. [**C1**]{}, 247 (1998). E. Ma and P. Roy, Phys. Rev. [**D52**]{}, R4780 (1995). S. P. Mikheyev and A. Yu. Smirnov, Yad. Fiz. [**42**]{}, 1441 (1985) \[Sov. J. Nucl. Phys. [**42**]{}, 913 (1985)\]; Nuovo Cim. [**C9**]{}, 17 (1986); L. Wolfenstein, Phys. Rev. [**D17**]{}, 2369 (1978); [**20**]{}, 2634 (1979). E. J. Chun, A. S. Joshipura, and A. Yu. Smirnov, Phys. Lett. [**B357**]{}, 608 (1995); Phys. Rev. [**D54**]{}, 4654 (1996). K. Benakli and A. Yu. Smirnov, Phys. Rev. Lett. [**79**]{}, 4314 (1997). Z. G. Berezhiani and R. N. Mohapatra, Phys. Rev. [**D52**]{}, 6607 (1995). R. Foot and R. R. Volkas, Phys. Rev. [**D52**]{}, 6595 (1995). E. Ma, Phys. Lett. [**B380**]{}, 286 (1996); E. Keith and E. Ma, Phys. Rev. [**D54**]{}, 3587 (1996). E. Keith, E. Ma, and B. Mukhopadhyaya, Phys. Rev. [**D55**]{}, 3111 (1997). C. Y. Cardall and G. M. Fuller, Phys. Rev. [**D53**]{}, 4421 (1996). See also the earlier work of S. M. Bilenky, A. Bottino, C. Giunti, and C. W. Kim, Phys. Lett. [**B356**]{}, 273 (1995); K. S. Babu, J. C. Pati, and F. Wilczek, [*ibid.*]{}, [**B359**]{}, 351 (1995); G. L. Fogli, E. Lisi, and G. Scioscia, Phys. Rev. [**D52**]{}, 5334 (1995). E. Ma and P. Roy, UCRHEP-T186 (September, 1997). M. M. Guzzo, A. Masiero, and S. T. Petcov, Phys. Lett. [**B260**]{}, 154 (1991); E. Roulet, Phys. Rev. [**D44**]{}. R935 (1991); V. Barger, R. J. N. Phillips, and K. Whisnant, Phys. Rev. [**D44**]{}, 1629 (1991); P. I. Krastev and J. N. Bahcall, hep-ph/9703267.

ArXiv

--- abstract: 'Let $2\leq m \leqs n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ the Lorentz–Sobolev space of order $m$ in the hyperbolic space $\mathbb H^n$. In this paper, we establish the following Adams inequality in the Lorentz–Sobolev space $W^m L^{\frac nm,q}(\mathbb H^n)$ $$\sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}\leq 1} \int_{\mathbb H^n} \Phi_{\frac nm,q}\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g \leqs \infty$$ for $q \in (1,\infty)$ if $m$ is even, and $q \in (1,n/m)$ if $m$ is odd, where $\beta_{n,m}^{q/(q-1)}$ is the sharp exponent in the Adams inequality under Lorentz–Sobolev norm in the Euclidean space. To our knowledge, much less is known about the Adams inequality under the Lorentz–Sobolev norm in the hyperbolic spaces. We also prove an improved Adams inequality under the Lorentz–Sobolev norm provided that $q\geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq \frac nm$ if $m$ is odd, $$\sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\na_g^m u\|_{\frac nm,q}^q -\lam \|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb H^n} \Phi_{\frac nm,q}\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g \leqs \infty$$ for any $0\leqs \lambda \leqs C(n,m,n/m)^q$ where $C(n,m,n/m)^q$ is the sharp constant in the Lorentz–Poincaré inequality. Finally, we establish a Hardy–Adams inequality in the unit ball when $m\geq 3$, $n\geq 2m+1$ and $q \geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq n/m$ if $m$ is odd $$\sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\na_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb B^n} \exp\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dx \leqs \infty.$$' author: - Van Hoang Nguyen title: 'The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms' --- [^1] [^2] [^3] Introduction ============ It is well-known that the Sobolev’s embedding theorems play the important roles in the analysis, geometry, partial differential equations, etc. Let $m\geq 1$, we we traditionally use the notation $$\na^m = \begin{cases} \Delta^{\frac m2} &\mbox{if $m$ is even,}\\ \na \Delta^{\frac{m-1}2} &\mbox{if $m$ is odd} \end{cases}$$ to denote the $m-$th derivatives. For a bounded domain $\Om\subset \R^n, n\geq 2$ and $1\leq p \leqs \infty$, we denote by $W^{m,p}_0(\Om)$ the usual Sobolev spaces which is the completion of $C_0^\infty(\Om)$ under the Dirichlet norm $\|\na^m u\|_{L^p(\Om)} = \Big(\int_\Om |\na^m u|^p dx \Big)^{\frac1p}$. The Sobolev inequality asserts that $W^{m,p}_0(\Om) \hookrightarrow L^q(\Om)$ for any $q \leq \frac{np}{n-mp}$ provided $mp \leqs n$. However, in the limits case $mp = n$ the embedding $W^{m,\frac nm}_0(\Om) \hookrightarrow L^\infty(\Om)$ fails. In this situation, the Moser–Trudinger inequality and Adams inequality are perfect replacements. The Moser–Trudinger inequality was proved independently by Yudovic [@Yudovic1961], Pohozaev [@Pohozaev1965] and Trudinger [@Trudinger67]. This inequality was then sharpened by Moser [@Moser70] in the following form $$\label{eq:Moserineq} \sup_{u\in W^{1,n}_0(\Om), \|\nabla u\|_{L^n(\Om)} \leq 1} \int_\Om e^{\alpha |u|^{\frac n{n-1}}} dx \leqs \infty$$ for any $\al \leq \al_{n}: = n \om_{n-1}^{\frac 1{n-1}}$ where $\om_{n-1}$ denotes the surface area of the unit sphere in $\R^n$. Furthermore, the inequality is sharp in the sense that the supremum in will be infinite if $\al \geqs \al_n$. The inequality was generalized to higher order Sobolev spaces $W^{m,\frac nm}_0(\Om)$ by Adams [@Adams] in the following form $$\label{eq:AMT} \sup_{u \in W^{m,n}_0(\Om), \, \int_\Om |\na^m u|^{\frac nm} dx \leq 1} \int_\Om e^{\al |u|^{\frac n{n-m}}} dx \leqs \infty,$$ for any $$\al \leq \al_{n,m}: = \begin{cases} \frac 1{\si_n}\Big(\frac{\pi^{n/2} 2^m \Gamma(\frac m2)}{\Gamma(\frac{n-m}2)}\Big)^{\frac n{n-m}} &\mbox{if $m$ is even},\\ \frac 1{\si_n}\Big(\frac{\pi^{n/2} 2^m \Gamma(\frac {m+1}2)}{\Gamma(\frac{n-m+1}2)}\Big)^{\frac n{n-m}} &\mbox{if $m$ is odd}, \end{cases}$$ where $\si_n = \om_{n-1}/n$ is the volume of the unit ball in $\R^n$. Moreover, if $\al \geqs \al_{n,m}$ then the supremum in becomes infinite though all integrals are still finite. The Moser-Trudinger inequality and Adams inequality play the role of the Sobolev embedding theorems in the limiting case $mp = n$. They have many applications to study the problems in analysis, geometry, partial differential equations, etc such as the Yamabe’s equation, the $Q-$curvature equations, especially the problems in partial differential equations with exponential nonlinearity, etc. There have been many generalizations of the Moser–Trudinger inequality and Adams inequality in literature. For examples, the Moser–Trudinger inequality and Adams inequality were established in the Riemannian manifolds in [@YangSuKong; @ManciniSandeep2010; @AdimurthiTintarev2010; @ManciniSandeepTintarev2013; @Bertrand; @Karmakar; @LuTang2013; @DongYang] and were established in the subRiemannian manifolds in [@CohnLu; @CohnLu1; @Balogh]. The singular version of the Moser–Trudinger inequality and Adams inequality was proved in [@AdimurthiSandeep2007; @LamLusingular]. The Moser–Trudinger inequality and Adams inequality were extended to unbounded domains and whole spaces in [@Ruf2005; @LiRuf2008; @RufSani; @AdimurthiYang2010; @LamLuHei; @Adachi00; @LamLuAdams; @LamLunew], and to fractional order Sobolev spaces in [@Martinazzi; @FM1; @FM2]. The improved version of the Moser–Trudinger inequality and Adams inequality were given in [@AdimurthiDruet2004; @Tintarev2014; @WangYe2012; @Nguyenimproved; @LuYangAiM; @Nguyen4; @delaTorre; @Mancini; @Yangjfa; @DOO; @NguyenCCM; @LuZhu; @LuYangHA; @LiLuYang]. An interesting question concerning to the Moser–Trudinger inequality and Adams inequality is whether or not the extremal functions exist. For this interesting topic, the reader may consult the papers [@Carleson86; @Flucher92; @Lin96; @Ruf2005; @LiRuf2008; @Chen; @LiYang; @LuZhu; @NguyenCCM; @LuYangAiM; @Nguyen4] and many other papers. Another generalization of the Moser–Trudinger inequality and Adams inequality is to establish the inequalities of same type in the Lorentz–Sobolev spaces. The Moser–Trudinger inequality and the Adams inequality in the Lorentz spaces was established by Alvino, Ferone and Trombetti [@Alvino1996] and Alberico [@Alberico] in the following form $$\label{eq:AMTLorentz} \sup_{u\in W^m L^{\frac nm,q}(\Om), \, \|\na^m u\|_{\frac nm,q} \leq 1} \int_{\Om} e^{\al |u|^{\frac q{q-1}}} dx < \infty$$ for any $\al \leq \beta_{n,m}^{\frac q{q-1}}$ with $$\beta_{n,m} = \begin{cases} \frac{\pi^{n/2} 2^m \Gamma(\frac m2)}{\si_n^{(n-m)/n} \Gamma(\frac{n-m}2)}&\mbox{if $m$ is even,}\\ \frac{\pi^{n/2} 2^m \Gamma(\frac {m+1}2)}{\si_n^{(n-m)/n} \Gamma(\frac{n-m+1}n)}&\mbox{if $m$ is odd.} \end{cases}$$ The constant $\beta_{n,m}$ is sharp in in the sense that the supremum will become infinite if $\al > \beta_{n,m}^{\frac q{q-1}}$. For unbounded domains in $\R^n$, the Moser–Trudinger inequality was proved by Cassani and Tarsi [@CassaniTarsi2009] (see Theorem $1$ and Theorem $2$ in [@CassaniTarsi2009]). In [@LuTang2016], Lu and Tang proved several sharp singular Moser–Trudinger inequalities in the Lorentz–Sobolev spaces which generalize the results in [@Alvino1996; @CassaniTarsi2009] to the singular weights. The singular Adams type inequalities in the Lorentz–Sobolev spaces were studied by the author in [@NguyenLorentz]. The motivation of this paper is to study the Adams inequalities in the hyperbolic spaces under the Lorentz–Sobolev norm. For $n\geq 2$, let us denote by $\H^n$ the hyperbolic space of dimension $n$, i.e., a complete, simply connected, $n-$dimensional Riemmanian manifold having constant sectional curvature $-1$. The aim in this paper is to generalize the main results obtained by the author in [@Nguyen2020a] to the higher order Lorentz–Sobolev spaces in $\H^n$. Before stating our results, let us fix some notation. Let $V_g, \na_g$ and $\Delta_g$ denote the volume element, the hyperbolic gradient and the Laplace–Beltrami operator in $\H^n$ with respect to the metric $g$ respectively. For higher order derivatives, we shall adopt the following convention $$\na_g^m \cdot = \begin{cases} \Delta_g^{\frac m2} \cdot &\mbox{if $m$ is even,}\\ \na_g (\Delta_g^{\frac{m-1}2} \cdot) &\mbox{if $m$ is odd.} \end{cases}$$ Furthermore, for simplicity, we write $|\na^m_g \cdot|$ instead of $|\na_g^m \cdot|_g$ when $m$ is odd if no confusion occurs. For $1\leq p, q\leqs \infty$, we denote by $L^{p,q}(\H^n)$ the Lorentz space in $\H^n$ and by $\|\cdot\|_{p,q}$ the Lorentz quasi-norm in $L^{p,q}(\H^n)$. When $p=q$, $\|\cdot\|_{p,p}$ is replaced by $\|\cdot\|_p$ the Lebesgue $L_p-$norm in $\H^n$, i.e., $\|f\|_p = (\int_{\H^n} |f|^p dV_g)^{\frac1p}$ for a measurable function $f$ on $\H^n$. The Lorentz–Sobolev space $W^m L^{p,q}(\H^n)$ is defined as the completion of $C_0^\infty(\H^n)$ under the Lorentz quasi-norm $\|\na_g^m u\|_{p,q}:=\| |\na_g^m u| \|_{p,q}$. In [@Nguyen2020a; @Nguyen2020b], the author proved the following Poincaré inequality in $W^1 L^{p,q}(\H^n)$ $$\label{eq:Poincare} \|\na_g^m u\|_{p,q}^q \geq C(n,m,p)^q \|u\|_{p,q}^q,\quad\forall\, u\in W^m L^{p,q}(\H^n).$$ provided $1\leqs q \leq p$ if $m$ is odd and for any $1\leqs q \leqs \infty$ if $m$ is even, where $$C(n,m,p) = \begin{cases} (\frac{(n-1)^2}{pp'})^{\frac m2} &\mbox{if $m$ is even,}\\ \frac {n-1}p (\frac{(n-1)^2}{pp'})^{\frac {m-1}2}&\mbox{if $m$ is odd,} \end{cases}$$ with $p' = p/(p-1)$. Furthermore, the constant $C(n,m,p)^q$ in is the best possible and is never attained. The inequality generalizes the result in [@NgoNguyenAMV] to the setting of Lorentz–Sobolev space. The Moser–Trudinger inequality in the hyperbolic spaces was firstly proved by Mancini and Sandeep [@ManciniSandeep2010] in the dimension $n =2$ (another proof of this result was given by Adimurthi and Tintarev [@AdimurthiTintarev2010]) and by Mancini, Sandeep and Tintarev [@ManciniSandeepTintarev2013] in higher dimension $n\geq 3$ (see [@FontanaMorpurgo2020] for an alternative proof) $$\label{eq:MThyperbolic} \sup_{u\in W^{1,n}(\H^n),\, \int_{\H^n} |\na_g u|_g^n dV_g \leq 1} \int_{\H^n} \Phi(\al_n |u|^{\frac n{n-1}}) dV_g < \infty,$$ where $\Phi(t) = e^t -\sum_{j=0}^{n-2} \frac{t^j}{j!}$. Lu and Tang [@LuTang2013] also established the sharp singular Moser–Trudinger inequality under the conditions $\|\na u\|_{L^n(\H^n)}^n + \tau \|u\|_{L^n(\H^n)}^n \leq 1$ for any $\tau >0$ (see Theorem $1.4$ in [@LuTang2013]). In [@NguyenMT2018], the author improves the inequality by proving the following inequality $$\label{eq:NguyenMT} \sup_{u\in W^{1,n}(\H^n),\, \int_{\H^n} |\na_g u|_g^n dV_g - \lam \int_{\H^n} |u|^n dV_g \leq 1} \int_{\H^n} \Phi(\al_n |u|^{\frac n{n-1}}) dV_g < \infty,$$ for any $\lambda < (\frac{n-1}n)^n$. The Adams inequality in the hyperbolic spaces were proved by Karmakar and Sandeep [@Karmakar] in the following form $$\sup_{u\in C_0^\infty(\H^{2n} \int_{\H^{2n}} P_nu \cdot u dV_g \leq 1} \int_{\H^{2n}} \Big(e^{ \al_{2n,n} u^2} -1\Big) dV_g < \infty.$$ where $P_k$ is the GJMS operator on the hyperbolic spaces $\H^{2n}$, i.e., $P_1 = -\Delta_g -n(n-1)$ and $$P_k = P_1(P_1+2)\cdots (P_1 + k(k-1)),\quad k\geq 2.$$ In recent paper, Fontana and Morpurgo [@FM2] established the following Adams inequality in the hyperbolic spaces $\H^n$, $$\label{eq:FM} \sup_{u\in W^{m,\frac nm}(\H^n), \int_{\H^n} |\na_g^m u|^{\frac nm} dV_g \leq 1} \int_{\H^n} \Phi_{\frac nm}(\al_{n,m} |u|^{\frac n{n-m}}) dV_g < \infty$$ where $$\Phi_{\frac nm}(t) = e^{t} -\sum_{j=0}^{j_{\frac nm} -2} \frac{t^j}{j!}, \quad\text{\rm and }\quad j_{\frac nm} = \min\{j\, :\, j \geq \frac nm\} \geq \frac nm.$$ In [@NgoNguyenRMI], Ngo and the author proved several Adams type inequalities in the hyperbolic spaces. To our knowledge, much less is known about the Trudinger–Moser inequality and Adams inequality under the Lorentz–Sobolev norm on complete noncompact Riemannian manifolds except Euclidean spaces. Recently, Yang and Li [@YangLi2019] proves a sharp Moser–Trudinger inequality in the Lorentz–Sobolev spaces defined in the hyperbolic spaces. More precisely, their result ([@YangLi2019 Theorem $1.6$]) states that for $1\leqs q \leqs \infty$ it holds $$\sup_{u\in W^1L^{n,q}(\H^n),\, \|\na_g u\|_{n,q} \leq 1} \int_{\H^n} \Phi_{n,q}(\al_{n,q} |u|^{\frac q{q-1}}) dV_g \leqs \infty,$$ where $$\Phi_{a,q}(t) =e^t - \sum_{j=0}^{j_{a,q} -2} \frac{t^j}{j!},\quad \text{\rm where}\,\, j_{a,q} = \min\{j\in \N\, :\, j \geqs 1+ a(q-1)/q\},$$ with $a \geqs 1$. The first aim in this paper is to establish the sharp Adams inequality in the hyperbolic spaces under the Lorentz–Sobolev norm which generalize the result of Yang and Li to higher order derivatives. Our fist result in this paper reads as follows. \[MAINI\] Let $n\geqs m \geq 2$ and $q \in (1,\infty)$. Then it holds $$\label{eq:AdamsLorentz} \sup_{u\in W^mL^{\frac nm,q}(\H^n),\, \|\na_g^m u\|_{\frac nm,q}\leq 1} \int_{\H^n} \Phi_{\frac nm,q}\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g \leqs \infty,$$ for any $q \in (1,\infty)$ if $m$ is even, or $1\leqs q \leq \frac nm$ if $m$ is odd. Futhermore, the constant $\beta_{n,m}^{\frac q{q-1}}$ is sharp in the sense that the supremum in will become infinite if $\beta_{n,m}^{\frac q{q-1}}$ is replaced by any larger constant. Let us make some comments on Theorem \[MAINI\]. When $q =\frac nm$, we obtain the inequality of Fontana and Morpurgo from Theorem \[MAINI\]. However, our approach is completely different with the one of Fontana and Morpurgo. Notice that in the case that $m$ is odd, we need an extra assumption Notice $q \leq \frac nm$ comparing with case that $m$ is even. This extra condition is a technical condition in our approach for which we can apply the Pólya–Szegö principle in the hyperbolic space (see Theorem \[PS\] below). This principle was proved by the author in [@Nguyen2020a] which generalizes the classical Pólya–Szegö principle in Euclidean space to the hyperbolic space. Note that when $m=1$, the extra condition is not need by the result of Yang and Li [@YangLi2019]. The approach of Yang and Li is based on an representation formula for function via Green’s function of the Laplace-Beltrami $-\Delta_g$ (similar with the one of Fontana and Morpurgo [@FM2]). Hence, we believe that the extra condition $q \leq \frac nm$ is superfluous when $m > 1$ is odd. One reasonable approach is to follow the one of Fontana and Morpurgo by using the representation formulas and estimates in [@FM2 Section $5$]. This problem is left for interesting reader. Next, we aim to improve the Lorentz–Adams inequality in Theorem \[MAINI\] in spirit of . In the case $m=1$, an analogue of under Lorentz–Sobolev norm was obtained by the author in [@Nguyen2020a Theorem $1.3$]. The result for $m > 1$ is given in the following theorem. \[MAINII\] Let $n > m\geq 2$ and $q \geq \frac{2n}{n-1}$. Suppose in addition that $q \leq \frac nm$ if $m$ is odd. Then we have $$\label{eq:improvedAL} \sup_{u\in W^mL^{\frac nm,q}(\H^n),\, \|\na_g^m u\|_{\frac nm,q}^q -\lam \|u\|_{\frac nm,q}^q \leq 1} \int_{\H^n} \Phi_{\frac nm,q}\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g \leqs \infty.$$ for any $ \lam \leqs C(n,m,\frac nm)^q$. Obviously, Theorem \[MAINII\] is stronger than Theorem \[MAINI\]. The extra condition $q \geq \frac{2n}{n-1}$ in Theorem \[MAINII\] is to apply a crucial point-wise estimate in [@NguyenPS2018 Lemma $2.1$]. Theorem \[MAINII\] is proved by using iteration method and some estimates in [@Nguyen2020b] which we will recall in Section §2 below. The Hardy–Moser–Trudinger inequality was proved by Wang and Ye (see [@WangYe2012]) in dimension $2$ $$\label{eq:WangYe} \sup_{u \in W^{1,2}_0(\B^2), \int_{\B^2} |\na u|^2 dx - \int_{\B^2} \frac{u^2}{(1-|x|^2)^2} dx \leq 1} \int_{\B^2} e^{4\pi u^2} dx < \infty.$$ The inequality is stronger than the classical Moser–Trudinger inequality in $\B^2$. It connects both the sharp Moser–Trudinger inequality in $\B^2$ and the sharp Hardy inequality in $\B^2$ $$\int_{\B^2} |\na u|^2 dx \geq \int_{\B^2} \frac{u^2}{(1-|x|^2)^2} dx, \quad u \in W^{1,2}_0(\B^2).$$ The higher dimensional version of was recently established by the author [@NguyenHMT] $$\sup_{u \in W^{1,n}_0(\B^n), \int_{\B^n} |\na u|^n dx - \lt(\frac{2(n-1)}n\rt)^n\int_{\B^n} \frac{|u|^n}{(1-|x|^2)^n} dx \leq 1} \int_{\B^2} e^{\al_n |u|^{\frac n{n-1}}} dx < \infty.$$ For higher order derivatives, the sharp Hardy–Adams inequality was proved by Lu and Yang [@LuYangHA] in dimension $4$ and by Li, Lu and Yang [@LiLuYang] in any even dimension. The approach in [@LuYangHA; @LiLuYang] relies heavily on the Hilbertian structure of the space $W^{\frac n2,2}_0(\B^n)$ with $n$ even for which the Fourier analysis in the hyperbolic spaces can be applied. Our next motivation in this paper is to establish the sharp Hardy–Adams inequality in any dimension. Our next result reads as follows. \[HARDYADAMS\] Let $m \geq 3$, $n \geq 2m+1$ and $q \geq \frac{2n}{n-1}$. Suppose in addition that $q \leq \frac nm$ if $m$ is odd. Then it holds $$\label{eq:HAineq} \sup_{u\in W^mL^{\frac nm,q}(\H^n),\, \|\na_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q \leq 1} \int_{\B^n} \exp\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dx \leqs \infty.$$ Notice that the condition $m \geq 3$ is crucial in our approach. Indeed, under this condition we can make some estimates for $\|\na_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q$ for which we can apply the results from Theorem \[MAINI\] and Theorem \[MAINII\]. We do not know an analogue of when $m=2$. When $q = \frac nm$, we obtain the following Hardy–Adams inequality $$\sup_{u\in W^{m,\frac nm}_0(\H^n),\, \int_{\H^n} |\na_g^m u|^{\frac nm} dV_g -C(n,m,\frac nm)^{\frac nm} \int_{\H^n} |u|^{\frac nm} dV_g \leq 1} \int_{\B^n} \exp\big(\al_{n,m} |u|^{\frac n{n-m}}\big) dx \leqs \infty.$$ The rest of this paper is organized as follows. In Section §2, we recall some facts on the hyperbolic spaces, the non-increasing rearrangement argument in the hyperbolic spaces and some important results from [@Nguyen2020b] which are used in the proof of Theorem \[MAINII\] and Theorem \[HARDYADAMS\]. The proof of Theorem \[MAINI\] is given in Section §3. Section §4 is devoted to prove Theorem \[MAINII\]. Finally, in Section §5 we provide the proof of Theorem \[HARDYADAMS\]. Preliminaries ============= We start this section by briefly recalling some basis facts on the hyperbolic spaces and the Lorentz–Sobolev space defined in the hyperbolic spaces. Let $n\geq 2$, a hyperbolic space of dimension $n$ (denoted by $\H^n$) is a complete , simply connected Riemannian manifold having constant sectional curvature $-1$. There are several models for the hyperbolic space $\H^n$ such as the half-space model, the hyperboloid (or Lorentz) model and the Poincaré ball model. Notice that all these models are Riemannian isometry. In this paper, we are interested in the Poincaré ball model of the hyperbolic space since this model is very useful for questions involving rotational symmetry. In the Poincaré ball model, the hyperbolic space $\H^n$ is the open unit ball $B_n\subset \R^n$ equipped with the Riemannian metric $$g(x) = \Big(\frac2{1- |x|^2}\Big)^2 dx \otimes dx.$$ The volume element of $\H^n$ with respect to the metric $g$ is given by $$dV_g(x) = \Big(\frac 2{1 -|x|^2}\Big)^n dx,$$ where $dx$ is the usual Lebesgue measure in $\R^n$. For $x \in B_n$, let $d(0,x)$ denote the geodesic distance between $x$ and the origin, then we have $d(0,x) = \ln (1+|x|)/(1 -|x|)$. For $\rho \geqs 0$, $B(0,\rho)$ denote the geodesic ball with center at origin and radius $\rho$. If we denote by $\na$ and $\Delta$ the Euclidean gradient and Euclidean Laplacian, respectively as well as $\la \cdot, \cdot\ra$ the standard scalar product in $\R^n$, then the hyperbolic gradient $\na_g$ and the Laplace–Beltrami operator $\Delta_g$ in $\H^n$ with respect to metric $g$ are given by $$\na_g = \Big(\frac{1 -|x|^2}2\Big)^2 \na,\quad \Delta_g = \Big(\frac{1 -|x|^2}2\Big)^2 \Delta + (n-2) \Big(\frac{1 -|x|^2}2\Big)\la x, \na \ra,$$ respectively. For a function $u$, we shall denote $\sqrt{g(\na_g u, \na_g u)}$ by $|\na_g u|_g$ for simplifying the notation. Finally, for a radial function $u$ (i.e., the function depends only on $d(0,x)$) we have the following polar coordinate formula $$\int_{\H^n} u(x) dx = n \sigma_n \int_0^\infty u(\rho) \sinh^{n-1}(\rho)\, d\rho.$$ It is now known that the symmetrization argument works well in the setting of the hyperbolic. It is the key tool in the proof of several important inequalities such as the Poincaré inequality, the Sobolev inequality, the Moser–Trudinger inequality in $\H^n$. We shall see that this argument is also the key tool to establish the main results in the present paper. Let us recall some facts about the rearrangement argument in the hyperbolic space $\H^n$. A measurable function $u:\H^n \to \R$ is called vanishing at the infinity if for any $t >0$ the set $\{|u| > t\}$ has finite $V_g-$measure, i.e., $$V_g(\{|u|> t\}) = \int_{\{|u|> t\}} dV_g < \infty.$$ For such a function $u$, its distribution function is defined by $$\mu_u(t) = V_g( \{|u|> t\}).$$ Notice that $t \to \mu_u(t)$ is non-increasing and right-continuous. The non-increasing rearrangement function $u^*$ of $u$ is defined by $$u^*(t) = \sup\{s > 0\, :\, \mu_u(s) > t\}.$$ The non-increasing, spherical symmetry, rearrangement function $u^\sharp$ of $u$ is defined by $$u^\sharp(x) = u^*(V_g(B(0,d(0,x)))),\quad x \in \H^n.$$ It is well-known that $u$ and $u^\sharp$ have the same non-increasing rearrangement function (which is $u^*$). Finally, the maximal function $u^{**}$ of $u^*$ is defined by $$u^{**}(t) = \frac1t \int_0^t u^*(s) ds.$$ Evidently, $u^*(t) \leq u^{**}(t)$. For $1\leq p, q < \infty$, the Lorentz space $L^{p,q}(\H^n)$ is defined as the set of all measurable function $u: \H^n \to \R$ satisfying $$\|u\|_{L^{p,q}(\H^n)}: = \lt(\int_0^\infty \lt(t^{\frac1p} u^*(t)\rt)^q \frac{dt}t\rt)^{\frac1q} < \infty.$$ It is clear that $L^{p,p}(\H^n) = L^p(\H^n)$. Moreover, the Lorentz spaces are monotone with respect to second exponent, namely $$L^{p,q_1}(\H^n) \subsetneq L^{p,q_2}(\H^n),\quad 1\leq q_1 < q_2 < \infty.$$ The functional $ u\to \|u\|_{L^{p,q}(\H^n)}$ is not a norm in $L^{p,q}(\H^n)$ except the case $q \leq p$ (see [@Bennett Chapter $4$, Theorem $4.3$]). In general, it is a quasi-norm which turns out to be equivalent to the norm obtained replacing $u^*$ by its maximal function $u^{**}$ in the definition of $\|\cdot\|_{L^{p,q}(\H^n)}$. Moreover, as a consequence of Hardy inequality, we have Given $p\in (1,\infty)$ and $q \in [1,\infty)$. Then for any function $u \in L^{p,q}(\H^n)$ it holds $$\label{eq:Hardy} \lt(\int_0^\infty \lt(t^{\frac1p} u^{**}(t)\rt)^q \frac{dt}t\rt)^{\frac1q} \leq \frac p{p-1} \lt(\int_0^\infty \lt(t^{\frac1p} u^*(t)\rt)^q \frac{dt}t\rt)^{\frac1q} = \frac p{p-1} \|u\|_{L^{p,q}(\H^n)}.$$ For $1\leq p, q \leqs \infty$ and an integer $m\geq 1$, we define the $m-$th order Lorentz–Sobolev space $W^mL^{p,q}(\H^n)$ by taking the completion of $C_0^\infty(\H^n)$ under the quasi-norm $$\|\na_g^m u\|_{p,q} := \| |\na_g^m u|\|_{p,q}.$$ It is obvious that $W^mL^{p,p}(\H^n) = W^{m,p}(\H^n)$ the $m-$th order Sobolev space in $\H^n$. In [@Nguyen2020a], the author established the following Pólya–Szegö principle in the first order Lorenz–Sobolev spaces $W^1L^{p,q}(\H^n)$ which generalizes the classical Pólya–Szegö principle in the hyperbolic space. \[PS\] Let $n\geq 2$, $1\leq q \leq p \leqs \infty$ and $u\in W^{1}L^{p,q}(\H^n)$. Then $u^\sharp \in W^{1}L^{p,q}(\H^n)$ and $$\|\na_g u^\sharp\|_{p,q} \leq \|\na_g u\|_{p,q}.$$ For $r \geq 0$, define $$\Phi(r) = n \int_0^r \sinh^{n-1}(s) ds, \quad r\geq 0,$$ and let $F$ be the function such that $$r = n \si_n \int_0^{F(r)} \sinh^{n-1}(s) ds, \quad r\geq 0,$$ i.e., $F(r) = \Phi^{-1}(r/\si_n)$. The following results was proved in [@Nguyen2020b] (see the Section §2). Let $n \geq 2$. Then it holds $$\label{eq:keyyeu} \sinh^{n}(F(t)) \geqs \frac t{\si_n},\quad t\geqs 0.$$ Furthermore, the function $$\vphi(t) =\frac{t}{\sinh^{n-1}(F(t))}$$ is strictly increasing on $(0,\infty)$, and $$\label{eq:keyyeu*} \lim_{t\to \infty} \varphi(t) = \frac{n \si_n}{n-1} > \frac{t}{\sinh^{n-1}(F(t))},\quad t >0.$$ It should be remark that under an extra condition $q \geq \frac{2n}{n-1}$, a stronger estimate which combines both and was established by the author in [@Nguyen2020a Lemma $2.1$] that $$\sinh^{q(n-1)}(F(t)) \geq \lt(\frac t{\si_n}\rt)^{q \frac{n-1}n} + \lt(\frac{n-1}n\rt)^q \lt(\frac t{\si_n}\rt)^q,\quad t \geqs 0.$$ Let $u \in C_0^\infty(\H^n)$ and $f = -\Delta_g u$. It was proved by Ngo and the author (see [@NgoNguyenAMV Proposition $2.2$]) that $$\label{eq:NgoNguyen} u^*(t) \leq v(t):= \int_t^\infty \frac{s f^{**}(s)}{(n \si_n \sinh^{n-1}(F(s)))^2} ds,\quad t\geqs 0.$$ The following results which were proved in [@Nguyen2020b; @Nguyen2020a] play the important role in the proof of our main results, \[L1\] Let $p\in (1,n)$ and $\frac{2n}{n-1} \leq q \leq p$. Then we have $$\label{eq:improvedLS1a} \|\na_g u\|_{p,q}^q - \lt(\frac{n-1}p\rt)^q \|u\|_{p,q}^q \geq \lt(\frac{n-p}p \si_n^{\frac1n}\rt)^q \|u\|_{p^*,q}^q,\quad u\in C_0^\infty(\H^n)$$ where $p' = p/(p-1)$, and \[L2\] Let $n\geq 2$, $p \in (1,n)$ and $q \in (1,\infty)$. If $p \in (1,\frac n2)$ then it holds $$\label{eq:LSorder2} \|\Delta_g u\|_{p,q}^q \geq \lt(\frac{n(n-2p)}{p p'} \si_n^{\frac 2n}\rt)^q \|u\|_{p_2^*,q}^q.$$ If $p\in (1,n)$ and $q \geq \frac{2n}{n-1}$ then we have $$\label{eq:improvedLS2} \|\Delta_g u\|_{p,q}^q - C(n,2,p)^q \|u\|_{p,q}^q \geq \lt(\frac{n^2 \si_n^{\frac2n}}{p'}\rt)^q \int_0^\infty |v'(t)|^q t^{q(\frac1p -\frac2n) + q -1} dt.$$ Furthermore, if $p\in (1,\frac n2)$ and $q \geq \frac{2n}{n-1}$ and $\frac{2n}{n-1} \leq q \leq p$ then we have $$\label{eq:improvedLS2a} \|\Delta_g u\|_{p,q}^q - C(n,2,p)^q \|u\|_{p,q}^q \geq \lt(\frac{n(n-2p)}{p p'} \si_n^{\frac 2n}\rt)^q \|u\|_{p_2^*,q}^q,\quad u \in C_0^\infty(\H^n).$$ Proposition \[L1\] follows from [@Nguyen2020a Theorem $1.2$] while Proposition \[L2\] follows from Theorem $2.8$ in [@Nguyen2020b]. Proof of Theorem \[MAINI\] ========================== In this section, we prove Theorem \[MAINI\]. The main point is the proof of the case $m=2$. For the case $m\geq 3$, the proof is based on the iteration argument by using the inequalities and below. We divide the proof of into three following cases:\ *Case 1: $m =2$.* It is enough to consider $u \in C_0^\infty(\H^n)$ with $\|\Delta_g u\|_{\frac n2,q} \leq 1$. Denote $f = -\Delta_g u$ and define $v$ by , then we have $u^* \leq v$. By [@Nguyen2020b Theorem $1.1$] , we have $\|u\|_{\frac n2,q}^q \leq C$. Here and in the sequel, we denote by $C$ a generic constant which does not depend on $u$ and whose value maybe changes on each line. For any $t\geqs 0$, we have $$\frac n{2q} u^*(t)^q t^{\frac{2q}n} \leq \int_0^t u^*(s)^q s^{\frac{2q}n-1} ds \leq \|u\|_{p,q}^q \leq C,$$ which yields $u^*(t) \leq C t^{-\frac 2n}$, $t\geqs 0$. Therefore, it is not hard to see that $$\Phi_{\frac n2,q}(\beta_{n,2}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) \leq C u^*(t)^{\frac{q}{q-1} (j_{\frac n2,q} -1)} \leq C t^{-\frac 2n \frac{q}{q-1} (j_{\frac n2,q} -1)},\quad \forall\, t\geq 1.$$ By the choice of $j_{\frac n2,q}$, we then have $$\label{eq:tach12} \int_1^\infty \Phi_{\frac n2,q}(\beta_{n,2}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt \leq C.$$ On the other hand, we have $$\begin{aligned} \label{eq:on01} \int_0^1 \Phi_{\frac n2,q}(\beta_{n,2}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt &\leq \int_0^1 \exp\Big(\beta_{n,2}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}\Big) dt \notag\\ &\leq \int_0^1 \exp\Big(\beta_{n,2}^{\frac q{q-1}} v(t)^{\frac q{q-1}}\Big) dt \notag\\ &= \int_0^\infty \exp\Big(-t + \beta_{n,2}^{\frac q{q-1}} v(e^{-t})^{\frac q{q-1}}) dt.\end{aligned}$$ Notice that $$v(e^{-t}) =\int_{e^{-t}}^\infty \frac{r}{(n\si_n \sinh^{n-1}(F(r)))^2} f^{**}(r) dr = \int_{-\infty}^t \frac{e^{-2(1-\frac1n)s}}{(n\si_n \sinh^{n-1}(F(e^{-s})))^2} e^{-\frac{2}ns}f^{**}(e^{-s}) ds.$$ Denote $$\phi(s) = \frac{n-2}{n} e^{-\frac 2n s} f^{**}(e^{-s}),$$ we then have $$\label{eq:boundnormorder2} \int_{\R}\phi(s)^q ds = \lt(\frac{n-2}n\rt)^q \int_0^\infty (f^{**}(t) t^{\frac2n})^q \frac{dt}t \leq 1,$$ here we used the Hardy inequality and $\|\Delta_g u\|_{L^{\frac n2,q}(\H^n)} \leq 1$. Define the function $$a(s,t) = \begin{cases} \be_{n,2} \frac{n}{n-2} \frac{e^{-2(1-\frac1n)s}}{(n\si_n \sinh^{n-1}(F(e^{-s})))^2} &\mbox{if $s \leq t$,}\\ 0&\mbox{if $s > t$.} \end{cases}$$ Using the inequality $\si_n \sinh^n(F(r)) \geq r$, we have for $0 \leq s \leq t$ $$\label{eq:boundby1order2} a(s,t) \leq \be_{n,2} \frac1{n(n-2) \si_n^{\frac2n}} = 1.$$ Moreover, for $t >0$ we have $$\begin{aligned} \int_{-\infty}^0 a(s,t)^{q'} ds + \int_t^\infty a(s,t)^{q'} ds& = \be_{n,2}^{q'} \lt(\frac n{n-2}\rt)^{q'} \int_{-\infty}^0 \lt(\frac{e^{-2(1-\frac1n)s}}{(n\si_n \sinh^{n-1}(F(e^{-s})))^2}\rt)^{q'} ds\\ &\leq \be_{n,2}^{q'} \lt(\frac n{n-2}\rt)^{q'} (n-1)^{-2q'} \int_{-\infty}^0 e^{\frac2n q'} ds\\ &= \be_{n,2}^{q'} \lt(\frac n{n-2}\rt)^{q'} (n-1)^{-2q'} \frac{n}{2q'},\end{aligned}$$ here we used $n\sigma_n \sinh^{n-1}(F(r)) \geq (n-1) r$. Hence $$\label{eq:dk2Adamsorder2} \sup_{t >0} \lt(\int_{-\infty}^0 a(s,t)^{q'} ds + \int_t^\infty a(s,t)^{q'} ds\rt)^{\frac1{q'}} \leq \lt(\beta_{n,2}^{q'} \lt(\frac n{n-2}\rt)^{q'} (n-1)^{-2q'} \frac{n}{2q'}\rt)^{\frac1{q'}}.$$ Notice that $$\label{eq:majoru*2} \be_{n,2}v(e^{-t}) \leq \int_{\R} a(s,t) \phi(s) ds.$$ With , , , and at hand, we can apply Adams’ Lemma [@Adams] to obtain $$\label{eq:tporder22} \int_0^1 \Phi_{\frac n2,q}(\beta_{n,2}^{q'} u^*(t)^{\frac q{q-1}}) dt \leq \int_0^\infty e^{-t + \beta_{n,2}^{q'} v(t)^{q'}} dt \leq C.$$ Combining and together, we arrive $$\int_{\R^n} \Phi_{\frac n2,q}(\be_{n,2}^{q'} |u|^{q'}) dx = \int_0^\infty \Phi_{\frac n2,q}(\be_{n,2}^{q'} (u^*(t))^{q'}) dt \leq C,$$ for any $u \in W^2L^{\frac n2,q}(\H^n)$ with $\|\Delta_g u\|_{L^{\frac n2,q}(\H^n)} \leq 1$. This proves for $m =2$.\ *Case 2: $m =2k$, $k\geq 2$.* To obtain the result in this case, we apply the iteration argument. Firstly, by iterating the inequality , we have that for $k\geq 1$, $q \in (1,\infty)$ and $p \in (1,\frac n{2k})$ $$\|\Delta_g^k u\|_{p,q}^q \geq S(n,2k,p)^q \|u\|_{p_{2k}^*,q}^q.$$ Hence, if $u \in W^{2k} L^{\frac n{2k},q}(\H^n)$ with $\|\Delta_g^k u\|_{\frac n{2k},q} \leq 1$, then we have $$S(n,2(k-1),\frac n{2k}) \|\Delta_g u\|_{\frac n2,q} \leq 1.$$ Define $w = S(n,2(k-1),\frac n{2k}) u$, then $\|w\|_{\frac n2,q} \leq 1$. Using the result in the *Case 1* with remark that $$\beta_{n,2k} = \beta_{n,2} S(n,2(k-1),\frac n{2k}),$$ we obtain $$\label{eq:Case1} \int_{\H^n} \Phi_{\frac n2,q}(\beta_{n,2k}^{q'} |u|^{q'}) dV_g \leq C.$$ By the Lorentz–Poincaré inequality , we have $\|u\|_{\frac n{2k},q}^q \leq C$. Similarly in the *Case 1*, we get $u^*(t) \leq C t^{-\frac{2k}n}$, $t\geqs 0$. Hence, for $t\geq 1$, it holds $$\Phi_{\frac n{2k},q}(\beta_{n,2k}^{q'} u^*(t)^{q'}) \leq C (u^*(t))^{q'(j_{\frac n{2k},q} -1)} \leq C t^{-\frac{2k}n q'(j_{\frac n{2k},q} -1)},$$ which implies $$\label{eq:tach12k} \int_1^\infty \Phi_{\frac n{2k} ,q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt \leq C$$ by the choice of $j_{\frac n{2k},q}$. Since $$\lim_{t\to \infty} \frac{\Phi_{\frac n{2k},q}(t)}{\Phi_{\frac n2,q}(t)} = 1,$$ then there exists $A$ such that $\Phi_{\frac n{2k},q}(t) \leq 2 \Phi_{\frac n2,q}(t)$ for $t \geq A$. Hence, we have $$\begin{aligned} \int_0^1 \Phi_{\frac n{2k},q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt & = \int_{\{t\in (0,1):u^*(t) \leqs A^{1/q'} \beta_{n,2k}^{-1}\}} \Phi_{\frac n{2k},q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt\\ &\quad + \int_{\{t\in (0,1):u^*(t) \geq A^{1/q'} \beta_{n,2k}^{-1}\}} \Phi_{\frac n{2k},q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt\\ &\leq C + 2\int_{\{t\in (0,1):u^*(t) \geq A^{1/q'} \beta_{n,2k}^{-1}\}} \Phi_{\frac n2,q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt\\ &\leq C + \int_0^1 \Phi_{\frac n2,q}(\beta_{n,2k}^{\frac q{q-1}} u^*(t)^{\frac q{q-1}}) dt\\ &\leq C\end{aligned}$$ here we have used . Combining the previous inequality together with proves the result in this case.\ *Case 3: $m =2k+1$, $k\geq 1$.* Let $f = -\Delta_g^{k} u$. Since $q \leq \frac n{2k+1}$, then it was proved in [@Nguyen2020a] (the formula after $(2.8)$ with $u$ replaced by $f$) that $$\|\na_g^m u\|_{\frac n{2k+1},q}^q = \|\na_g f\|_{\frac n{2k+1},q}^q \geq \int_0^\infty |(f^*)'(t)|^q (n \si_n \sinh^{n-1} (F(t)))^q t^{\frac{(2k+1) q}n -1} dt.$$ Using , we have $$\|\na_g^m u\|_{\frac n{2k+1},q}^q \geq n^q \si_n^{\frac qn} \int_0^\infty |(f^*)'(t)|^q t^{\frac{2kq}n + q -1} dt.$$ Applying the one-dimensional Hardy inequality, it holds $$\label{eq:Sob} \|\na_g^m u\|_{\frac n{2k+1},q}^q \geq (2k)^q \si_n^{\frac qn}\int_0^\infty |f^*(t)|^q t^{\frac{2kq}n -1} dt = (2k)^q \si_n^q \|\Delta_g^k u\|_{\frac n{2k},q}^q.$$ For any $u \in W^{2k+1} L^{\frac n{2k+1},q}(\H^n)$ with $\|\na_g^m u\|_{\frac n{2k+1},q} \leq 1$, define $w= 2k \si_n^{\frac1n} u$. By , we have $\|w\|_{\frac n{2k},q}^q \leq 1$. Using the result in the *Case 2* with remark that $$\beta_{n,2k+1} =2k \si_n^{\frac 1n} \beta_{n,2k},$$ we obtain $$\label{eq:Case2} \int_{\H^n} \Phi_{\frac n{2k},q} (\beta_{n,2k+1}^{q'} |u|^{q'}) dV_g \leq C.$$ Using together with the last arguments in the proof of the *Case 2* proves the result in this case.\ It remains to check the sharpness of constant $\beta_{n,m}^{\frac q{q-1}}$. To do this, we construct a sequence of test functions as follows $$v_j(x) = \begin{cases} \frac{(\ln j)^{1/q'}}{\beta_{n,m}} + \frac{n\beta_{n,m}}{2(\ln j)^{1/q}} \sum_{i=1}^{m-1} \frac{(1-j^{\frac 2n}|x|^2)^i}{i} &\mbox{if $0\leq |x| \leq j^{-\frac 1n}$,}\\ -\frac n{\beta_{n,m}} (\ln j)^{-1/q} \ln |x|&\mbox{if $j^{-\frac1n} \leqs |x| \leq 1$,}\\ \xi_j(x) &\mbox{if $1 \leqs |x| \leqs 2$}, \end{cases} \quad j\geq 2$$ where $\xi \in C_0^\infty(2\B^n)$ are radial function chosen such that $\xi_j = 0$ on $\pa \B^n$ and for $i=1,\ldots,m-1$ $$\frac{\pa^i \xi_j}{\pa r^i} \Big{|}_{\pa \B^n} = (-1)^i (i-1)! n\beta_{n,m}^{-1} (\ln j)^{-1/q},$$ and $\xi_j$, $|\na^l \xi_j|$ and $|\na^m \xi_j|$ are all $O((\ln j)^{-1/q})$ as $j\to \infty$. For $\ep \in (0,1/3)$ let us define $u_{\ep,j}(x) =v_j(x/\ep)$. Then $u_{\ep,j} \in W^m L^{\frac nm,q}(\H^n)$ has support contained in $\{|x| \leq 2\ep\}$. It is easy to check that $$|\na_g^m u_{\ep,j}(x)| \leq \lt(\frac{1-|x|^2}2\rt)^m C (\ep^{-1} j^{\frac1n})^m (\ln j)^{-1/q}\leq C2^{-m}(\ep^{-1} j^{\frac1n})^m (\ln j)^{-1/q}$$ for $|x| \leq \ep j^{-\frac1n}$, and $$|\na_g^m u_{\ep,j}(x)| \leq C \ep^{-m} (\ln j)^{-\frac1q}\lt(\frac{1-|x|^2}2\rt)^m \leq C2^{-m}\ep^{-m} (\ln j)^{-\frac1q}$$ for $|x|\in (\ep, 2\ep)$ with a positive constant $C$ independent of $\ep\leqs \frac13$ and $j$. Furthermore, we can check that $$\begin{aligned} |\na^m_g u_{\ep,j}(x)|&\leq \lt(\frac{1-|x|^2}2\rt)^m\lt((|x|^n \si_n)^{-\frac mn} + C |x|^{-m+1}\rt) (\ln j)^{-\frac1q}\\ & \leq 2^{-m}(\ln j)^{-\frac1q} \lt((|x|^n \si_n)^{-\frac mn} + C |x|^{-m+1}\rt)\end{aligned}$$ and $$\begin{aligned} |\na^m_g u_{\ep,j}(x)|&\geq \lt(\frac{1-|x|^2}2\rt)^m\lt((|x|^n \si_n)^{-\frac mn} - C |x|^{-m+1}\rt) (\ln j)^{-\frac1q}\\ & \geq \lt(\frac{1-\ep^2}2\rt)^{-m}(\ln j)^{-\frac1q} \lt((|x|^n \si_n)^{-\frac mn} - C |x|^{-m+1}\rt)\end{aligned}$$ for $|x| \in (\ep j^{-\frac1n}, \ep )$ with $\ep \geqs 0$ small enough where $C$ is a positive constant independent of $\ep$ and $j$. Define $$h_1(x) = \begin{cases} C2^{-m}(\ep^{-1} j^{\frac1n})^m (\ln j)^{-1/q}&\mbox{if $|x| \leq \ep j^{-\frac1n}$}\\ 2^{-m}(\ln j)^{-\frac1q} \lt((|x|^n \si_n)^{-\frac mn} + C |x|^{-m+1}\rt)&\mbox{if $|x| \in (\ep j^{-\frac1n}, \ep )$}\\ C2^{-m}\ep^{-m} (\ln j)^{-\frac1q}&\mbox{if $|x|\in (\ep, 2\ep)$}\\ 0&\mbox{if $|x| \in (2\ep,1)$}, \end{cases}$$ Then we have $0 \leq |\na_g^m u| \leq h_1$. Consequently, we get $0 \leq |\na^m_g u|^* \leq h_1^*$. Let us denote by $h_1^{*,e}$ the rearrangement function of $h_1$ with respect to Lebesgue measure. Since the support of $h_1$ is contained in $\ep \{|x| \leq \ep\}$, then we can easy check that $$h_1^*(t) \leq h_1^{*,e}\lt(\Big(\frac{1-\ep^2}2\Big)^n t\rt).$$ Consequently, we have $$\|\na_g^m u_{\ep,j}\|_{\frac nm,q}^q \leq \lt(\frac2{1-\ep^2}\rt)^{mq} \int_0^\infty h_1^{*,e}(t)^q t^{\frac{mq}n -1} dt$$ Notice that by enlarging the constant $C$ (which is still independent of $\ep$ and $j$), we can assume that $$C2^{-m}\ep^{-m} (\ln j)^{-\frac1q} \geq h_1\Big |_{\{|x| = \ep\}} =2^{-m}(\ln j)^{-\frac1q} \ep^{-m}\lt(\si_n^{-\frac mn} + C \ep \rt)$$ for $\ep \geqs 0$ small enough. For $j$ larger enough, we can chose $x_0$ with $\ep j^{-\frac1n} \leqs |x_0| \leq \ep$ such that $C2^{-m}\ep^{-m} (\ln j)^{-\frac1q} = h_1(x_0)$. It is easy to see that $c\ep \leq |x_0| \leq C \ep$ for constant $C, c \geqs 0$ independent of $\ep$ and $j$. We have $$h_1(x) \leq g(x) :=\begin{cases} h_1(x)&\mbox{if $|x| \leq |x_0| $}\\ C2^{-m}\ep^{-m} (\ln j)^{-\frac1q}&\mbox{if $|x|\in (|x_0|, 2\ep)$}\\ 0&\mbox{if $|x| \geq 2\ep$}. \end{cases}$$ Notice that $g$ is non-increasing radially symmetric function in $\B^n$, hence $g^{\sharp,e} = g$. Using the function $g$, we can prove that $$\int_0^\infty h_1^{*,e}(t)^q t^{\frac{mq}n -1} dt \leq 2^{-mq}(1 + C (\ln j)^{-1}).$$ Therefore, we have $$\|\na_g^m u_{\ep,j}\|_{\frac nm,q}^q \leq \lt(\frac1{1-\ep^2}\rt)^{mq}(1 + C (\ln j)^{-1})$$ Set $w_{\ep,j} = u_{\ep,j}/\|\na_g^m u_{\ep,j}\|_{\frac nm,q}$. For any $\beta \geqs \beta_{n,m}^{q'}$, we choose $\ep \geqs 0$ small enough such that $\gamma := \beta (1-\ep^2)^{\frac{mq}{q-1}}\geqs \be_{n,m}^{q'}$. Then we have $$\begin{aligned} \int_{\H^n} \Phi_{\frac nm,q}(\beta |w_{\ep,j}|^{q'}) dV_g &\geq \int_{\{|x|\leq \ep j^{-\frac1n}\}} \Phi_{\frac nm,q}\Big(\frac{\beta}{\|\na_g^m u_{\ep,j}\|_{\frac nm,q}^{q'}} |u_{\ep,j}|^{q'}\Big) dV_g\\ &\int_{\{|x|\leq \ep j^{-\frac1n}\}} \Phi_{\frac nm,q}\Big(\frac{\ga}{(1+ C (\ln j)^{-1})^{q'}} |u_{\ep,j}|^{q'}\Big) dV_g\\ &\geq 2^n \int_{\{|x|\leq \ep j^{-\frac1n}\}} \Phi_{\frac nm,q}\Big(\frac{\ga}{(1+ C (\ln j)^{-1})^{q'}} |u_{\ep,j}|^{q'}\Big) dx\\ &= 2^n \ep^{n}\int_{\{|x|\leq j^{-\frac1n}\}} \Phi_{\frac nm,q}\Big(\frac{\ga}{(1+ C (\ln j)^{-1})^{q'}} |v_{j}|^{q'}\Big) dx\\ &\geq 2^n \ep^{n}\int_{\{|x|\leq j^{-\frac1n}\}} \Phi_{\frac nm,q}\Big(\frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}} \Big) dx\\ &=2^n \ep^{n}\si_n \Phi_{\frac nm,q}\Big(\frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}} \Big) e^{-\ln j}.\end{aligned}$$ Since $$\lim_{j\to \infty} \frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}} = \infty,$$ then $$\Phi_{\frac nm,q}\Big(\frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}} \Big) \geq C e^{\frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}}}$$ for $j$ larger enough. Consequently, we get $$\int_{\H^n} \Phi_{\frac nm,q}(\beta |w_{\ep,j}|^{q'}) dV_g \geq 2^n \ep^n \si_n C e^{\frac{\ga}{\beta_{n,m}^{q'}}\frac{\ln j}{(1+ C (\ln j)^{-1})^{q'}} -\ln j} \to \infty$$ as $j\to \infty$ since $\ga \geqs \beta_{n,m}^{q'}$. This proves the sharpness of $\beta_{n,m}^{q'}$. The proof of Theorem \[MAINI\] is then completely finished. Proof of Theorem \[MAINII\] =========================== This section is devoted to prove Theorem \[MAINII\]. The proof is based on the inequalities and , the iteration argument and Theorem \[MAINII\] for $m\geq 3$. The case $m=2$ is proved by using inequality and the Moser–Trudinger inequality involving to the fractional dimension in Lemma \[MT\] below. Let $\theta \geqs 1$, we denote by $\lam_\theta$ the measure on $[0,\infty)$ of density $$d\lam_{\theta} = \theta \sigma_{\theta} x^{\theta -1} dx, \quad \sigma_{\theta} = \frac{ \pi^{\frac\theta 2}}{\Gamma(\frac\theta 2+ 1)}.$$ For $0\leqs R \leq \infty$ and $1\leq p \leqs \infty$, we denote by $L_\theta^p(0,R)$ the weighted Lebesgue space of all measurable functions $u: (0,R) \to \R$ for which $$\|u\|_{L^p_\theta(0,R)}= \lt(\int_0^R |u|^p d\lam_\theta\rt)^{\frac1p} \leqs \infty.$$ Besides, we define $$W^{1,p}_{\al,\theta}(0,R) =\Big\{u\in L^p_\theta(0,R)\, :\, u' \in L_\alpha^p(0,R),\,\, \lim_{x\to R^{-}} u(x) =0\Big\}, \quad \al, \theta \geqs 1.$$ In [@deOliveira], de Oliveira and do Ó prove the following sharp Moser–Trudinger inequality involving the measure $\lam_\theta$: suppose $0 \leqs R \leqs \infty$ and $\alpha \geq 2, \theta \geq 1$, then $$\label{eq:MTOO} D_{\al,\theta}(R) :=\sup_{u\in W^{1,\al}_{\al,\theta}(0,R),\, \|u'\|_{L^\alpha_\alpha(0,R)} \leq 1} \int_0^R e^{\mu_{\al,\theta} |u|^{\frac{\alpha}{\alpha -1}}} d\lam_\theta \leqs \infty$$ where $\mu_{\al,\theta} = \theta \alpha^{\frac1{\al -1}} \sigma_{\al}^{\frac1{\alpha -1}}$. Denote $D_{\al,\theta} = D_{\al,\theta}(1)$. It is easy to see that $D_{\al,\theta}(R) = D_{\al,\theta} R^\theta$. \[MT\] Let $\alpha \geqs 1$ and $q\geq 2$. There exists a constant $C_{\al,q} \geqs 0$ such that for any $u \in W^{1,q}_{q,\al}(0,\infty),$ $u' \leq 0$ and $\|u\|_{L^q_{\al}(0,\infty)}^q + \|u'\|_{L^q_q(0,\infty)}^q\leq 1$, it holds $$\label{eq:Abreu} \int_0^\infty \Phi_{\frac q\al,q}(\mu_{q,1} |u|^{\frac q{q -1}}) d\lam_1 \leq C_{\al,q}.$$ We follows the argument in [@Ruf2005]. Since $u' \leq 0$ then $u$ is a non-increasing function. Hence, for any $t \geqs 0$, it holds $$\label{eq:boundu} u(r)^q \leq \frac{1}{\si_\al r^\al} \int_0^r u(s)^q d\lam_\al \leq \frac{\int_0^\infty u(s)^q d\lam_\al}{\si_\al r^\al} \leq \frac{\|u\|_{L^q_\al(0,\infty)}^q}{\si_\al r^\al}.$$ For $R \geqs 0$, define $w(r) = u(r) - u(R)$ for $r \leq R$ and $w(r) =0$ for $r \geqs R$. Then $w \in W^{1,q}{q,q}(0,R)$ and $$\label{eq:on0R} \|w\|_{L^q_q(0,R)}^q = \int_0^R |u'(s)|^q d\lam_q \leq 1 - \|u\|_{L^q_\al(0,\infty)}^q.$$ For $r \leq R$, we have $u(r) = w(r) + u(R)$. Since $q \geq 2$, then there exists $C \geqs 0$ depending only on $q$ such that $$u(r)^{q'} \leq w(r)^{q'} + C w(r)^{q'-1} u(R) + u(R)^{q'}.$$ Applying Young’s inequality and , we get $$\begin{aligned} \label{eq:E1} u(r)^{q'} &\leq w(r)^{q'}\lt(1+ \frac{C}{q} u(R)^q\rt) + \frac{q-1}q + u(R)^{q'}\notag\\ &\leq w(r)^{q'}\lt(1+ \frac{C}{q\si_\al R^\al}\rt) + \frac{q-1}q + \lt(\frac1{\si_\al R^\al}\rt)^{q'-1}.\end{aligned}$$ Fix a $R \geq 1$ large enough such that $\frac{C}{q\si_\al R^\al} \leq 1$, and set $$v(r) = w(r) \lt(1+ \frac{C}{q\si_\al R^\al}\rt)^{\frac{q-1}q}.$$ Using and the choice of $R$, we can easily verify that $\|v\|_{L^q_q(0,R)}^q \leq 1$. Hence, applying , we get $$\label{eq:on0R1} \int_0^R e^{\mu_{q,1} |u|^{q'}} d\lam_1 \leq D_{q,1} R.$$ For $r \geq R$, we have $u(r) \leq \si_\al^{-\frac1q} R^{-\frac\al q}$, hence it holds $$\Phi_{\frac q\al,q} (\mu_{q,1} |u(r)|^{q'}) \leq C |u(r)|^{q'(j_{\al,q}-1)} \leq C r^{-\frac{\al}{q-1}(j_{\al,q} -1)}.$$ By the choice of $j_{\al,q}$, we have $$\label{eq:E2} \int_R^\infty \Phi_{\frac q\al,q}(\mu_{q,1} |u(r)|^{q'}) d\lam_1 \leq C.$$ Putting , , together and using $R\geq 1$, we get $$\begin{aligned} \int_0^\infty \Phi_{\frac q\al,q}(\mu_{q,1} |u|^{q'}) d\lam_1 &\leq \int_0^R \Phi_{\frac q\al,q}(\mu_{q,1} |u|^{q'}) d\lam_1 + \int_R^\infty \Phi_{\frac q\al,q}(\mu_{q,1} |u|^{q'}) d\lam_1 \\ &\leq \int_0^R \exp\Big(\mu_{q,1} |u|^{q'}\Big) d\lam_1 + C\\ &\leq \int_0^R \exp\Big(\mu_{q,1} v^{q'} + \mu_{q,1} \big(\frac{q-1}q + \si_\al^{-\frac1{q-1}}\big)\Big) d\lam_1 + C\\ &\leq \exp\Big(\mu_{q,1} \big(\frac{q-1}q + \si_\al^{-\frac1{q-1}}\big)\Big)D_{q,1} R + C\\ &\leq C.\end{aligned}$$ For any $\tau \geqs 0$ and $u \in W^{1,q}_{q,\al}(0,\infty),$ such that $u' \leq 0$ and $\tau \|u\|_{L^q_\alpha(0,\infty)}^q + \|u'\|_{L^q_q(0,\infty)}^q\leq 1$. Applying for function $u_\tau(x) = u(\tau^{-\frac1\al} x)$ and making the change of variables, we obtain $$\label{eq:Abreu1} \int_0^\infty \Phi_{\frac q \alpha,q}(\mu_{q,1} |u|^{q'}) d\lam_1 \leq C \tau^{-\frac 1\al}.$$ We are now ready to give the proof of Theorem \[MAINII\]. We divide the proof into the following cases.\ *Case 1: $m=2$.* Let $u \in C_0^\infty(\H^n)$ with $\|\Delta_g u\|_{\frac n2,q}^q - \lam \|u\|_{\frac n2,q}^q \leq 1$. Define $v$ by and $\tilde v(x) = v(V_g(B(0,d(0,x))))$, then $u^* \leq v$, $\|\Delta_g u\|_{\frac n2,q} = \|\Delta_g \tilde v\|_{\frac n2,q}$ and $\|u\|_{\frac n2,q} \leq \|\tilde v\|_{\frac n2,q}$. So, we have $$\|\Delta_g \tilde v\|_{\frac n2,q}^q -\lam \|\tilde v\|_{\frac n2,q}^q \leq 1.$$ We show that $\int_{\H^n} \Phi_{\frac n2,q}(\beta_{n,2} |\tilde v|^{q'}) dV_g \leq C.$ Set $\kappa = C(n,2,n/2)^q -\lam \geqs 0$. Applying the inequality for $\tilde v$, we get $$\lt(n(n-2) \si_n^{\frac2n}\rt)^q \int_0^\infty |v'(t)|^q t^{ q -1} dt + \kappa \int_0^\infty v(t)^q t^{\frac{2q}n -1} dt \leq 1.$$ Define $$w = \frac{n(n-2) \si_n^{\frac2n}}{(q \si_q)^{\frac1q}} v,\quad \tau = \frac{q \si_q}{(n(n-2) \si_n^{\frac2n})^q \frac{2q}n \si_{\frac{2q}n}}\kappa,$$ then, we have $$\int_0^\infty |w'|^q d\lam_q + \tau \int_0^\infty |w|^q d \lam_{\frac{2q}n} \leq 1.$$ Applying the inequality , we obtain $$\int_0^\infty \Phi_{\frac{n}2,q} (\mu_{q,1} w^{\frac q{q-1}}) d\lam_1 \leq C_{\frac{2q}n,q} \tau^{-\frac n{2q}}.$$ Notice that $$\int_{\H^n} \Phi_{\frac n2,q} (\beta_{n,2}^{q'} |\tilde v|^{q'}) dV_g = \frac12 \int_0^\infty \Phi_{\frac{n}2,q}(\beta_{n,2}^{q'} |v|^{q'}) d\lam_1=\frac12 \int_0^\infty \Phi_{\frac{n}2,q} (\mu_{q,1} w^{\frac q{q-1}}) d\lam_1.$$ Hence, it holds $$\int_{\H^n} \Phi_{\frac n2,q} (\beta_{n,m}^{q'} |\tilde v|^{q'}) dV_g \leq \frac12 C_{\frac{2q}n,q} \tau^{-\frac n{2q}}.$$ This completes the proof of this case.\ *Case 2: $m = 2k$, $k\geq 2$.* Denote $\tau = C(n,2k, \frac n{2k})^q -\lam \geqs 0$. We have $$1\geq \|\Delta^k_g u\|_{\frac n{2k},q}^q - \lam \|u\|_{\frac n{2k},q}^q \geq \tau \|u\|_{\frac n{2k},q}^q,$$ which yields $$\label{eq:normu} \|u\|_{\frac n{2k},q}^q \leq \tau^{-1}.$$ On the other hand, by the Lorentz–Poincaré inequality and the Poincaré–Sobolev inequality under Lorentz–Sobolev norm , we have $$\begin{aligned} \|\Delta^k_g u\|_{\frac n{2k},q}^q - \lam \|u\|_{\frac n{2k},q}^q &\geq \|\Delta^k_g u\|_{\frac n{2k},q}^q - C(n,2k,\frac{n}{2k})\|u\|_{\frac n{2k},q}^q + \tau \|u\|_{\frac n{2k},q}^q\\ &\geq \|\Delta^k_g u\|_{\frac n{2k},q}^q - C(n,2,\frac{n}{2k})\|\Delta^{k-1}_g u\|_{\frac n{2k},q}^q + \tau \|u\|_{\frac n{2k},q}^q\\ &\geq (2(k-1)(n-2k) \si_n^{\frac 2n})^q \|\Delta^{k-1}_g u\|_{\frac n{2(k-1)},q}^ q + \tau \|u\|_{\frac n{2k},q}^q.\end{aligned}$$ Set $ w = 2(k-1)(n-2k) \si_n^{\frac 2n} u $ we have $\|\Delta^{k-1}_g w\|_{\frac n{2(k-1)},q}^ q \leq 1$. Applying the Adams inequality , we obtain $$\int_{\H^n} \Phi_{n,2(k-1),q}(\beta_{n,2k}^{q'} |u|^{q'}) dV_g = \int_{\H^n} \Phi_{n,2(k-1),q}(\beta_{n,2(k-1)}^{q'} |w|^{q'}) \leq C,$$ here we use $$\beta_{n,2k} = 2(k-1)(n-2k) \si_n^{\frac 2n} \beta_{n,2(k-1)}.$$ Using and repeating the last argument in the proof of *Case 2* in the proof of Theorem \[MAINI\], we obtain in this case.\ *Case 3: $m =2k+1$, $k\geq 1$.* Denote $\tau = C(n,2k+1,\frac n{2k+1})^q -\tau \geqs 0$. Since $\frac{2n}{n-1} \leq q \leq \frac n{2k+1}$, then using the Lorentz–Poincaré inequality and the Poincaré–Sobolev inequality under Lorentz–Sobolev norm , we get $$\begin{aligned} 1&\geq \|\na_g \Delta_g^k u\|_{\frac n{2k+1},q}^q - \lam \|u\|_{\frac n{2k+1},q}^q \\ &\geq \|\na_g \Delta_g^k u\|_{\frac n{2k+1},q}^q - C(n,2k+1,\frac n{2k+1})^q \|u\|_{\frac n{2k+1},q}^q + \tau \|u\|_{\frac n{2k+1},q}^q\\ &\geq \|\na_g \Delta_g^k u\|_{\frac n{2k+1},q}^q - \lt(\frac{(2k+1)(n-1)}{n}\rt)^q \|\Delta_g^k u\|_{\frac n{2k+1},q}^q + \tau \|u\|_{\frac n{2k+1},q}^q\\ &\geq (2k \si_n^{\frac 1n})^q \|\Delta_g^k u\|_{\frac n{2k},q}^q + \tau \|u\|_{\frac n{2k+1},q}^q.\end{aligned}$$ We now can use the argument in the proof of *Case 2* to obtain the result in this case. The proof of Theorem \[MAINI\] is then completely finished. Proof of Theorem \[HARDYADAMS\] =============================== In this section, we provide the proof of Theorem \[HARDYADAMS\]. The proof uses the Lorentz–Poincaré inequality , the Poincaré–Sobolev inequality under Lorentz–Sobolev norm and , and the Adams type inequality . We divide the proof in two cases according to the facts that $m$ is even or odd.\ *Case 1: $m=2k$, $k\geq 2$.* Using the Lorentz–Poincaré inequality and the inequality , we have $$\begin{aligned} 1\geq \|\Delta^k_g u\|_{\frac n{2k},q}^q - C(n,2k,\frac{n}{2k})\|u\|_{\frac n{2k},q}^q &\geq \|\Delta^k_g u\|_{\frac n{2k},q}^q - C(n,2,\frac{n}{2k})\|\Delta^{k-1}_g u\|_{\frac n{2k},q}^q\notag\\ &\geq (2(k-1)(n-2k) \si_n^{\frac 2n})^q \|\Delta^{k-1}_g u\|_{\frac n{2(k-1)},q}^q.\end{aligned}$$ Let us define the function $w$ by $w = 2(k-1)(n-2k) \si_n^{\frac 2n} u$. Then we have $\|\Delta^{k-1}_g w\|_{\frac n{2(k-1)},q}^ q \leq 1$. Applying the Adams type inequality , we obtain $$\label{eq:aa0} \int_{\H^n} \Phi_{n,2(k-1),q}(\beta_{n,2k}^{q'} |u|^{q'}) dV_g = \int_{\H^n} \Phi_{n,2(k-1),q}(\beta_{n,2(k-1)}^{q'} |w|^{q'}) dV_g\leq C,$$ here we use $$\beta_{n,2k} = 2(k-1)(n-2k) \si_n^{\frac 2n} \beta_{n,2(k-1)}.$$ It follows from and the fact $\Phi_{n,2(k-1),q}(t) \geq C t^{j_{\frac{n}{2(k-1)},q}-1}$ that $$\int_0^\infty (u^*(t))^{q'(j_{\frac{n}{2(k-1)},q}-1)} dt = \int_{\H^n} |u|^{q'(j_{\frac{n}{2(k-1)},q}-1)} dV_g \leq C.$$ Using the non-increasing of $u^*$, we can easily verify that $$u^*(t) \leq C t^{-1/(q'(j_{\frac{n}{2(k-1)},q}-1))}$$ for any $t \geqs 0$. Let $x_0 \in \B^n$ such that $V_g(B(0,d(0,x_0))) = 1$. Since the function $h(x) = (1-|x|^2)^n$ is decreasing with respect to $d(0,|x|)$, then $h^\sharp = h$. Using Hardy–Littlewood inequality, we have $$\begin{aligned} \label{eq:aa1} \int_{\B^n} e^{\beta_{n,2k}^{q'} |u|^{q'}} dx = 2^{-n} \int_{\H^n} e^{\beta_{n,2k}^{q'} |u|^{q'}} h(x) dV_g& \leq 2^{-n} \int_{\H^n} e^{\beta_{n,2k}^{q'} |u^\sharp|^{q'}} h(x) dV_g\notag\\ &= 2^{-n} \int_0^\infty e^{\beta_{n,2k}^{q'} |u^*(t)|^{q'}} h(t) dt.\end{aligned}$$ For $t \geq 1$ we have $u^*(t) \leq C$, hence it holds $$\label{eq:aa2} 2^{-n}\int_1^\infty e^{\beta_{n,2k}^{q'} |u^*(t)|^{q'}} h(t) dt \leq C2^{-n} \int_1^\infty h(t) dt =C \int_{\{|x| \geq |x_0|\}} dx \leq C\si_n.$$ Notice that $$e^t = \Phi_{\frac n{2(k-1)},q}(t) + \sum_{j=0}^{j_{\frac{n}{2(k-1)},q} -2} \frac{t^j}{j!}.$$ Using Young’s inequality, we get $$e^t = \Phi_{\frac n{2(k-1)},q}(t) + C(1+ t^{j_{\frac{n}{2(k-1)},q} -2}).$$ Consequently, by using the previous inequality and the inequality and the fact $h\leq 1$, we obtain $$\begin{aligned} \label{eq:aa3} \int_0^1 e^{\beta_{n,2k}^{q'} |u^*(t)|^{q'}} h(t) dt&\leq \int_0^1 \Phi_{\frac n{2(k-1)},q}(\beta_{n,2k}^{q'} |u^*(t)|^{q'}) dt + C \int_0^1\lt(1 + (u^*(t))^{q'(j_{\frac{n}{2(k-1)},q} -2)}\rt) dt\notag\\ &\leq \int_0^\infty \Phi_{\frac n{2(k-1)},q}(\beta_{n,2k}^{q'} |u^*(t)|^{q'}) dt + C + C\int_0^1(u^*(t))^{q'(j_{\frac{n}{2(k-1)},q} -2)} dt\notag\\ &\leq \int_{\H^n} \Phi_{n,2(k-1),q}(\beta_{n,2k}^{q'} |u|^{q'}) dV_g + C + C \int_0^1 t^{-\frac{j_{\frac{n}{2(k-1)},q} -2}{j_{\frac{n}{2(k-1)},q} -1}} dt\notag\\ &\leq C.\end{aligned}$$ Combining , and we obtain the desired estimate.\ *Case 2: $m=2k+1$, $k\geq 1$.* Since $\frac{2n}{n-1} \leq q \leq \frac{n}{2k+1}$, then by using the Lorentz–Poincaré inequality and the Poincaré–Sobolev inequality under Lorentz–Sobolev norm , we get $$\begin{aligned} 1&\geq \|\na_g \Delta_g^k u\|_{\frac n{2k+1},q}^q - C(n,2k+1,\frac n{2k+1})^q \|u\|_{\frac n{2k+1},q}^q \\ &\geq \|\na_g \Delta_g^k u\|_{\frac n{2k+1},q}^q - \lt(\frac{(2k+1)(n-1)}{n}\rt)^q \|\Delta_g^k u\|_{\frac n{2k+1},q}^q\\ &\geq (2k \si_n^{\frac 1n})^q \|\Delta_g^k u\|_{\frac n{2k},q}^q.\end{aligned}$$ Setting $w = 2k \si_n^{\frac 1n} u$, we have $\|\Delta^k_g w\|_{\frac n{2k}, q}^q \leq 1$. Applying the Adams type inequality , we obtain $$\label{eq:bb0} \int_{\H^n} \Phi_{n,2k,q}(\beta_{n,2k+1}^{q'} |u|^{q'}) dV_g = \int_{\H^n} \Phi_{n,2k,q}(\beta_{n,2k}^{q'} |w|^{q'}) dV_g\leq C,$$ here we use $$\beta_{n,2k+1} = 2k \si_n^{\frac 1n} \beta_{n,2k}.$$ Similarly in the *Case 1*, the inequality yields $$\int_0^\infty (u^*(t))^{q'(j_{\frac{n}{2k},q}-1)} dt = \int_{\H^n} |u|^{q'(j_{\frac{n}{2k},q}-1)} dV_g \leq C,$$ which implies $$u^*(t) \leq C t^{-\frac1{q'(j_{\frac{n}{2k},q}-1)}},\quad t \geqs 0.$$ Repeating the last arguments in the proof of *Case 1*, we obtain the result in this case.\ The proof of Theorem \[HARDYADAMS\] is then completed. [10]{} S. Adachi and K. Tanaka. Trudinger type inequalities in [$\bold R^N$]{} and their best exponents. , 128(7):2051–2057, 2000. D. R. Adams. A sharp inequality of [J]{}. [M]{}oser for higher order derivatives. , 128(2):385–398, 1988. Adimurthi and O. Druet. Blow-up analysis in dimension 2 and a sharp form of [T]{}rudinger-[M]{}oser inequality. , 29(1-2):295–322, 2004. Adimurthi and K. Sandeep. A singular [M]{}oser-[T]{}rudinger embedding and its applications. , 13(5-6):585–603, 2007. Adimurthi and K. Tintarev. On a version of [T]{}rudinger-[M]{}oser inequality with [M]{}öbius shift invariance. , 39(1-2):203–212, 2010. Adimurthi and Y. Yang. An interpolation of [H]{}ardy inequality and [T]{}rundinger-[M]{}oser inequality in [$\Bbb R^N$]{} and its applications. , (13):2394–2426, 2010. A. Alberico. Moser type inequalities for higher-order derivatives in [L]{}orentz spaces. , 28(4):389–400, 2008. A. Alvino, V. Ferone, and G. Trombetti. Moser-type inequalities in [L]{}orentz spaces. , 5(3):273–299, 1996. Z. M. Balogh, J. J. Manfredi, and J. T. Tyson. Fundamental solution for the [$Q$]{}-[L]{}aplacian and sharp [M]{}oser-[T]{}rudinger inequality in [C]{}arnot groups. , 204(1):35–49, 2003. C. Bennett and R. Sharpley. , volume 129 of [*Pure and Applied Mathematics*]{}. Academic Press, Inc., Boston, MA, 1988. J. Bertrand and K. Sandeep. Adams inequality on pinched hadamard manifolds. , 2019. L. Carleson and S.-Y. A. Chang. On the existence of an extremal function for an inequality of [J]{}. [M]{}oser. , 110(2):113–127, 1986. D. Cassani and C. Tarsi. A [M]{}oser-type inequality in [L]{}orentz-[S]{}obolev spaces for unbounded domains in [$\Bbb R^N$]{}. , 64(1-2):29–51, 2009. L. Chen, G. Lu, and M. Zhu. Existence and nonexistence of extremals for critical adams inequalities in $\mathbb r^4$ and trudinger–moser inequalities in $\mathbb r^2$. , 2018. W. S. Cohn and G. Lu. Best constants for [M]{}oser-[T]{}rudinger inequalities on the [H]{}eisenberg group. , 50(4):1567–1591, 2001. W. S. Cohn and G. Z. Lu. Best constants for [M]{}oser-[T]{}rudinger inequalities, fundamental solutions and one-parameter representation formulas on groups of [H]{}eisenberg type. , 18(2):375–390, 2002. J. F. de Oliveira and J. a. M. do Ó. Trudinger-[M]{}oser type inequalities for weighted [S]{}obolev spaces involving fractional dimensions. , 142(8):2813–2828, 2014. A. DelaTorre and G. Mancini. Improved adams–type inequalities and their extremals in dimension $2m$. , 2017. J. a. M. do Ó and M. de Souza. A sharp inequality of [T]{}rudinger-[M]{}oser type and extremal functions in [$H^{1,n}(\Bbb{R}^n)$]{}. , 258(11):4062–4101, 2015. Y. Q. Dong and Q. H. Yang. An interpolation of [H]{}ardy inequality and [M]{}oser-[T]{}rudinger inequality on [R]{}iemannian manifolds with negative curvature. , 32(7):856–866, 2016. M. Flucher. Extremal functions for the [T]{}rudinger-[M]{}oser inequality in [$2$]{} dimensions. , 67(3):471–497, 1992. L. Fontana and C. Morpurgo. Sharp exponential integrability for critical [R]{}iesz potentials and fractional [L]{}aplacians on [$\Bbb R^n$]{}. , 167:85–122, 2018. L. Fontana and C. Morpurgo. Adams inequalities for [R]{}iesz subcritical potentials. , 192:111662, 32, 2020. L. Fontana and C. Morpurgo. Adams inequalities for [R]{}iesz subcritical potentials. , 192:111662, 32, 2020. V. I. Judovič. Some estimates connected with integral operators and with solutions of elliptic equations. , 138:805–808, 1961. D. Karmakar and K. Sandeep. Adams inequality on the hyperbolic space. , 270(5):1792–1817, 2016. N. Lam and G. Lu. Sharp [A]{}dams type inequalities in [S]{}obolev spaces [$W^{m,\frac{n}{m}} (\Bbb R^n)$]{} for arbitrary integer [$m$]{}. , 253(4):1143–1171, 2012. N. Lam and G. Lu. Sharp [M]{}oser-[T]{}rudinger inequality on the [H]{}eisenberg group at the critical case and applications. , 231(6):3259–3287, 2012. N. Lam and G. Lu. Sharp singular [A]{}dams inequalities in high order [S]{}obolev spaces. , 19(3):243–266, 2012. N. Lam and G. Lu. A new approach to sharp [M]{}oser-[T]{}rudinger and [A]{}dams type inequalities: a rearrangement-free argument. , 255(3):298–325, 2013. J. Li, G. Lu, and Q. Yang. Fourier analysis and optimal [H]{}ardy-[A]{}dams inequalities on hyperbolic spaces of any even dimension. , 333:350–385, 2018. X. Li and Y. Yang. Extremal functions for singular [T]{}rudinger-[M]{}oser inequalities in the entire [E]{}uclidean space. , 264(8):4901–4943, 2018. Y. Li and B. Ruf. A sharp [T]{}rudinger-[M]{}oser type inequality for unbounded domains in [$\Bbb R^n$]{}. , 57(1):451–480, 2008. K.-C. Lin. Extremal functions for [M]{}oser’s inequality. , 348(7):2663–2671, 1996. G. Lu and H. Tang. Best constants for [M]{}oser-[T]{}rudinger inequalities on high dimensional hyperbolic spaces. , 13(4):1035–1052, 2013. G. Lu and H. Tang. Sharp singular [T]{}rudinger-[M]{}oser inequalities in [L]{}orentz-[S]{}obolev spaces. , 16(3):581–601, 2016. G. Lu and Q. Yang. Sharp [H]{}ardy-[A]{}dams inequalities for bi-[L]{}aplacian on hyperbolic space of dimension four. , 319:567–598, 2017. G. Lu and Y. Yang. Adams’ inequalities for bi-[L]{}aplacian and extremal functions in dimension four. , 220(4):1135–1170, 2009. G. Lu and M. Zhu. A sharp [T]{}rudinger-[M]{}oser type inequality involving [$L^n$]{} norm in the entire space [$\Bbb{R}^n$]{}. , 267(5):3046–3082, 2019. G. Mancini and L. Martinazzi. Extremals for fractional moser–trudinger inequalities in dimension $1$ via harmonic extensions and commutator estimates. , 2019. G. Mancini and K. Sandeep. Moser-[T]{}rudinger inequality on conformal discs. , 12(6):1055–1068, 2010. G. Mancini, K. Sandeep, and C. Tintarev. Trudinger-[M]{}oser inequality in the hyperbolic space [${\Bbb H}^N$]{}. , 2(3):309–324, 2013. L. Martinazzi. Fractional [A]{}dams-[M]{}oser-[T]{}rudinger type inequalities. , 127:263–278, 2015. J. Moser. A sharp form of an inequality by [N]{}. [T]{}rudinger. , 20:1077–1092, 1970/71. Q. A. Ngô and V. H. Nguyen. Sharp adams–moser–trudinger type inequalities in the hyperbolic space. , 2016. Q. A. Ngô and V. H. Nguyen. Sharp constant for [P]{}oincaré-type inequalities in the hyperbolic space. , 44(3):781–795, 2019. V. H. Nguyen. A sharp adams inequality in dimension four and its extremal functions. , 2017. V. H. Nguyen. Improved [M]{}oser-[T]{}rudinger type inequalities in the hyperbolic space [$\Bbb{H}^n$]{}. , 168:67–80, 2018. V. H. Nguyen. Improved singular [M]{}oser-[T]{}rudinger and their extremal functions. , 2018. V. H. Nguyen. The sharp [P]{}oincaré-[S]{}obolev type inequalities in the hyperbolic spaces [$\Bbb{H}^n$]{}. , 462(2):1570–1584, 2018. V. H. Nguyen. Extremal functions for the [M]{}oser-[T]{}rudinger inequality of [A]{}dimurthi-[D]{}ruet type in [$W^{1,N}(\Bbb R^N)$]{}. , 21(4):1850023, 37, 2019. V. H. Nguyen. The sharp hardy-moser-trudinger inequality in dimension $n$. , 2019. V. H. Nguyen. The sharp [S]{}obolev type inequalities in the [L]{}orentz–[S]{}obolev spaces in the hyperbolic spaces. , 2019. V. H. Nguyen. The sharp higher order [L]{}orentz-[P]{}oincaré and [L]{}orentz-[S]{}obolev inequalities in the hyperbolic spaces. , 2020. V. H. Nguyen. Singular adams inequalities in [L]{}orentz–[S]{}obolev spaces. , 2020. S. I. Pohožaev. On the eigenfunctions of the equation [$\Delta u+\lambda f(u)=0$]{}. , 165:36–39, 1965. B. Ruf. A sharp [T]{}rudinger-[M]{}oser type inequality for unbounded domains in [$\Bbb R^2$]{}. , 219(2):340–367, 2005. B. Ruf and F. Sani. Sharp [A]{}dams-type inequalities in [$\Bbb{R}^n$]{}. , 365(2):645–670, 2013. C. Tintarev. Trudinger-[M]{}oser inequality with remainder terms. , 266(1):55–66, 2014. N. S. Trudinger. On imbeddings into [O]{}rlicz spaces and some applications. , 17:473–483, 1967. G. Wang and D. Ye. A [H]{}ardy-[M]{}oser-[T]{}rudinger inequality. , 230(1):294–320, 2012. Q. Yang and Y. Li. Trudinger-[M]{}oser inequalities on hyperbolic spaces under [L]{}orentz norms. , 472(1):1236–1252, 2019. Q. Yang, D. Su, and Y. Kong. Sharp [M]{}oser-[T]{}rudinger inequalities on [R]{}iemannian manifolds with negative curvature. , 195(2):459–471, 2016. Y. Yang. A sharp form of [M]{}oser-[T]{}rudinger inequality in high dimension. , 239(1):100–126, 2006. [^1]: Email: [[email protected]](mailto: Van Hoang Nguyen <[email protected]>). [^2]: 2010 *Mathematics Subject Classification*: 26D10, 46E35, 46E30, [^3]: *Key words and phrases*: Adams inequality, improved Adams inequality, Hardy–Adams inequality Lorentz–Sobolev space, hyperbolic spaces, rearrangement argument.

ArXiv

-1cm   \ [**Abstract**]{} The paper presents the $QCD$ description of the hard and semihard processes in the framework of the Wilson operator product expansion. The smooth transition between the cases of the soft and hard Pomerons is obtained. The recent measurements of the deep-inelastic (DIS) structure function (SF) $F_2$ by the $H1$ [@1] and $ZEUS$ [@2] collaborations open a new kinematical range to study proton structure. The new $HERA$ data show the strong increase of $F_2$ with decresing $x$. However, the data of the $NMC$ [@3] and $E665$ collaboration [@4] at small $x$ and smaller $Q^2$ is in the good agreement with the standard Pomeron or with the Donnachie-Landshoff picture where the Pomeron intercept: $\alpha_p = 1.08$, is very close to standard one. The interpritation of the fast changing of the intercept in the region of $Q^2$ between $Q^2=1GeV^2$ and $Q^2=10GeV^2$ (see Fig.3 in [@5]) is yet absent. There are the arguments in favour of that is one intercept (see [@6]) or the superposition of two different Pomeron trajectories, one having an intercept of $1.08$ and the one of $1.5$ (see Fig.4 in [@5]). The aim of this article is the possible “solution” of this problem in the framework of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [@6.5]. It is good known (see, for example, [@7]), that in the double-logarithmical approximation the DGLAP equation solution is the Bessel function, or $\exp{\sqrt{\phi (Q^2) ln(1/x)}}$, where $\phi (Q^2)$ is known $Q^2$-dependent function[^1]. However, we will seek the “solution”[^2] of DGLAP equation in the Regge form (we use the parton distributions (PD) multiplied by $x$ and neglect the nonsinglet quark distribution at small $x$): $$\begin{aligned} f_a (x,Q^2) \sim x^{-\delta} \tilde f_a(x,Q^2), ~~~~(a=q,g)~~(\alpha_p \equiv 1+\delta ) \label{1} \end{aligned}$$ where $\tilde f_a(x,Q^2)$ is nonsingular at $x \to 0$ and $\tilde f_a(x,Q^2) \sim (1-x)^{\nu}$ at $x \to 1$[^3]. The similar investigations were already done and the results are good known (see [@8], [@11]-[@15])[^4]. The aim of this letter is to expand these results to the range where $\delta \sim 0$ (and $Q^2$ is not large) following to the observed early (see [@12; @13])[^5] method to replace the Mellin convolution by a simple product. Of course, we understand that the Regge behaviour (\[1\]) is not in the agreement with the double-logarithmic solution, however the range, where $\delta \sim 0$ and the $Q^2$ values are nonlarge, is really the Regge regime and a “solution” of DGLAP equation in the form of (\[1\]) would be worthwhile. This “solution” may be understand as the solution of DGLAP equation together with the condition of its Regge asymptotic at $x \to 0$. Consider DGLAP equation and apply the method from [@13] to the Mellin convolution in its r.h.s. (in contrast with standard case, we use below $\alpha(Q^2)=\alpha_s(Q^2)/(4\pi)$): $$\begin{aligned} {&&\hspace*{-1cm}}\frac{d}{dt}f_a (x,t)~=~- \frac{1}{2} \sum_{i=a,b} \hat \gamma_{ai}(\alpha,x) \otimes f_a(x,t)~~~(a,b)=(q,g) \nonumber \\ {&&\hspace*{-1cm}}=~- \frac{1}{2} \sum_{i=a,b} \tilde \gamma_{ai}(\alpha,1+\delta)f_a(x,t)~+~O(x^{1-\delta})~~~ \Bigl( \gamma_{ab}(\alpha,n)=\alpha \gamma_{ab}^{(0)}(n)+\alpha^2 \gamma_{ab}^{(1)}(n)+...\Bigr) , \label{2} \end{aligned}$$ where $t=ln(Q^2/\Lambda ^2)$. The $\hat \gamma_{ab}(\alpha,x)$ are the spliting functions corresponding to the anomalous dimensions (AD) $\gamma_{ab}(\alpha,n) = \int^1_0 dx x^{n-1} \hat \gamma_{ab}(\alpha,x)$. Here the functions $\gamma_{ab}(\alpha,1+\delta)$ are the AD $\gamma_{ab}(\alpha,n)$ expanded from the integer argument “$n$” to the noninteger one “$1+\delta$”. The functions $\tilde \gamma_{ab}(\alpha,1+\delta)$ (marked lower as AD, too) can be obtained from the functions $\gamma_{ab}(\alpha,1+\delta)$ replacing the term $1/\delta$ by the one $1/\tilde \delta$: $$\begin{aligned} \frac{1}{\delta} \to \frac{1}{\tilde \delta}~=~\frac{1}{\delta} \Bigl( 1 - \varphi(x,\delta)x^{\delta} \Bigr) \label{3} \end{aligned}$$ This replacement (\[3\]) is appeared very naturally from the consideration the Mellin convolution at $x \to 0$ (see [@13]) and preserves the smooth and nonsingular transition to the case $\delta =0$, where $$\begin{aligned} \frac{1}{\tilde \delta}~=~ln\frac{1}{x} - \varrho(x) \label{4} \end{aligned}$$ The concrete form of the functions $\varphi(x,\delta)$ and $\varrho(x)$ depends strongly on the type of the behaviour of the PD $f_a(x,Q^2)$ at $x \to 0$ and in the case of the Regge regime (\[1\]) they are (see [@12; @13]): $$\begin{aligned} \varphi(x,\delta)~=~ \frac{\Gamma(\nu +1)\Gamma(1-\delta)}{\Gamma(\nu +1-\delta)}~~ \mbox{ and }~~ \varrho(x)~=~\Psi(\nu+1)-\Psi(1), \label{5} \end{aligned}$$ where $\Gamma(\nu+1)$ and $\Psi(\nu+1)$ are the Eulerian $\Gamma$- and $\Psi$-functions, respectively. As it can be seen, there is the correlation with the PD behaviour at large $x$. If $\delta$ is not small (i.e. $x^{\delta}>>1$), we can replace $1/ \tilde \delta$ to $1/\delta$ in the r.h.s. of Eq.(\[2\]) and obtain its solution in the form (hereafter $t_0=t(Q^2=Q^2_0)$): $$\begin{aligned} \frac{f_a(x,t)}{f_a(x,t_0)}~=~ \frac{M_a(1+\delta,t)}{M_a(1+\delta,t_0)}, \label{6} \end{aligned}$$ where $M_a(1+\delta,t)$ is the analytical expansion of the PD moments $M_a(n,t) = \int^1_0 dx x^{n-1} f_a(x,t)$ to the noninteger value “$n=1+\delta$”. This solution is good known one (see [@12] for the first two orders of the perturbation theory, [@14] for the first three orders and [@15] containing a resummation of all orders, respectively). Note that recently the fit of $HERA$ data was done in [@17] with the formula for PD $f_q(x,t)$ very close [^6] to (\[6\]) and the very well agreement (the $\chi^2$ per degree of freedom is $0.85$) is found at $\delta = 0.40 \pm 0.03$. There are also the fits [@17.5] of the another group using equations which are similar to (\[6\]) in the LO approximation. The news in our investigations are in the follows. Note that the $Q^2$-evolution of $M_a(1+\delta,t)$ contains the two: “+” and “$-$” components, i.e. $M_a(1+\delta,t)= \sum_{i= \pm} M_a^i(1+\delta,t)$, and in principle the every component evolves separately and may have the independent (and not equal) intercept. Here for the simplicity we restricte ourselves to the LO analysis and give NLO formulae lower without large intermediate equations. [**1.**]{} Consider DGLAP equation for the “+” and “$-$” parts (hereafter $s=ln(lnt/lnt_0)$): $$\begin{aligned} \frac{d}{ds} f_a^{\pm}(x,t)~=~- \frac{1}{2\beta_0} \tilde \gamma_{\pm}(\alpha,1+\delta_{\pm})f_a^{\pm}(x,t)~+~O(x^{1-\delta}), \label{7} \end{aligned}$$ where $$\gamma_{\pm}~=~ \frac{1}{2} \biggl[ \Bigl(\gamma_{gg}+\gamma_{qq} \Bigr)~\pm ~ \sqrt{ {\Bigl( \gamma_{gg}- \gamma_{qq} \Bigr)}^2 ~+~4\gamma_{qg}\gamma_{gq}} \biggr]$$ are the AD of the “$\pm $” components (see, for example, [@18]) The “$-$” component $\tilde \gamma_{-}(\alpha,1+\delta_-)$ does not contain the singular term (see [@12; @14] and lower) and its solution have the form: $$\begin{aligned} \frac{f_a^-(x,t)}{f_a^-(x,t_0)}~=~e^{-d_-(1+\delta_-)s}, \mbox{ where } d_{\pm}=\frac{\gamma_{\pm}(1+\delta_{\pm})}{2\beta_0} \label{8} \end{aligned}$$ The “+” component $\tilde \gamma_{+}(\alpha,1+\delta_+)$ contains the singular term and $f_a^+(x,t)$ have the solution similar (\[8\]) only for $x^{\delta_+}>>1$: $$\begin{aligned} \frac{f_a^+(x,t)}{f_a^+(x,t_0)}~=~e^{-d_+(1+\delta_+)s}, \mbox{ if } x^{\delta_+}>>1 \label{9} \end{aligned}$$ The both intersepts $1+\delta_+$ and $1+\delta_-$ are unknown and should be found, in principle, from the analysis of the experimental data. However there is the another way. From the small $Q^2$ (and small $x$) data of the $NMC$ [@3] and $E665$ collaboration [@4] we can conclude that the SF $F_2$ and hence the PD $f_a(x,Q^2)$ have the flat asymptotics for $x \to 0$ and $Q^2 \sim (1\div2)GeV^2$. Thus we know that the values of $\delta_+$ and $\delta_-$ is approximately zero at $Q^2 \sim 1GeV^2$. Consider Eqs.(\[7\]) with $\delta_{\pm}=0$ and with the boundary condition $f_a(x,Q^2_0)=A_a$ at $Q^2_0=1GeV^2$. For the “$-$” component we already have the solution: the Eq.(\[8\]) with $\delta_-=0$ and $d_-(1)=16f/(27\beta_0$), where $f$ is the number of the active quarks and $\beta_i$ are the coefficients in the $\alpha$-expansion of QCD $\beta$-function. For its “+” component Eq.(\[7\]) can be rewritten in the form (hereafter the index $1+\delta $ will be omitted in the case $\delta \to 0$): $$\begin{aligned} ln(\frac{1}{x})\frac{d}{ds}\delta_+(s)~+~ \frac{d}{ds} ln(A_a^+) ~=~- \frac{1}{2\beta_0} \biggr[ \hat \gamma_{+} \Bigl( ln(\frac{1}{x}) -\varrho(\nu) \Bigr) ~+~ \overline \gamma_+ \biggl] \label{10} \end{aligned}$$ where $\hat\gamma_{+}$ and $\overline \gamma_+$ are the coefficients of the singular and regular parts at $\delta \to 0$ of AD $\gamma^+(1+\delta)$: $$\gamma^+(1+\delta)~=~\hat\gamma^+ \frac{1}{\delta} ~+~ \overline\gamma^+,~~~\hat\gamma^+=-24,~\overline\gamma^+=22+ \frac{4f}{27}$$ The solution of Eq.([10]{}) is $$\begin{aligned} f_a^+(x,t)~=~A^+_a~x^{\hat d_+s}e^{-\overline d_+s}, \label{11} \end{aligned}$$ where $$\hat d_+ \equiv \frac{\hat \gamma^+}{2\beta_0} \simeq - \frac{4}{3},~~ \overline d_+ \equiv \frac{1}{2\beta_0} \Bigl( \overline \gamma_+ ~-~ \hat \gamma_+ \varrho(\nu)\Bigr) \simeq \frac{4}{3} \varrho(\nu) + \frac{101}{81}$$ Herefter the symbol $\simeq $ marks the case $f=3$. As it can be seen from (\[11\]) the flat form $\delta_+=0$ of the “+”-component of PD is very nonstable from the (perturbative) viewpoint, because $d(\delta_+)/ds \neq 0$, and for $Q^2 > Q_0^2$ we have already the nonzero power of $x$ (i.e. pomeron intercept $\alpha_p >1$). This is in the agreement with the experimental data. Let us note that the power of x is positive for $Q^2<Q^2_0$ that is in principle also supported by the $NMC$ [@3] data, but the use of this analysis to $Q^2<1GeV^2$ is open the question. Thus, we have the DGLAP equation solution for the “+” component at $Q^2$ is close to $Q^2_0=1GeV^2$, where Pomeron starts in its movement to the subcritical (or Lipatov [@19.5; @19.6]) regime and also for the large $Q^2$, where pomeron have the $Q^2$-independent intercept. In principle, the general solution of (\[7\]) should contain the smooth transition between these pictures but this solution is absent [^7]. We introduce the some “critical” value of $Q^2$: $Q^2_c$, where the solution (\[9\]) is replaced by the solution (\[11\]). The exact value of $Q^2_c$ may be obtained from the fit of experimental data. Thus, we have in the LO of the perturbation theory: $$\begin{aligned} {&&\hspace*{-1cm}}f_a(x,t)~=~ f_a^-(x,t)~+~ f_a^+(x,t) \nonumber \\ {&&\hspace*{-1cm}}f_a^-(x,t)~=~A^-_a~\exp{(- d_-s)} \nonumber \\ {&&\hspace*{-1cm}}f_a^+(x,t)~=~ \left\{ \begin{array}{ll} A^+_a x^{\hat d_+s}\exp{(-\overline d_+s)}, & \mbox{ if } Q^2 \leq Q^2_c \\ f_a^+(x,t_c) \exp{\Bigl(-d_+(1+\delta_c)(s-s_c)\Bigr)}, & \mbox{ if } Q^2>Q^2_c \end{array} \right. \label{12} \end{aligned}$$ where $$\begin{aligned} {&&\hspace*{-1cm}}t_c~=~t(Q^2_c),~~s_c~=~s(Q^2_c) \nonumber \\ {&&\hspace*{-1cm}}A^+_q~=~(1- \overline \alpha )A_q ~+~ \tilde \alpha A_g,~~ A^+_g~=~ \overline \alpha A_g ~-~ \varepsilon A_q \nonumber \\ {&&\hspace*{-1cm}}\mbox{and } A_a^-~=~A_a ~-~ A_a^+ \label{13} \end{aligned}$$ and the values of the coefficients $\overline \alpha$, $\tilde \alpha$ and $\varepsilon$ may be found, for example, in [@18]. Using the concrete AD values at $\delta =0$ and $f=3$, we have $$\begin{aligned} {&&\hspace*{-1cm}}A^+_q~ \approx ~\frac{1}{27}\frac{4A_q+9A_g}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \nonumber \\ {&&\hspace*{-1cm}}A^+_g~ \approx ~A_g~+~\frac{4}{9}A_q ~-~ \frac{4}{27}\frac{9A_g-A_q}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \label{14} \end{aligned}$$ Thus, the value of the “+”component of the quark PD is suppressed logarithmically that is in the qualitative agreement with the $HERA$ parametrizations of SF $F_2$ (see [@20.5; @21]) (in the LO $F_2(x,Q^2)~=~(2/9)f_q(x,Q^2)$ for $f=3$), where the magnitude connected with the factor $x^{-\delta}$ is $5 \div 10 \%$ from the flat (for $x \to 0$) magnitude. [**2.**]{} By analogy with the subsection [**1**]{} and knowing the NLO $Q^2$-dependence of PD moments, we obtain the following equations for the NLO $Q^2$-evolution of the both: ”+” and “$-$” PD components (hereafter $\tilde s=ln(\alpha(Q^2_0)/\alpha(Q^2)), p=\alpha(Q^2_0)-\alpha(Q^2)$): $$\begin{aligned} {&&\hspace*{-1cm}}f_a(x,t)~=~ f_a^-(x,t)~+~ f_a^+(x,t) \nonumber \\ {&&\hspace*{-1cm}}f_a^-(x,t)=~\tilde A^-_a~\exp{(- d_-\tilde s -d_{--}^ap)} \nonumber \\ {&&\hspace*{-1cm}}f_a^+(x,t)= \left\{ \begin{array}{ll} \tilde A^+_a x^{(\hat d_+\tilde s + \hat d_{++}^a p)}\exp{(-\overline d_+\tilde s -\overline d_{++}^ap)}, & \mbox{if } Q^2 \leq Q^2_c \\ f_a^+(x,t_c) \exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde s_c)-d_{++}^a(1+\delta_c)(p-p_c) \Bigr) }, & \mbox{if } Q^2>Q^2_c \end{array} \right. \label{15} \end{aligned}$$ where $$\begin{aligned} {&&\hspace*{-1cm}}\tilde s_c ~=~ \tilde s(Q^2_c),~ p_c~=~p(Q^2_c),~\alpha_0~=~\alpha(Q^2_0) ,~\alpha_c~=~\alpha(Q^2_c) \nonumber \\ {&&\hspace*{-1cm}}\tilde A^{\pm}_a~=~\Bigl(1~-~\alpha_0 K^a_{\pm} \Bigr) A^{\pm}_a ~+~ \alpha_0 K^a_{\pm} A^{\mp}_a \nonumber \\ {&&\hspace*{-1cm}}d_{++}^a~=~ \hat d_{++}^a \Bigl( ln(\frac{1}{x}) - \varrho(\nu) \Bigl) ~+~ \overline d_{++}^a, ~~ d^a_{++}~=~ \frac{\gamma_{\pm \pm}}{2\beta_0} ~-~ \frac{\gamma_{\pm} \beta_1}{2\beta^2_0} ~-~ K^a_{\pm} \nonumber \\ {&&\hspace*{-1cm}}\mbox{and }~~ K^q_{\pm}~=~ \frac{\gamma_{\pm \mp}}{2\beta_0 + \gamma_{\pm} - \gamma_{\mp}},~~ K^g_{\pm}~=~ K^q_{\pm} \frac{\gamma_{\pm}-\gamma^{(0)}_{qq}}{\gamma_{\mp}-\gamma^{(0)}_{qq}} \label{16} \end{aligned}$$ The NLO AD of the “$\pm$” components are connected with the NLO AD $\gamma^{(1)}_{ab}$. The corresponding formulae can be found in [@18]. Using the concrete values of the LO and NLO AD at $\delta =0$ and $f=3$, we obtain the following values for the NLO components from (\[15\]),(\[16\]) (note that we remail only the terms $\sim O(1)$ in the NLO terms) $$\begin{aligned} {&&\hspace*{-1cm}}d^q_{--}~=~ \frac{16}{81} \Big[ 2\zeta (3) + 9 \zeta (2) - \frac{779}{108} \Big] \approx 1.97, ~~ d^g_{--}~=~ d^q_{--}~+~ \frac{28}{81} \approx 2.32 \nonumber \\ {&&\hspace*{-1cm}}\hat d^q_{++}~=~ \frac{2800}{81} , ~~ \overline d^q_{++}~=~ 32 \Big[ \zeta (3) + \frac{263}{216}\zeta (2) - \frac{372607}{69984} \Big] \approx -67.82 \nonumber \\ {&&\hspace*{-1cm}}\hat d^g_{++}~=~ \frac{1180}{81} , ~~ \overline d^g_{++}~=~ \overline d^q_{++}~+~ \frac{953}{27} -12\zeta (2) \approx -52.26 \label{17} \end{aligned}$$ and $$\begin{aligned} {&&\hspace*{-1cm}}\tilde A^+_q~ \simeq ~\frac{20}{3} \alpha_0 \Bigl[ A_g + \frac{4}{9} A_q \Bigr] ~+~ \frac{1}{27}\frac{4A_q(1-7.67 \alpha_0)+9A_g(1-8.71 \alpha_0)}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \nonumber \\ {&&\hspace*{-1cm}}\tilde A^+_g~ \simeq ~ \Bigl(A_g ~+~\frac{4}{9}A_q \Bigr) \Bigl(1-\frac{80}{9}\alpha_0 \Bigr) ~-~ \frac{4}{27}\frac{9A_g-A_q}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \Bigl( 1+ \frac{692}{81}\alpha_0) \nonumber \\ {&&\hspace*{-1cm}}\mbox{and } \tilde A_a^-~=~A_a ~-~ \tilde A_a^+ \label{18} \end{aligned}$$ It is useful to change in Eqs.(\[15\])-(\[18\]) from the quark PD to the SF $F_2(x,Q^2)$, which is connected in NLO approximation with the PD by the following way (see [@18]): $$\begin{aligned} F_2(x,Q^2)~=~ \Bigl( 1+\alpha(Q^2)B_q(1+\delta) \Bigr) \delta^2_s f_q(x,Q^2) ~+~ \alpha(Q^2)B_g(1+\delta) \delta^2_s f_g(x,Q^2), \label{19} \end{aligned}$$ where $\delta ^2_s = \sum_{i=1}^f/f \equiv <e_f^2>$ is the average charge square of the active quarks: $\delta ^2_s$ = (2/9 and 5/18) for $f$ = (3 and 4), respectively. The NLO corrections lead to the appearence in the r.h.s. of Eqs.(\[15\]) of the additional terms $\Bigl( 1+\alpha B_{\pm} \Bigr)/\Bigl( 1+\alpha_0 B_{\pm} \Bigr)$ and the necessarity to transform $\tilde A^{\pm}_q$ to $C^{\pm} \equiv F_2^{\pm}(x,Q^2)$ into the input parts. The final results for $F_2(x,Q^2)$ are in the form: $$\begin{aligned} {&&\hspace*{-1cm}}F_2(x,t)~=~ F_2^-(x,t)~+~ F_2^+(x,t) \nonumber \\ {&&\hspace*{-1cm}}F_2^-(x,t)~=~ C^-~\exp{(- d_-\tilde s -d_{--}^qp)} (1+\alpha B^-)/(1+\alpha_0 B^-) \nonumber \\ {&&\hspace*{-1cm}}F_2^+(x,t)~=~ \left\{ \begin{array}{ll} C^+ x^{(\hat d_+\tilde s + \hat d_{++}^q p)}\exp{(-\overline d_+\tilde s -\overline d_{++}^qp)}(1+\alpha B^+)/(1+\alpha_0 B^+) , & \mbox{ if } Q^2 \leq Q^2_c \\ F_2^+(x,t_c) \exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde s_c)-d_{++}^q(1+\delta_c)(p-p_c) \Bigr) } & \\ \biggl(1+ \alpha B^+(1+\delta_c) \biggr)/ \biggl(1+ \alpha_c B^+(1+\delta_c) \biggr), & \mbox{ if } Q^2>Q^2_c \end{array} \right. \label{20} \end{aligned}$$ where $$B^{\pm}~=~B_q ~+~ \frac{\gamma_{\pm}}{\gamma^{(0)}_{qg}}B_g,~~ C^{\pm}~=~\tilde A^{\pm}_q (1+\alpha_0 B^{\pm})$$ with the substitution of $A_q$ by $C \equiv F_2(x,Q^2_0)$ into Eq.(\[18\]) $\tilde A^{\pm}_q$ according $$\begin{aligned} {&&\hspace*{-1cm}}C~=~ \Bigl( 1+\alpha_0 B_q \Bigr) \delta^2_s A_q ~+~ \alpha_0 B_g \delta^2_s A_g, \label{21} \end{aligned}$$ For the gluon PD the situation is more simple: in Eq.(\[18\]) it is necessary to replace $A_q$ by $C$ according (\[21\]). For the concrete values of the LO and NLO AD at $\delta =0$ and $f=3$, we have for $Q^2$-evolution of $F_2(x,Q^2)$ and the gluon PD: $$\begin{aligned} {&&\hspace*{-1cm}}F_2(x,t)~=~ F_2^-(x,t)~+~ F_2^+(x,t),~~ f_g(x,t)~=~ f_g^-(x,t)~+~ f_g^+(x,t) \nonumber \\ {&&\hspace*{-1cm}}F_2^-(x,t)~=~ C^-~\exp{(- \frac{32}{81} \tilde s -1.97p)}(1-\frac{8}{9} \alpha )/(1-\frac{8}{9} \alpha_0 ) \nonumber \\ {&&\hspace*{-1cm}}F_2^+(x,t)~=~ \left\{ \begin{array}{ll} C^+ x^{(-\frac{4}{3} \tilde s + \frac{2800}{81} p)} \exp{\Bigl(- \frac{4}{3}(\varrho(\nu)+\frac{101}{108}) \tilde s +(\frac{2800}{81} \varrho(\nu)-67.82)p \Bigr)} & \\ \Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha \Bigr)/ \Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha_0 \Bigr) , & \mbox{if } Q^2 \leq Q^2_c \\ F_2^+(x,t_c) \exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde s_c)-d_{++}^q(1+\delta_c)(p-p_c) \Bigr) } & \\ \biggl(1+ \alpha B^+(1+\delta_c) \biggr)/ \biggl(1+ \alpha_c B^+(1+\delta_c) \biggr), & \mbox{if } Q^2>Q^2_c \end{array} \right. \label{22} \\ {&&\hspace*{-1cm}}f_g^-(x,t)~=~ A_g^-~\exp{(- \frac{32}{81} \tilde s -2.32p)}(1-\frac{8}{9} \alpha )/(1-\frac{8}{9} \alpha_0 ) \nonumber \\ {&&\hspace*{-1cm}}f_g^+(x,t)~=~ \left\{ \begin{array}{ll} A_g^+ x^{(-\frac{4}{3} \tilde s + \frac{1180}{81} p)}\exp{\Bigl(- \frac{4}{3}(\varrho(\nu)+\frac{101}{108}) \tilde s +(\frac{1180}{81} \varrho(\nu)-52.26)p \Bigr)} & \\ \Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha \Bigr)/ \Bigl(1+6[ln(\frac{1}{x})-\varrho(\nu)-\frac{101}{108}] \alpha_0 \Bigr) , & \mbox{if } Q^2 \leq Q^2_c \\ f_g^+(x,t_c) \exp{\Bigl(-d_+(1+\delta_c)(\tilde s-\tilde s_c)+d_{++}^a(1+\delta_c)(p-p_c) \Bigr) } & \\ \biggl(1+ \alpha B^+(1+\delta_c) \biggr)/ \biggl(1+ \alpha_c B^+(1+\delta_c) \biggr), & \mbox{ if } Q^2>Q^2_c \end{array} \right. \label{23} \end{aligned}$$ where $$\begin{aligned} {&&\hspace*{-1cm}}\tilde C^+~ \simeq ~\frac{2}{27} \Biggl( 26\alpha_0 \Bigl[ A_g + 2C \Bigr] ~+~ \frac{A_g(1-9.74 \alpha_0)+2C(1-7.82 \alpha_0)}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \Biggr) \nonumber \\ {&&\hspace*{-1cm}}\mbox{and } C^-~=~C {}~-~ C^+ \label{24} \\ {&&\hspace*{-1cm}}\tilde A^+_g~ \simeq ~A_g \Bigl(1-\frac{28}{3}\alpha_0 \Bigr) {}~+~2C ~-~ \frac{2}{27}\frac{2A_g(1+ \frac{590}{81}\alpha_0)-C(1+ \frac{572}{81}\alpha_0)}{ln(\frac{1}{x})-\varrho (\nu) - \frac{85}{108}} \nonumber \\ {&&\hspace*{-1cm}}\mbox{and } \tilde A_g^-~=~A_g ~-~ \tilde A_g^+ \label{25} \end{aligned}$$ Let us give some conclusions following from Eqs.(\[24\])-(\[25\]). It is clearly seen that the NLO corrections reduce the LO contributions. Indeed, the value of the subcritical Pomeron intercept, which increases as $ln(\alpha_0/\alpha)$ in the LO, obtaines the additional term $ \sim (\alpha_0 - \alpha)$ with the large (and opposite in sign to the LO term) numerical coefficient. Note that this coefficient is different for the quark and gluon PD, that is in the agreement with the recent $MRS(G)$ fit in [@19] and the data analysis by $ZEUS$ group (see [@20]). The intercept of the gluon PD is larger then the quark PD one (see also [@19; @20]). However, the effective reduction of the quark PD is smaller (that is in the agreement with W.-K. Tung analysis in [@21]), because the quark PD part increasing at small $x$ obtains the additional ($ \sim \alpha_0$ but not $ \sim 1/lnx $) term, which is important at very small $x$. Note that there is the fourth quark threshold at $Q^2_{th} \sim 10 GeV^2$ and the $Q^2_{th}$ value may be larger or smaller to $Q^2_c$ one. Then, either the solution in the r.h.s. of Eqs. (\[20\],\[22\],\[23\]) before the critical point $Q^2_c$ and the one for $Q^2 > Q^2_c$ contain the threshold transition, where the values of all variables are changed from ones at $f=3$ to ones at $f=4$. The $\alpha(Q^2)$ is smooth because $\Lambda^{f=3}_{\overline{MS}} \to \Lambda^{f=4}_{\overline{MS}}$ (see also the recent experimental test of the flavour independence of strong interactions into [@22]). For simplicity here we suppose that $Q^2_{th} = Q^2_c$ and all changes initiated by threshold are done authomatically: the first (at $Q^2 \leq Q^2_c$) solutions contain $f=3$ and second (at $Q^2 > Q^2_c$) ones have $f=4$, respectively. For the “$-$” component we should use $Q^2_{th}=Q^2_c$, too. Note only that the Pomeron intercept $\alpha_p = 1~-~(d_+ \tilde s + \hat d^q_{++}p)$ increases at $Q^2=Q^2_{th}$, because $$\alpha_p ~-~1 ~=~ \left\{ \begin{array}{ll} \frac{4}{3} \tilde s(Q^2_{th},Q^2_0)~-~ \frac{2800}{81} p(Q^2_{th},Q^2_0) , & \mbox{ if } Q^2 \leq Q^2_c \\ 1.44 \tilde s(Q^2_{th},Q^2_0)~-~ 38.11 p(Q^2_{th},Q^2_0) ,& \mbox{ if } Q^2>Q^2_c \end{array} \right.$$ that agrees with results [@23] obtained in the framework of dual parton model. The difference $$\bigtriangleup \alpha_p ~=~ 0.11 \tilde s(Q^2_{th},Q^2_0) - 3.55 p(Q^2_{th},Q^2_0)$$ dependes from the values of $Q^2_{th}$ and $Q^2_0$. For $Q^2_{th}=10GeV^2$ and $Q^2_0=1GeV^2$ it is very small: $$\bigtriangleup \alpha_p ~=~ 0.012$$ [**3.**]{} Let us resume the obtained results. We have got the DGLAP equation “solution” having the Regge form (\[1\]) for the two cases: at small $Q^2$ ($Q^2 \sim 1GeV^2$), where SF and PD have the flat behaviour at small $x$, and at large $Q^2$, where SF $F_2(x,Q^2)$ fastly increases when $x \to 0$. The behaviour in the flat case is nonstable with the perturbative viewpoint because it leads to the production of the subcritical value of pomeron intercept at larger $Q^2$ and the its increase (like $4/3~ ln(\alpha (Q^2_0)/\alpha(Q^2)$ in LO) when the $Q^2$ value increases[^8]. The solution in the Lipatov Pomeron case corresponds to the well-known results (see [@12; @14; @17]) with $Q^2$-independent Pomeron intercept. The general “solution” should contains the smooth transition between these pictures. Unfortunately, it is impossible to obtain it in the case of the simple approximation (\[1\]), because the r.h.s. of DGLAP equation (\[7\]) contains the both: $\sim x^{-\delta}$ and $\sim Const$, terms. As a result, we used two above “solutions” gluing in some point $Q^2_c$. Note that our “solution” is some generation (or a application) of the solution of DGLAP equation in the momentum space. The last one have two: ”+” and “$-$” components. The above our conclusions are related to the “+” component, which is the basic Regge asymptotic. The Pomeron intercept corresponding to “$-$” component, is $Q^2$-independent and this component is the subasymptotical one at large $Q^2$. However, the magnitude of the “+” is suppressed like $1/ln(1/x)$ and $\alpha (Q^2_0)$, and the subasymptotical “$-$” component may be important. Indeed, it is observed experimentally (see [@20.5; @20]). Note, however, that the suppression $\sim \alpha(Q^2_0)$ is really very slight if we choose a small value of $Q^2_0$. Our “solution” in the form of Eqs.(\[22\])-(\[25\]) is in the very well agreement with the recent $MRS(G)$ fit [@19] and with the results of [@17] at $Q^2=15GeV^2$. As it can be seen from Eqs.(\[22\]),(\[23\]), in our formulae there is the dependence on the PD behaviour at large $x$. Following to [@25] we choose $\nu =5$ that agrees in the gluon case with the quark counting rule [@26]. This $\nu$ value is also close to the values obtained by $CCFR$ group [@27] ($\nu = 4$) and in the last $MRS(G)$ analysis [@19] ($\nu =6$). Note that this dependence is strongly reduced for the gluon PD in the form $$f_g(x,Q^2_0)~=~A_g(\nu)(1-x)^{\nu},$$ if we suppose that the proton’s momentum is carred by gluon, is $\nu$-independent. We used $A_g(5)=2.1$ and $F_2(x,Q^2_0)=0.3$ when $x \to 0$. For the quark PD the choise $\nu =3$ is more preferable, however the use of two different $\nu$ values complicates the analysis. Because the quark contribution to the “+” component is not large, we put $\nu =5$ to both: quark and gluon cases. Note also that the variable $\nu (Q^2)$ have (see [@28]) the $Q^2$-dependence determinated by the LO AD $\gamma^{(0)}_{NS}$. However this $Q^2$-dependence is proportional $s$ and it is not important in our analysis. Starting from $Q^2_0=1GeV^2$ (by analogy with [@23.6]) and from $Q^2_0=2GeV^2$, and using two values of QCD parameter $\Lambda$: more standard one ($\Lambda^{f=4}_{\overline {MS}}$ = 200 $MeV$) and ($\Lambda^{f=4}_{\overline {MS}}$= 255 $MeV$) obtained in [@19], we have the following values of the quark and gluon PD “intercepts” $\delta_a ~=~-(d_+ \tilde s +\hat d^q_{++}a)$ (here $\Lambda^{f=4}_{\overline {MS}}$ is marked as $\Lambda$): if $Q^2_0$ = 1 $GeV^2$ ------- ------------------- ------------------- ------------------- ----------------- $Q^2$ $\delta_q(Q^2)$ $ \delta_g(Q^2)$ $\delta_q(Q^2)$ $\delta_g(Q^2)$ $\Lambda =200MeV$ $\Lambda =200MeV$ $\Lambda =255MeV$ $\Lambda =255MeV$ 4 0.191 0.389 0.165 0.447 10 0.318 0.583 0.295 0.659 15 0.367 0.652 0.345 0.734 ------- ------------------- ------------------- ------------------- ----------------- if $Q^2_0$ = 2 $GeV^2$ ------- ------------------- ------------------- ------------------- ----------------- $Q^2$ $\delta_q(Q^2)$ $ \delta_g(Q^2)$ $\delta_q(Q^2)$ $\delta_g(Q^2)$ $\Lambda =200MeV$ $\Lambda =200MeV$ $\Lambda =255MeV$ $\Lambda =255MeV$ 4 0.099 0.175 0.097 0.198 10 0.226 0.368 0.227 0.410 15 0.275 0.438 0.278 0.486 ------- ------------------- ------------------- ------------------- ----------------- Note that these values of $\delta_a $ are above the ones from [@19]. Because we have the second (subasymptotical) part, the effective our “intercepts” have the smaller values. As a conclusion, we note that BFKL equation (and thus the value of Lipatov Pomeron intercept) was obtained in [@19.5] in the framework of perturbative QCD. The large-$Q^2$ $HERA$ experimental data are in the good agreement with Lipatov’s trajectory and thus with perturbative QCD. The small $Q^2$ data agrees with the standard Pomeron intercept $\alpha_p=1$ or with Donnachie-Landshoff pisture: $\alpha_p=1.08$. Perhaps, this range requires already the knowledge of nonperturbative QCD dynamics and perturbative solutions (including BFKL one) should be not applied here directly and are corrected by some nonperturbative contributions. In our analysis Eq.(\[1\]) can be considered as the nonperturbative (Regge-type) input at $Q^2_0 \sim 1GeV^2$. Above $Q^2_0$ the PD behaviour obeys DGLAP equation, Pomeron moves to the subcritical regime and tends to its perturbative value. After some $Q^2_c$, where its perturbative value was already attained, Pomeron intercept saves the permanent value. The application of this approach to analyse small $x$ data invites futher investigation. [99]{} H1 Collab.: T.Ahmed et al., $DESY$ preprint 95-006 (1995). ZEUS Collab.: M.Derrick et al., $DESY$ preprint 94-143 (1994). NM Collab.: P.Amadrus et al., [*Phys.Lett.*]{} [**B295**]{}, (1992) 159, [**B309**]{}, (1993) 222. E665 Collab.: in the B.Badelek’s report “Low $Q^2$, low $x$ in electroproduction. An overview.”. In Proceeding de Moriond on QCD and high energy hadron interactions (1995) Les Arc. A.Levy, $DESY$ preprint 95-003 (1995). J.D. Bjorken, In Proceeding of the International Workshop on DIS, Eilat, Izrael, Feb.1994. V.N.Gribov and L.N.Lipatov, [*Sov.J.Nucl.Phys.*]{} [**18**]{}, (1972) 438; L.N.Lipatov, [*Yad.Fiz.*]{} [**20**]{}, (1974) 181; G.Altarelli and G.Parisi, [*Nucl.Phys.*]{} [**B126**]{}, (1977) 298; Yu.L.Dokshitzer, [*ZHETF*]{} [**46**]{} (1977) 641. V.N.Gribov, E.M.Levin and M.G.Ryskin, [*Phys.Rep.*]{} [**100**]{} (1983) 1; E.M.Levin and M.G.Ryskin, [*Phys.Rep.*]{} [**189**]{} (1990) 267. V.I.Vovk, A.V.Kotikov and S.J.Maximov, [*Teor.Mat.Fiz.*]{} [**84**]{} (1990) 101; A.V.Kotikov, S.I.Maximov and I.S. Parobij, Preprint ITP-93-21E (1993) Kiev, [*Teor.Mat.Fiz.*]{} (1995) in press. M.Virchaux and A.Milsztain, [*Phys.Lett.*]{} [**B274**]{} (1992) 221. A.V.Kotikov, work in progress L.L.Enkovszky, A.V.Kotikov and F.Paccanoni, [ *Yad.Fiz.*]{} [**55**]{} (1993) 2205. A.V.Kotikov, [*Yad.Fiz.*]{} [**56**]{} (1993) N9, 217. A.V.Kotikov, [*Yad.Fiz.*]{} [**57**]{} (1994) 142; [*Phys.Rev.*]{} [**D49**]{} (1994) 5746. R.K.Ellis, E.Levin and Z.Kunst, [*Nucl.Phys.*]{} [**420B**]{} (1994) 514. R.K.Ellis, F.Hautmann and B.R.Webber, [*Phys.Lett.*]{} [**B348**]{} (1995) 582. N.N.Nikolaev, B.G.Zakharov and V.R.Zoller, [ *Phys.Lett.*]{} [**B328**]{}, (1994) 486; N.N.Nikolaev and B.G.Zakharov, [*Phys.Lett.*]{} [**B327**]{}, (1994) 149. C.Lopez and F.J.Yndurain, [*Nucl.Phys.*]{} [**171B**]{} (1980) 231; [**183B**]{} (1981) 157; A.M.Cooper-Sarkar, G.Ingelman, K.R.Long, R.G.Roberts and D.H.Saxon, [*Z.Phys.*]{} [**C39**]{} (1988) 281; A.V.Kotikov, $JINR$ preprints P2-88-139, E2-88-422 (1988) Dubna (unpublished). G.M.Frichter, D.W.McKay and J.P.Ralston, [*Phys.Rev.Lett.*]{} [**74**]{} (1995) 1508. M.Bertini, P.Desgrolard, M.Giffon, L.Jenkovszky and F.Paccanoni, Preprint LYCEN/9366 (1993). A.J.Buras, [*Rev.Mod.Phys.*]{} [**52**]{} (1980) 149. E.A.Kuraev, L.N.Lipatov and V.S.Fadin, [*ZHETF*]{} [**53**]{} (1976) 2018, [**54**]{} (1977) 128; Ya.Ya.Balitzki and L.N.Lipatov, [*Yad.Fiz.*]{} [**28**]{} (1978) 822; L.N.Lipatov, [*ZHETF*]{} [**63**]{} (1986) 904. M.Giafaloni, [*Nucl.Phys.*]{} [**B296**]{}, (1987) 249; S.Catani, F.Fiorani and G.Marchesini, [*Phys.Lett.*]{} [**B234**]{} (1990) 389, [*Nucl.Phys.*]{} [**B336**]{} (1990) 18; S.Catani, F.Fiorani, G.Marchesini and G.Oriani, [*Nucl.Phys.*]{} [**B361**]{} (1991) 645; G.Wolf, $DESY$ preprint 94-022 (1994). ZEUS Collab.: M.Derrick et al., [*Phys.Lett.*]{} [**B345**]{}, (1995) 576. A.D.Martin, W.S.Stirling and R.G.Roberts, Preprint RAL-95-021, DTP/95/14 (1995). W.K.Tung, [*Nucl.Phys.*]{} [**B315**]{} (1989) 378. SLD Collab.: K.Abe et al., preprint $SLAC$-PUB-6687 (1995), submitted to [*Phys.Rev.Lett.*]{}. A.Capella, U.Sukhatme,C.-I.Tan and J.Tran Thanh Van, [*Phys.Rep.*]{} [**236**]{} (1993) 225; [*Phys.Rev.*]{} [**D36**]{} (1987) 109. A.Capella, A.Kaidalov, C.Merino,and J.Tran Thanh Van, [*Phys.Lett.*]{} [**B337**]{} (1994) 358; M.Bertini, M.Giffon and E.Predazzi, Preprint LYCEN/9504 (1995). R.D.Ball and S.Forte, [*Phys.Lett.*]{} [**B336**]{} (1994) 77; preprints CERN-TH-7422-94 (1994), CERN-TH-95-1(1995). A.Donnachie and P.V.Landshoff, [*Nucl.Phys.*]{} [**B303**]{} (1988) 634. S.Brodsky and G.Farrar, [*Phys.Rev.Lett.*]{} [**31**]{} (1973) 1153; V.Matveev, R.Muradyan and A.Tavkhelidze, [ *Lett. Nouvo Cim.*]{} [**7**]{} (1973) D654.. CCFR Collab.: R.Z.Quintas et al., [*Phys.Rev.Lett.*]{} [**71**]{} (1993) 1307. D.J.Gross, [*Phys.Rev.Lett.*]{} [**32**]{} (1974) 1071. [^1]: More correctly, $\phi$ is $Q^2$-dependent for the solution of DGLAP equation with the boundary condition: $f_a(x,Q^2_0)= Const$ at $x \to 0$. In the case of the boundary condition: $f_a(x,Q^2_0) \sim \exp{\sqrt{ ln(1/x)}}$, $\phi$ is lost (see [@8]) its $Q^2$-dependence [^2]: We use the termin “solution” because we will work in the leading twist approximation in the range of $Q^2$: $Q^2>1GeV^2$, where the higher twist terms may give the sizeable contribution (see, for example, [@9]). Moreover, our “solution” is the Regge asymptotic with unknown parameters rather then the solution of DGLAP equation. The parameters are found from the agreement of the r.h.s. and l.h.s. of the equation. [^3]: Consideration of the more complicate behaviour in the form $x^{-\delta}(ln(1/x))^b I_{2g}(\sqrt{\phi ln(1/x)})$ is given in [@8] and will be considered in this content in the forthcomming article [@10] [^4]: In the double-logarithmical approximation the similar results were obtained in [@15.5] [^5]: The method is based on the earlier results [@16] [^6]: The used formula (Eq.(2) from [@17]) coincides with (\[6\]) in the leading order (LO) approximation, if we save only $f_g(x,Q^2)$ in the r.h.s. of (\[2\]) (or put $\gamma_{qq}=0$ and $\gamma_{qg}=0$ formally). Eq.(\[6\]) and Eq.(\[2\]) from [@17] have some differences in the next-to-leading order (NLO), which are not very important because they are corrections to the $\alpha$-correction. [^7]: The form $\exp \Bigl({ -s \tilde \gamma_+(1+\delta)/(2\beta_0)} \Bigr)$ coincides with the both solution: Eq.(\[9\]) if $x^{\hat d_+} >>1$ and Eq.(\[11\]) when $\delta =0$ but it is not the solution of DGLAP equation. [^8]: The Pomeron intercept value increasing with $Q^2$ was obtained also in [@23.5].

ArXiv

--- abstract: | Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude $a_P\sim 8\times 10^{-8}$ cm/s$^2$, directed towards the Sun. Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction. There is now further data for the Pioneer 10 orbit determination. The extended Pioneer 10 data set spans 3 January 1987 to 22 July 1998. \[For Pioneer 11 the shorter span goes from 5 January 1987 to the time of loss of coherent data on 1 October 1990.\] With these data sets and more detailed studies of all the systematics, we now give a result, of $a_P = (8.74 \pm 1.33) \times 10^{-8} ~~{\rm cm/s}^2$. (Annual/diurnal variations on top of $a_P$, that leave $a_P$ unchanged, are also reported and discussed.) author: - | [John D. Anderson]{},[^1]$^a$ [Philip A. Laing]{},[^2]$^b$ [Eunice L. Lau]{},[^3]$^a$\ [Anthony S. Liu]{},[^4]$^c$ [Michael Martin Nieto]{},[^5]$^d$ and [Slava G. Turyshev]{}[^6]$^a$\      date: 11 April 2002 title: Study of the anomalous acceleration of Pioneer 10 and 11 --- \[intro\]INTRODUCTION ===================== Some thirty years ago, on 2 March 1972, Pioneer 10 was launched on an Atlas/Centaur rocket from Cape Canaveral. Pioneer 10 was Earth’s first space probe to an outer planet. Surviving intense radiation, it successfully encountered Jupiter on 4 December 1973 [@science]-[@pioweb]. In trail-blazing the exploration of the outer solar system, Pioneer 10 paved the way for, among others, Pioneer 11 (launched on 5 April 1973), the Voyagers, Galileo, Ulysses, and the upcoming Cassini encounter with Saturn. After Jupiter and (for Pioneer 11) Saturn encounters, the two spacecraft followed hyperbolic orbits near the plane of the ecliptic to opposite sides of the solar system. Pioneer 10 was also the first mission to enter the edge of interstellar space. That major event occurred in June 1983, when Pioneer 10 became the first spacecraft to “leave the solar system” as it passed beyond the orbit of the farthest known planet. The scientific data collected by Pioneer 10/11 has yielded unique information about the outer region of the solar system. This is due in part to the spin-stabilization of the Pioneer spacecraft. At launch they were spinning at approximately 4.28 and 7.8 revolutions per minute (rpm), respectively, with the spin axes running through the centers of the dish antennae. Their spin-stabilizations and great distances from the Earth imply a minimum number of Earth-attitude reorientation maneuvers are required. This permits precise acceleration estimations, to the level of $10^{-8}$ cm/s$^2$ (single measurement accuracy averaged over 5 days). Contrariwise, a Voyager-type three-axis stabilized spacecraft is not well suited for a precise celestial mechanics experiment as its numerous attitude-control maneuvers can overwhelm the signal of a small external acceleration. In summary, Pioneer spacecraft represent an ideal system to perform precision celestial mechanics experiments. It is relatively easy to model the spacecraft’s behavior and, therefore, to study small forces affecting its motion in the dynamical environment of the solar system. Indeed, one of the main objectives of the Pioneer extended missions (post Jupiter/Saturn encounters) [@extended] was to perform accurate celestial mechanics experiments. For instance, an attempt was made to detect the presence of small bodies in the solar system, primarily in the Kuiper belt. It was hoped that a small perturbation of the spacecraft’s trajectory would reveal the presence of these objects [@jdakuiper]-[@pulsar]. Furthermore, due to extremely precise navigation and a high quality tracking data, the Pioneer 10 scientific program also included a search for low frequency gravitational waves [@anderson85; @anderson93]. Beginning in 1980, when at a distance of 20 astronomical units (AU) from the Sun the solar-radiation-pressure acceleration on Pioneer 10 [*away*]{} from the Sun had decreased to $< 5 \times 10^{-8}$ cm/s$^2$, we found that the largest systematic error in the acceleration residuals was a constant bias, $a_P$, directed [*toward*]{} the Sun. Such anomalous data have been continuously received ever since. Jet Propulsion Laboratory (JPL) and The Aerospace Corporation produced independent orbit determination analyses of the Pioneer data extending up to July 1998. We ultimately concluded [@anderson; @moriond], that there is an unmodeled acceleration, $a_P$, towards the Sun of $\sim 8 \times 10^{-8}$ cm/s$^2$ for both Pioneer 10 and Pioneer 11. The purpose of this paper is to present a detailed explanation of the analysis of the apparent anomalous, weak, long-range acceleration of the Pioneer spacecraft that we detected in the outer regions of the solar system. We attempt to survey all sensible forces and to estimate their contributions to the anomalous acceleration. We will discuss the effects of these small non-gravitational forces (both generated on-board and external to the vehicle) on the motion of the distant spacecraft together with the methods used to collect and process the radio Doppler navigational data. We begin with descriptions of the spacecraft and other systems and the strategies for obtaining and analyzing information from them. In Section \[pioneer\] we describe the Pioneer (and other) spacecraft. We provide the reader with important technical information on the spacecraft, much of which is not easily accessible. In Section \[Exp\_tech\] we describe how raw data is obtained and analyzed and in Section \[navigate\] we discuss the basic elements of a theoretical foundation for spacecraft navigation in the solar system. The next major part of this manuscript is a description and analysis of the results of this investigation. We first describe how the anomalous acceleration was originally identified from the data of all the spacecraft in Section \[results\] [@anderson; @moriond]. We then give our recent results in Section \[recent\_results\]. In the following three sections we discuss possible experimental systematic origins for the signal. These include systematics generated by physical phenomena from sources external to (Section \[ext-systema\]) and internal to (Section \[int-systema\]) the spacecraft. This is followed by Section \[Int\_accuracy\], where the accuracy of the solution for $a_P$ is discussed. In the process we go over possible numerical/calculational errors/systematics. Sections \[ext-systema\]-\[Int\_accuracy\] are then summarized in the total error budget of Section \[budget\]. We end our presentation by first considering possible unexpected physical origins for the anomaly (Section \[newphys\]). In our conclusion, Section \[disc\], we summarize our results and suggest venues for further study of the discovered anomaly. \[pioneer\]THE PIONEER AND OTHER SPACECRAFT =========================================== In this section we describe in some detail the Pioneer 10 and 11 spacecraft and their missions. We concentrate on those spacecraft systems that play important roles in maintaining the continued function of the vehicles and in determining their dynamical behavior in the solar system. Specifically we present an overview of propulsion and attitude control systems, as well as thermal and communication systems. Since our analysis addresses certain results from the Galileo and Ulysses missions, we also give short descriptions of these missions in the final subsection. General description of the Pioneer spacecraft {#sec:pio_description} --------------------------------------------- Although some of the more precise details are often difficult to uncover, the general parameters of the Pioneer spacecraft are known and well documented [@science]-[@pioweb]. The two spacecraft are identical in design [@design]. At launch each had a “weight” (mass) of 259 kg. The “dry weight” of the total module was 223 kg as there were 36 kg of hydrazine propellant [@mass; @gasuse]. The spacecraft were designed to fit within the three meter diameter shroud of an added third stage to the Atlas/Centaur launch vehicle. Each spacecraft is 2.9 m long from its base to its cone-shaped medium-gain antenna. The high gain antenna (HGA) is made of aluminum honeycomb sandwich material. It is 2.74 m in diameter and 46 cm deep in the shape of a parabolic dish. (See Figures \[fig:pio\_design\] and \[fig:trusters\].) -10pt The main equipment compartment is 36 cm deep. The hexagonal flat top and bottom have 71 cm long sides. The equipment compartment provides a thermally controlled environment for scientific instruments. Two three-rod trusses, 120 degrees apart, project from two sides of the equipment compartment. At their ends, each holds two SNAP-19 (Space Nuclear Auxiliary Power, model 19) RTGs (Radioisotope Thermoelectric Generators) built by Teledyne Isotopes for the Atomic Energy Commission. These RTGs are situated about 3 m from the center of the spacecraft and generate its electric power. \[We will go into more detail on the RTGs in Section \[int-systema\].\] A third single-rod boom, 120 degrees from the other two, positions a magnetometer about 6.6 m from the spacecraft’s center. All three booms were extended after launch. With the mass of the magnetometer being 5 kg and the mass of each of the four RTGs being 13.6 kg, this configuration defines the main moment of inertia along the $z$-spin-axis. It is about ${\cal I}_{\tt z} \approx 588.3$ kg m$^2$ [@vanallen]. \[Observe that this all left only about 164 kg for the main bus and superstructure, including the antenna.\] Figures \[fig:pio\_design\] and \[fig:trusters\] show the arrangement within the spacecraft equipment compartment. The majority of the spacecraft electrical assemblies are located in the central hexagonal portion of the compartment, surrounding a 16.5-inch-diameter spherical hydrazine tank. Most of the scientific instruments’ electronic units and internally-mounted sensors are in an instrument bay (“squashed” hexagon) mounted on one side of the central hexagon. The equipment compartment is in an aluminum honeycomb structure. This provides support and meteoroid protection. It is covered with insulation which, together with louvers under the platform, provides passive thermal control. \[An exception is from off-on control by thermal power dissipation of some subsystems. (See Sec. \[int-systema\]).\] Propulsion and attitude control systems {#sec:prop} --------------------------------------- Three pairs of these rocket thrusters near the rim of the HGA provide a threefold function of spin-axis precession, mid-course trajectory correction, and spin control. Each of the three thruster pairs develops its repulsive jet force from a catalytic decomposition of liquid hydrazine in a small rocket thrust chamber attached to the oppositely-directed nozzle. The resulted hot gas is then expended through six individually controlled thruster nozzles to effect spacecraft maneuvers. The spacecraft is attitude-stabilized by spinning about an axis which is parallel to the axis of the HGA. The nominal spin rate for Pioneer 10 is 4.8 rpm. Pioneer 11 spins at approximately 7.8 rpm because a spin-controlling thruster malfunctioned during the spin-down shortly after launch. \[Because of the danger that the thruster’s valve would not be able to close again, this particular thruster has not been used since.\] During the mission an Earth-pointing attitude is required to illuminate the Earth with the narrow-beam HGA. Periodic attitude adjustments are required throughout the mission to compensate for the variation in the heliocentric longitude of the Earth-spacecraft line. \[In addition, correction of launch vehicle injection errors were required to provide the desired Jupiter encounter trajectory and Saturn (for Pioneer 11) encounter trajectory.\] These velocity vector adjustments involved reorienting the spacecraft to direct the thrust in the desired direction. There were no anomalies in the engineering telemetry from the propulsion system, for either spacecraft, during any mission phase from launch to termination of the Pioneer mission in March 1997. From the viewpoint of mission operations at the NASA/Ames control center, the propulsion system performed as expected, with no catastrophic or long-term pressure drops in the propulsion tank. Except for the above-mentioned Pioneer 11 spin-thruster incident, there was no malfunction of the propulsion nozzles, which were only opened every few months by ground command. The fact that pressure was maintained in the tank has been used to infer that no impacts by Kuiper belt objects occurred, and a limit has been placed on the size and density distribution of such objects [@jdakuiper], another useful scientific result. For attitude control, a star sensor (referenced to Canopus) and two sunlight sensors provided reference for orientation and roll maneuvers. The star sensor on Pioneer 10 became inoperative at Jupiter encounter, so the sun sensors were used after that. For Pioneer 10, spin calibration was done by the DSN until 17 July 1990. From 1990 to 1993 determinations were made by analysts using data from the Imaging Photo Polarimeter (IPP). After the 6 July 1993 maneuver, there was not enough power left to support the IPP. But approximately every six months analysts still could get a rough determination using information obtained from conscan maneuvers [@conscan] on an uplink signal. When using conscan, the high gain feed is off-set. Thruster firings are used to spiral in to the correct pointing of the spacecraft antenna to give the maximum signal strength. To run this procedure (conscan and attitude) it is now necessary to turn off the traveling-wave-tube (TWT) amplifier. So far, the power and tube life-cycle have worked and the Jet Propulsion Laboratory’s (JPL) Deep Space Network (DSN) has been able to reacquire the signal. It takes about 15 minutes or so to do a maneuver. \[The magnetometer boom incorporates a hinged, viscous, damping mechanism at its attachment point, for passive nutation control.\] In the extended mission phase, after Jupiter and Saturn encounters, the thrusters have been used for precession maneuvers only. Two pairs of thrusters at opposite sides of the spacecraft have nozzles directed along the spin axis, fore and aft (See Figure \[fig:trusters\].) In precession mode, the thrusters are fired by opening one nozzle in each pair. One fires to the front and the other fires to the rear of the spacecraft [@rearfront], in brief thrust pulses. Each thrust pulse precesses the spin axis a few tenths of a degree until the desired attitude is reached. The two nozzles of the third thruster pair, no longer in use, are aligned tangentially to the antenna rim. One points in the direction opposite to its (rotating) velocity vector and the other with it. These were used for spin control. Thermal system and on-board power {#sec:onboard} --------------------------------- Early on the spacecraft instrument compartment is thermally controlled between $\approx$ $0$ F and 90 F. This is done with the aid of thermo-responsive louvers located at the bottom of the equipment compartment. These louvers are adjusted by bi-metallic springs. They are completely closed below $\sim40$ F and completely open above $\sim 85$ F. This allows controlled heat to escape in the equipment compartment. Equipment is kept within an operational range of temperatures by multi-layered blankets of insulating aluminum plastic. Heat is provided by electric heaters, the heat from the instruments themselves, and by twelve one-watt radioisotope heaters powered directly by non-fissionable plutonium ($^{238}_{~94}$Pu$ \rightarrow ^{234}_{~92}$U$+{}^4_2$He). $^{238}$Pu, with a half life time of 87.74 years, also provides the thermal source for the thermoelectric devices in the RTGs. Before launch, each spacecraft’s four RTGs delivered a total of approximately 160 W of electrical power [@tele; @Rconf]. Each of the four space-proven SNAP-19 RTGs converts 5 to 6 percent of the heat released from plutonium dioxide fuel to electric power. RTG power is greatest at 4.2 Volts; an inverter boosts this to 28 Volts for distribution. RTG life is degraded at low currents; therefore, voltage is regulated by shunt dissipation of excess power. The power subsystem controls and regulates the RTG power output with shunts, supports the spacecraft load, and performs battery load-sharing. The silver cadmium battery consists of eight cells of 5 ampere-hours capacity each. It supplies pulse loads in excess of RTG capability and may be used for sharing peak loads. The battery voltage is often discharged and charged. This can be seen by telemetry of the battery discharge current and charge current At launch each RTG supplied about 40 W to the input of the $\sim 4.2$ V Inverter Assemblies. (The output for other uses includes the DC bus at 28 V and the AC bus at 61 V) Even though electrical power degrades with time (see Section \[subsec:mainbus\]), at $-41$ F the essential platform temperature as of the year 2000 is still between the acceptable limits of $-63$ F to 180 F. The RF power output from the traveling-wave-tube amplifier is still operating normally. The equipment compartment is insulated from extreme heat influx with aluminized mylar and kapton blankets. Adequate warmth is provided by dissipation of 70 to 120 watts of electrical power by electronic units within the compartment; louvers regulating the release of this heat below the mounting platform maintain temperatures in the vicinity of the spacecraft equipment and scientific instruments within operating limits. External component temperatures are controlled, where necessary, by appropriate coating and, in some cases, by radioisotope or electrical heaters. The energy production from the radioactive decay obeys an exponential law. Hence, 29 years after launch, the radiation from Pioneer 10’s RTGs was about 80 percent of its original intensity. However the electrical power delivered to the equipment compartment has decayed at a faster rate than the $^{238}$Pu decays radioactively. Specifically, the electrical power first decayed very quickly and then slowed to a still fast linear decay [@lasher]. By 1987 the degradation rate was about $-2.6$ W/yr for Pioneer 10 and even greater for the sister spacecraft. This fast depletion rate of electrical power from the RTGs is caused by normal deterioration of the thermocouple junctions in the thermoelectric devices. The spacecraft needs 100 W to power all systems, including 26 W for the science instruments. Previously, when the available electrical power was greater than 100 W, the excess power was either thermally radiated into space by a shunt-resistor radiator or it was used to charge a battery in the equipment compartment. At present only about 65 W of power is available to Pioneer 10 [@theorypower]. Therefore, all the instruments are no longer able to operate simultaneously. But the power subsystem continues to provide sufficient power to support the current spacecraft load: transmitter, receiver, command and data handling, and the Geiger Tube Telescope (GTT) science instrument. As pointed out in Sec. \[subs:pioneer\], the science package and transmitter are turned off in extended cruise mode to provide enough power to fire the attitude control thrusters. Communication system -------------------- The Pioneer 10/11 communication systems use S-band ($\lambda\simeq 13$ cm) Doppler frequencies [@sband]. The communication uplink from Earth is at approximately 2.11 GHz. The two spacecraft transmit continuously at a power of eight watts. They beam their signals, of approximate frequency 2.29 GHz, to Earth by means of the parabolic 2.74 m high-gain antenna. Phase coherency with the ground transmitters, referenced to H-maser frequency standards, is maintained by means of an S-band transponder with the 240/221 frequency turnaround ratio (as indicated by the values of the above mentioned frequencies). The communications subsystem provides for: i) up-link and down-link communications; ii) Doppler coherence of the down-link carrier signal; and iii) generation of the conscan [@conscan] signal for closed loop precession of the spacecraft spin axis towards Earth. S-band carrier frequencies, compatible with DSN, are used in conjunction with a telemetry modulation of the down-link signal. The high-gain antenna is used to maximize the telemetry data rate at extreme ranges. The coupled medium-gain/omni-directional antenna with fore and aft elements respectively, provided broad-angle communications at intermediate and short ranges. For DSN acquisition, these three antennae radiate a non-coherent RF signal, and for Doppler tracking, there is a phase coherent mode with a frequency translation ratio of 240/221. Two frequency-addressable phase-lock receivers are connected to the two antenna systems through a ground-commanded transfer switch and two diplexers, providing access to the spacecraft via either signal path. The receivers and antennae are interchangeable through the transfer switch by ground command or automatically, if needed. There is a redundancy in the communication systems, with two receivers and two transmitters coupled to two traveling-wave-tube amplifiers. Only one of the two redundant systems has been used for the extended missions, however. At launch, communication with the spacecraft was at a data rate 256 bps for Pioneer 10 (1024 bps for Pioneer 11). Data rate degradation has been $-1.27$ mbps/day for Pioneer 10 ($-8.78$ mbps/day for Pioneer 11). The DSN still continues to provide good data with the received signal strength of about $-178$ dBm (only a few dB from the receiver threshold). The data signal to noise ratio is still mainly under 0.5 dB. The data deletion rate is often between 0 and 50 percent, at times more. However, during the test of 11 March 2000, the average deletion rate was about 8 percent. So, quality data are still available. Status of the extended mission {#subs:pioneer} ------------------------------ The Pioneer 10 mission officially ended on 31 March 1997 when it was at a distance of 67 AU from the Sun. (See Figure \[fig:pioneer\_path\].) At a now nearly constant velocity relative to the Sun of $\sim$12.2 km/s, Pioneer 10 will continue its motion into interstellar space, heading generally for the red star Aldebaran, which forms the eye of Taurus (The Bull) Constellation. Aldebaran is about 68 light years away and it would be expected to take Pioneer 10 over 2 million years to reach its neighborhood. A switch failure in the Pioneer 11 radio system on 1 October 1990 disabled the generation of coherent Doppler signals. So, after that date, when the spacecraft was $\sim 30$ AU away from the Sun, no useful data have been generated for our scientific investigation. Furthermore, by September 1995, its power source was nearly exhausted. Pioneer 11 could no longer make any scientific observations, and routine mission operations were terminated. The last communication from Pioneer 11 was received in November 1995, when the spacecraft was at distance of $\sim 40$ AU from the Sun. (The relative Earth motion carried it out of view of the spacecraft antenna.) The spacecraft is headed toward the constellation of Aquila (The Eagle), northwest of the constellation of Sagittarius, with a velocity relative to the Sun of $\sim$11.6 km/s Pioneer 11 should pass close to the nearest star in the constellation Aquila in about 4 million years [@pioweb]. (Pioneer 10 and 11 orbital parameters are given in the Appendix.) However, after mission termination the Pioneer 10 radio system was still operating in the coherent mode when commanded to do so from the Pioneer Mission Operations center at the NASA Ames Research Center (ARC). As a result, after 31 March 1997, JPL’s DSN was still able to deliver high-quality coherent data to us on a regular schedule from distances beyond 67 AU. Recently, support of the Pioneer spacecraft has been on a non-interference basis to other NASA projects. It was used for the purpose of training Lunar Prospector controllers in DSN coordination of tracking activities. Under this training program, ARC has been able to maintain contact with Pioneer 10. This has required careful attention to the DSN’s ground system, including the installation of advanced instrumentation, such as low-noise digital receivers. This extended the lifetime of Pioneer 10 to the present. \[Note that the DSN’s early estimates, based on instrumentation in place in 1976, predicted that radio contact would be lost about 1980.\] At the present time it is mainly the drift of the spacecraft relative to the solar velocity that necessitates maneuvers to continue keeping Pioneer 10 pointed towards the Earth. The latest successful precession maneuver to point the spacecraft to Earth was accomplished on 11 February 2000, when Pioneer 10 was at a distance from the Sun of 75 AU. \[The distance from the Earth was $\sim 76$ AU with a corresponding round-trip light time of about 21 hour.\] The signal level increased 0.5-0.75 dBm [@dBm] as a result of the maneuver. This was the seventh successful maneuver that has been done in the blind since 26 January 1997. At that time it had been determined that the electrical power to the spacecraft had degraded to the point where the spacecraft transmitter had to be turned off to have enough power to perform the maneuver. After 90 minutes in the blind the transmitter was turned back on again. So, despite the continued weakening of Pioneer 10’s signal, radio Doppler measurements were still available. The next attempt at a maneuver, on 8 July 2000, turned out in the end to be successful. Signal was tracked on 9 July 2001. Contact was reestablished on the 30th anniversary of launch, 2 March 2002. The Galileo and Ulysses missions and spacecraft {#othercraft} ----------------------------------------------- ### The Galileo mission {#galileocraft} The Galileo mission to explore the Jovian system [@johnson] was launched 18 October 1989 aboard the Space Shuttle Discovery. Due to insufficient launch power to reach its final destination at 5.2 AU, a trajectory was chosen with planetary flybys to gain gravity assists. The spacecraft flew by Venus on 10 February 1990 and twice by the Earth, on 8 December 1990 and on 8 December 1992. The current Galileo Millennium Mission continues to study Jupiter and its moons, and coordinated observations with the Cassini flyby in December 2000. The dynamical properties of the Galileo spacecraft are very well known. At launch the orbiter had a mass of 2,223 kg. This included 925 kg of usable propellant, meaning over 40% of the orbiter’s mass at launch was for propellant! The science payload was 118 kg and the probe’s total mass was 339 kg. Of this latter, the probe descent module was 121 kg, including a 30 kg science payload. The tensor of inertia of the spacecraft had the following components at launch: $J_{\tt xx}= 4454.7, J_{\tt yy}= 4061.2, J_{\tt zz}= 5967.6, J_{\tt xy}= -52.9, J_{\tt xz}= 3.21, J_{\tt yz}= -15.94$ in units of kg m$^2$. Based on the area of the sun-shade plus the booms and the RTGs we obtained a maximal cross-sectional area of 19.5 m$^2$. Each of the two of the Galileo’s RTGs at launch delivered of 285 W of electric power to the subsystems. Unlike previous planetary spacecraft, Galileo featured an innovative “dual spin” design: part of the orbiter would rotate constantly at about three rpm and part of the spacecraft would remain fixed in (solar system) inertial space. This means that the orbiter could easily accommodate magnetospheric experiments (which need to made while the spacecraft is sweeping) while also providing stability and a fixed orientation for cameras and other sensors. The spin rate could be increased to 10 revolutions per minute for additional stability during major propulsive maneuvers. Apparently there was a mechanical problem between the spinning and non-spinning sections. Because of this, the project decided to often use an all-spinning mode, of about 3.15 rpm. This was especially true close to the Jupiter Orbit Insertion (JOI), when the entire spacecraft was spinning (with a slower rate, of course). Galileo’s original design called for a deployable high-gain antenna (HGA) to unfurl. It would provide approximately 34 dB of gain at X-band (10 GHz) for a 134 kbps downlink of science and priority engineering data. However, the X-band HGA failed to unfurl on 11 April 1991. When it again did not deploy following the Earth fly-by in 1992, the spacecraft was reconfigured to utilize the S-band, 8 dB, omni-directional low-gain antenna (LGA) for downlink. The S-band frequencies are 2.113 GHz - up and 2.295 GHz - down, a conversion factor of 240/221 at the Doppler frequency transponder. This configuration yielded much lower data rates than originally scheduled, 8-16 bps through JOI [@LGA]. Enhancements at the DSN and reprogramming the flight computers on Galileo increased telemetry bit rate to 8-160 bps, starting in the spring of 1996. Currently, two types of Galileo navigation data are available, namely Doppler and range measurements. As mentioned before, an instantaneous comparison between the ranging signal that goes up with the ranging signal that comes down would yield an “instantaneous” two-way range delay. Unfortunately, an instantaneous comparison was not possible in this case. The reason is that the signal-to-noise ratio on the incoming ranging signal is small and a long integration time (typically minutes) must be used (for correlation purposes). During such long integration times, the range to the spacecraft is constantly changing. It is therefore necessary to “electronically freeze” the range delay long enough to permit an integration to be performed. The result represents the range at the moment of freezing [@anderson75; @Kinman92]. ### The Ulysses mission {#ulyssescraft} Ulysses was launched on 6 October 1990, also from the Space Shuttle Discovery, as a cooperative project of NASA and the European Space Agency (ESA). JPL manages the US portion of the mission for NASA’s Office of Space Science. Ulysses’ objective was to characterize the heliosphere as a function of solar latitude [@genU]. To reach high solar latitudes, its voyage took it to Jupiter on 8 February 1992. As a result, its orbit plane was rotated about 80 degrees out of the ecliptic plane. Ulysses explored the heliosphere over the Sun’s south pole between June and November, 1994, reaching maximum Southern latitude of 80.2 degrees on 13 September 1994. It continued in its orbit out of the plane of the ecliptic, passing perihelion in March 1995 and over the north solar pole between June and September 1995. It returned again to the Sun’s south polar region in late 2000. The total mass at launch was the sum of two parts: a dry mass of 333.5 kg plus a propellant mass of 33.5 kg. The tensor of inertia is given by its principal components $J_{\tt xx} =371.62, J_{\tt yy} = 205.51, J_{\tt zz} = 534.98$ in units kg m$^2$. The maximal cross section is estimated to be 10.056 m$^2$. This estimation is based on the radius of the antenna 1.65 m (8.556 m$^2$) plus the areas of the RTGs and part of the science compartment (yielding an additional $\approx$ 1.5 m$^2$). The spacecraft was spin-stabilized at 4.996 rpm. The electrical power is generated by modern RTGs, which are located much closer to the main bus than are those of the Pioneers. The power generated at launch was 285 W. Communications with the spacecraft are performed at X-band (for downlink at 20 W with a conversion factor of 880/221) and S-band (both for uplink 2111.607 MHz and downlink 2293.148 MHz, at 5 W with a conversion factor of 240/221). Currently both Doppler and range data are available for both frequency bands. While the main communication link is S-up/X-down, the S-down link was used only for radio-science purposes. Because of Ulysses’ closeness to the Sun and also because of its construction, any hope to model Ulysses for small forces might appear to be doomed by solar radiation pressure and internal heat radiation from the RTGs. However, because the Doppler signal direction is towards the Earth while the radiation pressure varies with distance and has a direction parallel the Sun-Ulysses line, in principle these effects could be separated. And again, there was range data. This all would make it easier to model non-gravitational acceleration components normal to the line of sight, which usually are poorly and not significantly determined. The Ulysses spacecraft spins at $\sim 5$ rpm around its antenna axis (4.996 rpm initially). The angle of the spin axis with respect to the spacecraft-Sun line varies from near zero at Jupiter to near 50 degrees at perihelion. Any on-board forces that could perturb the spacecraft trajectory are restricted to a direction along the spin axis. \[The other two components are canceled out by the spin.\] As the spacecraft and the Earth travel around the Sun, the direction from the spacecraft to the Earth changes continuously. Regular changes of the attitude of the spacecraft are performed throughout the mission to keep the Earth within the narrow beam of about one degree full width of the spacecraft–fixed parabolic antenna. \[Exp\_tech\]DATA ACQUISITION AND PREPARATION ============================================= Discussions of radio-science experiments with spacecraft in the solar system requires at least a general knowledge of the sophisticated experimental techniques used at the DSN complex. Since its beginning in 1958 the DSN complex has undergone a number of major upgrades and additions. This was necessitated by the needs of particular space missions. \[The last such upgrade was conducted for the Cassini mission when the DSN capabilities were extended to cover the Ka radio frequency bandwidth. For more information on DSN methods, techniques, and present capabilities, see [@dsn].\] For the purposes of the present analysis one will need a general knowledge of the methods and techniques implemented in the radio-science subsystem of the DSN complex. This section reviews the techniques that are used to obtain the radio tracking data from which, after analysis, results are generated. Here we will briefly discuss the DSN hardware that plays a pivotal role for our study of the anomalous acceleration. Data acquisition ---------------- The Deep Space Network (DSN) is the network of ground stations that are employed to track interplanetary spacecraft [@dsn; @dsn82]. There are three ground DSN complexes, at Goldstone, California, at Robledo de Chavela, outside Madrid, Spain, and at Tidbinbilla, outside Canberra, Australia. There are many antennae, both existing and decommissioned, that have been used by the DSN for spacecraft navigation. For our four spacecraft (Pioneer 10, 11, Galileo, and Ulysses), depending on the time period involved, the following Deep Space Station (DSS) antennae were among those used: (DSS 12, 14, 24) at the California antenna complex; (DSS 42, 43, 45, 46) at the Australia complex; and (DSS 54, 61, 62, 63) at the Spain complex. Specifically, the Pioneers used (DSS 12, 14, 42, 43, 62, 63), Galileo used (DSS 12, 14, 42, 43, 63), and Ulysses used (DSS 12, 14, 24, 42, 43, 46, 54, 61, 63). The DSN tracking system is a phase coherent system. By this we mean that an “exact” ratio exists between the transmission and reception frequencies; i.e., 240/221 for S-band or 880/221 for X-band [@sband]. (This is in distinction to the usual concept of coherent radiation used in atomic and astrophysics.) Frequency is an average frequency, defined as the number of cycles per unit time. Thus, accumulated phase is the integral of frequency. High measurement precision is attained by maintaining the frequency accuracy to 1 part per $10^{12}$ or better (This is in agreement with the expected Allan deviation for the S-band signals.) [ **The DSN Frequency and Timing System (FTS):  **]{} The DSN’s FTS is the source for the high accuracy just mentioned (see Figure \[fig:dsn\_block\]). At its center is an hydrogen maser that produces a precise and stable reference frequency [@barnes; @vessot74]. These devices have Allan deviations [@SFJ98] of approximately $3\times 10^{-15}$ to $1\times 10^{-15}$ for integration times of $10^2$ to $10^3$ seconds, respectively. -10pt These masers are good enough so that the quality of Doppler-measurement data is limited by thermal or plasma noise, and not by the inherent instability of the frequency references. Due to the extreme accuracy of the hydrogen masers, one can very precisely characterize the spacecraft’s dynamical variables using Doppler and range techniques. The FTS generates a 5 MHz and 10 MHz reference frequency which is sent through the local area network to the Digitally Controlled Oscillator (DCO). [**The Digitally Controlled Oscillator (DCO) and Exciter: **]{} Using the highly stable output from the FTS, the DCO, through digitally controlled frequency multipliers, generates the Track Synthesizer Frequency (TSF) of $\sim 22$ MHz. This is then sent to the Exciter Assembly. The Exciter Assembly multiplies the TSF by 96 to produce the S-band carrier signal at $\sim 2.2$ GHz. The signal power is amplified by Traveling Wave Tubes (TWT) for transmission. If ranging data are required, the Exciter Assembly adds the ranging modulation to the carrier. \[The DSN tracking system has undergone many upgrades during the 29 years of tracking Pioneer 10. During this period internal frequencies have changed.\] This S-band frequency is sent to the antenna where it is amplified and transmitted to the spacecraft. The onboard receiver tracks the up-link carrier using a phase lock loop. To ensure that the reception signal does not interfere with the transmission, the spacecraft (e.g., Pioneer) has a turnaround transponder with a ratio of 240/221. The spacecraft transmitter’s local oscillator is phase locked to the up-link carrier. It multiplies the received frequency by the above ratio and then re-transmits the signal to Earth. [**Receiver and Doppler Extractor: **]{} When the two-way [@way] signal reaches the ground, the receiver locks on to the signal and tunes the Voltage Control Oscillator (VCO) to null out the phase error. The signal is sent to the Doppler Extractor. At the Doppler Extractor the current transmitter signal from the Exciter is multiplied by 240/221 (or 880/241 for X-band)) and a bias, of 1 MHz for S-band or 5 MHz for X-band [@sband], is added to the Doppler. The Doppler data is no longer modulated at S-band but has been reduced as a consequence of the bias to an intermediate frequency of 1 or 5 MHz Since the light travel time to and from Pioneer 10 is long (more than 20 hours), the transmitted frequency and the current transmitted frequency can be different. The difference in frequencies are recorded separately and are accounted for in the orbit determination programs we discuss in Section \[results\]. [**Metric Data Assembly (MDA): **]{} The MDA consists of computers and Doppler counters where continuous count Doppler data are generated. The intermediate frequency (IF) of 1 or 5 MHz with a Doppler modulation is sent to the Metric Data Assembly (MDA). From the FTS a 10 pulse per second signal is also sent to the MDA for timing. At the MDA, the IF and the resulting Doppler pulses are counted at a rate of 10 pulses per second. At each tenth of a second, the number of Doppler pulses are counted. A second counter begins at the instant the first counter stops. The result is continuously-counted Doppler data. (The Doppler data is a biased Doppler of 1 MHz, the bias later being removed by the analyst to obtain the true Doppler counts.) The Range data (if present) together with the Doppler data is sent separately to the Ranging Demodulation Assembly. The accompanying Doppler data is used to rate aid (i.e., to “freeze” the range signal) for demodulation and cross correlation. [**Data Communication: **]{} The total set of tracking data is sent by local area network to the communication center. From there it is transmitted to the Goddard Communication Facility via commercial phone lines or by government leased lines. It then goes to JPL’s Ground Communication Facility where it is received and recorded by the Data Records Subsystem. Radio Doppler and range techniques {#Dopp_tech} ---------------------------------- Various radio tracking strategies are available for determining the trajectory parameters of interplanetary spacecraft. However, radio tracking Doppler and range techniques are the most commonly used methods for navigational purposes. The position and velocities of the DSN tracking stations must be known to high accuracy. The transformation from a Earth fixed coordinate system to the International Earth Rotation Service (IERS) Celestial System is a complex series of rotations that includes precession, nutation, variations in the Earth’s rotation ([UT1-UTC]{}) and polar motion. Calculations of the motion of a spacecraft are made on the basis of the range time-delay and/or the Doppler shift in the signals. This type of data was used to determine the positions, the velocities, and the magnitudes of the orientation maneuvers for the Pioneer, Galileo, and Ulysses spacecraft considered in this study. Theoretical modeling of the group delays and phase delay rates are done with the orbit determination software we describe in the next section. [**Data types:**]{} Our data describes the observations that are the basis of the results of this paper. We receive our data from DSN in closed-loop mode, i.e., data that has been tracked with phase lock loop hardware. (Open loop data is tape recorded but not tracked by phase lock loop hardware.) The closed-loop data constitutes our Archival Tracking Data File (ATDF), which we copy [@datatapes] to the National Space Science Data Center (NSSDC) on magnetic tape. The ATDF files are stored on hard disk in the RMDC (Radio Metric Data Conditioning group) of JPL’s Navigation and Mission Design Section. We access these files and run standard software to produce an Orbit Data File for input into the orbit determination programs which we use. (See Section \[results\].) The data types are two-way and three-way [@way] Doppler and two-way range. (Doppler and range are defined in the following two subsections.) Due to unknown clock offsets between the stations, three-way range is generally not taken or used. The Pioneer spacecraft only have two- and three-way S-band [@sband] Doppler. Galileo also has S-band range data near the Earth. Ulysses has two- and three-way S-band up-link and X-band [@sband] down-link Doppler and range as well as S-band up-link and S-band down-link, although we have only processed the Ulysses S-band up-link and X-band down-link Doppler and range. ### Doppler experimental techniques and strategy {#sec:doppler} In Doppler experiments a radio signal transmitted from the Earth to the spacecraft is coherently transponded and sent back to the Earth. Its frequency change is measured with great precision, using the hydrogen masers at the DSN stations. The observable is the DSN frequency shift [@drift] $$\Delta \nu(t)={\nu_0}\,\frac{1}{c}\frac{d \ell}{dt}, \label{eq:doppler}$$ where $\ell$ is the overall optical distance (including diffraction effects) traversed by a photon in both directions. \[In the Pioneer Doppler experiments, the stability of the fractional drift at the S-band is on the order of $\Delta \nu/\nu_0\simeq10^{-12}$, for integration times on the order of $10^3$ s.\] Doppler measurements provide the “range rate” of the spacecraft and therefore are affected by all the dynamical phenomena in the volume between the Earth and the spacecraft. Expanding upon what was discussed in Section \[data-acquisition\], the received signal and the transmitter frequency (both are at S-band) as well as a 10 pulse per second timing reference from the FTS are fed to the Metric Data Assembly (MDA). There the Doppler phase (difference between transmitted and received phases plus an added bias) is counted. That is, digital counters at the MDA record the zero crossings of the difference (i.e., Doppler, or alternatively the beat frequency of the received frequency and the exciter frequency). After counting, the bias is removed so that the true phase is produced. The system produces “continuous count Doppler” and it uses two counters. Every tenth of a second, a Doppler phase count is recorded from one of the counters. The other counter continues the counts. The recording alternates between the two counters to maintain a continuous unbroken count. The Doppler counts are at 1 MHz for S-band or 5 MHz for X-band. The wavelength of each S-band cycle is about 13 cm. Dividers or “time resolvers” further subdivide the cycle into 256 parts, so that fractional cycles are measured with a resolution of 0.5 mm. This accuracy can only be maintained if the Doppler is continuously counted (no breaks in the count) and coherent frequency standards are kept throughout the pass. It should be noted that no error is accumulated in the phase count as long as lock is not lost. The only errors are the stability of the hydrogen maser and the resolution of the “resolver.” Consequently, the JPL Doppler records are not frequency measurements. Rather, they are digitally counted measurements of the Doppler phase difference between the transmitted and received S-band frequencies, divided by the count time. Therefore, the Doppler observables, we will refer to, have units of cycles per second or Hz. Since total count phase observables are Doppler observables multiplied by the count interval T$_c$, they have units of cycles. The Doppler integration time refers to the total counting of the elapsed periods of the wave with the reference frequency of the hydrogen maser. The usual Doppler integrating times for the Pioneer Doppler signals refers to the data sampled over intervals of 10 s, 60 s, 600 s, or 1980 s. ### Range measurements A range measurement is made by phase modulating a signal onto the up-link carrier and having it echoed by the transponder. The transponder demodulates this ranging signal, filters it, and then re-modulates it back onto the down-link carrier. At the ground station, this returned ranging signal is demodulated and filtered. An instantaneous comparison between the outbound ranging signal and the returning ranging signal that comes down would yield the two-way delay. Cross correlating the returned phase modulated signal with a ground duplicate yields the time delay. (See [@anderson75] and references therein.) As the range code is repeated over and over, an ambiguity can exist. The orbit determination programs are then used to infer (some times with great difficulty) the number of range codes that exist between a particular transmitted code and its own corresponding received code. Thus, the ranging data are independent of the Doppler data, which represents a frequency shift of the radio carrier wave without modulation. For example, solar plasma introduces a group delay in the ranging data but a phase advance in the Doppler data. Ranging data can also be used to distinguish an actual range change from a fictitious range change seen in Doppler data that is caused by a frequency error [@falsedop]. The Doppler frequency integrated over time (the accumulated phase) should equal the range change except for the difference introduced by charged particles ### Inferring position information from Doppler It is also possible to infer the position in the sky of a spacecraft from the Doppler data. This is accomplished by examining the diurnal variation imparted to the Doppler shift by the Earth’s rotation. As the ground station rotates underneath a spacecraft, the Doppler shift is modulated by a sinusoid. The sinusoid’s amplitude depends on the declination angle of the spacecraft and its phase depends upon the right ascension. These angles can therefore be estimated from a record of the Doppler shift that is (at least) of several days duration. This allows for a determination of the distance to the spacecraft through the dynamics of spacecraft motion using standard orbit theory contained in the orbit determination programs. Data preparation {#Data_edit} ---------------- In an ideal system, all scheduled observations would be used in determining parameters of physical interest. However, there are inevitable problems that occur in data collection and processing that corrupt the data. So, at various stages of the signal processing one must remove or “edit” corrupted data. Thus, the need arises for objective editing criteria. Procedures have been developed which attempt to excise corrupted data on the basis of objective criteria. There is always a temptation to eliminate data that is not well explained by existing models, to thereby “improve” the agreement between theory and experiment. Such an approach may, of course, eliminate the very data that would indicate deficiencies in the [*a priori*]{} model. This would preclude the discovery of improved models. In the processing stage that fits the Doppler samples, checks are made to ensure that there are no integer cycle slips in the data stream that would corrupt the phase. This is done by considering the difference of the phase observations taken at a high rate (10 times a second) to produce Doppler. Cycle slips often are dependent on tracking loop bandwidths, the signal to noise ratios, and predictions of frequencies. Blunders due to out-of-lock can be determined by looking at the original tracking data. In particular, cycle slips due to loss-of-lock stand out as a 1 Hz blunder point for each cycle slipped. If a blunder point is observed, the count is stopped and a Doppler point is generated by summing the preceding points. Otherwise the count is continued until a specified maximum duration is reached. Cases where this procedure detected the need for cycle corrections were flagged in the database and often individually examined by an analyst. Sometimes the data was corrected, but nominally the blunder point was just eliminated. This ensures that the data is consistent over a pass. However, it does not guarantee that the pass is good, because other errors can affect the whole pass and remain undetected until the orbit determination is done. To produce an input data file for an orbit determination program, JPL has a software package known as the Radio Metric Data Selection, Translation, Revision, Intercalation, Processing and Performance Evaluation Reporting (RMD-STRIPPER) Program. As we discussed in Section \[sec:doppler\], this input file has data that can be integrated over intervals with different durations: 10 s, 60 s, 600 s and 1980 s. This input Orbit Determination File (ODFILE) obtained from the RMDC group is the initial data set with which both the JPL and The Aerospace Corporation groups started their analyses. Therefore, the initial data file already contained some common data editing that the RMDC group had implemented through program flags, etc. The data set we started with had already been compressed to 60 s. So, perhaps there were some blunders that had already been removed using the initial STRIPPER program. The orbit analyst manually edits the remaining corrupted data points. Editing is done either by plotting the data residuals and deleting them from the fit or plotting weighted data residuals. That is, the residuals are divided by the standard deviation assigned to each data point and plotted. This gives the analyst a realistic view of the data noise during those times when the data was obtained while looking through the solar plasma. Applying an “$N$-$\sigma$” ($\sigma$ is the standard deviation) test, where $N$ is the choice of the analyst (usually 4-10) the analyst can delete those points that lie outside the $N$-$\sigma$ rejection criterion without being biased in his selection. The $N$-$\sigma$ test, implemented in CHASMP, is very useful for data taken near solar conjunction since the solar plasma adds considerable noise to the data. This criterion later was changed to a similar criteria that rejects all data with residuals in the fit extending for more than $\pm 0.025$ Hz from the mean. Contrariwise, the JPL analysis edits only very corrupted data; e.g., a blunder due to a phase lock loss, data with bad spin calibration, etc. Essentially the Aerospace procedure eliminates data in the tails of the Gaussian probability frequency distribution whereas the JPL procedure accepts this data. If needed or desired, the orbit analyst can choose to perform an additional data compression of the original navigation data. The JPL analysis does not apply any additional data compression and uses all the original data from the ODFILE as opposed to Aerospace’s approach. Aerospace makes an additional compression of data within CHASMP. It uses the longest available data integration times which can be composed from either summing up adjacent data intervals or by using data spans with duration $\ge 600$ s. (Effectively Aerospace prefers 600 and 1980 second data intervals and applies a low-pass filter.) The total count of corrupted data points is about 10% of the total raw data points. The analysts’ judgments play an important role here and is one of the main reasons that JPL and Aerospace have slightly different results. (See Sections \[results\]and \[recent\_results\].) In Section \[results\]we will show a typical plot (Figure \[fig:aerospace\] below) with outliers present in the data. Many more outliers are off the plot. One would expect that the two different strategies of data compression used by the two teams would result in significantly different numbers of total data points used in the two independent analyses. The influence of this fact on the solution estimation accuracy will be addressed in Section \[recent\_results\] below. Data weighting {#dataweight} -------------- Considerable effort has gone into accurately estimating measurement errors in the observations. These errors provide the data weights necessary to accurately estimate the parameter adjustments and their associated uncertainties. To the extent that measurement errors are accurately modeled, the parameters extracted from the data will be unbiased and will have accurate sigmas assigned to them. Both JPL and Aerospace assign a standard uncertainty of 1 mm/s over a 60 second count time for the S–band Pioneer Doppler data. (Originally the JPL team was weighting the data by 2 mm/s uncertainty.) A change in the DSN antenna elevation angle also directly affects the Doppler observables due to tropospheric refraction. Therefore, to correct for the influence of the Earth’s troposphere the data can also be deweighted for low elevation angles. The phenomenological range correction is given as $$\sigma= \sigma_{\tt nominal} \left(1+\frac{18}{(1+\theta_E)^2}\right), \label{eq:sig_aer0}$$ where $\sigma_{\tt nominal}$ is the basic standard deviation (in Hz) and $\theta_E$ is the elevation angle in degrees [@cane]. Each leg is computed separately and summed. For Doppler the same procedure is used. First, Eq. (\[eq:sig\_aer0\]) is multiplied by $\sqrt{60 \,{\rm s}/T_c}$, where $T_c$ is the count time. Then a numerical time differentiation of Eq. (\[eq:sig\_aer0\]) is performed. That is, Eq. (\[eq:sig\_aer0\]) is differenced and divided by the count time, $T_c$. (For more details on this standard technique see Refs. [@Moyer71]-[@MuhlemanAnderson81].) There is also the problem of data weighting for data influenced by the solar corona. This will be discussed in Section \[corona+wt\]. Spin calibration of the data {#spincalibrate} ---------------------------- The radio signals used by DSN to communicate with spacecraft are circularly polarized. When these signals are reflected from spinning spacecraft antennae a Doppler bias is introduced that is a function of the spacecraft spin rate. Each revolution of the spacecraft adds one cycle of phase to the up-link and the down-link. The up-link cycle is multiplied by the turn around ratio 240/221 so that the bias equals (1+240/221) cycles per revolution of the spacecraft. High-rate spin data is available for Pioneer 10 only up to July 17, 1990, when the DSN ceased doing spin calibrations. (See Section \[sec:prop\].) After this date, in order to reconstruct the spin behavior for the entire data span and thereby account for the spin bias in the Doppler signal, both analyses modeled the spin by performing interpolations between the data points. The JPL interpolation was non-linear with a high-order polynomial fit of the data. (The polynomial was from second up to sixth order, depending on the data quality.) The CHASMP interpolation was linear between the spin data points. After a maneuver in mid-1993, there was not enough power left to support the IPP. But analysts still could get a rough determination approximately every six months using information obtained from the conscan maneuvers. No spin determinations were made after 1995. However, the archived conscan data could still yield spin data at every maneuver time if such work was approved. Further, as the phase center of the main antenna is slightly offset from the spin axis, a very small (but detectable) $\sin$e-wave signal appears in the high-rate Doppler data. In principle, this could be used to determine the spin rate for passes taken after 1993, but it has not been attempted. Also, the failure of one of the spin-down thrusters prevented precise spin calibration of the Pioneer 11 data. Because the spin rate of the Pioneers was changing over the data span, the calibrations also provide an indication of gas leaks that affect the acceleration of the spacecraft. A careful look at the records shows how this can be a problem. This will be discussed in Sections \[spinhistory\] and \[sec:gleaks\]. \[navigate\]BASIC THEORY OF SPACECRAFT NAVIGATION ================================================= Accuracy of modern radio tracking techniques has provided the means necessary to explore the gravitational environment in the solar system up to a limit never before possible [@massprog]. The major role in this quest belongs to relativistic celestial mechanics experiments with planets (e.g., passive radar ranging) and interplanetary spacecraft (both Doppler and range experiments). Celestial mechanics experiments with spacecraft have been carried out by JPL since the early 1960’s [@anderson74; @dsn86]. The motivation was to improve both the ephemerides of solar system bodies and also the knowledge of the solar system’s dynamical environment. This has become possible due to major improvements in the accuracy of spacecraft navigation, which is still a critical element for a number of space missions. The main objective of spacecraft navigation is to determine the present position and velocity of a spacecraft and to predict its future trajectory. This is usually done by measuring changes in the spacecraft’s position and then, using those measurements, correcting (fitting and adjusting) the predicted spacecraft trajectory. In this section we will discuss the theoretical foundation that is used for the analysis of tracking data from interplanetary spacecraft. We describe the basic physical models used to determine a trajectory, given the data. Relativistic equations of motion -------------------------------- The spacecraft ephemeris, generated by a numerical integration program, is a file of spacecraft positions and velocities as functions of ephemeris (or coordinate) time ([ET]{}). The integrator requires the input of various parameters. These include adopted constants ($c$, $G$, planetary mass ratios, etc.) and parameters that are estimated from fits to observational data (e.g., corrections to planetary orbital elements). The ephemeris programs use equations for point-mass relativistic gravitational accelerations. They are derived from the variation of a time-dependent, Lagrangian-action integral that is referenced to a non-rotating, solar-system, barycentric, coordinate frame. In addition to modeling point-mass interactions, the ephemeris programs contain equations of motion that model terrestrial and lunar figure effects, Earth tides, and lunar physical librations [@Newhall83]-[@Standish95a]. The programs treat the Sun, the Moon, and the nine planets as point masses in the isotropic, parameterized post-Newtonian, N-body metric with Newtonian gravitational perturbations from large, main-belt asteroids. Responding to the increasing demand of the navigational accuracy, the gravitational field in the solar system is modeled to include a number of relativistic effects that are predicted by the different metric theories of gravity. Thus, within the accuracy of modern experimental techniques, the parameterized post-Newtonian (PPN) approximation of modern theories of gravity provides a useful starting point not only for testing these predictions, but also for describing the motion of self-gravitating bodies and test particles. As discussed in detail in [@Will93], the accuracy of the PPN limit (which is slow motion and weak field) is adequate for all foreseeable solar system tests of general relativity and a number of other metric theories of gravity. (For the most general formulation of the PPN formalism, see the works of Will and Nordtvedt [@Will93; @WillNordtvedt72].) For each body $i$ (a planet or spacecraft anywhere in the solar system), the point-mass acceleration is written as [@Moyer71; @Moyer00; @Newhall83; @estabrook69; @Moyer81] $$\begin{aligned} \nonumber \ddot{\bf r}_i&=&\sum_{j\not=i}\frac{\mu_j({\bf r}_j-{\bf r}_i)} {r^3_{ij}}\Bigg(1-\frac{2(\beta+\gamma)}{c^2}\sum_{k\not=i} \frac{\mu_k}{r_{ik}}-\frac{2\beta-1}{c^2}\sum_{k\not=j} \frac{\mu_k}{r_{jk}}-\frac{3}{2c^2}\Big[\frac{({\bf r}_j-{\bf r}_i)\dot{\bf r}_j}{r_{ij}}\Big]^2+\frac{1}{2c^2} ({\bf r}_j-{\bf r}_i)\ddot{\bf r}_j-\frac{2(1+\gamma)}{c^2} \dot{\bf r}_i\dot{\bf r}_j+\\ &+&\gamma\left(\frac{v_i}{c}\right)^2+ (1+\gamma)\left(\frac{v_j}{c}\right)^2\Bigg)+ \frac{1}{c^2}\sum_{j\not=i}\frac{\mu_j}{r^3_{ij}} \Big([{\bf r}_i-{\bf r}_j)]\cdot[(2+2\gamma)\dot{\bf r}_i-(1+2\gamma) \dot{\bf r}_j]\Big)(\dot{\bf r}_i-\dot{\bf r}_j)+ \frac{3+4\gamma}{2c^2}\sum_{j\not=i} \frac{\mu_j\ddot{\bf r}_j}{r_{ij}} \label{eq:rdotdot}\end{aligned}$$ where $\mu_i$ is the “gravitational constant” of body [*i*]{}. It actually is its mass times the Newtonian constant: $\mu_i={\it G}m_i$. Also, ${\bf r}_i(t)$ is the barycentric position of body $i$, $r_{ij}=|{\bf r}_j-{\bf r}_i|$ and $v_i=|\dot{\bf r}_i|$. For planetary motion, each of these equations depends on the others. So they must be iterated in each step of the integration of the equations of motion. The barycentric acceleration of each body $j$ due to Newtonian effects of the remaining bodies and the asteroids is denoted by $\ddot{\bf r}_j$. In Eq. (\[eq:rdotdot\]), $\beta$ and $\gamma$ are the PPN parameters [@Will93; @WillNordtvedt72]. General relativity corresponds to $\beta = \gamma = 1$, which we choose for our study. The Brans-Dicke theory is the most famous among the alternative theories of gravity. It contains, besides the metric tensor, a scalar field $\varphi$ and an arbitrary coupling constant $\omega$, related to the two PPN parameters as $\gamma= \frac{1+\omega}{2+\omega}, ~\beta=1$. Equation (\[eq:rdotdot\]) allows the consideration of any problem in celestial mechanics within the PPN framework. Light time solution and time scales {#sec:time_scales} ----------------------------------- In addition to planetary equations of motion Eq. (\[eq:rdotdot\]), one needs to solve the relativistic light propagation equation in order to get the solution for the total light time travel. In the solar system, barycentric, space-time frame of reference this equation is given by: $$\begin{aligned} \nonumber t_2-t_1&=&\frac{r_{21}}{c}+\frac{(1+\gamma)\mu_\odot}{c^3} \ln\bigg[\frac{r_1^\odot+r_2^\odot+r_{12}^\odot} {r_1^\odot+r_2^\odot-r_{12}^\odot}\bigg]+\\ &+& \sum_{i} \frac{(1+\gamma)\mu_i}{c^3} \ln\bigg[\frac{r_1^i+r_2^i+r_{12}^i} {r_1^i+r_2^i-r_{12}^i}\bigg], \label{eq:lt}\end{aligned}$$ where $\mu_\odot$ is the gravitational constant of the Sun and $\mu_i$ is the gravitational constant of a planet, an outer planetary system, or the Moon. $r_1^\odot, r_2^\odot and r_{12}^\odot$ are the heliocentric distances to the point of RF signal emission on Earth, to the point of signal reflection at the spacecraft, and the relative distance between these two points. Correspondingly, $r_1^i, r_2^i,$ and $r_{12}^i$ are similar distances relative to a particular $i$-th body in the solar system. In the spacecraft light time solution, $t_1$ refers to the transmission time at a tracking station on Earth, and $t_2$ refers to the reflection time at the spacecraft or, for one-way [@way] data, the transmission time at the spacecraft. The reception time at the tracking station on Earth or at an Earth satellite is denoted by $t_3$. Hence, Eq. (\[eq:lt\]) is the up-leg light time equation. The corresponding down-leg light time equation is obtained by replacing subscripts as follows: $1\rightarrow 2 $ and $2\rightarrow 3 $. (See the details in [@Moyer00].) The spacecraft equations of motion relative to the solar system barycenter are essentially the same as given by Eq. (\[eq:rdotdot\]). The gravitational constants of the Sun, planets and the planetary systems are the values associated with the solar system barycentric frame of reference, which are obtained from the planetary ephemeris [@Moyer81]. We treat a distant spacecraft as a point-mass particle. The spacecraft acceleration is integrated numerically to produce the spacecraft ephemeris. The ephemeris is interpolated at the ephemeris time ([ET]{}) value of the interpolation epoch. This is the time coordinate $t$ in Eqs. (\[eq:rdotdot\]) and (\[eq:lt\]), i.e., $t\equiv\,{\tt ET}$. As such, ephemeris time means coordinate time in the chosen celestial reference frame. It is an independent variable for the motion of celestial bodies, spacecraft, and light rays. The scale of [ET]{} depends upon which reference frame is selected and one may use a number of time scales depending on the practical applications. It is convenient to express [ET]{} in terms of International Atomic Time ([TAI]{}). [TAI]{} is based upon the second in the International System of Units ([SI]{}). This second is defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom [@exp_cat]. The differential equation relating ephemeris time ([ET]{}) in the solar system barycentric reference frame to [TAI]{} at a tracking station on Earth or on Earth satellite can be obtained directly from the Newtonian approximation to the N-body metric [@Moyer81]. This expression has the form $$\begin{aligned} \frac{d \,{\tt TAI}}{d\, \tt ET}= 1-\frac{1}{c^2}\Big(U-\langle U\rangle + \frac{1}{2}v^2-\frac{1}{2}\langle v^2 \rangle\Big) +{\cal O}(\frac{1}{c^{4}}), \label{eq:tai_et}\end{aligned}$$ where $U$ is the solar system gravitational potential evaluated at the tracking station and $v$ is the solar system barycentric velocity of the tracking station. The brackets $\langle ~\rangle$ on the right side of Eq. (\[eq:tai\_et\]) denote long-time average of the quantity contained within them. This averaging amounts to integrating out periodic variations in the gravitational potential, $U$, and the barycentric velocity, $v^2$, at the location of a tracking station. The desired time scale transformation is then obtained by using the planetary ephemeris to calculate the terms in Eq. (\[eq:tai\_et\]). The vector expression for the ephemeris/coordinate time ([ET]{}) in the solar system barycentric frame of reference minus the [TAI]{} obtained from an atomic clock at a tracking station on Earth has the form [@Moyer81] $$\begin{aligned} {\tt ET-TAI} &=& 32.184~{\rm s}+ \frac{2}{c^2}(\dot{\bf r}^\odot_{\tt B}\cdot {\bf r}^\odot_{\tt B})+ \frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_{\tt B} \cdot {\bf r}^{\tt B}_{\tt E})+ \nonumber\\ &+&\frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_{\tt E}\cdot {\bf r}^{\tt E}_A)+\frac{\mu_J}{c^2(\mu_\odot+\mu_J)} (\dot{\bf r}^\odot_J\cdot {\bf r}^\odot_J)+\nonumber\\ &+& \frac{\mu_{Sa}}{c^2(\mu_\odot+\mu_{Sa})} (\dot{\bf r}^\odot_{Sa}\!\cdot{\bf r}^\odot_{Sa})+ \frac{1}{c^2}(\dot{\bf r}^{\tt SSB}_\odot\!\!\cdot {\bf r}^\odot_{\tt B}), \label{eq:time}\hskip 14pt\end{aligned}$$ where ${\bf r}^j_i$ and $\dot{\bf r}^j_i$ position and velocity vectors of point $i$ relative to point $j$ (they are functions of [ET]{}); superscript or subscript ${\tt SSB}$ denotes solar system barycenter; $\odot$ stands for the Sun; ${\tt B}$ for the Earth-Moon barycenter; $ E, J, Sa $ denote the Earth, Jupiter, and Saturn correspondingly, and $A$ is for the location of the atomic clock on Earth which reads [TAI]{}. This approximated analytic result contains the clock synchronization term which depends upon the location of the atomic clock and five location-independent periodic terms. There are several alternate expressions that have up to several hundred additional periodic terms which provide greater accuracies than Eq. (\[eq:time\]). The use of these extended expressions provide transformations of [ET]{} – [TAI]{} to accuracies of 1 ns [@Moyer00]. For the purposes of our study the Station Time ([ST]{}) is especially significant. This time is the atomic time [TAI]{} at a DSN tracking station on Earth, [ST]{}=[TAI]{}$_{\tt station}$. This atomic time scale departs by a small amount from the “reference time scale.” The reference time scale for a DSN tracking station on Earth is the Coordinated Universal Time ([UTC]{}). This last is standard time for 0$^\circ$ longitude. (For more details see [@Moyer00; @exp_cat].) All the vectors in Eq. (\[eq:time\]) except the geocentric position vector of the tracking station on Earth can be interpolated from the planetary ephemeris or computed from these quantities. Universal Time ([UT]{}) is the measure of time which is the basis for all civil time keeping. It is an observed time scale. The specific version used in JPL’s Orbit Determination Program (ODP) is [UT1]{}. This is used to calculate mean sidereal time, which is the Greenwich hour angle of the mean equinox of date measured in the true equator of date. Observed [UT1]{} contains 41 short-term terms with periods between 5 and 35 days. They are caused by long-period solid Earth tides. When the sum of these terms, [$\Delta$UT1]{}, is subtracted from [UT1]{} the result is called [UT1R]{}, where [R]{} means regularized. Time in any scale is represented as seconds past 1 January 2000, 12$^{\tt h}$, in that time scale. This epoch is J2000.0, which is the start of the Julian year 2000. The Julian Date for this epoch is JD 245,1545.0. Our analyses used the standard space-fixed J2000 coordinate system, which is provided by the International Celestial Reference Frame (ICRF). This is a quasi-inertial reference frame defined from the radio positions of 212 extragalactic sources distributed over the entire sky [@Ma98]. The variability of the earth-rotation vector relative to the body of the planet or in inertial space is caused by the gravitational torque exerted by the Moon, Sun and planets, displacements of matter in different parts of the planet and other excitation mechanisms. The observed oscillations can be interpreted in terms of mantle elasticity, earth flattening, structure and properties of the core-mantle boundary, rheology of the core, underground water, oceanic variability, and atmospheric variability on time scales of weather or climate. Several space geodesy techniques contribute to the continuous monitoring of the Earth’s rotation by the International Earth Rotation Service (IERS). Measurements of the Earth’s rotation presented in the form of time developments of the so-called Earth Orientation Parameters ([EOP]{}). Universal time ([UT1]{}), polar motion, and the celestial motion of the pole (precession/nutation) are determined by Very Long-Baseline Interferometry (VLBI). Satellite geodesy techniques, such as satellite laser ranging (SLR) and using the Global Positioning System (GPS), determine polar motion and rapid variations of universal time. The satellite geodesy programs used in the IERS allow determination of the time variation of the Earth’s gravity field. This variation reflects the evolutions of the Earth’s shape and of the distribution of mass in the planet. The programs have also detected changes in the location of the center of mass of the Earth relative to the crust. It is possible to investigate other global phenomena such as the mass redistributions of the atmosphere, oceans, and solid Earth. Using the above experimental techniques, Universal time and polar motion are available daily with an accuracy of about 50 picoseconds (ps). They are determined from VLBI astrometric observations with an accuracy of 0.5 milliarcseconds (mas). Celestial pole motion is available every five to seven days at the same level of accuracy. These estimations of accuracy include both short term and long term noise. Sub-daily variations in Universal time and polar motion are also measured on a campaign basis. In summary, this dynamical model accounts for a number of post-Newtonian perturbations in the motions of the planets, the Moon, and spacecraft. Light propagation is correct to order $c^{-2}$. The equations of motion of extended celestial bodies are valid to order $c^{-4}$. Indeed, this dynamical model has been good enough to perform tests of general relativity [@anderson75; @Will93; @WillNordtvedt72]. Standard modeling of small, non-gravitational forces {#sec:syst0} ---------------------------------------------------- In addition to the mutual gravitational interactions of the various bodies in the solar system and the gravitational forces acting on a spacecraft as a result of presence of those bodies, it is also important to consider a number of non-gravitational forces which are important for the motion of a spacecraft. (Books and lengthy reports have been written about practically all of them. Consult Ref. [@milani; @longuski] for a general introduction.) The Jet Propulsion Laboratory’s ODP accounts for many sources of non-gravitational accelerations. Among them, the most relevant to this study, are: i) solar radiation pressure, ii) solar wind pressure, iii) attitude-control maneuvers together with a model for unintentional spacecraft mass expulsion due to gas leakage of the propulsion system. We can also account for possible influence of the interplanetary media and DSN antennae contributions to the spacecraft radio tracking data and consider the torques produced by above mentioned forces. The Aerospace CHASMP code uses a model for gas leaks that can be adjusted to include the effects of the recoil force due to emitted radio power and anisotropic thermal radiation of the spacecraft. In principle, one could set up complicated engineering models to predict at least some of the effects. However, their residual uncertainties might be unacceptable for the experiment, in spite of the significant effort required. In fact, a constant acceleration produces a linear frequency drift that can be accounted for in the data analysis by a single unknown parameter. The figure against which we compare the effects of non-gravitational accelerations on the Pioneers’ trajectories is the expected error in the acceleration error estimations. This is on the order of $$\sigma_0 \sim 2\times 10^{-8} ~~{\rm cm/s^2}, \label{eq:req}$$ where $\sigma_0$ is a single determination accuracy related to acceleration measurements averaged over number of days. This would contribute to our result as $\sigma_N\sim\sigma_0/\sqrt{N}$. Thus, if no systematics are involved then $\sigma_N$ will just tend to zero as time progresses. Therefore, the important thing is to know that these effects (systematics) are not too large, thereby overwhelming any possibly important signal (such as our anomalous acceleration). This will be demonstrated in Sections \[ext-systema\] and \[int-systema\]. Solar corona model and weighting {#corona+wt} -------------------------------- The electron density and density gradient in the solar atmosphere influence the propagation of radio waves through the medium. So, both range and Doppler observations at S-band are affected by the electron density in the interplanetary medium and outer solar corona. Since, throughout the experiment, the closest approach to the center of the Sun of a radio ray path was greater than 3.5 $R_\odot$, the medium may be regarded as collisionless. The [*one way*]{} time delay associated with a plane wave passing through the solar corona is obtained [@MuhlemanAnderson81; @anderson74; @MuhlemanEspositoAnderson77] by integrating the group velocity of propagation along the ray’s path, $\ell$: $$\begin{aligned} \nonumber \Delta t &=& \pm \frac{1}{2c\,n_{\tt crit}(\nu)} \int_\oplus^{SC}d\ell~n_e(t, {\bf r}), \\ n_{\tt crit}(\nu) &=& 1.240\times 10^4~ \Big(\frac{\nu}{1~{\rm MHz}}\Big)^2~~{\rm cm}^{-3}, \label{eq:sol_plasma} \end{aligned}$$ where $n_e(t, {\bf r})$ is the free electron density in the solar plasma, $c$ is the speed of light, and $n_{\tt crit}(\nu)$ is the critical plasma density for the radio carrier frequency $\nu$. The plus sign is applied for ranging data and the minus sign for Doppler data [@slavanote]. Therefore, in order to calibrate the plasma contribution, one should know the electron density along the path. One usually decomposes the electron density, $n_e$, into a static, steady-state part, $\overline{n}_e(\bf{r})$, plus a fluctuation $\delta n_e(t, {\bf r})$, i.e., $n_e(t, {\bf r})= \overline{n}_e({\bf r})+ \delta n_e(t, {\bf r})$. The physical properties of the second term are hard to quantify. But luckily, its effect on Doppler observables and, therefore, on our results is small. (We will address this issue in Sec. \[solarwind\].) On the contrary, the steady-state corona behavior is reasonably well known and several plasma models can be found in the literature [@MuhlemanEspositoAnderson77]-[@bird]. Consequently, while studying the effect of a systematic error from propagation of the S-band carrier wave through the solar plasma, both analyses adopted the following model for the electron density profile [@MuhlemanAnderson81]: $$\begin{aligned} n_e(t, {\bf r})= A\Big(\frac{R_\odot}{r}\Big)^2+ B\Big(\frac{R_\odot}{r}\Big)^{2.7} e^{-\left[\frac{\phi}{\phi_0}\right]^2}+ C\Big(\frac{R_\odot}{r}\Big)^6. \label{corona_model_content}\end{aligned}$$ $r$ is the heliocentric distance to the immediate ray trajectory and $\phi$ is the helio-latitude normalized by the reference latitude of $\phi_0=10^\circ$. The parameters $r$ and $\phi$ are determined from the trajectory coordinates on the tracking link being modeled. The parameters $A, B, C$ are parameters chosen to describe the solar electron density. (They are commonly given in two sets of units, meters or cm$^{-3}$ [@scunits].) They can be treated as stochastic parameters, to be determined from the fit. But in both analyses we ultimately chose to use the values determined from the recent solar corona studies done for the Cassini mission. These newly obtained values are: $A= 6.0\times 10^3, B= 2.0\times 10^4, C= 0.6\times 10^6$, all in meters [@Ekelund]. \[This is what we will refer to as the “Cassini corona model.”\] Substitution of Eq. (\[corona\_model\_content\]) into Eq. (\[eq:sol\_plasma\]) results in the following steady-state solar corona contribution to the range model that we used in our analysis: $$\begin{aligned} \nonumber \Delta_{\tt SC}{\rm range}&=& \pm \Big(\frac{\nu_0}{\nu}\Big)^2\bigg[ A\Big(\frac{R_\odot}{\rho}\Big)F+ \\ &+& B\Big(\frac{R_\odot}{\rho}\Big)^{1.7} e^{-\left[\frac{\phi}{\phi_0}\right]^2}+ C\Big(\frac{R_\odot}{\rho}\Big)^{5}\bigg]. \hskip 10pt \label{corona_model}\end{aligned}$$ $\nu_0$ and $\nu$ are a reference frequency and the actual frequency of radio-wave \[for Pioneer 10 analysis $\nu_0=2295$ MHz\], $\rho$ is the impact parameter with respect to the Sun and $F$ is a light-time correction factor. For distant spacecraft this function is given as follows: $$\begin{aligned} F&=&F(\rho, r_T, r_E)=\\\nonumber &=&\frac{1}{\pi}\Bigg\{ {\sf ArcTan}\Big[\frac{\sqrt{r_T^2-\rho^2}}{\rho}\Big]+ {\sf ArcTan}\Big[\frac{\sqrt{r_E^2-\rho^2}}{\rho}\Big]\Bigg\}, \label{eq:weight_doppler*}\end{aligned}$$ where $r_T$ and $r_E$ are the heliocentric radial distances to the target and to the Earth, respectively. Note that the sign of the solar corona range correction is negative for Doppler and positive for range. The Doppler correction is obtained from Eq. (\[corona\_model\]) by simple time differentiation. Both analyses use the same physical model, Eq. (\[corona\_model\]), for the steady-state solar corona effect on the radio-wave propagation through the solar plasma. Although the actual implementation of the model in the two codes is different, this turns out not to be significant. (See Section \[Ext\_accuracy\].) CHASMP can also consider the effect of temporal variation in the solar corona by using the recorded history of solar activity. The change in solar activity is linked to the variation of the total number of sun spots per year as observed at a particular wavelength of the solar radiation, $\lambda$=10.7 cm. The actual data corresponding to this variation is given in Ref. [@F10-7]. CHASMP averages this data over 81 days and normalizes the value of the flux by 150. Then it is used as a time-varying scaling factor in Eq. (\[corona\_model\]). The result is referred to as the “F10.7 model.” Next we come to corona data weighting. JPL’s ODP does not apply corona weighting. On the other hand, Aerospace’s CHASMP can apply corona weighting if desired. Aerospace uses a standard weight augmented by a weight function that accounts for noise introduced by solar plasma and low elevation. The weight values are adjusted so that i) the post-fit weighted sum of the squares is close to unity and ii) approximately uniform noise in the residuals is observed throughout the fit span. Thus, the corresponding solar-corona weight function is: $$\sigma_{\tt r}= \frac{k}{2} \Big(\frac{\nu_0}{\nu}\Big)^2 \Big(\frac{R_\odot}{\rho}\Big)^{\frac{3}{2}}, \label{weightrange}$$ where, for range data, $k$ is an input constant nominally equal to 0.005 light seconds, $\nu_0$ and $\nu$ are a reference frequency and the actual frequency, $\rho$ is the trajectory’s impact parameter with respect to the Sun in km, and $R_\odot$ is the solar radius in km [@Muhleman]. The solar-corona weight function for Doppler is essentially the same, but obtained by numerical time differentiation of Eq. (\[weightrange\]). Modeling of maneuvers {#model-maneuvers} --------------------- There were 28 Pioneer 10 maneuvers during our data interval from 3 January 1987 to 22 July 1998. Imperfect coupling of the hydrazine thrusters used for the spin orientation maneuvers produced integrated velocity changes of a few millimeters per second. The times and durations of each maneuver were provided by NASA/Ames. JPL used this data as input to ODP. The Aerospace team used a slightly different approach. In addition to the original data, CHASMP used the spin-rate data file to help determine the times and duration of maneuvers. The CHASMP determination mainly agreed with the data used by JPL. \[There were minor variations in some of the times, one maneuver was split in two, and one extraneous maneuver was added to Interval II to account for data not analyzed (see below).\] Because the effect on the spacecraft acceleration could not be determined well enough from the engineering telemetry, JPL included a single unknown parameter in the fitting model for each maneuver. In JPL’s ODP analysis, the maneuvers were modeled by instantaneous velocity increments at the beginning time of each maneuver (instantaneous burn model). \[Analyses of individual maneuver fits show the residuals to be small.\] In the CHASMP analysis, a constant acceleration acting over the duration of the maneuver was included as a parameter in the fitting model (finite burn model). Analyses of individual maneuver fits show the residuals are small. Because of the Pioneer spin, these accelerations are important only along the Earth-spacecraft line, with the other two components averaging out over about 50 revolutions of the spacecraft over a typical maneuver duration of 10 minutes. By the time Pioneer 11 reached Saturn, the pattern of the thruster firings was understood. Each maneuver caused a change in spacecraft spin and a velocity increment in the spacecraft trajectory, immediately followed by two to three days of gas leakage, large enough to be observable in the Doppler data [@null81]. Typically the Doppler data is time averaged over 10 to 33 minutes, which significantly reduces the high-frequency Doppler noise. The residuals represent our fit. They are converted from units of Hz to Doppler velocity by the formula [@drift] $$[\Delta v]_{\tt DSN} = \frac{c}{2} \frac{[\Delta \nu]_{\tt DSN}}{\nu_0}, \label{hztodoppler}$$ where $\nu_0$ is the downlink carrier frequency, $\sim 2.29$ GHz, $\Delta \nu$ is the Doppler residual in Hz from the fit, and $c$ is the speed of light. As an illustration, consider the fit to one of the Pioneer 10 maneuvers, \# 17, on 22 December 1993, given in Figure \[fig:man17\]. This was particularly well covered by low-noise -10pt -10pt Doppler data near solar opposition. Before the start of the maneuver, there is a systematic trend in the residuals which is represented by a cubic polynomial in time. The standard error in the residuals is 0.095 mm/s. After the maneuver, there is a relatively small velocity discontinuity of $-0.90 \pm 0.07$ mm/s. The discontinuity arises because the model fits the entire data interval. In fact, the residuals increase after the maneuver. By 11 January 1994, 19 days after the maneuver, the residuals are scattered about their pre-maneuver mean of $-0.15$ mm/s. For purposes of characterizing the gas leak immediately after the maneuver, we fit the post-maneuver residuals by a two-parameter exponential curve, $$\Delta v = -v_0 \exp\Big[-\frac{t}{\tau}\,\Big] - 0.15 ~~~{\rm mm/s}.$$ The best fit yields $v_0 = 0.808$ mm/s and the time constant $\tau$ is 13.3 days, a reasonable result. The time derivative of the exponential curve yields a residual acceleration immediately after the maneuver of 7.03 $\times$ 10$^{-8}$ cm/s$^{2}$. This is close to the magnitude of the anomalous acceleration inferred from the Doppler data, but in the *opposite* direction. However the gas leak rapidly decays and becomes negligible after 20 days or so. Orbit determination procedure {#sec:OD} ----------------------------- Our orbit determination procedure first determines the spacecraft’s initial position and velocity in a data interval. For each data interval, we then estimate the magnitudes of the orientation maneuvers, if any. The analyses are modeled to include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, and bias and drift in the Doppler and range (if available). Planetary coordinates and solar system masses are obtained using JPL’s Export Planetary Ephemeris DE405, where DE stands for the Development Ephemeris. \[Earlier in the study, DE200 was used. See Section \[subsec:accel\].\] We include models of precession, nutation, sidereal rotation, polar motion, tidal effects, and tectonic plates drift. Model values of the tidal deceleration, nonuniformity of rotation, polar motion, Love numbers, and Chandler wobble are obtained observationally, by means of Lunar and Satellite Laser Ranging (LLR and SLR) techniques and VLBI. Previously they were combined into a common publication by either the International Earth Rotation Service (IERS) or by the United States Naval Observatory (USNO). Currently this information is provided by the ICRF. JPL’s Earth Orientation Parameters (EOP) is a major source contributor to the ICRF. The implementation of the J2000.0 reference coordinate system in CHASMP involves only rotation from the Earth-fixed to the J2000.0 reference frame and the use of JPL’s   DE200   planetary ephemeris [@Laing91]. The rotation from   J2000.0   to Earth-fixed is computed from a series of rotations which include precession, nutation, the Greenwich hour angle, and pole wander. Each of these general categories is also a multiple rotation and is treated separately by most software. Each separate rotation matrix is chain multiplied to produce the final rotation matrix. CHASMP, however, does not separate precession and nutation. Rather, it combines them into a single matrix operation. This is achieved by using a different set of angles to describe precession than is used in the ODP. (See a description of the standard set of angles in [@Lieske76].) These angles separate luni-solar precession from planetary precession. Luni-solar precession, being the linear term of the nutation series for the nutation in longitude, is combined with the nutation in longitude from the DE200 ephemeris tape [@Standish82]. Both JPL’s ODP and The Aerospace Corporation’s CHASMP use the JPL/Earth Orientation Parameters (EOP) values. This could be a source of common error. However the comparisons between EOP and IERS show an insignificant difference. Also, only secular terms, such as precession, can contribute errors to the anomalous acceleration. Errors in short period terms are not correlated with the anomalous acceleration. Parameter estimation strategies {#sec:PE} ------------------------------- During the last few decades, the algorithms of orbital analysis have been extended to incorporate Kalman-filter estimation procedure that is based on the concept of “process noise” (i.e., random, non-systematic forces, or random-walk effects). This was motivated by the need to respond to the significant improvement in observational accuracy and, therefore, to the increasing sensitivity to numerous small perturbing factors of a stochastic nature that are responsible for observational noise. This approach is well justified when one needs to make accurate predictions of the spacecraft’s future behavior using only the spacecraft’s past hardware and electronics state history as well as the dynamic environment conditions in the distant craft’s vicinity. Modern navigational software often uses Kalman filter estimation since it more easily allows determination of the temporal noise history than does the weighted least-squares estimation. To take advantage of this while obtaining JPL’s original results [@anderson; @moriond] discussed in Section \[results\], JPL used batch-sequential methods with variable batch sizes and process noise characteristics. That is, a batch-sequential filtering and smoothing algorithm with process noise was used with ODP. In this approach any small anomalous forces may be treated as stochastic parameters affecting the spacecraft trajectory. As such, these parameters are also responsible for the stochastic noise in the observational data. To better characterize these noise sources, one splits the data interval into a number of constant or variable size batches and makes assumptions on possible statistical properties of these noise factors. One then estimates the mean values of the unknown parameters within the batch and also their second statistical moments. Using batches has the advantage of dealing with a smaller number of experimental data segments. We experimented with a number of different constant batch sizes; namely, 0, 5, 30, and 200 day batch sizes. (Later we also used 1 and 10 day batch sizes.) In each batch one estimates the same number of desired parameters. So, one expects that the smaller the batch size the larger the resulting statistical errors. This is because a smaller number of data points is used to estimate the same number of parameters. Using the entire data interval as a single batch while changing the process noise [*a priori*]{} values is expected in principle (see below) to yield a result identical to the least-squares estimation. In the single batch case, it would produce only one solution for the anomalous acceleration. There is another important parameter that was taken into account in the statistical data analysis reported here. This is the expected correlation time for the underlying stochastic processes (as well as the process noise) that may be responsible for the anomalous acceleration. For example, using a zero correlation time is useful in searches for an $a_P$ that is generated by a random process. One therefore expects that an $a_P$ estimated from one batch is statistically independent (uncorrelated) from those estimated from other batches. Also, the use of finite correlation times indicates one is considering an $a_P$ that may show a temporal variation within the data interval. We experimented with a number of possible correlation times and will discuss the corresponding assumptions when needed. In each batch one estimates solutions for the set of desired parameters at a specified epoch within the batch. One usually chooses to report solutions corresponding to the beginning, middle, or end of the batch. General coordinate and time transformations (discussed in Section \[sec:time\_scales\]) are then used to report the solution in the epoch chosen for the entire data interval. One may also adjust the solutions among adjacent batches by accounting for possible correlations. This process produces a smoothed solution for the set of solved-for parameters. More details on this so called “batch–sequential algorithm with smoothing filter” are available in Refs. [@Moyer71]-[@Gelb]. Even without process noise, the inversion algorithms of the Kalman formulation and the weighted least-squares method seem radically different. But as shown in [@sherman], if one uses a single batch for all the data and if one uses certain assumptions about, for instance, the process noise and the smoothing algorithms, then the two methods are mathematically identical. When introducing process noise, an additional process noise matrix is also added into the solution algorithm. The elements of this matrix are chosen by the user as prescribed by standard statistical techniques used for navigational data processing. For the recent results reported in Section \[recent\_results\], JPL used both the batch-sequential and the weighted least-squares estimation approaches. JPL originally implemented only the batch-sequential method, which yielded the detection (at a level smaller than could be detected with any other spacecraft) of an annual oscillatory term smaller in size than the anomalous acceleration [@moriond]. (This term is discussed in Section \[annualterm\].) The recent studies included weighted least-squares estimation to see if this annual term was a calculational anomaly. The Aerospace Corporation uses only the weighted least-squares approach with its CHASMP software. A $\chi^2$ test is used as an indicator of the quality of the fit. In this case, the anomalous acceleration is treated as a constant parameter over the entire data interval. To solve for $a_P$ one estimates the statistical weights for the data points and then uses these in a general weighted least-squares fashion. Note that the weighted least-squares method can obtain a result similar to that from a batch-sequential approach (with smoothing filter, zero correlation time and without process noise) by cutting the data interval into smaller pieces and then looking at the temporal variation among the individual solutions. As one will see in the following, in the end, both programs yielded very similar results. The differences between them can be mainly attributed to (other) systematics. This gives us confidence that both programs and their implemented estimation algorithms are correct to the accuracy of this investigation. \[results\]ORIGINAL DETECTION OF THE ANOMALOUS ACCELERATION =========================================================== Early JPL studies of the anomalous Pioneer Doppler residuals {#subsec:accel} ------------------------------------------------------------ As mentioned in the introduction, by 1980 Pioneer 10 was at 20 AU, so the solar radiation pressure acceleration had decreased to $< 5\times 10^{-8}$ cm/s$^2$. Therefore, a search for unmodeled accelerations (at first with the further out Pioneer 10) could begin at this level. With the acceptance of a proposal of two of us (JDA and ELL) to participate in the Heliospheric Mission on Pioneer 10 and 11, such a search began in earnest [@jpl]. The JPL analysis of unmodeled accelerations used the JPL’s Orbit Determination Program (ODP) [@Moyer71]-[@Moyer00]. Over the years the data continually indicated that the largest systematic error in the acceleration residuals is a constant bias of $a_P \sim (8\pm 3) \times 10^{-8}$ cm/s$^2$, directed [*toward*]{} the Sun (to within the beam-width of the Pioneers’ antennae [@sunearth]). As stated previously, the analyses were modeled to include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, together with bias and drift in the Doppler signal. Planetary coordinates and the solar system masses were taken from JPL’s Export Planetary Ephemeris DE405, referenced to ICRF. The analyses used the standard space-fixed J2000 coordinate system with its associated JPL planetary ephemeris DE405 (or earlier, DE200). The time-varying Earth orientation in J2000 coordinates is defined by a 1998 version of JPL’s EOP file, which accounts for the inertial precession and nutation of the Earth’s spin axis, the geophysical motion of the Earth’s pole with respect to its spin axis, and the Earth’s time varying spin rate. The three-dimensional locations of the tracking stations in the Earth’s body-fixed coordinate system (geocentric radius, latitude, longitude) were taken from a set recommended by ICRF for JPL’s DE405. Consider ${\nu}_{\tt obs}$, the frequency of the re-transmitted signal observed by a DSN antennae, and $\nu_{\tt model}$, the predicted frequency of that signal. The observed, two-way anomalous effect can be expressed to first order in $v/c$ as [@drift] $$\begin{aligned} \nonumber \left[\nu_{\tt obs}(t)- \nu_{\tt model}(t)\right]_{\tt DSN} = - \nu_{0}\frac{2a_P~t}{c}, \\ \nu_{\tt model} = \nu_{0}\left[1 - \frac{2v_{\tt model}(t)}{c}\right]. \label{eq:delta_nu}\end{aligned}$$ Here, $\nu_{0}$ is the reference frequency, the factor $2$ is because we use two- and three-way data [@way]. $v_{\tt model}$ is the modeled velocity of the spacecraft due to the gravitational and other large forces discussed in Section \[navigate\]. (This velocity is outwards and hence produces a red shift.) We have already included the sign showing that $a_P$ is inward. (Therefore, $a_P$ produces a slight blue shift on top of the larger red shift.) By DSN convention [@drift], the first of Eqs. (\[eq:delta\_nu\]) is $[\Delta \nu_{\tt obs} - \Delta \nu_{\tt model}]_{\tt usual} = - [\Delta \nu_{\tt obs} - \Delta \nu_{\tt model}]_{\tt DSN}$. Over the years the anomaly remained in the data of both Pioneer 10 and Pioneer 11 [@bled]. (See Figure \[fig:forces\].) -20pt In order to model any unknown forces acting on Pioneer 10, the JPL group introduced a stochastic acceleration, exponentially correlated in time, with a time constant that can be varied. This stochastic variable is sampled in ten-day batches of data. We found that a correlation time of one year produces good results. We did, however, experiment with other time constants as well, including a zero correlation time (white noise). The result of applying this technique to 6.5 years of Pioneer 10 and 11 data is shown in Figure \[fig:correlation\]. The plotted points represent our determination of the stochastic variable at ten-day sample intervals. We plot the stochastic variable as a function of heliocentric distance, not time, because that is more fundamental in searches for trans-Neptunian sources of gravitation. -10pt -20pt As possible “perturbative forces” to explain this bias, we considered gravity from the Kuiper belt, gravity from the galaxy, spacecraft “gas leaks,” errors in the planetary ephemeris, and errors in the accepted values of the Earth’s orientation, precession, and nutation. We found that none of these mechanisms could explain the apparent acceleration, and some were three orders of magnitude or more too small. \[We also ruled out a number of specific mechanisms involving heat radiation or “gas leaks,” even though we feel these are candidates for the cause of the anomaly. We will return to this in Sections \[ext-systema\] and \[int-systema\].\] We concluded [@anderson], from the JPL-ODP analysis, that there is an unmodeled acceleration, $a_P$, towards the Sun of $(8.09\pm0.20)\times 10^{-8}$ cm/s$^2$ for Pioneer 10 and of $(8.56\pm 0.15) \times 10^{-8}$ cm/s$^2$ for Pioneer 11. The error was determined by use of a five-day batch sequential filter with radial acceleration as a stochastic parameter subject to white Gaussian noise ($\sim$ 500 independent five-day samples of radial acceleration) [@tap]. No magnitude variation of $a_P$ with distance was found, within a sensitivity of $\sigma_0=2\times10^{-8}$ cm/s$^2$ over a range of 40 to 60 AU. All our errors are taken from the covariance matrices associated with the least–squares data analysis. The assumed data errors are larger than the standard error on the post–fit residuals. \[For example, the Pioneer S–band Doppler error was set at 1 mm/s at a Doppler integration time of 60 s, as opposed to a characteristic $\chi^2$ value of 0.3 mm/s.\] Consequently, the quoted errors are realistic, not formal, and represent our attempt to include systematics and a reddening of the noise spectrum by solar plasma. Any spectral peaks in the post-fit Pioneer Doppler residuals were not significant at a 90% confidence level [@anderson]. First Aerospace study of the apparent Pioneer acceleration {#subsec:aero} ---------------------------------------------------------- With no explanation of this data in hand, our attention focused on the possibility that there was some error in JPL’s ODP. To investigate this, an analysis of the raw data was performed using an independent program, The Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP) [@chasmp] – one of the standard Aerospace orbit analysis programs. CHASMP’s orbit determination module is a development of a program called POEAS (Planetary Orbiter Error Analysis Study program) that was developed at JPL in the early 1970’s independently of JPL’s ODP. As far as we know, not a single line of code is common to the two programs [@poeas]. Although, by necessity, both ODP and CHASMP use the same physical principles, planetary ephemeris, and timing and polar motion inputs, the algorithms are otherwise quite different. If there were an error in either program, they would not agree. Aerospace analyzed a Pioneer 10 data arc that was initialized on 1 January 1987 at 16 hr (the data itself started on 3 January) and ended at 14 December 1994, 0 hr. The raw data set was averaged to 7560 data points of which 6534 points were used. This CHASMP analysis of Pioneer 10 data also showed an unmodeled acceleration in a direction along the radial toward the Sun [@aero]. The value is $(8.65 \pm 0.03) \times 10^{-8}$ cm/s$^{2}$, agreeing with JPL’s result. The smaller error here is because the CHASMP analysis used a batch least-squares fit over the whole orbit [@tap; @chasmp], not looking for a variation of the magnitude of $a_P$ with distance. Without using the apparent acceleration, CHASMP shows a steady frequency drift [@drift] of about $-6 \times 10^{-9}$ Hz/s, or 1.5 Hz over 8 years (one-way only). (See Figure \[fig:aerospace\].) This equates to a clock acceleration, $-a_t$, of $-2.8\times 10^{-18}$ s/s$^{2}$. The identity with the apparent Pioneer acceleration is $$a_t \equiv a_P/c. \label{asubt}$$ The drift in the Doppler residuals (observed minus computed data) is seen in Figure \[fig:pio10best\_fit\]. -10pt -10pt The drift is clear, definite, and cannot be removed without either the added acceleration, $a_P$, or the inclusion in the data itself of a frequency drift, i.e., a “clock acceleration” $a_t$. If there were a systematic drift in the atomic clocks of the DSN or in the time-reference standard signals, this would appear like a non-uniformity of time; i.e., all clocks would be changing with a constant acceleration. We now have been able to rule out this possibility. (See Section \[sec:timemodel\].) Continuing our search for an explanation, we considered the possibilities: i) that the Pioneer 10/11 spacecraft had internal systematic properties, undiscovered because they are of identical design, and ii) that the acceleration was due to some not-understood viscous drag force (proportional to the approximately constant velocity of the Pioneers). Both these possibilities could be investigated by studying spin-stabilized spacecraft whose spin axes are not directed towards the Sun, and whose orbital velocity vectors are far from being radially directed. Two candidates were Galileo in its Earth-Jupiter mission phase and Ulysses in Jupiter-perihelion cruise out of the plane of the ecliptic. As well as Doppler, these spacecraft also yielded a considerable quantity of range data. By having range data one can tell if a spacecraft is accumulating a range effect due to a spacecraft acceleration or if the orbit determination process is fooled by a Doppler frequency rate bias. Galileo measurement analysis {#galileo} ---------------------------- We considered the dynamical behavior of Galileo’s trajectory during its cruise flight from second Earth encounter (on 8 December 1992) to arrival at Jupiter. \[This period ends just before the Galileo probe release on 13 July 1995. The probe reached Jupiter on 7 December 1995.\] During this time the spacecraft traversed a distance of about 5 AU with an approximately constant velocity of 7.19(4) km/s. A quick JPL look at limited Galileo data (241 days from 8 January 1994 to 6 September 1994) demonstrated that it was impossible to separate solar radiation effects from an anomalous constant acceleration. The Sun was simply too close and the radiation cross-section too large. The nominal value obtained was $\sim 8 \times 10^{-8}$ cm/s$^2$. The Aerospace’s analysis of the Galileo data covered the same arc as JPL and a second arc from 2 December 1992 to 24 March 1993. The analysis of Doppler data from the first arc resulted in a determination for $a_P$ of $\sim (8 \pm 3) \times 10^{-8}$ cm/s$^2$, a value similar to that from Pioneer 10. But the correlation with solar pressure was so high (0.99) that it is impossible to decide whether solar pressure is a contributing factor [@aT]. The second data arc was 113 days long, starting six days prior to the second Earth encounter. This solution was also too highly correlated with solar pressure, and the data analysis was complicated by many mid-course maneuvers in the orbit. The uncertainties in the maneuvers were so great, a standard null result could not be ruled out. -10pt -10pt However, there was an additional result from the data of this second arc. This arc was chosen for study because of the availability of ranging data. It had 11596 Doppler points of which 10111 were used and 5643 range points of which 4863 used. The two-way range change and time integrated Doppler are consistent (see Figure \[fig:galileo\_range\]) to $\sim 4$ m over a time interval of one day. For comparison, note that for a time of $t=1$ day, $(a_Pt^2/2)\sim 3$ m. For the apparent acceleration to be the result of hardware problems at the tracking stations, one would need a linear frequency drift at all the DSN stations, a drift that is not observed. Ulysses measurement analysis {#ulysses} ---------------------------- ### JPL’s analysis An analysis of the radiation pressure on Ulysses, in its out-of-the-ecliptic journey from 5.4 AU near Jupiter in February 1992 to the perihelion at 1.3 AU in February 1995, found a varying profile with distance [@uly]. The orbit solution requires a periodic updating of the solar radiation pressure. The radio Doppler and ranging data can be fit to the noise level with a time-varying solar constant in the fitting model [@mcelrath]. We obtained values for the time-varying solar constant determined by Ulysses navigational data during this south polar pass [@uly]. The inferred solar constant is about 40 percent larger at perihelion (1.3 AU) than at Jupiter (5.2 AU), a physical impossibility! We sought an alternative explanation. Using physical parameters of the Ulysses spacecraft, we first converted the time-varying values of the solar constant to a positive (i.e., outward) radial spacecraft acceleration, $a_r$, as a function of heliocentric radius. Then we fit the values of $a_r$ with the following model: $$a_r = \frac{\mathcal{K}f_\odot A}{c M}\frac{\cos\theta(r)}{r^2} - a_{P(U)}, \label{Armodel_corr}$$ where $r$ is the heliocentric distance in AU, $M$ is the total mass of the spacecraft, $f_\odot=1367 ~{\rm W/m}^{2}$(AU)$^2$ is the (effective-temperature Stefan-Boltzmann) “solar radiation constant” at 1 AU, $A$ is the cross-sectional area of the spacecraft and $\theta(r)$ is the angle between the direction to the Sun at distance $r$ and orientation of the antennae. \[For the period analyzed $\theta(r)$ was almost a constant. Therefore its average value was used which corresponded to $\langle{\cos\theta(r)}\rangle\approx 0.82$.\] Optical parameters defining the reflectivity and emissivity of the spacecraft’s surface were taken to yield $\mathcal{K}\approx 1.8$. (See Section \[solarP\] for a discussion on solar radiation pressure.) Finally, the parameter $a_{P(U)}$ was determined by linear least squares. The best–fit value was obtained $$a_{P(U)} = (12 \pm 3)\times 10^{-8}~~{\rm cm/s}^2, \label{SandA}$$ where both random and systematic errors are included. So, by interpreting this time variation as a true $r^{-2}$ solar pressure plus a constant radial acceleration, we found that Ulysses was subjected to an unmodeled acceleration towards the Sun of (12 $\pm$ 3) $\times 10^{-8}$ cm/s$^{2}$. Note, however, that the determined constant $a_{P(U)}$ is highly correlated with solar radiation pressure (0.888). This shows that the constant acceleration and the solar-radiation acceleration are not independently determined, even over a heliocentric distance variation from 5.4 to 1.3 AU. ### Aerospace’s analysis {#sec:AUlysses} The next step was to perform a detailed calculation of the Ulysses orbit from near Jupiter encounter to Sun perihelion, using CHASMP to evaluate Doppler and ranging data. The data from 30 March 1992 to 11 August 1994 was processed. It consisted of 50213 Doppler points of which 46514 were used and 9851 range points of which 8465 were used. Such a calculation would in principle allow a more precise and believable differentiation between an anomalous constant acceleration towards the Sun and systematics. Solar radiation pressure and radiant heat systematics are both larger on Ulysses than on the Pioneers. However, this calculation turned out to be a much more difficult than imagined. Because of a failed nutation damper, an inordinate number of spacecraft maneuvers were required (257). Even so, the analysis was completed. But even though the Doppler and range residuals were consistent as for Galileo, the results were disheartening. For an unexpected reason, any fit is not significant. The anomaly is dominated by (what appear to be) gas leaks [@ulygas]. That is, after each maneuver the measured anomaly changes. The measured anomalies randomly change sign and magnitude. The values go up to about an order of magnitude larger than $a_P$. So, although the Ulysses data was useful for range/Doppler checks to test models (see Section \[sec:timemodel\]), like Galileo it could not provide a good number to compare to $a_P$. \[recent\_results\]RECENT RESULTS ================================= Recent changes to our strategies and orbit determination programs, leading to new results, are threefold. First, we have added a longer data arc for Pioneer 10, extending the data studied up to July 1998. The entire data set used (3 Jan. 1987 to 22 July 1998) covers a heliocentric distance interval from 40 AU to 70.5 AU [@AU]. \[Pioneer 11 was much closer in (22.42 to 31.7 AU) than Pioneer 10 during its data interval (5 January 1987 to 1 October 1990).\] For later use in discussing systematics, we here note that in the ODP calculations, masses used for the Pioneers were $M_{Pio~10}=251.883$ kg and $M_{Pio~11}=239.73$ kg. CHASMP used 251.883 kg for both [@gasuse]. As the majority of our results are from Pioneer 10, we will make $M_0 = 251.883$ kg to be our nominal working mass. Second, and as we discuss in the next subsection, we have studied the spin histories of the craft. In particular, the Pioneer 10 history exhibited a very large anomaly in the period 1990.5 to 1992.5. This led us to take a closer look at any possible variation of $a_P$ among the three time intervals: The JPL analysis defined the intervals as I (3 Jan. 1987 to 17 July 1990); II (17 July 1990 to 12 July 1992) bounded by 49.5 to 54.8 AU; and III (12 July 1992 to 22 July 1998). (CHASMP used slightly different intervals [@I/II]) The total updated data set now consists of 20,055 data points for Pioneer 10. (10,616 data points were used for Pioneer 11.) This helped us to better understand the systematic due to gas leaks, which is taken up in Section \[sec:gleaks\]. Third, in looking at the detailed measurements of $a_P$ as a function of time using ODP, we found an anomalous oscillatory annual term, smaller in size than the anomalous acceleration [@moriond]. As mentioned in Section \[sec:PE\], and as will be discussed in detail in Section \[annualterm\], we wanted to make sure this annual term was not an artifact of our computational method. For the latest results, JPL used both the batch-sequential and the least-squares methods. All our recent results obtained with both the JPL and The Aerospace Corporation software have given us a better understanding of systematic error sources. At the same time they have increased our confidence in the determination of the anomalous acceleration. We present a description and summary of the new results in the rest of this section. Analysis of the Pioneer spin history {#spinhistory} ------------------------------------ Both Pioneers 10 and 11 were spinning down during the respective data intervals that determined their $a_P$ values. Because any changes in spacecraft spin must be associated with spacecraft torques (which for lack of a plausible external mechanism we assume are internally generated), there is also a possibility of a related internally generated translational force along the spin axis. Therefore, it is important to understand the effects of the spin anomalies on the anomalous acceleration. In Figures \[fig:pioneer\_spin\] and \[fig:pio11spin\] we show the spin histories of the two craft during the periods of analysis. -5pt Consider Pioneer 10 in detail. In time Interval I there is a slow spin down at an average rate (slope) of $\sim(-0.0181\pm 0.0001)$ rpm/yr. Indeed, a closer look at the curve (either by eye or from an expanded graph) shows that the spin down is actually slowing with time (the curve is flattening). This last feature will be discussed in Sections \[subsec:katz\] and \[subsec:mainbus\]. Every time thrusters are used, there tends to be a short-term leakage of gas until the valves set (perhaps a few days later). But there can also be long-term leakages due to some mechanism which does not quickly correct itself. The major Pioneer 10 spin anomaly that marks the boundary of Intervals I and II, is a case in point. During this interval there was a major factor of $\sim 4.5$ increase in the average spin-rate change to $\sim(-0.0861\pm0.0009)$ rpm/yr. One also notices kinks during the interval. Few values of the Pioneer 10 spin rate were obtained after mid-1993, so the long-term spin-rate change is not well-determined in Interval III. But from what was measured, there was first a short-term transition region of about a year where the spin-rate change was $\sim-0.0160$ rpm/yr. Then things settled down to a spin-rate change of about $\sim(-0.0073 \pm 0.0015)$ rpm/yr, which is small and less than that of interval I. The effects of the maneuvers on the values of $a_P$ will allow an estimation of the gas leak systematic in Section \[sec:gleaks\]. Note, however, that in the time periods studied, only orientation maneuvers were made, not trajectory maneuvers. Shortly after Pioneer 11 was launched on 5 April 1973, the spin period was 4.845 s. A spin precession maneuver on 18 May 1973 reduced the period to 4.78 s and afterwards, because of a series of precession maneuvers, the period lengthened until it reached 5.045 s at encounter with Jupiter in December 1974. The period was fairly constant until 18 December 1976, when a mid-course maneuver placed the spacecraft on a Saturn-encounter trajectory. Before the maneuver the period was 5.455 s, while after the maneuver it was 7.658 s. At Saturn encounter in December 1979 the period was 7.644 s, little changed over the three-year post maneuver cruise phase. At the start of our data interval on 5 January 1987, the period was 7.321 s, while at the end of the data interval in October 1990 it was 7.238 s. Although the linear fit to the Pioneer 11 spin rate shown in Figure \[fig:pio11spin\] is similar to that for Pioneer 10 in Interval I, $\sim(-0.0234\pm 0.0003)$ rpm/yr, the causes appear to be very different. (Remember, although identical in design, Pioneers 10 and 11 were not identical in quality [@design].) Unlike Pioneer 10, the spin period for Pioneer 11 was primarily affected at the time of spin precession maneuvers. One sees that at maneuvers the spin period decreases very quickly, while in between maneuvers the spin rate actually tends to [*increase*]{} at a rate of $\sim(+0.0073 \pm 0.0003)$ rpm/yr (perhaps due to a gas leak in the opposite direction). All the above observations aid us in the interpretation of systematics in the following three sections. Recent results using JPL software {#jplresults} --------------------------------- The latest results from JPL are based on an upgrade, *Sigma*, to JPL’s ODP software [@sigma]. *Sigma*, developed for NASA’s Cassini Mission to Saturn, eliminates structural restrictions on memory and architecture that were imposed 30 years ago when JPL space navigation depended solely on a Univac 1108 mainframe computer. Five ODP programs and their interconnecting files have been replaced by the single program *Sigma* to support filtering, smoothing, and mapping functions. Program/Estimation method Pio 10 (I) Pio 10 (II) Pio 10 (III) Pio 11 -------------------------------- ------------------ ---------------- --------------- --------------- *Sigma*, [WLS]{}, $$&$$ $$&$$ no solar corona model $8.02\pm0.01$ $ 8.65\pm0.01$ $7.83\pm0.01$ $8.46\pm0.04$ *Sigma*, [WLS]{}, $$&$$ $$&$$ with solar corona model $8.00\pm0.01$ $8.66\pm0.01$ $7.84\pm0.01$ $8.44\pm0.04$ *Sigma*, [BSF]{}, 1-day batch, with solar corona model $7.82\pm0.29$ $8.16\pm0.40$ $7.59\pm0.22$ $8.49\pm0.33$ CHASMP, [WLS]{}, no solar corona model $8.25\pm0.02$ $8.86\pm0.02$ $7.85\pm0.01$ $8.71\pm0.03$ CHASMP, [WLS]{}, with solar corona model $8.22 \pm 0.02 $ $8.89\pm0.02$ $7.92\pm0.01$ $8.69\pm0.03$ CHASMP, [WLS]{}, with corona, weighting, and F10.7 $8.25\pm0.03$ $8.90\pm0.03$ $7.91\pm0.01$ $8.91\pm0.04$ We used *Sigma* to reduce the Pioneer 10 (in three time intervals) and 11 Doppler of the unmodeled acceleration, $a_P$, along the spacecraft spin axis. As mentioned, the Pioneer 10 data interval was extended to cover the total time interval 3 January 1987 to 22 July 1998. Of the total data set of 20,055 Pioneer 10 Doppler points, JPL used $\sim$19,403, depending on the initial conditions and editing for a particular run. Of the available 10,616 (mainly shorter time-averaged) Pioneer 11 data points, 10,252 were used (4919 two-way and 5333 three-way). We wanted to produce independent (i.e., uncorrelated) solutions for $a_P$ in the three Pioneer 10 segments of data. The word independent solution in our approach means only the fact that data from any of the three segments must not have any information (in any form) passed onto it from the other two intervals while estimating the anomaly. We moved the epoch from the beginning of one data interval to the next by numerically integrating the equations of motion and not iterating on the data to obtain a better initial conditions for this consequent segment. Note that this numerical iteration provided us only with an *a priori* estimate for the initial conditions for the data interval in question. Other parameters included in the fitting model were the six spacecraft heliocentric position and velocity coordinates at the 1987 epoch of 1 January 1987, 01:00:00 [ET]{}, and 84 (i.e., $28\times 3$) instantaneous velocity increments along the three spacecraft axes for 28 spacecraft attitude (or spin orientation) maneuvers. If these orientation maneuvers had been performed at exactly six month intervals, there would have been 23 maneuvers over our 11.5 year data interval. But in fact, five more maneuvers were performed than expected over this 11.5 year interval giving a total of 28 maneuvers in all. As noted previously, in fitting the Pioneer 10 data over 11.5 years we used the standard space-fixed J2000 coordinate system with planetary ephemeris DE405, referenced to ICRF. The three-dimensional locations of the tracking stations in the Earth’s body-fixed coordinate system (geocentric radius, latitude, longitude) were taken from a set recommended by ICRF for JPL’s DE405. The time-varying Earth orientation in J2000 coordinates was defined by a 1998 version of JPL’s EOP file. This accounted for the geophysical motion of the Earth’s pole with respect to its spin axis and the Earth’s time varying spin rate. JPL used both the weighted least-squares ([WLS]{}) and the batch-sequential filter ([BSF]{}) algorithms for the final calculations. In the first three rows of Table \[resulttable\] are shown the ODP results for i) [WLS]{} with no corona, ii) [WLS]{} with the Cassini corona model, and iii) [BSF]{} with the Cassini corona model. Observe that the [WLS]{} acceleration values for Pioneer 10 in Intervals I, II, and III are larger or smaller, respectively, just as the spin-rate changes in these intervals are larger or smaller, respectively. This indicates that the small deviations may be due to a correlation with the large gas leak/spin anomaly. We will argue this quantitatively in Section \[sec:gleaks\]. For now we just note that we therefore expect the number from Interval III, $a_P= 7.83 \times 10^{-8}$cm/s$^2$, to be close to our basic (least perturbed) JPL result for Pioneer 10. We also note that the statistical errors and the effect of the solar corona are both small for [WLS]{}, and will be handled in our error budget. In Figure \[ODPall\] we show ODP/*Sigma* [WLS]{} Doppler residuals for the entire Pioneer 10 data set. The residuals were obtained by first solving for $a_P$ with no corona in each of the three Now look at the batch-sequential results in row 3 of Table \[resulttable\]. First, note that the statistical Intervals independently and then subtracting these solutions (given in Table \[resulttable\]) from the fits within the corresponding data intervals. -10pt One can easily see the very close agreement with the CHASMP residuals of Figure \[fig:pio10best\_fit\], which go up to 14 December 1994. The Pioneer 11 number is significantly higher. A deviation is not totally unexpected since the data was relatively noisy, was from much closer in to the Sun, and was taken during a period of high solar activity. We also do not have the same handle on spin-rate change effects as we did for Pioneer 10. We must simply take the number for what it is, and give the basic JPL result for Pioneer 11 as $a_P= 8.46 \times 10^{-8}$ cm/s$^2$. Now look at the batch-sequential results in row 3 of Table \[resulttable\]. First, note that the statistical errors are an order of magnitude larger than for [WLS]{}. This is not surprising since: i) the process noise significantly affects the precision, ii) [BSF]{} smoothes the data and the data from the various intervals is more correlated than in [WLS]{}. The effects of all this are that all four numbers change so as to make them all closer to each other, but yet all the numbers vary by less than $2 \sigma$ from their [WLS]{} counterparts. Finally, there is the annual term. It remains in the data (for both Pioneers 10 and 11). A representation of it can be seen in a 1-day batch-sequential averaged over all 11.5 years. It yielded a result $a_P= (7.77 \pm 0.16) \times 10^{-8}$ cm/s$^2$, consistent with the other numbers/errors, but with an added annual oscillation. In the following subsection we will compare JPL results showing the annual term with the counterpart Aerospace results. We will argue in Section \[annualterm\] that this annual term is due to the inability to model the angles of the Pioneers’ orbits accurately enough. \[Note that this annual term is not to be confused with a small oscillation seen in Figure \[fig:aerospace\] that can be caused by mispointing towards the spacecraft by the fit programs.\] Recent results using The Aerospace Corporation software {#aerospaceresults} ------------------------------------------------------- As part of an ongoing upgrade to CHASMP’s accuracy, Aerospace has used Pioneer 10 and 11 as a test bed to confirm the revision’s improvement. In accordance with the JPL results of Section \[jplresults\], we used the new version of CHASMP to concentrate on the Pioneer 10 and 11 data. The physical models are basically the same ones that JPL used, but the techniques and methods used are largely different. (See Section \[Ext\_accuracy\].) The new results from the Aerospace Corporation’s software are based on first improving the Planetary Ephemeris and Earth orientation and spacecraft spin models required by the program. That is: i) the spin data file has been included with full detail; ii) a newer JPL Earth Orientation Parameters file was used; iii) all IERS tidal terms were included; iv) plate tectonics were included; v) DE405 was used; vi) no [*a priori*]{} information on the solved for parameters was included in the fit; vii) Pioneer 11 was considered, viii) the Pioneer 10 data set used was extended to 14 Feb. 1998. Then the Doppler data was refitted. Beginning with this last point: CHASMP uses the same original data file, but it performs an additional data compression. This compression combines the longest contiguous data composed of adjacent data intervals or data spans with duration $\ge 600$ s (effectively it prefers 600 and 1980 second data intervals). It ignores short-time data points. Also, Aerospace uses an N-$\sigma$/fixed boundary rejection criteria that rejects all data in the fit with a residual greater than $\pm 0.025$ Hz. These rejection criteria resulted in the loss of about 10 % of the original data for both Pioneers 10 and 11. In particular, the last five months of Pioneer 10 data, which was all of data-lengths less than 600 s, was ignored. Once these data compression/cuts were made, CHASMP used 10,499 of its 11,610 data points for Pioneer 10 and 4,380 of its 5,137 data points for Pioneer 11. Because of the spin-anomaly in the Pioneer 10 data, the data arc was also divided into three time intervals (although the I/II boundary was taken as 31 August 1990 [@I/II]). In what was especially useful, the Aerospace analysis uses direct propagation of the trajectory data and solves for the parameter of interest only for the data within a particular data interval. That means the three interval results were truly independent. Pioneer 11 was fit as a single arc. Three types of runs are listed, with: i) no corona; ii) with Cassini corona model of Sections \[corona+wt\] and \[sec:corona\]; and iii) with the Cassini corona model, but added are corona data weighting (Section \[corona+wt\]) and the time-variation called “F10.7” [@F10-7]. (The number 10.7 labels the wavelength of solar radiation, $\lambda$=10.7 cm, that, in our analysis, is averaged over 81 days.) The results are given in rows 4-6 of Table \[resulttable\]. The no corona results (row 4) are in good agreement with the *Sigma* results of the first row. This is especially true for the extended-time Interval III values for Pioneer 10, which interval had clean data. However there is more disagreement with the values for Pioneer 10 in Intervals I and II and for Pioneer 11. These three data sets all were noisy and underwent more data-editing. Therefore, it is significant that the deviations between *Sigma* and CHASMP in these arcs are all similar, but small, between $0.20$ to $0.25$ of our units. As before, the effect of the solar corona is small, even with the various model variations. But most important, the numbers from *Sigma* and CHASMP for Pioneer 10 Interval III are in excellent agreement. Further, CHASMP also found the annual term. (Recall that CHASMP can also look for a temporal variation by calculating short time averages.) Results on the time variation in $a_P$ can be seen in Figure \[fig:rec\_res\_comb\]. Although there could possibly be $a_P$ variations of $\pm 2\times10^{-8}$ cm/s$^2$ on a 200-day time scale, a comparison of the variations with the error limits shown in Figure \[fig:rec\_res\_comb\] indicate that our measurements of these variations are not statistically significant. The 5-day averages of $a_P$ from ODP (using the batch-sequential method) are not reliable at solar conjunction in the middle (June) of each year, and hence should be ignored there. The CHASMP 200-day averages suppress the solar conjunction bias inherent in the ODP 5-day averages, and they reliably indicate a constant value of $a_P$. Most encouraging, these results clearly indicate that the obtained solution is consistent, stable, and its mean value does not strongly depend on the estimation procedure used. The presence of the small annual term on top of the obtained solution is apparent. Our solution, before systematics, for the anomalous acceleration {#final_sol} ---------------------------------------------------------------- From Table \[resulttable\] we can intuitively draw a number of conclusions:\ A) The effect of the corona is small. This systematic will be analyzed in Section \[solarwind\].\ B) The numerical error is small. This systematic will be analyzed in Section \[leastsquares\].\ C) The differences between the *Sigma* and CHASMP Pioneer 10 results for Interval I and Interval II, respectively, we attribute to two main causes: especially i) the different data rejection techniques of the two analyses but also ii) the different maneuver simulations. Both of these effects were especially significant in Interval II, where the data arc was small and a large amount of noisy data was present. Also, to account for the discontinuity in the spin data that occurred on 28 January 1992 (see Figure \[fig:pioneer\_spin\]), Aerospace introduced a fictitious maneuver for this interval. Even so, the deviation in the two values of $a_P$ was relatively small, namely $0.23$ and $0.21$, respectively, $\times 10^{-8}$ cm/s$^2$.\ D) The changes in $a_P$ in the different Intervals, correlated with the changes in spin-rate change, are likely (at least partially) due to gas leakage. This will be discussed in Section \[sec:gleaks\]. But independent of the origin, this last correlation between shifts in $a_P$ and changes in spin rate actually allows us to calculate the best “experimental” base number for Pioneer 10. To do this, assume that the spin-rate change is directly contributing to an anomalous acceleration offset. Mathematically, this is saying that in any interval $i=\mathrm{I,II,III}$, for which the spin-rate change is an approximate constant, one has $$a_{P}(\ddot{\theta}) = a_{P(0)} - \kappa~\ddot{\theta}, \label{gasleakeq}$$ where $\kappa$ is a constant with units of length and $ a_{P(0)} \equiv a_P(\ddot{\theta}=0)$ is the Pioneer acceleration without any spin-rate change. One now can fit the data to Eq. (\[gasleakeq\]) to obtain solutions for $\kappa$ and $a_{P(0)}$. The three intervals $i=\mathrm{I,II,III}$ provide three data combinations $\{a_{P(i)}(\ddot{\theta}), \ddot{\theta}_i\}$. We take our base number, with which to reference systematics, to be the weighted average of the *Sigma* and CHASMP results for $a_{P(0)}$ when no corona model was used. Start first with the *Sigma* Pioneer 10 solutions in row one of Table \[resulttable\] and the Pioneer 10 spin-down rates given in Section \[jplresults\] and Figure \[fig:pioneer\_spin\]: $a_{P(i)}^{\tt Sigma}=(8.02\pm 0.01,~ 8.65\pm 0.01,~7.83\pm 0.01)$ in units of $10^{-8}$ cm/s$^2$ and $\ddot{\theta}_{i}=-(0.0181\pm 0.0001,~0.0861\pm 0.0009,~0.0073\pm 0.0015)$ in units of rpm/yr, where $$\begin{aligned} 1~\mathrm{rpm/year}&=& 5.281\times10^{-10}~\mathrm{rev/s}^2 \nonumber\\ &=& 3.318\times10^{-9}~\mathrm{radians/s}^2. \end{aligned}$$ With these data we use the maximum likelihood and minimum variance approach to find the optimally weighted least-squares solution for $a_{P(0)}$: $$\begin{aligned} a^{\tt Sigma}_{P(0)} &=& (7.82\pm 0.01) \times 10^{-8}~\mathrm{cm/s}^2, \label{eq:sol_sigma_a} %\\ \kappa^{\tt Sigma} &=& (29.2 \pm 0.7)~ \mathrm{cm}. %\label{eq:sol_sigma_k} \end{aligned}$$ with solution for the parameter $\kappa$ obtained as $\kappa^{\tt Sigma} = (29.2 \pm 0.7)~ \mathrm{cm}$. Similarly, for CHASMP one takes the values for $a_P$ from row four of Table \[resulttable\]: $a^{\tt CHASMP}_{P(i)}=(8.25\pm0.02,~ 8.86\pm0.02,~7.85\pm0.01)$ and uses them with the same $\ddot{\theta}_{i}$ as above. The solution for $a_{P(0)}$ in this case is $$\begin{aligned} a^{\tt CHASMP}_{P(0)} & = &(7.89\pm 0.02) \times 10^{-8}~\mathrm{cm/s}^2 \label{eq:sol_chasmp}, %\\\kappa^{\tt CHASMP} &= &(32.1 \pm 1.0)~ \mathrm{cm}.\end{aligned}$$ together with $\kappa^{\tt CHASMP} = (34.7 \pm 1.1)~ \mathrm{cm}$. The solutions for *Sigma* and CHASMP are similar, 7.82 and 7.89 in our units. We take the weighted average of these two to yield our base line “experimental” number for $a_P$: $$\begin{aligned} a_{P({\tt exper)}}^{\tt Pio10} &=& (7.84\pm 0.01)~\times~10^{-8}~\mathrm{cm/s}^2. \label{pio10lastresult}\end{aligned}$$ \[The weighted average constant $\kappa$ is $\kappa_0 =(30.7\pm 0.6)$ cm.\] For Pioneer 11, we only have the one 3$\frac{3}{4}$ year data arc. The weighted average of the two programs’ no corona results is $(8.62\pm 0.02) \times 10^{-8}$ cm/s$^2$. We observed in Section \[spinhistory\] that between maneuvers (which are accounted for - see Section \[model-maneuvers\]) there is actually a spin rate [*increase*]{} of $\sim(+0.0073 \pm 0.0003)$ rpm/yr. If one uses this spin-up rate and the Pioneer 10 value for $\kappa_0=30.7$ cm given above, one obtains a spin-rate change corrected value for $a_P$. We take this as the experimental value for Pioneer 11: $$a_{P({\tt exper)}}^{\tt Pio 11}= (8.55\pm 0.02) \times 10^{-8} ~{\rm cm/s}^2. \label{pio11lastresult}$$ \[ext-systema\]SOURCES OF SYSTEMATIC ERROR EXTERNAL TO THE SPACECRAFT ===================================================================== We are concerned with possible systematic acceleration errors that could account for the unexplained anomalous acceleration directed toward the Sun. There exist detailed publications describing analytic recipes developed to account for non-gravitational accelerations acting on spacecraft. (For a summary see Milani et al. [@milani].) With regard to the specific Pioneer spacecraft, possible sources of systematic acceleration have been discussed before for Pioneer 10 and 11 at Jupiter [@null76] and Pioneer 11 at Saturn [@null81]. External forces can produce three vector components of spacecraft acceleration, unlike forces generated on board the spacecraft, where the two non-radial components (i.e., those that are effectively perpendicular to the spacecraft spin) are canceled out by spacecraft rotation. However, non-radial spacecraft accelerations are difficult to observe by the Doppler technique, which measures spacecraft velocity along the Earth-spacecraft line of sight. But with several years of Doppler data, it is in principle possible to detect systematic non-radial acceleration components [@sunearth]. With our present analysis [@sunearth] we find that the Doppler data yields only one significant component of unmodeled acceleration, and that any acceleration components perpendicular to the spin axis are small. This is because in the fitting we tried including three unmodeled acceleration constants along the three spacecraft axes (spin axis and two orthogonal axes perpendicular to the spin axis). The components perpendicular to the spin axis had values consistent with zero to a 1-$\sigma$ accuracy of 2 $\times$ 10$^{-8}$ cm/s$^{2}$ and the radial component was equal to the reported anomalous acceleration. Further, the radial acceleration was not correlated with the other two unmodeled acceleration components. Although one could in principle set up complicated engineering models to predict all or each of the systematics, often the uncertainty of the models is too large to make them useful, despite the significant effort required. A different approach is to accept our ignorance about a non-gravitational acceleration and assess to what extent these can be assumed a constant bias over the time scale of all or part of the mission. (In fact, a constant acceleration produces a linear frequency drift that can be accounted for in the data analysis by a single unknown parameter.) In fact, we will use both approaches. In most orbit determination programs some effects, like the solar radiation pressure, are included in the set of routinely estimated parameters. Nevertheless we want to demonstrate their influence on Pioneer’s navigation from the general physics standpoint. This is not only to validate our results, but also to be a model as to how to study the influence of the other physical phenomena that are not yet included in the standard navigational packages for future more demanding missions. Such missions will involve either spacecraft that will be distant or spacecraft at shorter distances where high-precision spacecraft navigation will be required. In this section we will discuss possible systematics (including forces) generated external to the spacecraft which might significantly affect our results. These start with true forces due to (1) solar-radiation pressure and (2) solar wind pressure. We go on to discuss (3) the effect of the solar corona and its mismodeling, (4) electro-magnetic Lorentz forces, (5) the influence of the Kuiper belt, (6) the phase stability of the reference atomic clocks, and (7) the mechanical and phase stability of the DSN antennae, together with influence of the station locations and troposphere and ionosphere contributions. Direct solar radiation pressure and mass {#solarP} ---------------------------------------- There is an exchange of momentum when solar photons impact the spacecraft and are either absorbed or reflected. Models for this solar pressure effect were developed before either Pioneer 10 or 11 were launched [@rad] and have been refined since then. The models take into account various parts of the spacecraft exposed to solar radiation, primarily the high-gain antenna. It computes an acceleration directed away from the Sun as a function of spacecraft orientation and solar distance. The models for the acceleration due to solar radiation can be formulated as $$a_{\tt s.p.}(r)=\frac{\mathcal{K} f_\odot A }{c~M} \frac{ \cos\theta(r)}{r^2}. \label{eq:srp}$$ $f_\odot=1367 ~{\rm W/m}^{2}$(AU)$^2$ is the (effective-temperature Stefan-Boltzmann) “solar radiation constant” at 1 AU from the Sun and $A$ is the effective size of the craft as seen by the Sun [@solar_irr]. (For Pioneer the area was taken to be the antenna dish of radius 1.73 m.) $\theta$ is the angle between the axis of the antenna and the direction of the Sun, $c$ is the speed of light, $M$ is the mass of the spacecraft (taken to be 251.883 for Pioneer 10), and $r$ is the distance from the Sun to the spacecraft in AU. $\mathcal{K}$ [@Lambda] is the [*effective*]{} [@effect] absorption/reflection coefficient. For Pioneer 10 the simplest approximately correct model yields $\mathcal{K}_{0}=1.71$ [@effect]. Eq. (\[eq:srp\]) provides a good model for analysis of the effect of solar radiation pressure on the motion of distant spacecraft and is accounted for by most of the programs used for orbit determination. However, in reality the absorptivities, emissivities, and effective areas of spacecraft parts parameters which, although modeled by design, are determined by calibration early in the mission [@sunparam]. One determines the magnitude of the solar-pressure acceleration at various orientations using Doppler data. (The solar pressure effect can be distinguished from gravity’s $1/r^2$ law because $\cos\theta$ varies [@massprog].) The complicated set of program input parameters that yield the parameters in Eq. (\[eq:srp\]) are then set for later use [@sunparam]. Such a determination of the parameters for Pioneer 10 was done, soon after launch and later. When applied to the solar radiation acceleration in the region of Jupiter, this yields (from a 5 % uncertainty in $a_{\tt s.p.}$ [@null76]) $$\begin{aligned} a_{\tt s.p.}(r={\tt 5.2 AU})&=& (70.0 \pm 3.5) \times 10^{-8}~{\rm cm/s}^2, \nonumber\\ \mathcal{K}_{\tt 5.2} &=& 1.77. \label{aspS}\end{aligned}$$ The second of Eqs. (\[aspS\]) comes from putting the first into Eq. (\[eq:srp\]). Note, specifically, that in a fit a too high input mass will be compensated for by a higher effective $\mathcal{K}$. Because of the $1/r^2$ law, by the time the craft reached 10 AU the solar radiation acceleration was $18.9\times 10^{-8}$ cm/s$^2$ going down to 0.39 of those units by 70 AU. Since this systematic falls off as $r^{-2}$, it can bias the Doppler determination of a constant acceleration at some level, even though most of the systematic is correctly modeled by the program itself. By taking the average of the $r^{-2}$ acceleration curves over the Pioneer distance intervals, we estimate that the systematic error from solar-radiation pressure in units of 10$^{-8}$ ${\rm cm/s}^2$ is 0.001 for Pioneer 10 over an interval from 40 to 70 AU, and 0.006 for Pioneer 11 over an interval from 22 to 32 AU. However, this small uncertainty is not our main problem. In actuality, since the parameters were fit the mass has decreased with the consumption of propellant. Effectively, the $1/r^2$ systematic has changed its normalization with time. If not corrected for, the difference between the original $1/r^2$ and the corrected $1/r^2$ will be interpreted as a bias in $a_P$. Unfortunately, exact information on gas usage is unavailable [@gasuse]. Therefore, in dealing with the effect of the temporal mass variation during the entire data span (i.e. nominal input mass vs. actual mass history [@mass; @gasuse]) we have to address two effects on the solutions for the anomalous acceleration $a_P$. They are i) the effect of mass variation from gas consumption and ii) the effect of an incorrect input mass [@mass; @gasuse]. To resolve the issue of mass variation uncertainty we performed a sensitivity analysis of our solutions to different spacecraft input masses. We simply re-did the no-corona, WLS runs of Table \[resulttable\] with a range of different masses. The initial wet weight of the package was 259 kg with about 36 kg of consumable propellant. For Pioneer 10, the input mass in the program fit was 251.883 kg, roughly corresponding to the mass after spin-down. By our data period, roughly half the fuel (18 kg) was gone so we take 241 kg as our nominal Pioneer 10 mass. Thus, the effect of going from 251.883 kg to 241 kg we take to be our bias correction for Pioneer 10. We take the uncertainty to be given by one half the effect of going from plus to minus 9 kg (plus or minus a quarter tank) from the nominal mass of 241 kg. For the three intervals of Pioneer 10 data, using ODP/*Sigma* yields the following changes in the accelerations: $$\begin{aligned} \delta a^{\tt mass }_P &=& [(0.040 \pm 0.035),~(0.029 \pm 0.025), ~~~~~~~~~~~~~~~~\nonumber \\ &~& ~~~~~~~~~~~~~~~ (0.020 \pm 0.017)]~\times 10^{-8}~ \mathrm{cm/s}^2. \nonumber\end{aligned}$$ As expected,these results make $a_P$ larger. For our systematic bias we take the weighted average of $\delta a^{\tt mass }_P$ for the three intervals of Pioneer 10. The end result is $$a_{\tt s.p.}= (0.03~ \pm~ 0.01) \times 10^{-8}~ \mathrm{cm/s}^2.$$ For Pioneer 11 we did the same except our bias point was 3/4 of the fuel gone (232 kg). Therefore the bias results by going from the input mass of 239.73 to 232 kg. The uncertainty is again defined by $\pm$ 9 kg. The result for Pioneer 11 is more sensitive to mass changes, and we find using ODP/*Sigma* $$a_{\tt s.p.}= (0.09~ \pm~ 0.21) \times 10^{-8}~ \mathrm{cm/s}^2.$$ The bias number is three times larger than the similar number for Pioneer 10, and the uncertainty much larger. We return to this difference in Section \[twospace\]. The previous analysis also allowed us to perform consistency checks on the effective values of $\mathcal{K}$ which the programs were using. By taking $[r_{\mathrm{min}}r_{\mathrm{max}}]^{-1}= [\int(dr/r^2)/\int dr]$ for the inverse distance squared of a data set, varying the masses, and determining the shifts in $a_P$ we could determine the values of $\mathcal{K}$ implied, We found: $\mathcal{K}_{\tt Pio-10(I)}^{\tt ODP} \approx 1.72$; $\mathcal{K}_{\tt Pio-11}^{\tt ODP} \approx 1.82$; $\mathcal{K}_{\tt Pio-10(I)}^{\tt CHASMP} \approx 1.74$; and $\hat{\mathcal{K}}_{\tt Pio-11)}^{\tt CHASMP} \approx 1.84$. \[The hat over the last $\mathcal{K}$ indicates it was multiplied by (237.73/251.883) because CHASMP uses 259.883 kg instead of 239.73 kg for the input mass.\] All these values of $\mathcal{K}$ are in the region expected and are clustered around the value $\mathcal{K}_{\tt 5.2}$ in Eq. (\[aspS\]). Finally, if you take the average values of $\mathcal{K}$ for Pioneers 10 and 11 (1.73, 1.83), multiply these numbers by the input masses (251.883, 239.73) kg, and divide them by our nominal masses (241, 232) kg, you obtain (1.87, 1.89), indicating our choice of nominal masses was well motivated. The solar wind {#solarwind} -------------- The acceleration caused by the solar wind has the same form as Eq. (\[eq:srp\]), with $f_\odot$ replaced by $m_pv^3n$, where $n \approx5$ cm$^{-3}$ is the proton density at 1 AU and $v\approx400$ km/s is the speed of the wind. Thus, $$\begin{aligned} \sigma_{\tt s.w.}(r)&=&\mathcal{K}_{\tt s.w.}\frac{m_pv^3\,n\, A\cos\theta}{cM \,r^2}\nonumber\\ &\approx& 1.24\times10^{-13} \left(\frac{20 ~\rm AU}{r}\right)^2~{\rm cm/s}^2. \label{eq:sw}\end{aligned}$$ Because the density can change by as much as 100%, the exact acceleration is unpredictable. But there are measurements [@solar_irr] showing that it is about 10$^{-5}$ times smaller than the direct solar radiation pressure. Even if we make the very conservative assumption that the solar wind contributes only 100 times less force than the solar radiation, its smaller contribution is completely negligible. The effects of the solar corona and models of it {#sec:corona} ------------------------------------------------ As we saw in the previous Section \[solarwind\], the effect of the solar wind pressure is negligible for distant spacecraft motion in the solar system. However, the solar corona effect on propagation of radio waves between the Earth and the spacecraft needs to be analyzed in more detail. Initially, to study the sensitivity of $a_P$ to the solar corona model, we were also solving for the solar corona parameters $A$, $B$, and $C$ of Eq. (\[corona\_model\_content\]) in addition to $a_P$. However, we realized that the Pioneer Doppler data is not precise enough to produce credible results for these physical parameters. In particular, we found that solutions could yield a value of $a_P$ which was changed by of order 10 % even though it gave unphysical values of the parameters (especially $B$, which previously had been poorly defined even by the Ulysses mission [@bird]). \[By “unphysical” we mean electron densities that were either negative or positive with values that are vastly different from what would be expected.\] Therefore, as noted in Section \[corona+wt\], we decided to use the newly obtained values for $A$, $B$, and $C$ from the Cassini mission and use them as inputs for our analyses: $A= 6.0\times 10^3, B= 2.0\times 10^4, C= 0.6\times 10^6$, all in meters [@Ekelund]. This is the “Cassini corona model.” The effect of the solar corona is expected to be small for Doppler and large for range. Indeed it is small for *Sigma*. For ODP/*Sigma*, the time-averaged effect of the corona was small, of order $$\sigma_{\tt corona} = \pm 0.02~ \times~ 10^{-8}~ \mathrm{cm/s}^2,$$ as might be expected. We take this number to be the error due to the corona. What about the results from CHASMP. Both analyses use the same physical model for the effect of the steady-state solar corona on radio-wave propagation through the solar plasma (that is given by Eq. (\[corona\_model\])). However, there is a slight difference in the actual implementation of the model in the two codes. ODP calculates the corona effect only when the Sun-spacecraft separation angle as seen from the Earth (or Sun-Earth-spacecraft angle) is less then $\pi/2$. It sets the corona contribution to zero in all other cases. Earlier CHASMP used the same model and got a small corona effect. Presently CHASMP calculates an approximate corona contribution for all the trajectory. Specific attention is given to the region when the spacecraft is at opposition from the Sun and the Sun-Earth-spacecraft angle $\sim \pi$. There CHASMP’s implementation truncates the code approximation to the scaling factor $F$ in Eq. (\[corona\_model\]). This is specifically done to remove the fictitious divergence in the region where “impact parameter” is small, $\rho \rightarrow 0$. However, both this and also the more complicated corona models (with data-weighting and/or “F10.7” time variation) used by CHASMP produce small deviations from the no-corona results. Our decision was to incorporate these small deviations between the two results due to corona modeling into our overall error budget as a separate item: $$\sigma_{\tt corona\_model} = \pm 0.02~\times~ 10^{-8} ~~{\rm cm/s}^2.$$ This number could be discussed in Section \[Int\_accuracy\], on computational systematics. Indeed, that is where it will be listed in our error budget. Electro-magnetic Lorentz forces ------------------------------- The possibility that the spacecraft could hold a charge, and be deflected in its trajectory by Lorentz forces, was a concern for the magnetic field strengths at Jupiter and Saturn. However, the magnetic field strength in the outer solar system is on the order of $<1~\gamma~(\gamma=10^{-5}$ Gauss). This is about a factor of $10^5$ times smaller than the magnetic field strengths measured by the Pioneers at their nearest approaches to Jupiter: 0.185 Gauss for Pioneer 10 and 1.135 Gauss for the closer in Pioneer 11 [@edsmith]. Also, there is an upper limit to the charge that a spacecraft can hold. For the Pioneers that limit produced an upper bound on the Lorentz acceleration at closest approach to Jupiter of $20 \times 10^{-8}$ cm/s$^{2}$ [@null76]. With the interplanetary field being so much lower than at Jupiter, we conclude that the electro-magnetic force on the Pioneer spacecraft in the outer solar system is at worst on the order of $10^{-12}$ cm/s$^{2}$, completely negligible [@lorentz]. Similarly, the magnetic torques acting on the spacecraft were about a factor of $10^{-5}$ times smaller than those acting on Earth satellites, where they are a concern. Therefore, for the Pioneers any observed changes in spacecraft spin cannot be caused by magnetic torques. The Kuiper belt’s gravity {#sec:kuiper} ------------------------- From the study of the resonance effect of Neptune upon Pluto, two primary mass concentration resonances of 3:2 and 2:1 were discovered [@malhotra], corresponding to 39.4 AU and 47.8 AU, respectively. Previously, Boss and Peale had derived a model for a non-uniform density distribution in the form of an infinitesimally thin disc extending from 30 AU to 100 AU in the ecliptic plane [@liupeale]. We combined the results of Refs. [@malhotra] and [@liupeale] to determine if the matter in the Kuiper belt could be the source of the anomalous acceleration of Pioneer 10 [@liudust]. We specifically studied three distributions, namely: i) a uniform distribution, ii) a 2:1 resonance distribution with a peak at 47.8 AU, and iii) a 3:2 resonance distribution with a peak at 39.4 AU. Figure \[fig:pioneer\_kb\] exhibits the resulting acceleration felt by Pioneer 10, from 30 to 65 AU which encompassed our data set at the time. We assumed a total mass of one Earth mass, which is significantly larger than standard estimates. Even so, the accelerations are only on the order of $10^{-9}$ cm/s$^2$, which is two orders of magnitude smaller than the observed effect. (See Figure \[fig:pioneer\_kb\].) Further, the accelerations are not constant across the data range. Rather, they show an increasing effect as Pioneer 10 approaches the belt and a decreasing effect as Pioneer 10 recedes from the belt, even with a uniform density model. For these two reasons, we excluded the dust belt as a source for the Pioneer effect. More recent infrared observations have ruled out more than 0.3 Earth mass of Kuiper Belt dust in the trans-Neptunian region [@backman; @teplitzinfra]. Therefore, we can now place a limit of $\pm 3 \times 10^{-10}$ cm/s$^2$ for the contribution of the Kuiper belt. Finally, we note that searches for gravitational encounters of Pioneer with large Kuiper-belt objects have so far not been successful [@gio]. Phase and frequency stability of clocks {#sec:clocs} --------------------------------------- After traversing the mechanical components of the antenna, the radio signal enters the DSN antenna feed and passes through a series of amplifiers, filters, and cables. Averaged over many experiments, the net effect of this on the calculated dynamical parameters of a spacecraft should be very small. We expect instrumental calibration instabilities to contribute $0.2\times10^{-8} $ cm/s$^2$ to the anomalous acceleration on a 60 s time interval. Thus, in order for the atomic clocks [@vessot_clocks] to have caused the Pioneer effect, all the atomic clocks used for signal referencing clocks would have had to have drifted in the same manner as the local DSN clocks. In Section \[results\] we observed that without using the apparent anomalous acceleration, the CHASMP residuals show a steady frequency drift [@drift] of about $-6 \times 10^{-9}$ Hz/s, or 1.5 Hz over 8 years (one-way only). This equates to a clock acceleration, $-a_t$, of $-2.8\times 10^{-18}$ s/s$^{2}$. (See Eq. (\[asubt\]) and Figure \[fig:aerospace\].) To verify that it is actually not the clocks that are drifting, we analyzed the calibration of the frequency standards used in the DSN complex. The calibration system itself is referenced to Hydrogen maser atomic clocks. Instabilities in these clocks are another source of instrumental error which needs to be addressed. The local reference is synchronized to the frequency standards generated either at the National Institute of Standards and Technology (NIST), located in Boulder, Colorado or at the U. S. Naval Observatory (USNO), Washington, DC. These standards are presently distributed to local stations by the Global Positioning System (GPS) satellites. \[During the pre-GPS era, the station clocks used signals from WWV to set the Cesium or Hydrogen masers. WWV, the radio station which broadcasts time and frequency services, is located in Fort Collins, CO.\] While on a track, the station is “free-running,” i.e., the frequency and timing data are generated locally at the station. The Allan variances are about $10^{-13}$ for Cesium and $10^{-15}$ for Hydrogen masers. Therefore, over the data-pass time interval, the data accuracy is on the order of one part in 1000 GHz or better. Long-term frequency stability tests are conducted with the exciter/transmitter subsystems and the DSN’s radio-science open-loop subsystem. An uplink signal generated by the exciter is translated at the antenna by a test translator to a downlink frequency. (See Section \[Exp\_tech\].) The downlink signal is then passed through the RF-IF downconverter present at the antenna and into the radio science receiver chain [@dsn]. This technique allows the processes to be synchronized in the DSN complex based on the frequency standards whose Allan variances are of the order of $\sigma_y \sim 10^{-14}-10^{-15}$ for integration time in the range from 10 s to 10$^3$ s. For the S-band frequencies of the Pioneers, the corresponding Allan variances are 1.3 $\times$ 10$^{-12}$ and 1.0 $\times$ 10$^{-12}$, respectively, for a 10$^3$ s Doppler integration time. Phase-stability testing characterizes stability over very short integration times; that is, spurious signals whose frequencies are very close to the carrier (frequency). The phase noise region is defined to be frequencies within 100 kHz of the carrier. Both amplitude and phase variations appear as phase noise. Phase noise is quoted in dB relative to the carrier, in a 1 Hz band at a specified deviation from the carrier; for example, dBc-Hz at 10 Hz. Thus, for the frequency 1 Hz, the noise level is at $-51$ dBc and 10 Hz corresponds to $-60$ dBc. This was not significant for our study. Finally, the influence of the clock stability on the detected acceleration, $a_P$, may be estimated based on the reported Allan variances for the clocks, $\sigma_y$. Thus, the standard ‘single measurement’ error on acceleration as derived by the time derivative of the Doppler frequency data is $(c \sigma_y)/\tau$, where the Allan variance, $\sigma_y$, is calculated for 1000 s Doppler integration time, and $\tau$ is the signal averaging time. This formula provides a good rule of thumb when the Doppler power spectral density function obeys a $1/f$ flicker-noise law, which is approximately the case when plasma noise dominates the Doppler error budget. Assume a worst case scenario, where only one clock was used for the whole 11 years study. (In reality each DSN station has its own atomic clock.) To estimate the influence of that one clock on the reported accuracy of the detected anomaly $a_P$, combine $\sigma_y={\Delta\nu}/{\nu_0}$, the fractional Doppler frequency shift from the reference frequency of $\nu_0=\sim 2.29$ GHz, with the estimate for the Allan variance, $\sigma_y =1.3 \times 10^{-12}$. This yields a number that characterizes the upper limit for a frequency uncertainty introduced in a single measurement by the instabilities in the atomic clock: $\sigma_\nu=\nu_0\sigma_y=2.98\times10^{-3}$ Hz for a 10$^3$ Doppler integration time. In order to derive an estimate for the total effect, recall that the Doppler observation technique is essentially a continuous count of the total number of complete frequency circles during observational time. Within a year one can have as many as $N\approx3.156\times10^3$ independent single measurements of the clock with duration $10^3$ seconds. This yields an upper limit for the contribution of atomic clock instability on the frequency drift of $\sigma_{\tt clock} = {\sigma_\nu}/{\sqrt{N}} \approx 5.3\times 10^{-5}$ Hz/year. But in Section \[subsec:aero\] we noted that the observed $a_P$ corresponds to a frequency drift of about 0.2 Hz/year, so the error in $a_P$ is about $0.0003 \times 10^{-8}$ cm/s$^2$. Since all data is not integrated over 1,000 seconds and is data is not available for all time, we increase the numerical factor to $0.001$, which is still negligible to us. \[But further, this upper limit for the error becomes even smaller if one accounts for the number of DSN stations and corresponding atomic clocks that were used for the study.\] Therefore, we conclude that the clocks are not a contributing factor to the anomalous acceleration at a meaningfully level. We will return to this issue in Section \[sec:timemodel\] where we will discuss a number of phenomenological time models that were used to fit the data. DSN antennae complex {#sec:dsn_complex} -------------------- The mechanical structures which support the reflecting surfaces of the antenna are not perfectly stable. Among the numerous effects influencing the DSN antennae performance, we are only interested in those whose behavior might contribute to the estimated solutions for $a_P$. The largest systematic instability over a long period is due to gravity loads and the aging of the structure. As discussed in [@SoversJacobs96], antenna deformations due to gravity loads should be absorbed almost entirely into biases of the estimated station locations and clock offsets. Therefore, they will have little effect on the derived solutions for the purposes of spacecraft navigation. One can also consider ocean loading, wind loading, thermal expansion, and aging of the structure. We found none of these can produce the constant drift in the Doppler frequency on a time scale comparable to the Pioneer data. Also, routine tests are performed by DSN personnel on a regular basis to access all the effects that may contribute to the overall performance of the DSN complex. The information is available and it shows all parameters are in the required ranges. Detailed assessments of all these effect on the astrometric VLBI solutions were published in [@SFJ98; @SoversJacobs96]. The results for the astrometric errors introduced by the above factors may be directly translated to the error budget for the Pioneers, scaled by the number of years. It yields a negligible contribution. Our analyses also estimated errors introduced by a number of station-specific parameters. These include the error due to imperfect knowledge in a DSN station location, errors due to troposphere and ionosphere models for different stations, and errors due to the Faraday rotation effects in the Earth’s atmosphere. Our analysis indicates that at most these effects would produce a distance- and/or time-dependent drifts that would be easily noticeable in the radio Doppler data. What is more important is that none of the effects would be able to produce a constant drift in the Doppler residuals of Pioneers over such a long time scale. The updated version of the ODP, [*Sigma*]{}, routinely accounts for these error factors. Thus, we run covariance analysis for the whole set of these parameters using both [*Sigma*]{} and CHASMP. Based on these studies we conclude that mechanical and phase stability of the DSN antennae together with geographical locations of the antennae, geophysical and atmospheric conditions on the antennae site have negligible effects on our solutions for $a_P$. At most their contributions are at the level of $\sigma_{\tt DSN}\leq10^{-5}a_P$. \[int-systema\]SOURCES OF SYSTEMATIC ERROR INTERNAL TO THE SPACECRAFT ===================================================================== In this section we will discuss the forces that may be generated by spacecraft systems. The mechanisms we consider that may contribute to the found constant acceleration, $a_P$, and that may be caused by the on-board mechanisms include: (1) the radio beam reaction force, (2) RTG heat reflecting off the spacecraft, (3) differential emissivity of the RTGs, (4) non-isotropic radiative cooling of the spacecraft, (5) expelled Helium produced within the RTG, (6) thruster gas leakage, and (7) the difference in experimental results from the two spacecraft. Radio beam reaction force {#radioantbeam} ------------------------- The Pioneer navigation does not require that the spacecraft constantly beam its radio signal, but instead it does so only when it is requested to do so from the ground control. Nevertheless, the recoil force due to the emitted radio-power must also be analyzed. The Pioneers have a total nominal emitted radio power of eight Watts. It is parameterized as $$P_{\tt rp}~=\int_0^{\theta_{\tt max}} d\theta~ \sin\theta~ {\cal P}(\theta),$$ ${\cal P}(\theta)$ being the antenna power distribution. The radiated power has been kept constant in time, independent of the coverage from ground stations. That is, the radio transmitter is always on, even when not received by a ground station. The recoil from this emitted radiation produces an acceleration bias, $b_{\tt rp}$, on the spacecraft away from the Earth of $$b_{\tt rp}= \frac{\beta \,P_{\tt rp}}{Mc}. \label{eq:rp}$$ $M$ is taken to be the Pioneer mass when half the fuel is gone [@mass]. $\beta$ is the fractional component of the radiation momentum that is going in a direction opposite to $a_P$: $$\beta =\frac{1}{P_{\tt rp}} {\int_0^{\theta_{\tt max}} d\theta~ \sin\theta~\cos\theta~ {\cal P}(\theta)}. \label{radiopower}$$ Ref [@piodoc] describes the HGA and shows its downlink antenna pattern in Fig. 3.6-13. (Thermal antenna expansion mismodeling is thought to be negligible.) The gain is given as $(33.3 \pm 0.4)$ dB at zero (peak) degrees. The intensity is down by a factor of two ($-3$ dB) at 1.8 degrees. It is down a factor of 10 ($-10$ dB) at 2.7 degrees and down by a factor of 100 ($-20$ dB) at 3.75 degrees. \[The first diffraction minimum is at a little over four degrees.\] Therefore, the pattern is a very good conical beam. Further, since $\cos [3.75^\circ] = 0.9978$, we can take $\beta = (0.99 \pm 0.01)$, yielding $b_{\tt rp}=1.10$. Finally, taking the error for the nominal 8 Watts power to be given by the 0.4 dB antenna error ($0.10$) and the error due to the uncertainty in our nominal mass ($0.04$), we arrive at the result $$a_{\tt rp} =b_{\tt rp} \pm \sigma_{\tt rp} =(1.10 \pm 0.11)\times 10^{-8} ~{\rm cm/s}^2.$$ RTG heat reflecting off the spacecraft {#subsec:katz} -------------------------------------- It has been argued that the anomalous acceleration seen in the Pioneer spacecraft is due to anisotropic heat reflection off of the back of the spacecraft high-gain antennae, the heat coming from the RTGs [@katz]. Before launch, the four RTGs had a total thermal fuel inventory of 2580 W (now $\sim$ 2070 W). They produced a total electrical power of 160 W (now $\sim$ 65 W). Presently $\sim 2000$ W of RTG heat must be dissipated. Only $\sim63$ W of directed power could explain the anomaly. Therefore, in principle there is enough power to explain the anomaly this way. However, there are two reasons that preclude such a mechanism, namely: i\) [The spacecraft geometry:]{} The RTGs are located at the end of booms, and rotate about the spacecraft in a plane that contains the approximate base of the antenna. From the closest axial center point of the RTGs, the antenna is seen nearly “edge on” (the longitudinal angular width is 24.5$^o$). The total solid angle subtended is $\sim$ 1-2% of $4\pi$ steradians [@s]. Even though a more detailed calculation yields a value of 1.5% [@ss], even taking the higher bound of 2% means this proposal could provide at most $\sim 40$ W. But there is more [@heatreflect]. ii\) [The RTGs’ radiation pattern:]{} The above estimate was based on the assumption that the RTGs are spherical black bodies. But they are not. The main bodies of the RTGs are cylinders and they are grouped in two packages of two. Each package has the two cylinders end to end extending away from the antenna. Every RTG has six fins separated by equal angles of 60 degrees that go radially out from the cylinder. Presumably this results in a symmetrical radiation of thermal power into space. Thus, the fins are “edge on” to the antenna (the fins point perpendicular to the cylinder axes). The largest opening angle of the fins is seen only by the narrow-angle parts of the antenna’s outer edges. Ignoring these edge effects, only $\sim$2.5% of the surface area of the RTGs is facing the antenna. This is a factor 10 less than that from integrating the directional intensity from a hemisphere: $[(\int^{\tt h.sph.}d\Omega\cos\theta)/(4\pi)]=1/4$. So, one has only 4 W of directed power. This suggests a systematic bias of $\sim0.55 \times 10^{-8}$ cm/s$^2$. Even adding an uncertainty of the same size yields a systematic for heat reflection of $$a_{\tt h.r.}= (-0.55 \pm 0.55) \times 10^{-8}~\mathrm{ cm/s}^2.$$ But there are reasons to consider this an upper bound. The Pioneer SNAP 19 RTGs have larger fins than the earlier test models and the packages were insulated so that the end caps have lower temperatures. This results in lower radiation from the end caps than from the cylinder/fins [@tele; @Rconf]. As a result, even though this is not exact, we can argue that the vast majority of the heat radiated by the RTGs is symmetrically directed to space unobscured by the antenna. Further, for this mechanism to work one still has to assume that the energy hitting the antenna is completely reradiated in the direction of the spin axis [@heatreflect]. Finally, if this mechanism were the cause, ultimately an unambiguous decrease in the size of $a_P$ should be seen because the RTGs’ radioactively produced radiant heat is decreasing. As noted previously, the heat produced is now about 80% of the original magnitude. In fact, one would similarly expect a decrease of about $0.75\times 10^{-8}$ cm/s$^2$ in $a_P$ over the 11.5 year Pioneer 10 data interval if this mechanism were the origin of $a_P$. So, even though a complete thermal/physical model of the spacecraft might be able to ascertain if there are any other unsuspected heat systematics, we conclude that this particular mechanism does not provide enough power to explain the Pioneer anomaly [@uskatz]. In addition to the observed constancy of the anomalous acceleration, any explanation involving thermal radiation must also discuss the absence of a disturbance to the spin of the spacecraft. There may be a small correlation of the spin angular acceleration with the anomalous linear acceleration. However, as described in Section \[recent\_results\], the linear acceleration is much more constant than the spin. This suggests that most of the linear acceleration is not caused by whatever disturbs the spin, thermal or not. However, a careful look at the Interval I results of Figure \[fig:pioneer\_spin\] shows that the nearly steady, background spin-rate change of about $6 \times 10^{-5}$ rpm/day is slowly decreasing. In principle this could be caused by heat. The spin-rate change produced by the torque of radiant power directed against the rotation with a lever arm $d$ is $$\ddot{\theta} = \frac {P~d}{c~{\cal I}_{\tt z}}, \label{tdd}$$ where ${\cal I}_{\tt z}$ is the moment of inertia, 588.3 kg m$^2$ [@vanallen]. We take a base unit of $\ddot{\theta}_0$ for a power of one Watt and a lever arm of one meter. This is $$\begin{aligned} \ddot{\theta}_0 &=& 5.63 \times 10^{-12}~\mathrm{rad/s^2} =4.65 \times 10^{-6}~\mathrm{ rpm/day} = \nonumber\\ &=& 1.71 \times 10^{-3}~\mathrm{ rpm/yr}. \end{aligned}$$ So, about 13 Watt-meters of directed power could cause the base spin-rate change. It turns out that such sources could, in principle, be available. There are $3\times 3 = 9$ radioisotope heater units (RHUs) with one Watt power to heat the Thruster Cluster Assembly (TCA). (See pages 3.4-4 and 3.8-1–3.8-17 of Ref. [@piodoc].) The units are on the edge of the antenna of radius 1.37 m, in the housings of the TCAs which are approximately 180$^\circ$ apart from each other. At one position there are six RHUs and at the other position there are three. An additional RHU is near the sun sensor which is located near the second assembly. The final RHU is located at the magnetometer, 6.6 meters out from the center of the spacecraft. The placement gives an “ideal” rotational asymmetry of two Watts. But note, the real asymmetry should be less, since these RHUs do not radiate only in one direction. Even one Watt unidirected at the magnetometer, is not enough to cause the baseline spin rate decrease. Further, since the base line is decreasing faster than what would come from the change cause by radioactive decay decrease, one cannot look for this effect or some complicated RTG source as the entire origin of the baseline change. One would suspect a very small gas leak or a combination of this and heat from the powered bus. (See Section \[subsec:mainbus\].) Indeed, the factor $1/c$ in Eq. (\[tdd\]) is a manifestation of the energy-momentum conservation power needed to produce $\ddot{\theta}$ by heat vs. massive particles. But in any event, this baseline spin-rate change is not significantly correlated with the anomalous acceleration, so we do not have to pursue it further. Differential emissivity of the RTGs {#differemit} ----------------------------------- Another suggestion related to the RTGs is the following [@slusher]: during the early parts of the missions, there might have been a differential change of the radiant emissivity of the solar-pointing sides of the RTGs with respect to the deep-space facing sides. Note that, especially closer in the Sun, the inner sides were subjected to the solar wind. Contrariwise, the outer sides were sweeping through the solar-system dust cloud. Therefore, it can be argued that these two processes could have caused the effect. However, other information seems to make it difficult for this explanation to work. The six fins of each RTG, designed to “provide the bulk of the heat rejection capacity,” were fabricated of HM21A-T8 magnesium alloy plate [@tele]. The metal, after being specially prepared, was coated with two to three mils of zirconia in a sodium silicate binder to provide a high emissivity $(\sim 0.9)$ and low absorptivity $(\sim 0.2)$. Depending on how symmetrically fore-and-aft they radiated, the relative fore-and-aft emissivity of the alloy would have had to have changed by $\sim10$% to account for $a_P$ (see below). Given our knowledge of the solar wind and the interplanetary dust (see Section \[sec:know\]), we find that this amount of a radiant change would be difficult to explain, even if it were of the right sign. (In fact, even the brace bars holding the RTGs were built such that radiation is roughly fore/aft symmetric,) We also have “visual” evidence from the Voyager spacecraft. As mentioned, the Voyagers are not spin-stabilized. They have imaging video cameras attached [@camera]. The cameras are mounted on a scan platform that is pointed under both celestial and inertial attitude control modes [@plate]. The cameras [*do not*]{} have lens covers [@hansen]. During the outward cruise calibrations, the cameras were sometimes pointed towards an imaging target plate mounted at the rear of the spacecraft. But most often they were pointed all over the sky at specific star fields in support of ultraviolet spectrometer observations. Meanwhile, the spacecraft antennae were pointed towards Earth. Therefore, at an angle, the lenses were sometimes hit by the solar wind and sometimes by the interplanetary dust. Even so, there was no noticeable deterioration of the images received, even when Voyager 2 reached Neptune [@neptune]. We infer, therefore, that this mechanism can not explain the Pioneer effect. It turned out that the greatest radiation damage occurred during the flybys. The peak Pioneer 10 radiation flux near Jupiter was about 10,000 times that of Earth for electrons (1,000 times for protons). Pioneer 11 experienced an even higher radiation flux and also went by Saturn [@piopr2]. (We return to this in Section \[twospace\].) Therefore, if radiation damage was a problem, one should have seen an approximately uniform change in emissivity during flyby. Since the total heat flux, $\cal{F}$, from the RTGs was a constant over a flyby, there would have been a change in the RTG surface temperature manifested by the radiation formula ${\cal{F}} \propto \epsilon_1T_1^4 = \epsilon_2T_2^4$, the $\epsilon_i$ being the emissivities of the fin material. There are several temperature sensors mounted at RTG fin bases. They measured average temperatures of approximately 330 F, roughly 440 K. Therefore, a 10% change in the [*total average*]{} emissivity would have produced a temperature change of $\sim$12.2 K $=$ 22 F. Such a change would have been noticed. (Measurements would be compared from, say, 30 days before and after flyby to eliminate the flyby power/thermal distortions.) Since (see below) a 10% [*differential*]{} fore/aft emissivity could cause the Pioneer effect, the lack of observation of a 10% [*total average*]{} emissivity change limits the size of the differential emissivity systematic. To obtain a reasonable estimate of the uncertainty, consider if one side (fore or aft) of the RTGs had its emissivity changed by 1% with respect to the other side. In a simple cylindrical model of the RTGs, with 2000 W power (here we presume only radial emission with no loss out the sides), the ratio of power emitted by the two sides would be 0.99 = 995/1005, or a differential emission between the half cylinders of 10 W. Therefore, the fore/aft asymmetry towards the normal would be $[10~{\mathrm{W}}] \times \int_0^\pi [\sin \phi]d\phi/\pi \approx 6.37$ W. If one does a more sophisticated fin model, with 4 of the 12 fins facing the normal (two flat and two at 30$^\circ$), one gets a number of 6.12 W. We take this to yield our uncertainty, $$\sigma_{\tt d.e.} = 0.85 \times 10^{-8} ~ {\mathrm{cm/s}}^2.$$ Note that $10~ \sigma_{\tt d.e.}$ almost equals our final $a_P$. This is the origin of our previous statement that $\sim 10$% differential emissivity (in the correct direction) would be needed to explain $a_P$. Finally, we want to comment on the significance of radioactive decay for this mechanism. Even acknowledging the Interval jumps due to gas leaks (see below), we reported a one-day batch-sequential value (before systematics) for $a_P$, averaged over the entire 11.5 year interval, of $a_P = (7.77 \pm 0.16)\times 10^{-8}$ cm/s$^2$. From radioactive decay, the value of $a_P$ should have decreased by $0.75$ of these units over 11.5 years. This is 5 times the above variance, which is very large with batch sequential. Even more stringently, this bound is good for [*all*]{} radioactive heat sources. So, what if one were to argue that emissivity changes occurring before 1987 were the cause of the Pioneer effect? There still should have been a decrease in $a_P$ with time since then, which has not been observed. We will return to these points in Section \[twospace\]. Non-isotropic radiative cooling of the spacecraft {#subsec:mainbus} ------------------------------------------------- It has also been suggested that the anomalous acceleration seen in the Pioneer 10/11 spacecraft can be, “explained, at least in part, by non-isotropic radiative cooling of the spacecraft [@murphy].” So, the question is, does “at least in part” mean this effect comes near to explaining the anomaly? We argue it does not [@usmurphy]. Consider radiation of the main-bus electrical systems power from the spacecraft rear. For the Pioneers, the aft has a louver system, and “the louver system acts to control the heat rejection of the radiating platform. A bimetallic spring, thermally coupled radiatively to the platform, provides the motive force for altering the angle of each blade. In a closed position (below 40 F) the heat rejection of the platform is minimized by virtue of the blockage of the blades while open fin louvers provide the platform with a nearly unobstructed view of space [@piodoc].” If these louvers were open (above $\sim$ 88 F) and all the diminishing electrical-power heat was radiated only out of the louvers, this mechanism could produce a significant effect. However, by nine AU the actuator spring temperature had already reached $\sim$40 F [@piodoc]. This means the louver doors were closed (i.e., the louver angle was zero) from where we obtained our data. Thus, from that time on of the radiation properties, the contribution of the thermal radiation to the Pioneer anomalous acceleration should be small. Although one might speculate that a louver stuck, there are 30 louvers on each craft. They clearly worked as designed, or else the temperature of the crafts’ interiors would have fallen to disastrous levels. As shown in Figure \[epower\], in 1984 Pioneer 10 was at about 33 AU and the power was about 105 W. (Always reduce the total power numbers by 8 W to account for the radio beam power.) In (1987, 1992, 1996) the spacecraft was at $\sim$(41, 55, 65) AU and the power was $\sim$(95, 82, 73) W. The louvers were inactive, and no decrease in $a_P$ was seen. In fact, during the entire 11.5 year period from 1987 to 1998 the electrical power decreased from around 95 W to around 68 W, a change of 27 W. Since we already have noted that about $\sim 65$ W is needed to cause our effect, such a large decrease in the “source” of the acceleration would have been seen. But as shown in Section \[recent\_results\], it was not. Even the small differences in the three intervals are most likely to be from gas leaks (as will be demonstrated in Section \[sec:gleaks\]). Later a double modification of this idea was given. It was first suggested that “most, if not all, of the unmodeled acceleration” of Pioneer 10 and 11 is due to an essentially constant supply of heat coming from the central compartment, directed out the front of the craft through the closed louvers [@scheffer]-(a). However, when one studies the electrical power history in both parts (instruments and experimental) of the central compartment, there is no constancy of heat. (See the details in [@usscheffer].) Indeed during our data period the heat from this compartment decreased from about 73 W to about 57 Watts, or a factor of 1.26. This is inconsistent with the constancy of our result. Further, if one looks at the earlier, very roughly analyzed [@earlydata] data in Figure \[fig:correlation\] one sees nothing close to the internal power change of 93 to 57 W (a factor of 1.6) [@usscheffer]. To address this inconsistency a second modification [@scheffer]-b,c was made. It was arbitrarily argued that there was an incorrect determination of the reflection/absorption coefficients by a large factor. But these coefficients are known to 5%. If they were as poorly determined as speculated, the mission would have failed early on. (Further discussion is in [@usscheffer].) We conclude that neither the original proposal [@murphy] nor the modification [@scheffer] can explain the anomalous Pioneer acceleration [@usmurphy; @usscheffer]. A bound on the constancy of $a_P$ comes from first noting the 11.5 year 1-day batch-sequential result, sensitive to time variation: $a_P = (7.77 \pm 0.16)\times 10^{-8}$ cm/s$^2$. Also given the constancy of the earlier imprecise date, it is conservative to take three times this error to be our systematic uncertainty for radiative cooling of the craft, $\sigma_{\tt r.c.}= \pm 0.48 \times 10^{-8}$ cm/s$^2$. Although doubtful, one can also speculate that some mechanism like this might be involved with the baseline spin-rate change discussed in Section \[subsec:katz\]. In 1986-7, Pioneer 10 power was about 97 W, decreasing at about 2.5-3.0 W/yr. If you take a lever arm of 0.71 meters (the hexagonal bus size), this is more than enough to provide the 13 W-meters necessary to produce the baseline spin-rate change of Figure \[fig:pioneer\_spin\]. Further for the first three years the decrease about matches the bus power loss rate. Then after the complex changes associated with the end of 1989 to 1990, there is a decrease in the base rate with a continued similar slope. Perhaps the “baseline" rate is indeed from the heat of the bus being vented to the side. But the much larger gas leaks would be on top of the baseline. Expelled Helium produced within the RTGs {#subsec:helium} ---------------------------------------- Another possible on-board systematic is from the expulsion of the He being created in the RTGs from the $\alpha$-decay of $^{238}$Pu. To make this mechanism work, one would need that the He leakage from the RTGs be preferentially directed away from the Sun, with a velocity large enough to cause the acceleration. The SNAP-19 Pioneer RTGs were designed in a such a way that the He pressure has not been totally contained within the Pioneer heat source over the life of RTGs [@tele]. Instead, the Pioneer heat source contains a pressure relief device which allows the generated He to vent out of the heat source and into the thermoelectric converter. (The strength member and the capsule clad contain small holes to permit He to escape into the thermoelectric converter.) The thermoelectric converter housing-to-power output receptacle interface is sealed with a viton O-ring. The O-ring allows the helium gas within the converter to be released by permeation to the space environment throughout the mission life of the Pioneer RTGs. Information on the fuel pucks [@puck] shows that they each have heights of 0.212 inches with diameters of 2.145 inches. With 18 in each RTG and four RTGs per mission, this gives a total volume of fuel of about 904 cm$^3$. The fuel is PMC Pu conglomerate. The amount of $^{238}$Pu in this fuel is about 5.8 kg. With a half life of 87.74 years, that means the rate of He production (from Pu decay) is about 0.77 gm/year, assuming it all leaves the cermet. Taking on operational temperature on the RTG surface of 320 F = 433 K, implies a $3kT/2$ helium velocity of 1.22 km/s. (The possible energy loss coming out of the viton is neglected for helium.) Using this in the rocket equation, $$a(t) = -v(t) \frac{d}{dt} \Big[\ln M(t)\Big]$$ with our nominal Pioneer mass with half the fuel gone [*and the assumption*]{} that the gas is all unidirected, yields a maximal bound on the possible acceleration of $1.16 \times 10^{-8}$ cm/s$^2$. So, we can rule out helium permeating through the O-rings as the cause of $a_P$ although it is a systematic to be dealt with. Of course, the gas is not totally unidirected. As one can see by looking at Figures \[fig:trusters\] and III-2 of [@tele]: the connectors with the O-rings are on the RTG cylinder surfaces, on the ends of the cylinders where the fins are notched. They are equidistant (30 degrees) from two of the fins. The placement is exactly at the “rear” direction of the RTG cylinders, i.e., at the position closest to the Sun/Earth. The axis through the O-rings is parallel to the spin-axis. The O-rings, sandwiched by the receptacle and connector plates, “see” the outside world through an angle of about 90$^\circ$ in latitude [@hefromrtg]. (Overhead of the O-rings is towards the Sun.) In longitude the O-rings see the direction of the bus and space through about 90$^\circ$, and “see” the fins through most of the rest of the longitudinal angle. If one assumes a single elastic reflection, one can estimate the fraction of the bias away from the Sun. (Indeed, multiple and back reflections will produce an even greater bias. Therefore, we feel this approximation is justified.) This estimate is $(3/4) \sin30^\circ$ times the average of the heat momentum component parallel to the shortest distance to the RTG fin. Using this, we find the bias would be $0.31 \times 10^{-8}$ cm/s$^2$. This bias effectively increases the value of our solution for $a_P$, which we hesitate to accept given all the true complications of the real system. Therefore we take the systematic expulsion to be $a_{\tt He} = (0.15 \pm 0.16) \times 10^{-8}$ cm/s$^2$. Propulsive mass expulsion due to gas leakage {#sec:gleaks} -------------------------------------------- The effect of propulsive mass expulsion due to gas leakage has to be assessed. Although this effect is largely unpredictable, many spacecraft have experienced gas leaks producing accelerations on the order of $10^{-7} ~{\rm cm/s^2}$. \[The reader will recall the even higher figure for Ulysses found in Section \[sec:AUlysses\].\] As noted previously, gas leaks generally behave differently after each maneuver. The leakage often decreases with time and becomes negligibly small. Gas leaks can originate from Pioneer’s propulsion system, which is used for mid-course trajectory maneuvers, for spinning-up or -down the spacecraft, and for orientation of the spinning spacecraft. The Pioneers are equipped with three pairs of hydrazine thrusters which are mounted on the circumference of the Earth-pointing high gain antenna. Each pair of thrusters forms a Thruster Cluster Assembly (TCA) with two nozzles aligned in opposition to each other. For attitude control, two pairs of thrusters can be fired forward or aft and are used to precess the spinning antenna (See Section \[sec:prop\].) The other pair of thrusters is aligned parallel to the rim of the antenna with nozzles oriented in co- and contra-rotation directions for spin/despin maneuvers. During both observing intervals for the two Pioneers, there were no trajectory or spin/despin maneuvers. So, in this analysis we are mainly concerned with precession (i.e., orientation or attitude control) maneuvers only. (See Section \[sec:prop\].) Since the valve seals in the thrusters can never be perfect, one can ask if the leakages through the hydrazine thrusters could be the cause of the anomalous acceleration, $a_P$. However, when we investigate the total computational accuracy of our solution in Section \[Ext\_accuracy\], we will show that the currently implemented models of propulsion maneuvers may be responsible for an uncertainty in $a_P$ only at the level of $\pm0.01\times 10^{-8}$ cm/s$^2$. Therefore, the maneuvers themselves are the main contributors neither to the total error budget nor to the gas leak uncertainty, as we now detail The serious uncertainty comes from the possibility of undetected gas leaks. We will address this issue in some detail. First consider the possible action of gas leaks originating from the spin/despin TCA. Each nozzle from this pair of thrusters is subject to a certain amount of gas leakage. But only a differential leakage from the two nozzles would produce an observable effect causing the spacecraft to either spin-down or spin-up [@leaks]. So, to obtain a gas leak uncertainty (and we emphasize “uncertainty” vs. “error” because we have no other evidence) let us ask how large a differential force is needed to cause the spin-down or spin-up effects observed? Using the moment of inertia about the spin axis, ${\cal I}_{\tt z}=\sim 588.3$ kg$\cdot$m${^2}$ [@vanallen], and the antenna radius, ${\cal{R}}=1.37$ m, as the lever arm, one can calculate that the differential force needed to torque the spin-rate change, $\ddot{\theta}_i$, in Intervals $i=$I,II,III is $$\begin{aligned} F_{\ddot{\theta}_i} &=& \frac{{\cal I}_{\tt z}{\ddot{\theta}_i}}{{\cal{R}}} =\big(2.57, ~12.24, ~1.03\big) \times 10^{-3}~~{\rm dynes}.\hskip 20pt \label{FthetaI} \end{aligned}$$ It is possible that a similar mechanism of undetected gas leakage could be responsible for the net differential force acting in the direction along the line of sight. In other words, what if there were some undetected gas leakage from the thrusters oriented along the spin axis of the spacecraft that is causing $a_P$? How large would this have to be? A force of ($M= 241$ kg) $$F_{a_P}= M \, a_P =21.11\times10^{-3}~{\rm dynes}$$ would be needed to produce our final unbiased value of $a_P$. (See Section \[budget\]. That is, one would need even more force than is needed to produce the anomalously high rotational gas leak of Interval II. Furthermore, the differential leakage to produce this $a_P$ would have had to have been constant over many years and in the same direction for both spacecraft, without being detected as a spin-rate change. That is possible, but certainly not demonstrated. Furthermore if the gas leaks hypothesis were true, one would expect to see a dramatic difference in $a_P$ during the three Intervals of Pioneer 10 data. Instead an almost 500 % spin-down rate change between Intervals I and II resulted only in a less than 8% change in $a_P$. Given the small amount of information, we propose to [*conservatively*]{} take as our gas leak uncertainties the acceleration values that would be produced by differential forces equal to $$\begin{aligned} F_{a_P(i)\tt g.l.}&\simeq & \pm \sqrt{2}F_{\ddot{\theta}_i} = \\ &=& \big(\pm 3.64,~\pm 17.31,~\pm 1.46\big) \times 10^{-3}~~{\rm dynes}.\nonumber \label{eq:diff}\end{aligned}$$ The argument for this is that, in the root sum of squares sense, one is accounting for the differential leakages from the two pairs of thrusters with their nozzles oriented along the line of sight direction. This directly translates into the acceleration errors introduced by the leakage during the three intervals of Pioneer 10 data, $$\begin{aligned} \sigma(a_{P(i) \tt g.l.})&=& \pm F_{a_P(i)\tt g.l.}/M =\\ &=&\big(\pm 1.51,~\pm 7.18,~\pm 0.61\big)\times10^{-8}~{\rm cm/s}^2. \nonumber\end{aligned}$$ Assuming that these errors are uncorrelated and are normally distributed around zero mean, we find the gas leak uncertainty for the entire Pioneer 10 data span to be $$\sigma_{\tt g.l.} = \pm 0.56 \times 10^{-8}~~{\rm cm/s}^2. \label{gluncertC}$$ This is one of our largest uncertainties. The data set from Pioneer 11 is over a much smaller time span, taken when Pioneer 11 was much closer to the Sun (off the plane of the ecliptic), and during a maximum of solar activity. For Pioneer 11 the main effects of gas leaks occurred at the maneuvers, when there were impulsive lowerings of the spin-down rate. These dominated the over-all spin rate change of $\ddot{\theta}_{11}= -0.0234$ rpm/yr. (See Figure \[fig:pio11spin\].) But in between maneuvers the spin rate was actually [*increasing*]{}. One can argue that this explains the higher value for $a_{P(11)}$ in Table \[resulttable\] as compared to $a_{P(10)}$. Unfortunately, one has no [*a priori*]{} way of predicting the effect here. We do not know that the same specific gas leak mechanism applied here as did in the case of Pioneer 10 and there is no well-defined interval set as there is for Pioneer 10. Therefore, although we feel this “spin up” may be part of the explanation of the higher value of $a_P$ for Pioneer 11, we leave the different numbers as a separate systematic for the next subsection. At this point, we must conclude that the gas leak mechanism for explaining the anomalous acceleration seems very unlikely, because it is hard to understand why it would affect Pioneer 10 and 11 at the same level (given that both spacecraft had different quality of propulsion systems, see Section \[sec:prop\]). One also expects a gas leak would obey the rules of a Poisson distribution. That clearly is not true. Instead, our analyses of different data sets indicate that $a_P$ behaves as a constant bias rather than as a random variable. (This is clearly seen in the time history of $a_P$ obtained with batch-sequential estimation.) Variation between determinations from the two spacecraft {#twospace} -------------------------------------------------------- Finally there is the important point that we have two “experimental” results from the two spacecraft, given in Eqs. (\[pio10lastresult\]) and (\[pio11lastresult\]): 7.84 and 8.55, respectively, in units of $10^{-8}$ cm/s$^2$. If the Pioneer effect is real, and not a systematic, these numbers should be approximately equal. The first number, 7.84, is for Pioneer 10. In Section \[final\_sol\] we obtained this number by correlating the values of $a_P$ in the three data Intervals with the different spin-down rates in these Intervals. The weighted correlation between a shift in $a_P$ and the spin-down rate is $\kappa_0 =(30.7\pm 0.6)$ cm. (We argued in the previous Section \[sec:gleaks\] that this correlation is the manifestation of the rotational gas leak systematic.) Therefore, this number represents the entire 11.5 year data arc of Pioneer 10. Similarly, Pioneer 11’s number, 8.55, represents a 3$\frac{3}{4}$ year data arc. Even though the Pioneer 11 number may be less reliable since the craft was so much closer to the Sun, we calculate the time-weighted average of the experimental results from the two craft: $[(11.5)(7.84) + (3.75)(8.55)]/(15.25) = 8.01$ in units of $10^{-8}$ cm/s$^2$. This implies a bias of $b_{\tt 2\_craft}=+0.17\times10^{-8}$ cm/s$^2$ with respect to the Pioneer 10 experimental result $a_{P({\tt exper})}$. We also take this number to be our two spacecraft uncertainty. This means $$\begin{aligned} a_{\tt 2-craft}&=&b_{\tt 2-craft}\pm \sigma_{\tt 2\_craft} = \nonumber\\ &=& (0.17 \pm 0.17)~\times~10^{-8}~\mathrm{cm/s}^2.\end{aligned}$$ The difference between the two craft could be due to different gas leakage. But it also could be due to heat emitted from the RTGs. In particular, the two sets of RTGs have had different histories and so might have different emissivities. Pioneer 11 spent more time in the inner solar system (absorbing radiation). Pioneer 10 has swept out more dust in deep space. Further, Pioneer 11 experienced about twice as much Jupiter/Saturn radiation as Pioneer 10. Further, note that $[a_{P({\tt exper)}}^{\tt Pio11} - a_{P({\tt exper)}}^{\tt Pio10}]$ and the uncertainty from differential emissivity of the RTGs, $\sigma_{\tt d.e.}$, are of the same size: 0.71 and 0.85 $\times10^{-8}$ cm/s$^2$. It could therefore be argued that Pioneer 11’s offset from Pioneer 10 comes from Pioneer 11 having obtained twice as large a differential emissivity bias as Pioneer 10. Then our final value of $a_P$, given in Section \[budget\], would be reduced by about $0.7$ of our units since $\sigma_{\tt d.e.}$ would have become mainly a negative bias, $b_{\tt d.e.}$. This would make the final number closer to $8 \times10^{-8}$ cm/s$^2$. Because this model and our final number are consistent, we present this observation only for completeness and as a possible reason for the different results of the two spacecraft. \[Int\_accuracy\]COMPUTATIONAL SYSTEMATICS ========================================== Given the very large number of observations for the same spacecraft, the error contribution from observational noise is very small and not a meaningful measure of uncertainty. It is therefore necessary to consider several other effects in order to assign realistic errors. Our first consideration is the statistical and numerical stability of of the calculations. We then go on to the cumulative influence of all modeling errors and editing decisions. Finally we discuss the reasons for and significance of the annual term. Besides the factors mentioned above, we will discuss in this section errors that may be attributed to the specific hardware used to run the orbit determination computer codes, together with computational algorithms and statistical methods used to derive the solution. Numerical stability of least-squares estimation {#leastsquares} ----------------------------------------------- Having presented estimated solutions along with their formal statistics, we should now attempt to characterize the true accuracy of these results. Of course, the significance of the results must be assessed on the basis of the expected measurement errors. These expected errors are used to weight a least-squares adjustment to parameters which describe the theoretical model. \[Examination of experimental systematics from sources both external to and also internal to the spacecraft was covered in Sections \[ext-systema\]-\[int-systema\].\] First we look at the numerical stability of the least squares estimation algorithm and the derived solution. The leading computational error source turns out to be subtraction of similar numbers. Due to the nature of floating point arithmetic, two numbers with high order digits the same are subtracted one from the other results in the low order digits being lost. This situation occurs with time tags on the data. Time tags are referenced to some epoch, such as say 1 January 1 1950 which is used by CHASMP. As more than one billion seconds have passed since 1950, time tags on the Doppler data have a start and end time that have five or six common leading digits. Doppler signal is computed by a differenced range formulation (see Section \[Dopp\_tech\]). This noise in the time tags causes noise in the computed Doppler at the 0.0006 Hz level for both Pioneers. This noise can be reduced by shifting the reference epoch closer to the data or increasing the word length of the computation, however, it is not a significant error source for this analysis. In order to guard against possible computer compiler and/or hardware errors we ran orbit determination programs on different computer platforms. JPL’s ODP resides on an HP workstation. The Aerospace Corporation ran the analysis on three different computer architectures: (i) Aerospace’s DEC 64-bit RISC architecture workstation (Alphastation 500/266), (ii) Aerospace’s DEC 32-bit CISC architecture workstation (VAX 4000/60), and (iii) Pentium Pro PC. Comparisons of computations performed for CHASMP in the three machine show consistency to 15 digits which is just sufficient to represent the data. While this comparison does not eliminate the possibility of systematic errors that are common to both systems, it does test the numerical stability of the analysis on three very different computer architectures. The results of the individual programs were given in Sections \[results\]and \[recent\_results\]. In a test we took the JPL results for a batch-sequential [*Sigma*]{} run with 50-day averages of the anomalous acceleration of Pioneer 10, $a_P$. The data interval was from January 1987 to July 1998. We compared this to an Aerospace determination using CHASMP, where the was split into 200 day intervals, over a shorter data interval ending in 1994. As seen in Figure \[fig:rec\_res\_comb\], the results basically agree. Given the excellent agreement in these implementations of the modeling software, we conclude that differences in analyst choices (parameterization of clocks, data editing, modeling options, etc.) give rise to coordinate discrepancies only at the level of $0.3$ cm. This number corresponds to an uncertainty in estimating the anomalous acceleration on the order of $8\times 10^{-12}$ cm/s$^2$. But there is a slightly larger error to contend with. In principle the STRIPPER can give output to 16 significant figures. From the beginning the output was-rounded off to 15 and later to 14 significant figures. When Block 5 came on near the beginning of 1995, the output was rounded off to 13 significant figures. Since the Doppler residuals are 1.12 mm/s this last truncation means an error of order 0.01 mm/s. If we divide this number by 2 for an average round off, this translates to $\pm 0.04\times10^{-8}$ cm/s$^2$. The roundoff occurred in approximately all the data we added for this paper. This is the cleanest 1/3 of the Pioneer 10 data. Considering this we take the uncertainty to be $$\sigma_{\tt num} \pm 0.02 \times 10^{-8} ~~{\rm cm/s}^2. \label{eq:num_st}$$ It needs to be stressed that such tests examine only the accuracy of implementing a given set of model codes, without consideration of the inherent accuracy of the models themselves. Numerous external tests, which we have been discussing in the previous three sections, are possible for assessing the accuracy of the solutions. Comparisons between the two software packages enabled us to evaluate the implementations of the theoretical models within a particular software. Likewise, the results of independent radio tracking observations obtained for the different spacecraft and analysis programs have enabled us to compare our results to infer realistic error levels from differences in data sets and analysis methods. Our analysis of the Galileo and Ulysses missions (reported in Sections \[galileo\] and \[ulysses\]) was done partially for this purpose. Accuracy of consistency/model tests {#Ext_accuracy} ----------------------------------- #### Consistency of solutions: A code that models the motion of solar system bodies and spacecraft includes numerous lengthy calculations. Therefore, the software used to obtain solutions from the Doppler data is, of necessity, very complex. To guard against potential errors in the implementation of these models, we used two software packages; JPL’s ODP/*Sigma* modeling software [@Moyer71; @Moyer81] and The Aerospace Corporation’s POEAS/CHASMP software package [@chasmp; @poeas]. The differences between the JPL and Aerospace orbit determination program results are now examined. As discussed in Section \[sec:OD\], in estimating parameters the CHASMP code uses a standard variation of parameters method whereas ODP uses the Cowell method to integrate the equations of motion and the variational equations. In other words, CHASMP integrates six first-order differential equation, using the Adams-Moulton predictor-corrector method in the orbital elements. Contrariwise, ODP integrates three second-order differential equations for the accelerations using the Gauss-Jackson method. (For more details on these methods see Ref. [@herrick].) As seen in our results of Sections \[results\]and \[recent\_results\], agreement was good; especially considering that each program uses independent methods, models, and constants. Internal consistency tests indicate that a solution is consistent at the level of one part in $10^{15}$. This implies an acceleration error on the order of no more then one part in $10^{4}$ in $a_P$. #### Earth orientation parameters: In order to check for possible problems with Earth orientation, CHASMP was modified to accept Earth orientation information from three different sources. (1) JPL’s STOIC program that outputs [UT1R-UTC]{}, (2) JPL’s Earth Orientation Parameter files ([UT1-UTC]{}), and (3) The International Earth Rotation Service’s Earth Orientation Parameter file ([UT1-UTC]{}). We found that all three sources gave virtually identical results and changed the value of $a_P$ only in the 4th digit [@Folkner]. #### Planetary ephemeris: Another possible source of problems is the planetary ephemeris. To explore this a fit was first done with CHASMP that used DE200. The solution of that fit was then used in a fit where DE405 was substituted for DE200. The result produced a small annual signature before the fit. After the fit, the maneuver solutions changed a small amount (less then 10%) but the value of the anomalous acceleration remained the same to seven digits. The post-fit residuals to DE405 were virtually unchanged from those using DE200. This showed that the anomalous acceleration was unaffected by changes in the planetary ephemeris. This is pertinent to note for the following subsection. To reemphasize the above, a small “annual term” can be introduced by changing the planetary ephemerides. This annual term can then be totally taken up by changing the maneuver estimations. Therefore, in principle, any possible mismodeling in the planetary ephemeris could be at least partially masked by the maneuver estimations. #### Differences in the codes’ model implementations: The impact of an analyst’s choices is difficult to address, largely because of the time and expense required to process a large data set using complex models. This is especially important when it comes to data editing. It should be understood that small differences are to be expected as models differ in levels of detail and accuracy. The analysts’ methods, experience, and judgment differ. The independence of the analysis of JPL and Aerospace has been consistently and strictly maintained in order to provide confidence on the validity of the analyses. Acknowledging such difficulties, we still feel that using the very limited tests given above is preferable to an implicit assumption that all analysts’ choices were optimally made. Another source for differences in the results presented in Table \[resulttable\] is the two codes’ modeling of spacecraft re-orientation maneuvers. ODP uses a model that solves for the resulted change in the Doppler observable $\Delta v$ (instantaneous burn model). This is a more convenient model for Doppler velocity measurements. CHASMP models the change in acceleration, solves for $\Delta a$ (finite burn model), and only then produces a solution for $\Delta v$. Historically, this was done in order to incorporate range observations (for Galileo and Ulysses) into the analysis. Our best handle on this is the no-corona results, especially given that the two critical Pioneer 10 Interval III results differed by very little, $0.02 \times 10^{-8}$ cm/s$^2$. This data is least affected by maneuver modeling, data editing, corona modeling, and spin calibration. Contrariwise, for the other data, the differences were larger. The Pioneer Interval I and II results and the Pioneer 11 results differed, respectively, by (0.21, 0.23, 0.25) in units of $10^{-8}$ cm/s$^2$. In these intervals models of maneuvers and data editing were crucial. Assuming that these errors are uncorrelated, we compute their combined effect on anomalous acceleration $a_P$ as $$\sigma_{\tt consist/model} = \pm 0.13 \times 10^{-8}~\textrm{cm/s}^2.$$ #### Mismodeling of maneuvers: A small contribution to the error comes from a possible mismodeling of the propulsion maneuvers. In Section \[model-maneuvers\] we found that for a typical maneuver the standard error in the residuals is $\sigma_0\sim0.095$ mm/s. Then we would expect that in the period between two maneuvers, which on average is $\tau=$ 11.5/28 year, the effect of the mismodeling would produce a contribution to the acceleration solution with a magnitude on the order of $\delta a_{\tt man}= {\sigma_0}/{\tau} = 0.07 \times 10^{-8}$ cm/s$^2$. Now let us assume that the errors in the Pioneer Doppler residuals are normally distributed around zero mean with the standard deviation of $\delta a_{\tt man}$ that constitute a single measurement accuracy. Then, since there are $N=28$ maneuvers in the data set, the total error due to maneuver mismodeling is $$\sigma_{\tt man} = \frac{\delta a_{\tt man}}{\sqrt{N}} = 0.01 \times 10^{-8} ~~{\rm cm/s}^2. \label{gluncertA}$$ #### Mismodeling of the solar corona: Finally, recall that our number for mismodeling of the solar corona, $ \pm 0.02 \times 10^{-8}$ cm/s$^2$, was already explained in Section \[sec:corona\]. Apparent annual/diurnal periodicities in the solution {#annualterm} ----------------------------------------------------- In Ref. [@moriond] we reported, in addition to the constant anomalous acceleration term, a possible annual sinusoid. If approximated by a simple sine wave, the amplitude of this oscillatory term is about $1.6 \times 10^{-8}$ cm/s$^2$. The integral of a sine wave in the acceleration, $a_P$, with angular velocity $\omega$ and amplitude $A_0$ yields the following first-order Doppler amplitude in two-way fractional frequency: $$\frac{\Delta \nu}{\nu} = \frac{2A_0}{c~ \omega}. \label{lasttwo}$$ The resulting Doppler amplitude for the annual angular velocity $\sim 2 \times 10^{-7}$ rad/s is $\Delta \nu/\nu$ = 5.3 $\times$ 10$^{-12}$. At the Pioneer downlink S-band carrier frequency of $\sim 2.29$ GHz, the corresponding Doppler amplitude is 0.012 Hz (i.e. 0.795 mm/s). This term was first seen in ODP using the [BSF]{} method. As we discussed in Section \[sec:PE\], treating $a_P$ as a stochastic parameter in JPL’s batch–sequential analysis allows one to search for a possible temporal variation in this parameter. Moreover, when many short interval times were used with least-squares CHASMP, the effect was also observed. (See Figure \[fig:rec\_res\_comb\] in Section \[recent\_results\].) The residuals obtained from both programs are of the same magnitude. In particular, the Doppler residuals are distributed about zero Doppler velocity with a systematic variation $\sim$ 3.0 mm/s on a time scale of $\sim$ 3 months. More precisely, the least-squares estimation residuals from both ODP/*Sigma* and CHASMP are distributed well within a half-width taken to be 0.012 Hz. (See, for example, Figure \[fig:pio10best\_fit\].) Even the general structures of the two sets of residuals are similar. The fact that both programs independently were able to produce similar post-fit residuals gives us confidence in the solutions. With this confidence, we next looked in greater detail at the acceleration residuals from solutions for $a_P$. Consider Figure \[annualresiduals\], which shows the $a_P$ residuals from a value for $a_P$ of $(7.77\pm 0.16) \times 10^{-8}$ cm/s$^2$. The data was processed using ODP/*Sigma* with a batch-sequential filter and smoothing algorithm. The solution for $a_P$ was obtained using 1-day batch sizes. Also shown are the maneuver times. At early times the annual term is largest. During Interval II, the interval of the large spin-rate change anomaly, coherent oscillation is lost. During Interval III the oscillation is smaller and begins to die out. In attempts to understand the nature of this annual term, we first examined a number of possible sources, including effects introduced by imprecise modeling of maneuvers, the solar corona, and the Earth’s troposphere. We also looked at the influence of the data editing strategies that were used. We concluded that these effects could not account for the annual term. Then, given that the effect is particularly large in the out-of-the-ecliptic voyage of Pioneer 11 [@moriond], we focused on the possibility that inaccuracies in solar system modeling are the cause of the annual term in the Pioneer solutions. In particular, we looked at the modeling of the Earth orbital orientation and the accuracy of the planetary ephemeris. #### Earth’s orientation: We specifically modeled the Earth orbital elements $\Delta p$ and $\Delta q$ as stochastic parameters. ($\Delta p$ and $\Delta q$ are two of the Set III elements defined by Brouwer and Clemence [@bc].) *Sigma* was applied to the entire Pioneer 10 data set with $a_P$, $\Delta p$, and $\Delta q$ determined as stochastic parameters sampled at an interval of five days and exponentially correlated with a correlation time of 200 days. Each interval was fit independently, but with information on the spacecraft state (position and velocity) carried forward from one interval to the next. Various correlation times, 0-day, 30-day, 200-day, and 400-day, were investigated. The [*a priori*]{} error and process noise on $\Delta p$ and $\Delta q$ were set equal to 0, 5, and 10 $\mu$rad in separate runs, but only the 10 $\mu$rad case removed the annual term. This value is at least three orders of magnitude too large a deviation when compared to the present accuracy of the Earth orbital elements. It is most unlikely that such a deviation is causing the annual term. Furthermore, changing to the latest set of EOP has very little effect on the residuals. \[We also looked at variations of the other four Set III orbital elements, essentially defining the Earth’s orbital shape, size, and longitudinal phase angle. They had little or no effect on the annual term.\] #### Solar system modeling: We concentrated on Interval III, where the spin anomaly is at a minimum and where $a_P$ is presumably best determined. Further, this data was partially taken after the DSN’s Block 5 hardware implementation from September 1994 to August 1995. As a result of this implementation the data is less noisy than before. Over Interval III the annual term is roughly in the form of a sine wave. (In fact, the modeling error is not strictly a sine wave. But it is close enough to a sine wave for purposes of our error analysis.) The peaks of the sinusoid are centered on conjunction, where the Doppler noise is at a maximum. Looking at a CHASMP set of residuals for Interval III, we found a 4-parameter, nonlinear, weighted, least-squares fit to an annual sine wave with the parameters amplitude $v_{\tt a.t.}=(0.1053\pm 0.0107)$ mm/s, phase $(-5.3^\circ \pm 7.2^\circ$), angular velocity $\omega_{\tt a.t}=(0.0177 \pm 0.0001$) rad/day, and bias ($0.0720 \pm 0.0082$) mm/s. The weights eliminate data taken inside of solar quadrature, and also account for different Doppler integration times $T_c$ according to $\sigma = (0.765 {\rm ~mm/s})\,[(60$ s$)/T_c]^{1/2}$. This rule yields post-fit weighted RMS residuals of 0.1 mm/s. The amplitude, $v_{\tt a.t.}$, and angular velocity, $\omega_{\tt a.t.}$, of the annual term results in a small acceleration amplitude of $a_{\tt a.t.}=v_{\tt a.t.}\omega_{\tt a.t.} = (0.215 \pm 0.022) \times 10^{-8}$ cm/s$^2$. We will argue below that the cause is most likely due to errors in the navigation programs’ determinations of the direction of the spacecraft’s orbital inclination to the ecliptic. A similar troubling modeling error exists on a much shorter time scale that is most likely an error in the spacecraft’s orbital inclination to the Earth’s equator. We looked at CHASMP acceleration residuals over a limited data interval, from 23 November 1996 to 23 December 1996, centered on opposition where the data is least affected by solar plasma. As seen in Figure \[opp96\], there is a significant diurnal term in the Doppler residuals, with period approximately equal to the Earth’s sidereal rotation period ($23^{\rm h}56^{\rm m}04^{\rm s}$.0989 mean solar time). After the removal of this diurnal term, the RMS Doppler residuals are reduced to amplitude 0.054 mm/s for $T_c = 660$ s ($\sigma_\nu/\nu = 2.9 \times 10^{-13}$ at $T_c = 1000$ s). The amplitude of the diurnal oscillation in the fundamental Doppler observable, $v_{\tt d.t.}$, is comparable to that in the annual oscillation, $v_{\tt a.t.}$, but the angular velocity, $\omega_{\tt d.t.}$, is much larger than $\omega_{\tt a.t.}$. This means the magnitude of the apparent angular acceleration, $a_{\tt d.t.}=v_{\tt d.t.}\omega_{\tt d.t.} = (100.1 \pm 7.9) \times 10^{-8}$ cm/s$^2$, is large compared to $a_P$. Because of the short integration times, $T_c=660$ s, and long observing intervals, $T\sim 1$ yr, the high frequency, diurnal, oscillation signal averages out to less than $0.03\times 10^{-8}$ cm/s$^2$ over a year. This intuitively helps to explain why the apparently noisy acceleration residuals still yield a precise value of $a_P$. Further, all the residuals from CHASMP and ODP/*Sigma* are essentially the same. Since ODP and CHASMP both use the same Earth ephemeris and the same Earth orientation models, this is not surprising. This is another check that neither program introduces serious modeling errors of its own making. Due to the long distances from the Sun, the spin-stabilized attitude control, the long continuous Doppler data history, and the fact that the spacecraft communication systems utilize coherent radio-tracking, the Pioneers allow for a very sensitive and precise positioning on the sky. For some cases, the Pioneer 10 coherent Doppler data provides accuracy which is even better than that achieved with VLBI observing natural sources. In summary, the Pioneers are simply much more sensitive detectors of a number of solar system modeling errors than other spacecraft. The annual and diurnal terms are very likely different manifestations of the same modeling problem. The magnitude of the Pioneer 10 post-fit weighted RMS residuals of $\approx 0.1$ mm/s, implies that the spacecraft angular position on the sky is known to $\le 1.0$ milliarcseconds (mas). (Pioneer 11, with $\approx 0.18$ mm/s, yields the result $\approx 1.75$ mas.) At their great distances, the trajectories of the Pioneers are not gravitationally affected by the Earth. (The round-trip light time is now $\sim 24 $ hours for Pioneer 10.) This suggests that the sources of the annual and diurnal terms are both Earth related. Such a modeling problem arises when there are errors in any of the parameters of the spacecraft orientation with respect to the chosen reference frame. Because of these errors, the system of equations that describes the spacecraft’s motion in this reference frame is under-determined and its solution requires non-linear estimation techniques. In addition, the whole estimation process is subject to Kalman filtering and smoothing methods. Therefore, if there are modeling errors in the Earth’s ephemeris, the orientation of the Earth’s spin axis (precession and nutation), or in the station coordinates (polar motion and length of day variations), the least-squares process (which determines best-fit values of the three direction cosines) will leave small diurnal and annual components in the Doppler residuals, like those seen in Figures \[annualresiduals\]-\[opp96\]. Orbit determination programs are particularly sensitive to an error in a poorly observed direction [@melbourne]. If not corrected for, such an error could in principle significantly affect the overall navigational accuracy. In the case of the Pioneer spacecraft, navigation was performed using only Doppler tracking, or line-of-sight observations. The other directions, perpendicular to the line-of-sight or in the plane of the sky, are poorly constrained by the data available. At present, it is infeasible to precisely parameterize the systematic errors with a physical model. That would have allowed one to reduce the errors to a level below those from the best available ephemeris and Earth orientation models. A local empirical parameterization is possible, but not a parameterization over many months. We conclude that for both Pioneer 10 and 11, there are small periodic errors in solar system modeling that are largely masked by maneuvers and by the overall plasma noise. But because these sinusoids are essentially uncorrelated with the constant $a_P$, they do not present important sources of systematic error. The characteristic signature of $a_P$ is a linear drift in the Doppler, not annual/diurnal signatures [@myles]. #### Annual/diurnal mismodeling uncertainty: We now estimate the annual term contribution to the error budget for $a_P$. First observe that the standard errors for radial velocity, $v_r$, and acceleration, $a_r$, are essentially what one would expect for a linear regression. The caveat is that they are scaled by the root sum of squares (RSS) of the Doppler error and unmodeled sinusoidal errors, rather than just the Doppler error. Further, because the error is systematic, it is unrealistic to assume that the errors for $v_r$ and $a_r$ can be reduced by a factor 1/$\sqrt{N}$, where $N$ is the number of data points. Instead, averaging their correlation matrix over the data interval, $T$, results in the estimated systematic error of $$\begin{aligned} \sigma_{a_r}^2 = \frac{12}{T^2}~\sigma_{v_r}^2 = \frac{12}{T^2}~\Big(\sigma_{T}^2 + \sigma_{v_{\tt a.t.}}^2+\sigma_{v_{\tt d.t.}}^2\Big). \label{syserror}\end{aligned}$$ $\sigma_{T}=0.1$ mm/s is the Doppler error averaged over $T$ (not the standard error on a single Doppler measurement). $\sigma_{v_{\tt a.t.}}$ and $\sigma_{v_{\tt d.t.}}$ are equal to the amplitudes of corresponding unmodeled annual and diurnal sine waves divided by $\sqrt{2}$. The resulting RSS error in radial velocity determination is about $\sigma_{v_r}= (\sigma_{T}^2 + \sigma_{v_{\tt a.t.}}^2+ \sigma_{v_{\tt d.t.}}^2)^{1/2}=0.15$ mm/s for both Pioneer 10 and 11. Our four interval values of $a_P$ were determined over time intervals of longer than a year. At the same time, to detect an annual signature in the residuals, one needs at least half of the Earth’s orbit complete. Therefore, with $T = 1/2$ yr, Eq. (\[syserror\]) results in an acceleration error of $$\sigma_{{\tt a/d}} = \frac{0.50~~{\rm mm/s}}{T} = 0.32~\times 10^{-8}~{\mathrm{cm/s}}^2. \label{aderror}$$ We use this number for the systematic error from the annual/diurnal term. \[budget\]ERROR BUDGET AND FINAL RESULT ======================================= It is important to realize that our experimental observable is a Doppler frequency shift, i.e., $\Delta \nu (t)$. \[See Figure \[fig:aerospace\] and Eq. (\[eq:delta\_nu\]).\] In actual fact it is a cycle count. We *interpret* this as an apparent acceleration experienced by the spacecraft. However, it is possible that the Pioneer effect is not due to a real acceleration. (See Section \[newphys\].) Therefore, the question arises “In what units should we report our errors?” The best choice is not clear at this point. For reasons of clarity we chose units of acceleration. ------------ ------------------------------------------------------- ----------------------- ----------------------- Item Description of error budget constituents Bias      Uncertainty $10^{-8} ~\rm cm/s^2$ $10^{-8} ~\rm cm/s^2$ 1 [Systematics generated external to the spacecraft:]{} a\) Solar radiation pressure and mass $+0.03$ $\pm 0.01$ b\) Solar wind $ \pm < 10^{-5}$ c\) Solar corona $ \pm 0.02$ d\) Electro-magnetic Lorentz forces $\pm < 10^{-4}$ e\) Influence of the Kuiper belt’s gravity $\pm 0.03$ f\) Influence of the Earth orientation $\pm 0.001$ g\) Mechanical and phase stability of DSN antennae $\pm < 0.001$ h\) Phase stability and clocks $\pm <0.001$ i\) DSN station location $\pm < 10^{-5}$ j\) Troposphere and ionosphere $\pm < 0.001$ \[10pt\] 2 [On-board generated systematics:]{} a\) Radio beam reaction force $+1.10$ $\pm 0.11$ b\) RTG heat reflected off the craft $-0.55$ $\pm 0.55$ c\) Differential emissivity of the RTGs $\pm 0.85$ d\) Non-isotropic radiative cooling of the spacecraft $\pm 0.48$ e\) Expelled Helium produced within the RTGs $+0.15$ $\pm 0.16$ f\) Gas leakage $\pm 0.56$ g\) Variation between spacecraft determinations $+0.17$ $\pm 0.17$ \[10pt\] 3 [Computational systematics:]{} a\) Numerical stability of least-squares estimation $\pm0.02$ b\) Accuracy of consistency/model tests $\pm0.13$ c\) Mismodeling of maneuvers $\pm 0.01$ d\) Mismodeling of the solar corona $\pm 0.02$ e\) Annual/diurnal terms $\pm 0.32$ \[10pt\] Estimate of total bias/error $+0.90$ $\pm 1.33$ ------------ ------------------------------------------------------- ----------------------- ----------------------- The tests documented in the preceding sections have considered various potential sources of systematic error. The results of these tests are summarized in Table \[error\_budget\], which serves as a systematic “error budget.” This budget is useful both for evaluating the accuracy of our solution for $a_P$ and also for guiding possible future efforts with other spacecraft. In our case it actually is hard to totally distinguish “experimental” error from “systematic error.” (What should a drift in the atomic clocks be called?) Further, there is the intractable mathematical problem of how to handle combined experimental and systematic errors. In the end we have decided to treat them all in a least squares [*uncorrelated*]{} manner. The results of our analyses are summarized in Table \[error\_budget\]. There are two columns of results. The first gives a bias, $b_P$, and the second gives an uncertainty, $\pm \sigma_P$. The constituents of the error budget are listed separately in three different categories: 1) systematics generated external to the spacecraft; 2) on-board generated systematics, and 3) computational systematics. Our final result then will become some average $$a_P = a_{P({\tt exper)}}~ + b_P ~\pm \sigma_P,$$ where, from Eq. (\[pio10lastresult\]), $a_{P({\tt exper)}} = (7.84\pm 0.01) \times 10^{-8}$ cm/s$^2$. The least significant factors of our error budget are in the first group of effects, those external to the spacecraft. From the table one sees that some are near the limit of contributing. But in totality, they are insignificant. As was expected, the on-board generated systematics are the largest contributors to our total error budget. All the important constituents are listed in the second group of effects in Table \[error\_budget\]. Among these effects, the radio beam reaction force produces the largest bias to our result, $1.10\times 10^{-8}$ cm/s$^2$. It makes the Pioneer effect larger. The largest bias/uncertainty is from RTG heat reflecting off the spacecraft. We argued for an effect as large as $(-0.55 \pm 0.55) \times 10^{-8}$ cm/s$^2$. Large uncertainties also come from differential emissivity of the RTGs, radiative cooling, and gas leaks, $\pm 0.85$, $\pm 0.48$, and $\pm 0.56$, respectively, $\times 10^{-8}$ cm/s$^2$. The computational systematics are listed in the third group of Table \[error\_budget\]. Therefore, our final value for $a_P$ is $$\begin{aligned} a_P &=& (8.74 \pm 1.33) \times 10^{-8}~{\rm cm/s}^2 \nonumber\\ &\sim& (8.7 \pm 1.3) \times 10^{-8}~{\rm cm/s}^2.\end{aligned}$$ The effect is clearly significant and remains to be explained. \[newphys\]POSSIBLE PHYSICAL ORIGINS OF THE SIGNAL ================================================== A new manifestation of known physics? {#sec:know} ------------------------------------- With the anomaly still not accounted for, possible effects from applications of known physics have been advanced. In particular, Crawford [@crawford] suggested a novel new effect: a gravitational frequency shift of the radio signals that is proportional to the distance to the spacecraft and the density of dust in the intermediate medium. In particular, he has argued that the gravitational interaction of the S-band radio signals with the interplanetary dust may be responsible for producing an anomalous acceleration similar to that seen by the Pioneer spacecraft. The effect of this interaction is a frequency shift that is proportional to the distance and the square root of the density of the medium in which it travels. Similarly, Didon, Perchoux, and Courtens [@courtens] proposed that the effect comes from resistance of the spacecraft antennae as they transverse the interplanetary dust. This is related to more general ideas that an asteroid or comet belt, with its associated dust, might cause the effect by gravitational interactions (see Section \[sec:kuiper\]) or resistance to dust particles. However, these ideas have problems with known properties of the interplanetary medium that were outlined in Section \[sec:kuiper\]. In particular, infrared observations rule out more than 0.3 Earth mass from Kuiper Belt dust in the trans-Neptunian region [@backman; @teplitzinfra]. Ulysses and Galileo measurements in the inner solar system find very few dust grains in the $10^{-18}-10^{-12}$ kg range [@dust]. The density varies greatly, up and down, within the belt (which precludes a constant force) and, in any event, the density is not large enough to produce a gravitational acceleration on the order of $a_P$ [@malhotra]-[@liudust]. One can also speculate that there is some unknown interaction of the radio signals with the solar wind. An experimental answer could be given with two different transmission frequencies. Although the main communication link on the Ulysses mission is S-up/X-down mode, a small fraction of the data is S-up/S-down. We had hoped to utilize this option in further analysis. However, using them in our attempt to study a possible frequency dependent nature of the anomaly, did not provide any useful results. This was in part due to the fact that X-band data (about 1.5 % of the whole data available) were taken only in the close proximity to the Sun, thus prohibiting the study of a possible frequency dependence of the anomalous acceleration. Dark matter or modified gravity? {#sec:DMgr} -------------------------------- It is interesting to speculate on the unlikely possibility that the origin of the anomalous signal is new physics [@photon]. This is true even though the probability is that some “standard physics” or some as-yet-unknown systematic will be found to explain this “acceleration.” The first paradigm is obvious. “Is it dark matter or a modification of gravity?” Unfortunately, neither easily works. If the cause is dark matter, it is hard to understand. A spherically-symmetric distribution of matter which goes as $\rho \sim r^{-1}$ produces a constant acceleration [*inside*]{} the distribution. To produce our anomalous acceleration even only out to 50 AU would require the total dark matter to be greater than $3 \times 10^{-4} M_\odot$. But this is in conflict with the accuracy of the ephemeris, which allows only of order a few times $10^{-6} M_\odot$ of dark matter even within the orbit of Uranus [@ephem]. (A 3-cloud neutrino model also did not solve the problem [@jgscold].) Contrariwise, the most commonly studied possible modification of gravity (at various scales) is an added Yukawa force [@physrep]. Then the gravitational potential is $$V(r) = -\frac{GMm}{(1+\alpha)r}\left[1 +\alpha e^{-r/\lambda}\right], \label{V}$$ where $\alpha$ is the new coupling strength relative to Newtonian gravity, and $\lambda$ is the new force’s range. Since the radial force is $F_r = -d_r V(r) =ma$, the power series for the acceleration yields an inverse-square term, no inverse-$r$ term, then a constant term. Identifying this last term as the Pioneer acceleration yields $$a_P = -\frac{\alpha {a_1}}{2(1+\alpha)} \frac{r_1^2}{\lambda^2}, \label{solution}$$ where $a_1$ is the Newtonian acceleration at distance $r_1 =1$ AU. (Out to 65 AU there is no observational evidence of an $r$ term in the acceleration.) Eq. (\[solution\]) is the solution curve; for example, $\alpha = -1 \times 10^{-3}$ for $\lambda = 200$ AU. It is also of interest to consider some of the recent proposals to modify gravity, as alternatives to dark matter [@milgrom]-[@mil]. Consider Milgrom’s proposed modification of gravity [@mil], where the gravitational acceleration of a massive body is $a \propto 1/r^2$ for some constant $a_0 \ll a$ and $a \propto 1/r$ for $a_0 \gg a$. Depending on the value of $H$, the Hubble constant, $a_0 \approx a_P$! Indeed, as a number of people have noted, $$a_H = cH \rightarrow 8 \times 10^{-8}~ {\rm cm/s}^2, \label{hubble}$$ if $H = 82$ km/s/Mpc. Of course, there are (fundamental and deep) theoretical problems if one has a new force of the phenomenological types of those above. Even so, the deep space data piques our curiosity. In fact, Capozziello et al. [@Capozzielloetal] note the Pioneer anomaly in their discussion of astrophysical structures as manifestations of Yukawa coupling scales. This ties into the above discussion. However, any universal gravitational explanation for the Pioneer effect comes up against a hard experimental wall. The anomalous acceleration is too large to have gone undetected in planetary orbits, particularly for Earth and Mars. NASA’s Viking mission provided radio-ranging measurements to an accuracy of about 12 m [@reasenberg; @mg6]. If a planet experiences a small, anomalous, radial acceleration, $a_A$, its orbital radius $r$ is perturbed by $$\Delta r =-\frac{{\it l}^6 a_A}{(GM_\odot)^4} \rightarrow - \frac{r~ a_A}{a_N} , \label{deltar}$$ where [*l*]{} is the orbital angular momentum per unit mass and $a_N$ is the Newtonian acceleration at $r$. (The right value in Eq. (\[deltar\]) holds in the circular orbit limit.) For Earth and Mars, $\Delta r$ is about $-21$ km and $-76$ km. However, the Viking data determines the difference between the Mars and Earth orbital radii to about a 100 m accuracy, and their sum to an accuracy of about 150 m. The Pioneer effect is not seen. Further, a perturbation in $r$ produces a perturbation to the orbital angular velocity of $$\Delta \omega = \frac{2{\it l}a_A}{GM_\odot} \rightarrow \frac{2 \dot{\theta}~ a_A}{a_N}.$$ The determination of the synodic angular velocity $(\omega_E - \omega_M)$ is accurate to 7 parts in 10$^{11}$, or to about 5 ms accuracy in synodic period. The only parameter that could possibly mask the spacecraft-determined $a_R$ is $(GM_\odot)$. But a large error here would cause inconsistencies with the overall planetary ephemeris [@ephem; @Standish92]. \[Also, there would be a problem with the advance of the perihelion of Icarus [@sanmil].\] We conclude that the Viking ranging data limit any unmodeled radial acceleration acting on Earth and Mars to no more than $0.1 \times 10^{-8}$ cm/s$^2$. Consequently, if the anomalous radial acceleration acting on spinning spacecraft is gravitational in origin, it is [*not*]{} universal. That is, it must affect bodies in the 1000 kg range more than bodies of planetary size by a factor of 100 or more. This would be a strange violation of the Principle of Equivalence [@pe]. (Similarly, the $\Delta \omega$ results rule out the universality of the $a_t$ time-acceleration model. In the age of the universe, $T$, one would have $a_t T^2/2 \sim 0.7~T$.) A new dark matter model was recently proposed by Munyaneza and Viollier [@MunyanezaViollier] to explain the Pioneer anomaly. The dark matter is assumed to be gravitationally clustered around the Sun in the form of a spherical halo of a degenerate gas of heavy neutrinos. However, although the resulting mass distribution is consistent with constraints on the mass excess within the orbits of the outer planets previously mentioned, it turns out that the model fails to produce a viable mechanism for the detected anomalous acceleration. New suggestions stimulated by the Pioneer effect {#sec:neww} ------------------------------------------------ Due to the fact that the size of the anomalous acceleration is of order $cH$, where $H$ is the Hubble constant (see Eq. (\[hubble\])), the Pioneer results have stimulated a number of new physics suggestions. For example, Rosales and Sánchez-Gomez [@rosales] propose that $a_P$ is due to a local curvature in light geodesics in the expanding spacetime universe. They argue that the Pioneer effect represents a new cosmological Foucault experiment, since the solar system coordinates are not true inertial coordinates with respect to the expansion of the universe. Therefore, the Pioneers are mimicking the role that the rotating Earth plays in Foucault’s experiment. Therefore, in this picture the effect is not a “true physical effect” and a coordinate transformation to the co-moving cosmological coordinate frame would entirely remove the Pioneer effect. From a similar viewpoint, Guruprasad [@guru] finds accommodation for the constant term while trying to explain the annual term as a tidal effect on the physical structure of the spacecraft itself. In particular, he suggests that the deformations of the physical structure of the spacecraft (due to external factors such as the effective solar and galactic tidal forces) combined with the spin of the spacecraft are directly responsible for the detected annual anomaly. Moreover, he proposes a hypothesis of the planetary Hubble’s flow and suggests that Pioneer’s anomaly does not contradict the existing planetary data, but supports his new theory of relativistically elastic space-time. [Ø]{}stvang [@ostvang] further exploits the fact that the gravitational field of the solar system is not static with respect to the cosmic expansion. He does note, however, that in order to be acceptable, any non-standard explanation of the effect should follow from a general theoretical framework. Even so, [Ø]{}stvang still presents quite a radical model. This model advocates the use of an expanded PPN-framework that includes a direct effect on local scales due to the cosmic space-time expansion. Belayev [@belayev] considers a Kaluza-Klein model in 5 dimensions with a time-varying scale factor for the compactified fifth dimension. His comprehensive analysis led to the conclusion that a variation of the physical constants on a cosmic time scale is responsible for the appearance of the anomalous acceleration observed in the Pioneer 10/11 tracking data. Modanese [@modan] considers the effect of a scale-dependent cosmological term in the gravitational action. It turns out that, even in the case of a static spherically-symmetric source, the external solution of his modified gravitational field equations contains a non-Schwartzschild-like component that depends on the size of the test particles. He argues that this additional term may be relevant to the observed anomaly. Mansouri, Nasseri and Khorrami [@MansouriNasseriKhorrami] argue that there is an effective time variation in the Newtonian gravitational constant that in turn may be related to the anomaly. In particular, they consider the time evolution of $G$ in a model universe with variable space dimensions. When analyzed in the low energy limit, this theory produces a result that may be relevant to the long-range acceleration discussed here. A similar analysis was performed by Sidharth [@Sidharth], who also discussed cosmological models with a time-varying Newtonian gravitational constant. Inavov [@ivanov] suggests that the Pioneer anomaly is possibly the manifestation of a superstrong interaction of photons with single gravitons that form a dynamical background in the solar system. Every gravitating body would experience a deceleration effect from such a background with a magnitude proportional to Hubble’s constant. Such a deceleration would produce an observable effect on a solar system scale. All these ideas produce predictions that are close to Eq. (\[hubble\]), but they certainly must be judged against discussions in the following two subsections. In a different framework, Foot and Volkas [@foot], suggest the anomaly can be explained if there is mirror matter of mirror dust in the solar system. this could produce a drag force and not violate solar-system mass constraints. Several scalar-field ideas have also appeared. Mbelek and Lachièze-Rey [@rey] have a model based on a long-range scalar field, which also predicts an oscillatory decline in $a_P$ beyond about 100 AU. This model does explain the fact that $a_P$ stays approximately constant for a long period (recall that Pioneer 10 is now past 70 AU). From a similar standpoint Calchi Novati et al. [@novati] discuss a weak-limit, scalar-tensor extension to the standard gravitational model. However, before any of these proposals can be seriously considered they must explain the precise timing data for millisecond binary pulsars, i.e., the gravitational radiation indirectly observed in PSR 1913+16 by Hulse and Taylor [@millipulsar]. Furthermore, there should be evidence of a distance-dependent scalar field if it is uniformly coupled to ordinary matter. Consoli and Siringo [@consoli_siringo] and Consoli [@consoli] consider the Newtonian regime of gravity to be the long wavelength excitation of a scalar condensate from electroweak symmetry breaking. They speculate that the self-interactions of the condensate could be the origins of both Milgrom’s inertia modification [@milgrom; @mil] and also of the Pioneer effect. Capozziello and Lambiase [@CapozzielloLambiase] argue that flavor oscillations of neutrinos in the Brans-Dicke theory of gravity may produce a quantum mechanical phase shift of neutrinos. Such a shift would produce observable effects on astrophysical/cosmological length and time scales. In particular, it results in a variation of the Newtonian gravitational constant and, in the low energy limit, might be relevant to our study. Motivated by the work of Mannheim [@mann; @mann2], Wood and Moreau [@moreau] investigated the theory of conformal gravity with dynamical mass generation. They argue that the Higgs scalar is a feature of the theory that cannot be ignored. In particular, within this framework they find one can reproduce the standard gravitational dynamics and tests within the solar system, and yet the Higgs fields may leave room for the Pioneer effect on small bodies. In summary, as highly speculative as all these ideas are, it can be seen that at the least the Pioneer anomaly is influencing the phenomenological discussion of modern gravitational physics and quantum cosmology [@BertolamiNunes]. Phenomenological time models {#sec:timemodel} ---------------------------- Having noted the relationships $a_P = c~ a_t$ of Eq. (\[asubt\]) and that of Eq. (\[hubble\]), we were motivated to try to think of any (purely phenomenological) “time” distortions that might fortuitously fit the CHASMP Pioneer results shown in Figure \[fig:aerospace\]. In other words, are Eqs. (\[hubble\]) and/or (\[asubt\]) indicating something? Is there any evidence that some kind of “time acceleration” is being seen? The Galileo and Ulysses spacecraft radio tracking data was especially useful. We examined numerous “time” models searching for any (possibly radical) solution. It was thought that these models would contribute to the definition of the different time scales constructed on the basis of Eq. (\[eq:time\]) and discussed in the Section \[sec:time\_scales\]. The nomenclature of the standard time scales [@Moyer81]-[@exp_cat] was phenomenologically extended in our hope to find a desirable quality of the trajectory solution for the Pioneers. In particular we considered: i\) [Drifting Clocks.]{} This model adds a constant acceleration term to the Station Time ([ST]{}) clocks, i.e., in the [ST-UTC]{} (Universal Time Coordinates) time transformation. The model may be given as follows: $$\Delta{\tt ST}={\tt ST}_{\tt received}-{\tt ST}_{\tt sent} ~~\rightarrow ~~\Delta{\tt ST}+\frac{1}{2}a_{\tt clocks} \cdot\Delta{\tt ST}^2$$ where ${\tt ST}_{\tt received}$ and ${\tt ST}_{\tt sent}$ are the atomic proper times of sending and receiving the signal by a DSN antenna. The model fit Doppler well for Pioneer 10, Galileo, and Ulysses but failed to model range data for Galileo and Ulysses. ii\) [Quadratic Time Augmentation.]{} This model adds a quadratic-in-time augmentation to the [TAI-ET]{} (International Atomic Time – Ephemeris Time) time transformation, as follows $${\tt ET} ~~\rightarrow ~~ {\tt ET}+\frac{1}{2}a_{\tt ET}\cdot{\tt ET}^2.$$ The model fits Doppler fairly well but range very badly. iii\) [Frequency Drift.]{} This model adds a constant frequency drift to the reference S-band carrier frequency: $$\nu_{\tt S-band}(t)=\nu_{0} \Big(1+\frac{a_{\tt fr.drift}\cdot{\tt TAI}}{c} \Big).$$ The model also fits Doppler well but again fits range poorly. iv\) [Speed of Gravity]{}. This model adds a “light time” delay to the actions of the Sun and planets upon the spacecraft: $$v_{\tt grav}=c \Big(1+\frac{a_{\tt sp.grav} \cdot|\vec{r}_{\tt body}-\vec{r}_{\tt Pioneer}|}{c^2} \Big).$$ The model fits Pioneer 10 and Ulysses well. But the Earth flyby of Galileo fit was terrible, with Doppler residuals as high as 20 Hz. All these models were rejected due either to poor fits or to inconsistent solutions among spacecraft. Quadratic in time model ----------------------- There was one model of the above type that was especially fascinating. This model adds a quadratic in time term to the light time as seen by the DSN station. Take any labeled time ${\tt T}_a$ to be $${\tt T}_a = t_a - t_0 \rightarrow t_a - t_0 + \frac{1}{2}a_t\left(t_a^2 - t_0^2\right).$$ Then the light time is $$\begin{aligned} \Delta{\tt TAI}&=&{\tt TAI}_{\tt received}-{\tt TAI}_{\tt sent} ~~\rightarrow \nonumber\\ &&\hskip -30pt \rightarrow~~ \Delta{\tt TAI}+\frac{1}{2}a_{\tt quad}\cdot \Big( {\tt TAI}_{\tt received}^2-{\tt TAI}_{\tt sent}^2\Big). \label{eq:aqt}\end{aligned}$$ It mimics a line of sight acceleration of the spacecraft, and could be thought of as an [*expanding space*]{} model. Note that $a_{\tt quad}$ affects only the data. This is in contrast to the $a_t$ of Eq. (\[asubt\]) that affects both the data and the trajectory. This model fit both Doppler and range very well. Pioneers 10 and 11, and Galileo have similar solutions although Galileo solution is highly correlated with solar pressure; however, the range coefficient of the quadratic is negative for the Pioneers and Galileo while positive for Ulysses. Therefore we originally rejected the model because of the opposite signs of the coefficients. But when we later appreciated that the Ulysses anomalous acceleration is dominated by gas leaks (see Section \[sec:AUlysses\]), which makes the different-sign coefficient of Ulysses meaningless, we reconsidered it. The fact that the Pioneer 10 and 11, Galileo, and Ulysses are spinning spacecraft whose spin axis are periodically adjusted so as to point towards Earth turns out to make the quadratic in time model and the constant spacecraft acceleration model highly correlated and therefore very difficult to separate. The quadratic in time model produces residuals only slightly ($\sim20\%$) larger than the constant spacecraft acceleration model. However, when estimated together with no [*a priori*]{} input [ i.e.]{}, based only the tracking data, even though the correlation between the two models is 0.97, the value $a_{\tt quad}$ determined for the quadratic in time model is zero while the value for the constant acceleration model $a_P$ remains the same as before. The orbit determination process clearly prefers the constant acceleration model, $a_P$, over that the quadratic in time model, $a_{\tt quad}$ of Eq. (\[eq:aqt\]). This implies that a real acceleration is being observed and not a pseudo acceleration. We have not rejected this model as it may be too simple in that the motions of the spacecraft and the Earth may need to be included to produce a true expanding space model. Even so, the numerical relationship between the Hubble constant and $a_P$, which many people have observed (cf. Section \[sec:neww\]), remains an interesting conjecture. \[disc\]CONCLUSIONS =================== In this paper we have discussed the equipment, theoretical models, and data analysis techniques involved in obtaining the anomalous Pioneer acceleration $a_P$. We have also reviewed the possible systematic errors that could explain this effect. These included computational errors as well as experimental systematics, from systems both external to and internal to the spacecraft. Thus, based on further data for the Pioneer 10 orbit determination (the extended data spans 3 January 1987 to 22 July 1998) and more detailed studies of all the systematics, we can now give a total error budget for our analysis and a latest result of $a_P = (8.74 \pm 1.33) \times 10^{-8}$ cm/s$^2$. This investigation was possible because modern radio tracking techniques have provided us with the means to investigate gravitational interactions to an accuracy never before possible. With these techniques, relativistic solar-system celestial mechanical experiments using the planets and interplanetary spacecraft provide critical new information. Our investigation has emphasized that effects that previously thought to be insignificant, such as rejected thermal radiation or mass expulsion, are now within (or near) one order of magnitude of possible mission requirements. This has unexpectedly emphasized the need to carefully understand all systematics to this level. In projects proposed for the near future, such as a Doppler measurement of the solar gravitational deflection using the Cassini spacecraft [@GAB] and the Space Interferometry Mission [@sim], navigation requirements are more stringent than those for current spacecraft. Therefore, all the effects we have discussed will have to be well-modeled in order to obtain sufficiently good trajectory solutions. That is, a better understanding of the nature of these extra small forces will be needed to achieve the stringent navigation requirements for these missions. Currently, we find no mechanism or theory that explains the anomalous acceleration. What we can say with some confidence is that the anomalous acceleration is a line of sight constant acceleration of the spacecraft toward the Sun [@sunearth]. Even though fits to the Pioneers appear to match the noise level of the data, in reality the fit levels are as much as 50 times above the fundamental noise limit of the data. Until more is known, we must admit that the most likely cause of this effect is an unknown systematic. (We ourselves are divided as to whether “gas leaks” or “heat” is this “most likely cause.”) The arguments for “gas leaks” are: i) All spacecraft experience a gas leakage at some level. ii) There is enough gas available to cause the effect. iii) Gas leaks require no new physics. However, iv) it is unlikely that the two Pioneer spacecraft would have gas leaks at similar rates, over the entire data interval, especially when the valves have been used for so many maneuvers. \[Recall also that one of the Pioneer 11 thrusters became inoperative soon after launch. (See Section \[sec:prop\].)\] v) Most importantly, it would require that these gas leaks be precisely pointed towards the front [@rearfront] of the spacecraft so as not to cause a large spin-rate changes. But vi) it could still be true anyway. The main arguments for “heat” are: i) There is so much heat available that a small amount of the total could cause the effect. ii) In deep space the spacecraft will be in approximate thermal equilibrium. The heat should then be emitted at an approximately constant rate, deviating from a constant only because of the slow exponential decay of the Plutonium heat source. It is hard to resist the notion that this heat somehow must be the origin of the effect. However, iii) there is no solid explanation in hand as to how a specific heat mechanism could work. Further, iv) the decrease in the heat supply over time should have been seen by now. Further experiment and analysis is obviously needed to resolve this problem. On the Pioneer 10 experimental front, there now exists data up to July 2000. Further, there exists archived high-rate data from 1978 to the beginning of our data arc in Jan 1987 that was not used in this analysis. Because this early data originated when the Pioneers were much closer in to the Sun, greater effort would be needed to perform the data analyses and to model the systematics. As Pioneer 10 continues to recede into interstellar space, its signal is becoming dimmer. Even now, the return signal is hard to detect with the largest DSN antenna. However, with appropriate instrumentation, the 305-meter antenna of the Arecibo Observatory in Puerto Rico will be able to detect Pioneer’s signal for a longer time. If contact with Pioneer 10 can be maintained with conscan maneuvers, such further extended data would be very useful, since the spacecraft is now so far from the Sun. Other spacecraft can also be used in the study of $a_P$. The radio Doppler and range data from the Cassini mission could offer a potential contribution. This mission was launched on 15 October 1997. The potential data arc will be the cruise phase from after the Jupiter flyby (30 December 2000) to the vicinity of Saturn (just before the Huygens probe release) in July 2004. Even though the Cassini spacecraft is in three-axis-stabilization mode, using on-board active thrusters, it was built with very sophisticated radio-tracking capabilities, with X-band being the main navigation frequency. (There will also be S- and K-band links.) Further, during much of the cruise phase, reaction wheels will be used for stabilization instead of thrusters. Their use will aid relativity experiments at solar conjunction and gravitational wave experiments at solar opposition. (Observe, however, that the relatively large systematic from the close in Cassini RTGs will have to be accounted for.) Therefore, Cassini could yield important orbit data, independent of the Pioneer hyperbolic-orbit data. A similar opportunity may exist, out of the plane of the ecliptic, from the proposed Solar Probe mission. Under consideration is a low-mass module to be ejected during solar flyby. On a longer time scale, the reconsidered Pluto/Kuiper mission (with arrival at Pluto around 2029) could eventually provide high-quality data from very deep space. All these missions might help test our current models of precision navigation and also provide a new test for the anomalous $a_P$. In particular, we anticipate that, given our analysis of the Pioneers, in the future precision orbital analysis may concentrate more on systematics. That is, data/systematic modeling analysis may be assigned more importance relative to the astronomical modeling techniques people have concentrated on for the past 40 years [@tuck]-[@bartlett]. Finally, we observe that if no convincing explanation is to be obtained, the possibility remains that the effect is real. It could even be related to cosmological quantities, as ha\]s been intimated. \[See Eq. (\[asubt\]) and Sec. \[newphys\], especially the text around Eq. (\[hubble\]).\] This possibility necessitates a cautionary note on phenomenology: At this point in time, with the limited results available, there is a phenomenological equivalence between the $a_P$ and $a_t$ points of view. But somehow, the choice one makes affects one’s outlook and direction of attack. If one has to consider new physics one should be open to both points of view. In the unlikely event that there is new physics, one does not want to miss it because one had the wrong mind set. Firstly we must acknowledge the many people who have helped us with suggestions, comments, and constructive criticisms. Invaluable information on the history and status of Pioneer 10 came from Ed Batka, Robert Jackson, Larry Kellogg, Larry Lasher, David Lozier, and Robert Ryan. E. Myles Standish critically reviewed the manuscript and provided a number of important insights, especially on time scales, solar system dynamics and planetary data analysis. We also thank John E. Ekelund, Jordan Ellis, William Folkner, Gene L. Goltz, William E. Kirhofer, Kyong J. Lee, Margaret Medina, Miguel Medina, Neil Mottinger, George W. Null, William L. Sjogren, S. Kuen Wong, and Tung-Han You of JPL for their aid in obtaining and understanding DSN Tracking Data. Ralph McConahy provided us with very useful information on the history and current state of the DSN complex at Goldstone. R. Rathbun and A. Parker of TRW provided information on the mass of the Pioneers. S. T. Christenbury of Teledyne-Brown, to whom we are very grateful, supplied us with critical information on the RTGs. Information was also supplied by G. Reinhart of LANL, on the RTG fuel pucks, and by C. J. Hansen of JPL, on the operating characteristics of the Voyager image cameras. We thank Christopher S. Jacobs of JPL for encouragement and stimulating discussions on present VLBI capabilities. Further guidance and information were provided by John W. Dyer, Alfred S. Goldhaber, Jack G. Hills, Timothy P. McElrath, Irwin I. Shapiro, Edward J. Smith, and Richard J. Terrile. Edward L. Wright sent useful observations on the RTG emissivity analysis. We also thank Henry S. Fliegel, Gary B. Green, and Paul Massatt of The Aerospace Corporation for suggestions and critical reviews of the present manuscript. This work was supported by the Pioneer Project, NASA/Ames Research Center, and was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. P.A.L. and A.S.L. were supported by a grant from NASA through the Ultraviolet, Visible, and Gravitational Astrophysics Program. M.M.N. acknowledges support by the U.S. DOE. Finally, the collaboration especially acknowledges the contributions of our friend and colleague, Tony Liu, who passed away while the manuscript was nearing completion. APPENDIX ======== In Table \[poeas00479\] we give the hyperbolic orbital parameters for Pioneer 10 and Pioneer 11 at epoch 1 January 1987, 01:00:00 [UTC]{}. The semi-major axis is $a$, $e$ is the eccentricity, $I$ is the inclination, $\Omega$ is the longitude of the ascending node, $\omega$ is the argument of the perihelion, $M_0$ is the mean anomaly, $f_0$ is the true anomaly at epoch, and $r_0$ is the heliocentric radius at the epoch. The direction cosines of the spacecraft position for the axes used are $(\alpha, \,\beta, \,\gamma)$. These direction cosines and angles are referred to the mean equator and equinox of J2000. The ecliptic longitude $\ell_0$ and latitude $b_0$ are also listed for an obliquity of 23$^\circ$26$^{'}$21.$^{''}$4119. The numbers in parentheses denote realistic standard errors in the last digits. Parameter Pioneer 10 Pioneer 11 -------------------------- --------------------- ---------------------- $a$ \[km\] $-1033394633(4)$ $-1218489295(133)$ \[1pt\] $e$ $1.733593601(88)$ $2.147933251(282)$ \[1pt\] $I$ \[Deg\] $26.2488696(24)$ $9.4685573(140)$ \[1pt\] $\Omega$ \[Deg\] $-3.3757430(256)$ $35.5703012(799)$ \[1pt\] $\omega$ \[Deg\] $-38.1163776(231)$ $-221.2840619(773)$ \[1pt\] $M_0$ \[Deg\] $259.2519477(12)$ $109.8717438(231)$ \[1pt\] $f_0$ \[Deg\] $112.1548376(3)$ $81.5877236(50)$ \[1pt\] $r_0$ \[km\] $5985144906(22)$ $3350363070(598)$ \[1pt\] $\alpha$ $0.3252905546(4)$ $-0.2491819783(41)$ \[1pt\] $\beta$ $0.8446147582(66)$ $-0.9625930916(22)$ \[1pt\] $\gamma$ $0.4252199023(133)$ $-0.1064090300(223)$ \[1pt\] $\ell_0$ \[Deg\] $70.98784378(2)$ $-105.06917250(31)$ \[1pt\] $b_0$ \[Deg\] $3.10485024(85)$ $16.57492890(127)$ \[1pt\] : Orbital parameters for Pioneer 10 and Pioneer 11 at epoch 1 January 1987, 01:00:00 [UTC]{}. \[poeas00479\] [99]{} See the special issue of Science [**183**]{}, No. 4122, 25 January 1974; specifically, J. D. Anderson, G. W. Null, and S. K. Wong, Science [**183**]{}, 322 (1974). R. O. Fimmel, W. Swindell, and E. Burgess, [*Pioneer Odyssey: Encounter with a Giant*]{}, NASA document No. SP-349 (NASA, Washington D.C., 1974). R. O. Fimmel, J. Van Allen, and E. Burgess, [*Pioneer: First to Jupiter, Saturn, and beyond*]{}, NASA report NASA–SP-446 (NASA, Washington D.C., 1980). , Pioneer Project NASA/ARC document No. PC-202 (NASA, Washington, D.C., 1971). , Revised, NASA/ARC document No. PC-1001 (NASA, Washington, D.C., 1994). For web summaries of Pioneer, go to: [http://quest.arc.nasa.gov/pioneer10]{},\ [http://spaceprojects.arc.nasa.gov/\ Space\_Projects/ pioneer/PNhome.html]{} J. D. Anderson, E. L. Lau, K. Scherer, D. C. Rosenbaum, and V. L. Teplitz, [Icarus]{} [**131**]{}, 167 (1998). J. D. Anderson, E. L. Lau, T. P. Krisher, D. A. Dicus, D. C. Rosenbaum, and V. L. Teplitz, [Astrophys. J.]{} [**448**]{}, 885 (1995). K. Scherer, H. Fichtner, J. D. Anderson, and E. L. Lau, [ Science]{} [**278**]{}, 1919 (1997). J. D. Anderson and B. Mashoon, [Astrophys. J.]{} [**290**]{}, 445 (1985). J. D. Anderson, J. W. Armstrong, and E. L.Lau, [Astrophys. J.]{} [**408**]{}, 287 (1993). J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Phys. Rev. Lett. [**81**]{}, 2858 (1998). Eprint gr-qc/9808081. S. G. Turyshev, J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, and M. M. Nieto, in: [*Gravitational Waves and Experimental Gravity, Proceedings of the [XVIIIth]{} Moriond Workshop of the Rencontres de Moriond*]{}, ed. by J. Dumarchez and J. Tran Thanh Van (World Pub., Hanoi, 2000), pp. 481-486. Eprint gr-qc/9903024. There were four Pioneers built of this particular design. After testing, the best components were placed in Pioneer 10. (This is probably why Pioneer 10 has lasted so long.) The next best were placed in Pioneer 11. The third best were placed in the “proof test model.” Until recently, the structure and many components of this model were included in an exhibit at the National Air and Space Museum. The other model eventually was dismantled. We thank Robert Ryan of JPL for telling us this. Figures given for the mass of the entire Pioneer package range from under 250 kg to over 315 kg. However, we eventually found that the total (“wet”) weight at launch was 259 kg (571 lbs), including 36 kg of hydrazine (79.4 lbs). Credit and thanks for these numbers are due to Randall Rathbun, Allen Parker, and Bruce A. Giles of TRW, who checked and rechecked for us including going to their launch logs. Consistent total mass with lower fuel (27 kg) numbers were given by Larry Kellogg of NASA/Ames. (We also thank V. J. Slabinski of USNO who first asked us about the mass.) Information about the gas usage is by this time difficult to find or lost. During the Extended Mission the collaboration was most concerned with power to the craft. The folklore is that most of Pioneer 11’s propellant was used up going to Saturn and used very little for Pioneer 10. In particular, a Pioneer 10 nominal input mass of 251.883 kg and a Pioneer 11 mass of 239.73 kg were used by the JPL program and the Aerospace program used 251.883 for both. The 251 number approximates the mass lost during spin down, and the 239 number models the greater fuel usage. These numbers were not changed in the programs. For reference, we will use 241 kg, the mass with half the fuel used, as our number with which to calibrate systematics. We take this number from Ref. [@piodoc], where the design, boom-deployed moment of inertia is given as 433.9 slug (ft)$^2$ (= 588.3 kg m$^2$). This should be a little low since we know a small amount of mass was added later in the development. A much later order-of-magnitude number 770 kg m$^2$ was obtained with a too large mass [@mass; @gasuse]. See J. A. Van Allen, [*Episodic Rate of Change in Spin Rate of Pioneer 10,*]{} Pioneer Project Memoranda, 20 March 1991 and 5 April 1991. Both numbers are dominated by the RTGs and magnetometer at the ends of long booms. Conscan stands for conical scan. The receiving antenna is moved in circles of angular size corresponding to one half of the beam-width of the incoming signal. This procedure, possibly iterated, allows the correct pointing direction of the antenna to be found. When coupled with a maneuver, it can also be used to find the correct pointing direction for the spacecraft antenna. The precession maneuvers can be open-loop, for orientation towards or away from Earth-pointing, or closed-loop, for homing on the uplink radio-frequency transmission from the Earth. When a Pioneer antenna points toward the Earth, this defines the “rear” direction on the spacecraft. The equipment compartment placed on the other side of of the antenna defines the “front” direction on the spacecraft. (See Figure \[fig:trusters\].) , Teledyne report IESD 2873-172, June, 1973, tech. report No. DOE/ET/13512-T1; DE85017964, gov. doc. \# E 1.9, and S. T. Christenbury, private communications. F. A. Russo, in: [*Proceedings of the 3rd RTG Working Group Meeting*]{} (Atomic Energy Commission, Washington, DC, 1972), ed. by P. A. O’Rieordan, papers \# 15 and 16. L. Lasher, Pioneer Project Manager, recently reminded us (March 2000) that not long after launch, the electrical power had decreased to about 155 W, and degraded from there. \[Plots of the available power with time are available.\] This is a “theoretical value,” which does not account for inverter losses, line losses, and such. It is interesting to note that at mission acceptance, the total “theoretical” power was 175 Watts. We take the S-band to be defined by the frequencies 1.55-5.20 GHz. We take the X-band to be defined by the frequencies 5.20-10.90 GHz. It turns out there is no consistent international definition of these bands. The definitions vary from field to field, with geography, and over time. The above definitions are those used by radio engineers and are consistent with the DSN usage. (Some detailed band definitions can be found at [http://www.eecs.wsu.edu/$\sim$hudson/ Teaching/ee432/spectrum.htm]{}.) \[We especially thank Ralph McConahy of DSN Goldstone on this point.\] dBm is used by radio engineers as a measure of received power. It stands for decibels in milliwatts. For a description of the Galileo mission see T. V. Johnson, C. M. Yeats, and R. Young, [Space Sci. Rev.]{} [**60**]{}, 3–21 (1992). For a description of the trajectory design see L. A. D’Amario, L. E. Bright, and A. A. Wolf, [Space Sci. Rev.]{} [**60**]{}, 22–78 (1992). The LGA was originally supposed to “trickle” down low-rate engineering data. It was also to be utilized in case a fault resulted in the spacecraft “safing” and shifting to a Sun-pointed attitude, resulting in loss of signal from the HGA. \[“Safing” refers to a spacecraft entering the so called “safe mode.” This happens in case of an emergency when systems are shut down.\] J. D. Anderson, P. B. Esposito, W. Martin, C. L. Thornton, and D. O. Muhleman, [Astrophys. J.]{} [**200**]{}, 221 (1975). P. W. Kinman, IEEE Transactions on Microwave Theory and Techniques [**40**]{}, 1199 (1992). For descriptions of the Ulysses mission see E. J. Smith and R. G. Marsden, Sci. Am. [**278**]{}, No. 1, 74 (1998); B. M. Bonnet, Alexander von Humboldt Magazin, No. 72, 27 (1998). A technical description, with a history and photographs, of the Deep Space Network can be found at [http://deepspace.jpl.nasa.gov/dsn/]{}. The document describing the radio science system is at [http://deepspace.jpl.nasa.gov/dsndocs/810-5/ 810-5.html]{}. N. A. Renzetti, J. F. Jordan, A. L. Berman, J. A. Wackley, T. P. Yunck, [*The Deep Space Network – An Instrument for Radio Navigation of Deep Space Probes*]{}, Jet Propulsion Laboratory Technical Report 82-102 (1982). J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, [IEEE Transactions on Instrumentation and Measurement]{} [**20**]{}, 105 (1971). R. F. C. Vessot, in: [*Experimental Gravitation*]{}, ed. B. Bertotti, ( Academic Press, New York and London, 1974), p.111. O. J. Sovers, J. L. Fanselow, and C. S. Jacobs, [Rev. Mod. Phys.]{}, [**70**]{}, 1393 (1998). One-way data refers to a transmission and reception, only. Two-way data is a transmission and reception, followed by a retransmission and reception at the original transmission site. This would be, for example, a transmission from a radio antenna on Earth to a spacecraft and then a retransmission back from the spacecraft to the same antenna. Three-way refers to the same as two-way, except the final receiving antenna is different from the original transmitting antenna. Much, but not all, of the data we used has been archived. Since the Extended Pioneer Mission is complete, the resources have not been available to properly convert the entire data set to easily accessible format. The JPL and DSN convention for Doppler frequency shift is $(\Delta \nu)_{\tt DSN} = \nu_0 - \nu$, where $\nu$ is the measured frequency and $\nu_0$ is the reference frequency. It is positive for a spacecraft receding from the tracking station (red shift), and negative for a spacecraft approaching the station (blue shift), just the opposite of the usual convention, $(\Delta \nu)_{\tt usual} = \nu - \nu_0$. In consequence, the velocity shift, $\Delta v = v - v_0$, has the same sign as $(\Delta \nu)_{\tt DSN}$ but the opposite sign to $(\Delta \nu)_{\tt usual}$. Unless otherwise stated, we will use the DSN frequency shift convention in this paper. We thank Matthew Edwards for asking us about this. As we will come to in Section \[sec:timemodel\], this property allowed us to test and reject several phenomenological models of the anomalous acceleration that fit Doppler data well but failed to fit the range data. D. L. Cain, JPL Technical Report (1966). T. D. Moyer, [*Mathematical Formulation of the Double Precision Orbit Determination Program (DPODP)*]{}, Jet Propulsion Laboratory Technical Report 32-1527 (1971). T. D. Moyer, [*Formulation for Observed and Computed Values of Deep Space Network (DSN) Data Types for Navigation*]{}, JPL Publication 00-7 (October 2000). A. Gelb, ed. [*Applied Optimal Estimation*]{} (M.I.T. Press, Cambridge, MA, 1974). D. O. Muhleman and J. D. Anderson, [Astrophys. J.]{} [**247**]{}, 1093 (1981). Once in deep space, all major forces on the spacecraft are gravitational. The Principle of Equivalence holds that the inertial mass ($m_I$) and the gravitational mass ($m_G$) are equal. This means the mass of the craft should cancel out in the dynamical gravitational equations. As a result, the people who designed early deep-space programs were not as worried as we are about having the correct mass. When non-gravitational forces were modeled, an incorrect mass could be accounted for by modifying other constants. For example, in the solar radiation pressure force the effective sizes of the antenna and the albedo could take care of mass inaccuracies. J. D. Anderson, in: [*Experimental Gravitation*]{}, ed. B. Bertotti (New York and London, Academic Press, 1974), p.163. J. D. Anderson, G. S. Levy, and N. A. Renzetti, “Application of the Deep Space Network (DSN) to the testing of general relativity,” in [*Relativity in Celestial Mechanics and Astrometry*]{}, eds. J. Kovalevsky and V. A. Brumberg. (Kluwer Academic, Dordrecht, Boston, 1986), p. 329. X X Newhall, E. M. Standish, and J. G. Williams, Astron. Astrophys. [**125**]{}, 150 (1983). E. M. Standish, Jr., X X Newhall, J. G. Williams, and D. K. Yeomans, “Orbital ephemeris of the Sun, Moon, and Planets,” in: Ref. [@exp_cat], p. 279. Also see E. M. Standish, Jr. and R. W. Hellings, Icarus [**80**]{}, 326 (1989). E. M. Standish, Jr., X X Newhall, J. G. Williams, and W. M. Folkner, [*JPL Planetary and Lunar Ephemeris, DE403/LE403*]{}, Jet Propulsion Laboratory Internal IOM No. 314.10-127 (1995). C. M. Will, [*Theory and Experiment in Gravitational Physics*]{}, (Rev. Ed.) (Cambridge University Press, Cambridge, 1993). C. M. Will and K. Nordtvedt, Jr, [Astrophys. J.]{} [**177**]{}, 757 (1972). F. E. Estabrook, [Astrophys. J.]{} [**158**]{}, 81 (1969). T. D. Moyer, Parts. 1 and 2, Celest. Mech. [**23**]{}, 33, 57 (1981). P. K. Seidelmann, ed., [*Explanatory Supplement to the Astronomical Almanac*]{} (University Science Books, Mill Valley, CA, 1992). C. Ma, E. F. Arias, T. M. Eubanks, A. L. Fey, A.-M. Gontier, C. S. Jacobs, O. J. Sovers, B. A. Archinal, and P. Charlot, Astron. J. [**116**]{}, 516 (1998). A. Milani, A. M. Nobili, and P. Farinella, [*Non-Gravitational Perturbations and Satellite Geodesy*]{}, (Adam Hilger, Bristol, 1987). See, especially, p. 125. J. M. Longuski, R. E. Todd, and W. W. König, J. Guidance, Control, and Dynamics, [**15**]{}, 545 (1992). D.O. Muhleman, P.B. Esposito, and J. D. Anderson, Astrophys. J. [**211**]{}, 943 (1977). The propagation speed for the Doppler signal is the phase velocity, which is greater than $c$. Hence, the negative sign in Eq. (\[eq:sol\_plasma\]) applies. The ranging signal propagates at the group velocity, which is less than $c$. Hence, there the positive sign applies. B.-G. Anderssen and S. G. Turyshev, JPL Internal IOM 1998-0625, and references therein. M. K. Bird, H. Volland, M. Pätzold, P. Edenhofer , S. W. Asmar and J. P. Brenkle, [Astrophys. J.]{} [**426**]{}, 373 (1994). The units conversion factor for $A,B,C$ from m to cm$^{-3}$ is $2 N_c(S)/R_\odot = 0.01877$, where $N_c(S) =1.240 \times 10^4 \nu^2$ is the S band (in MHz) critical plasma density, and $R_\odot$ is the radius of the Sun. These values of parameters $A, B, C$ were kindly provided to us by John E. Ekelund of JPL. They represent the best solution for the solar corona parameters obtained during his simulations of the solar conjunction experiments that will be performed with the Cassini mission spacecraft in 2001 and 2002. This model is explained and described at\ [http://science.msfc.nasa.gov/ssl/pad/solar/ predict.htm]{} These come from the adjustment in the system of data weights (inverse of the variance on each measurement) for Mariner 6/7 range measurements. Private communication by Inter-office Memorandum from D. O. Muhleman of Caltech to P. B. Esposito of JPL, dated 7 July 1971. G. W. Null, E. L. Lau, E. D. Biller, and J. D. Anderson, [Astron. J.]{} [**86**]{}, 456 (1981). P. A. Laing, “Implementation of J2000.0 reference frame in CHASMP,” The Aerospace Corporation’s Internal Memorandum \# 91(6703)-1. January 28, 1991. J. H. Lieske, [ Astron. Astrophys.]{} [**73**]{}, 282 (1979). Also, see [*FK5/J2000.0 for DSM Applications,*]{} Applied Technology Associates, 6 June 1985. E. M. Standish, [Astron. Astrophys.]{} [**114**]{}, 297 (1982) J. Sherman and W. Morrison. Ann. Math. Stat. [**21**]{}, 124 (1949) J. D. Anderson, Quarterly Report to NASA/Ames Research Center, [*Celestial Mechanics Experiment, Pioneer 10/11,*]{} 22 July 1992. Also see the later quarterly report for the period 1 Oct. 1992 to 31 Dec. 1992, dated 17 Dec. 1992, Letter of Agreement ARC/PP017. This last, specifically, contains the present Figure \[fig:correlation\]. We only measure Earth-spacecraft Doppler frequency and, as we will discuss in Sec. \[radioantbeam\], the down link antenna yields a conical beam of width 3.6 degrees at half-maximum power. Therefore, between Pioneer 10’s past and present (May 2001) distances of 20 to 78 AU, the Earth-spacecraft line and Sun-spacecraft line are so close that one can not resolve whether the force direction is towards the Sun or if the force direction is towards the Earth. If we could have used a longer arc fit that started earlier and hence closer, we might have able to separate the Sun direction from the Earth direction. A preliminary discussion of these results appeared in M. M. Nieto, T. Goldman, J. D. Anderson, E. L. Lau, and J. Pérez-Mercader, in: [*Proc. Third Biennial Conference on Low-Energy Antiproton Physics, LEAP’94*]{}, ed. by G. Kernel, P. Krizan, and M. Mikuz (World Scientific, Singapore, 1995), p. 606. Eprint hep-ph/9412234. Since both the gravitational and radiation pressure forces become so large close to the Sun, the anomalous contribution close to the Sun in Figures \[fig:forces\] and \[fig:correlation\] is meant to represent only what anomaly can be gleaned from the data, not a measurement. B. D. Tapley, in [*Recent Advances in Dynamical Astronomy*]{}, eds. B. D. Tapley and V. Szebehely (Reidel, Boston, 1973), p.396. P. A. Laing, [*Thirty Years of CHASMP*]{}, Aerospace report (in preparation). P. A. Laing, [*Software Specification Document, Radio Science Subsystem, Planetary Orbiter Error Analysis Study Program (POEAS)*]{}, Jet Propulsion Laboratory Technical Report DUK-5127-OP-D, 19 February 1981. POEAS was originally developed to support the Mariner Mars program. P. A. Laing and A. S. Liu. NASA Interim Technical Report, Grant NAGW-4968, 4 October 1996. Galileo is less sensitive to either an $a_P$- or an $a_t= a_P/c$-model effect than the Pioneers. Pioneers have a smaller solar pressure and a longer light travel time. Sensitivity to a clock acceleration is proportional to the light travel time squared. T. McElrath, private communication. T. P. McElrath, S. W. Thurman, and K. E. Criddle, in [*Astrodynamics 1993*]{}, edited by A. K. Misra, V. J. Modi,R. Holdaway, and P. M. Bainum (Univelt, San Diego CA, 1994), Ad. Astodynamical Sci. [**85**]{}, Part II, p. 1635, paper No. AAS 93-687. The gas leaks found in the Pioneers are about an order of magnitude too small to explain $a_P$. Even so, we feel that some systematic or combination of systematics (such as heat or gas leaks) will likely explain the anomaly. However, such an explanation has yet to be demonstrated. We will discuss his point further in Sections \[recent\_results\] and \[int-systema\]. More information on the “Heliocentric Trajectories for Selected Spacecraft, Planets, and Comets,” can be found at [http://nssdc.gsfc.nasa.gov/space/ helios/heli.html]{}. ODP/*Sigma* took the Interval I/II boundary as 22 July 1990, the date of a maneuver. CHASMP took this boundary date as 31 August 1990, when a clear anomaly in the spin data was seen. We have checked, and these choices produce less than one percent differences in the results. J. A. Estefan, L. R. Stavert, F. M. Stienon, A. H. Taylor, T. M. Wang, and P. J. Wolff, [*Sigma User’s Guide. Navigation Filtering/Mapping Program*]{}, JPL document 699-FSOUG/NAV-601 (Revised: 14 Dec. 1998). G. W. Null, [Astron. J.]{} [**81**]{}, 1153 (1976). R. M. Georgevic, [*Mathematical model of the solar radiation force and torques acting on the components of a spacecraft,*]{} JPL Technical Memorandum 33-494 (1971). Data is available at [http://www.ngdc.noaa.gov/stp/ SOLAR/IRRADIANCE/irrad.html]{} For an ideal flat surface facing the Sun, $\mathcal{K} = (\alpha + 2\epsilon) = (1 +2\mu + 2 \nu)$. $\alpha$ and $\epsilon$ are, respectively, the absorption and reflection coefficients of the spacecraft’s surface. ODP uses the second formulation in terms of reflectivity coefficients, ODP’s input $\mu$ and $\nu$ for Pioneer 10, are obtained from design information and early fits to the data. (See the following paragraph.) These numbers by themselves yield $\mathcal{K_0}= 1.71$. When a first (negative) correction is made for the antenna’s parabolic surface, $\mathcal{K}\rightarrow 1.66$. There are complicating effects that modify the ideal antenna. The craft actually has multiple different-shaped surfaces (such as the RTGs), that are composed of different materials oriented at different angles to the spin axis, and which degrade with time. But far from the Sun, and given $M$ and $A$, the sum of all such corrections can be subsumed, for our purposes, in an [*effective*]{} $\mathcal{K}$. It is still expected to be of order 1.7. Eq. (\[eq:srp\]) is combined with information on the spacecraft surface geometry and it’s local orientation to determine the magnitude of its solar radiation acceleration as it faces the Sun. As with other non-gravitational forces, an incorrect mass in modeling the solar radiation pressure force can be accounted for by modifying other constants such as the effective sizes of the antenna and the albedo. E. J. Smith, L. Davis, Jr., D. E. Jones, D. S. Colburn, P. J. Coleman, Jr., P. Dyal, and C. P. Sonnett, Science [**183**]{}, 306 (1974); [*ibid*]{}. [**188**]{}, 451 (1975). This result was obtained from a limit for positive charge on the spacecraft [@null76]. No measurement dealt with negative charge, but such a charge would have to be proportionally larger to have a significant effect. R. Malhotra, Astron. J. [**110**]{}, 420 (1995); [*ibid.*]{} [**111**]{}, 504 (1996). A. P. Boss and S. J. Peale, Icarus [**27**]{}, 119 (1976). A. S. Liu, J. D. Anderson, and E. Lau, Proc. AGU (Fall Meeting, San Francisco, 16-18 December 1996), paper \# SH22B-05. G. E. Backman, A. Dasgupta, and R. E. Stencel, [Astrophys. J.]{} [**450**]{}, L35 (1995). Also see S. A. Stern, Astron. Astrophys. [**310**]{}, 999 (1996). V. L. Teplitz, S. A. Stern, J. D. Anderson, D. Rosenbaum, R. J. Scalise, and P. Wentzler, [Astrophys. J.]{} [**516**]{}, 425 (1999). J. D. Anderson, G. Giampieri, P. A. Laing, and E. L. Lau, work in progress. R. F. C. Vessot, “Space experiments with high stability clocks,” in proceedings of the “Workshop on the Scientific Applications of Clocks in Space,” (November 7-8, 1996. Pasadena, CA). Edited by L. Maleki. JPL Publication 97-15 (JPL, Pasadena, CA, 1997), p. 67. O. J. Sovers and C. S. Jacobs, [*Observational Model and Parameter Partials for the JPL VLBI Parameter Estimation Software “MODEST” - 1996*]{}, Jet Propulsion Laboratory Technical Report 83-39, Rev. 6 (1996). J. I. Katz, Phys. Rev. Lett. [**83**]{}, 1892 (1999). There is an intuitive way to understand this. Set up a coordinate system at the closest axial point of an RTG pair. Have the antenna be in the (+z,-x) direction, and the RTG pair in the positive x direction. Then from the RTGs the antenna is in 1/4 of a sphere (positive z and negative x). The ‘antenna occupies about 1/3 of 180 degrees in azimuthal angle. Its form is the base part of the parabola. Thus, it resembles a “flat” triangle of the same width, producing another factor of $\sim (1/2-2/3)$ compared to the angular size of a rectangle. It occupies of order (1/4-1/3) of the latitudinal-angle phase space angle of 90$^o$. This yields a total reduction factor of $\sim(1/96-2/108)$, or about 1 to 2 %. The value of 1.5% is obtained by doing an explicit calculation of the solid angle subtended by the antenna from the middle of the RTG modules using the Pioneer’s physical dimensions. V. J. Slabinski of USNO independently obtained a figure of 1.6%. Our high estimate of 40 W is not compromised by imprecise geometry. If the RTGs were completely in the plane of the top of the dish, then the maximum factor multiplying the 40 W directed power would be $\kappa_z = 1$. This would presume all the energy was reflected and/or absorbed and re-emitted towards the rear of the craft. (If the RTGs were underneath the antenna, then the total factor could ideally go as high as “2”, from adding the RTG heat going out the opposite direction.) The real situation is that the average sine of the latitudinal angle up to the antenna from the RTGs is about 0.3. This means that the heat gong out the opposite direction might cause an effective factor $\kappa_z$ to go as high as 1.3. However, the real reflection off of the antenna is not straight backwards. It is closer to 45$^o$. The absorbed and re-emitted radiation is also at an angle to the rotation axis, although smaller. (This does not even consider reflected/reemitted heat that does not go directly backwards but rather bounces off of the central compartment.) So, the original estimate of $\kappa=1$ is a good bound. J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Phys. Rev. Lett. [**83**]{}, 1893 (1999). We acknowledge R. E. Slusher of Bell Labs for raising this possibility. B. A. Smith, G. A. Briggs, G. E. Danielson, A. F. Cook, II, M. E. Davies, G. E. Hunt, H. Masursky, L. A. Soderblom, T. C. Owen, C. Sagan, and V. E. Suomi, Space Sci. Rev. [**21**]{}, 103 (1977). C. E. Kohlhase and P. A. Penzo, Space Sci. Rev. [**21**]{}, 77 (1977). We are grateful to C. J. Hansen of JPL, who kindly provided us with operational information on the Voyager video cameras. B. A. Smith, L. A. Soderblom, D. Banfield, C. Barnet, T. Basilevsky, R. F. Beebe, K. Bollinger, J. M. Boyce, A. Brahic, G. A. Briggs, R. H. Brown, C. Chyba, S. A. Collins, T. Colvin, A. F. Cook, II, D. Crisp, S. K. Croft, D. Cruikshank, J. N. Cuzzi, G. E. Danielson, M. E. Davies, E. De Jong, L. Dones, D. Godfrey, J. Goguen, I. Grenier, V. R. Haemmerle, H. Hammel, C. J. Hansen, C. P. Helfenstein, C. Howell, G. E. Hunt, A. P. Ingersoll, T. V. Johnson, J. Kargel, R. Kirk, D. I. Kuehn, S. Limaye, H. Masursky, A. McEwen, D. Morrison, T. Owen, W. Owen, J. B. Pollack, C. C. Porco, K. Rages, P. Rogers, D. Rudy, C. Sagan, J. Schwartz, E. M. Shoemaker, M. Showalter, B. Sicardy, D. Simonelli, J. Spencer, L. A. Sromovsky, C. Stoker, R. G. Strom, V. E. Suomi, S. P. Synott, R. J. Terrile, P. Thomas, W. R. Thompson, A. Verbiscer, and J. Veverka, [ Science]{} [**246**]{}, 1432 [1989]{}. E. M. Murphy, Phys. Rev. Lett. [**83**]{}, 1890 (1999). J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, [ Phys. Rev. Lett.]{} [**83**]{}, 1891 (1999). L. K. Scheffer, (a) eprint gr-qc/0106010, the original modification; (b) eprint gr-qc/0107092; (c) eprint gr-qc/0108054. J. D. Anderson, P. A. Laing, E. L. Lau, M. M. Nieto, and S. G. Turyshev, eprint gr-qc/0107022. These results were not treated for systematics, used different time-evolving estimation procedures, were done by three separate JPL navigation specialists, separated and smoothed by one of us [@jpl], and definitely not analyzed with the care of our recent run (1987.0 to 1998.5). In particular, the first two Pioneer 11 points, included in the early memos [@jpl], were after Pioneer 11 encountered Jupiter and then was going back across the central solar system to encounter Saturn. T. K. Keenan, R. A. Kent, and R. N. R. Milford, [*Data Sheets for PMC Radioisotopic Fuel*]{}, Los Alamos Report LA-4976-MS (1972), available from NTIS. We thank Gary Reinhart for finding this data for us. Diagrams showing the receptacle and the bayonet coupling connector were made by the Deutsch Company of Banning, CA. (The O-ring was originally planned to be silicon.) Diagrams of the receptacle as mounted on the RTGs were made by Teledyne Isotopes (now Teledyne Brown Engineering). Once again we gratefully acknowledge Ted Christenbury for obtaining these documents for us. In principle, many things could be the origin of some spin down: structural deformations due to adjustments or aging, thermal radiation, leakage of the helium from the RTGs, etc. But in the case of Pioneer spacecraft none of these provide an explanation for the spin history exhibited by the Pioneer 10, especially the large unexpected changes among the Intervals I, II, and III. S. Herrick, [*Astodynamics*]{} (Van Nostrand Reinhold Co., London, New York, 1971-72). Vols. 1-2, We thank William Folkner of JPL for his assistance in producing several test files and invaluable advice. D. Brouwer and G. M. Clemence, [*Methods of Celestial Mechanics*]{} (Academic Press, New York, 1961). W.G. Melbourne, Scientific American [**234**]{}, No. 6, 58 (1976). We thank E. Myles Standish of JPL, who encouraged us to address in greater detail the nature of the annual/diurnal terms seen in the Pioneer Doppler residuals. (This work is currently under way.) He also kindly provided us with the accuracies from his internal JPL solar system ephemeris, which is continually under development. new, soon to be published, solar system ephemeris. D. F. Crawford, eprint astro-ph/9904150. N. Didon, J. Perchoux, and E. Courtens, Université de Montpellier preprint (1999). D. A. Gurnett, J. A. Ansher, W. S. Kurth, and L. J. Granroth, Geophys. Res. Lett. [**24**]{}, 3125 (1997); M. Landgraf, K. Augustsson, E. Grün, and A. S. Gustafson, [ Science]{} [**286**]{}, 239 (1999). Pioneer 10 data yielded another fundamental physics result, a limit on the rest mass of the photon. See L. Davis, Jr., A. S. Goldhaber, and M. M. Nieto, [ Phys. Rev. Lett.]{} [**35**]{}, 1402 (1975). G. J. Stephenson, Jr., T. Goldman, and B. H. J. McKellar, [ Int. J. Mod. Phys. A]{} [**13**]{}, 2765 (1998), hep-ph/9603392. M. M. Nieto and T. Goldman, [ Phys. Rep.]{} [**205**]{}, 221 (1991); [**216**]{}, 343 (1992), and references therein. J. Bekenstein and M. Milgrom, [Astrophys. J.]{} [**286**]{}, 7 (1984); M. Milgrom and J. Bekenstein, in: [*Dark Matter in the Universe*]{}, eds. J. Kormendy and G. R. Knapp (Kluwer Academic, Dordrecht, Boston, 1987), p. 319; M. Milgrom, [ La Recherche]{} [**19**]{}, 182 (1988). P. D. Mannheim, [Astrophys. J.]{} [**419**]{}, 150 (1993). Also see K. S. Wood and R. J. Nemiroff, [Astrophys. J.]{} [**369**]{}, 54 (1991). K. G. Begeman, A. H. Broeils, and R. H. Sanders, Mon. Not. R. Astron. Soc. [**249**]{}, 523 (1991); T. G. Breimer and R. H. Sanders, Astron. Astrophys. [**274**]{}, 96 (1993). M. Milgrom, [Ann. Phys. (NY)]{} [**229**]{}, 384 (1994). Also see astro-ph/0112069. S. Capozziello, S. De Martino, S. De Siena, and F. Illuminati, Mod. Phys. Lett. A [**16**]{}, 693 (2001). Eprint gr-qc/0104052. Also see eprint gr-qc/9901042. R. D. Reasenberg, I. I. Shapiro, P. E. MacNeil, R. B. Goldstein, J. C. Breidenthal, J. P. Brenkle, D. L. Cain, T. M. Kaufman, T. A. Komarek, and A. I. Zygielbaum, [Astrophys. J.]{} [**234**]{}, L219 (1979). J. D. Anderson, J. K. Campbell, R. F. Jurgens, E. L. Lau, X X Newhall, M. A. Slade III, and E. M. Standish, Jr., in: [*Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity*]{}, Part A, ed. H. Sato and T. Nakamura, (World Scientific, Singapore, 1992), p. 353. R. H. Sanders, private communication to M. Milgrom (1998). The Principle of Equivalence figure of merit is $a_P/a_N$. This is worse than for laboratory experiments (comparing small objects) or for the Nordtvedt Effect (large objects of planetary size) [@Will93]. It again emphasizes that the Earth and Mars do not change positions due to $a_P$. F. Munyaneza and R. D. Viollier, eprint astro-ph/9910566. J. L. Rosales and J. L. Sánchez-Gomez, eprint gr-qc/9810085. V. Guruprasad, eprints astro-ph/9907363, gr-qc/0005014, gr-qc/0005090. D. [Ø]{}stvang, eprint gr-qc/9910054. W. B. Belayev, eprint gr-qc/9903016. G. Modanese, Nucl. Phys. B [**556**]{}, 397 (1999). Eprint gr-qc/9903085. R. Mansouri, F. Nasseri and M. Khorrami, Phys. Lett. A [**259**]{},194 (1999). Eprint gr-qc/9905052. B. G. Sidharth, Nuovo Cim. [**B115**]{}, 151 (2000). Eprint astro-ph/9904088. M. A. Ivanov, Gen. Rel. and Grav. [**33**]{}, 479 (2001). eprint astro-ph/0005084. Also see eprint gr-qc/0009043, a contribution to the SIGRAV/2000 Congress. R. Foot and R. R. Volkas, Phys. Lett. B [**517**]{}, 13 (2001). Eprint gr-qc/0108051. J. P. Mbelek and M. Lachièze-Rey, eprint gr-qc/9910105. S. Calchi Novati, S. Capozziello, and G. Lambiase, Grav. Cosmol. [**6**]{}, 173 (2000). Eprint astro-ph/0005104. R. A. Hulse and J. H. Taylor, Astrophys. J. [**195**]{}, L51 (1975); J. H. Taylor and J. M. Weisberg, Astrophys. J. [**253**]{}, 908 (1982). M. Consoli and F. Siringo, eprint hep-ph/9910372. M. Consoli, eprint hep-ph/0002098. S. Capozziello and G. Lambiase, Mod. Phys. Lett. A [**14**]{}, 2193 (1999). Eprint gr-qc/9910026. P. D. Mannheim and D. Kazanas, Astrophys. J. [**342**]{}, 635 (1989); P. D. Mannheim, Gen. Rel. Grav. [**25**]{}, 697 (1993); P. D. Mannheim, Astrophys. J. [**479**]{}, 659 (1997). J. Wood and W. Moreau, eprint gr-qc/0102056. O. Bertolami and F. M. Nunes, Phys. Lett. B [**452**]{}, 108 (1999). Eprint hep-ph/9902439. L. Iess, G Giampieri, J. D. Anderson, and B. Bertotti, [Class. Quant. Grav.]{} [**16**]{}, 1487 (1999). R. Danner and S. Unwin, eds., [*SIM Interferometry Mission: Taking the Measure of the Universe*]{}, NASA document JPL 400-811 (1999). Also see [http://sim.jpl.nasa.gov/]{} The situation may be analogous to what happened in the 1980’s to geophysical exploration. Mine and tower gravity experiments seemed to indicate anomalous forces with ranges on the order of km [@stacey]. But later analyses showed that the experiments had been so precise that small inhomogeneities in the field surveys had introduced anomalies in the results at this newly precise level [@bartlett]. But the very important positive outcome was that geophysicists realized the point had been reached where more precise studies of systematics were necessary. F. D. Stacey, G. J. Tuck, G. J. Moore, S. C. Holding, B. D. Goodwin, and R. Zhou, Rev. Mod. Phys. [**59**]{}, 157 (1987); D. H. Eckhardt, C. Jekeli, A. R. Lazarewicz, A. J. Romaides, and R. W. Sands, Phys. Rev. Lett. [**60**]{}, 2567 (1988). Measurements were more often taken at easily accessible sites, such as roads, rather than at more inaccessible cites at different heights, such as mountain sides or marshes. See D. F. Bartlett and W. L. Tew, Phys. Rev. D [**40**]{}, 673 (1989); [*. ibid.*]{}, J. Geophys. Res. \[Solid Earth Planet\] [**95**]{}, 17363 (1990); C. Jekeli, D. H. Eckhardt, and A. J. Romaides, Phys. Rev. Lett. [**64**]{}, 1204 (1990). For a review, see Section 4 of Ref. [@physrep]. [^1]: Electronic address: [email protected] [^2]: Electronic address: [email protected] [^3]: Electronic address: [email protected] [^4]: Deceased (13 November 2000). [^5]: Electronic address: [email protected] [^6]: [email protected]

ArXiv

--- abstract: 'A large solar pore with a granular light bridge was observed on October 15, 2008 with the IBIS spectrometer at the Dunn Solar Telescope and a 69-min long time series of spectral scans in the lines Ca II 854.2 nm and Fe I 617.3 nm was obtained. The intensity and Doppler signals in the Ca II line were separated. This line samples the middle chromosphere in the core and the middle photosphere in the wings. Although no indication of a penumbra is seen in the photosphere, an extended filamentary structure, both in intensity and Doppler signals, is observed in the Ca II line core. An analysis of morphological and dynamical properties of the structure shows a close similarity to a superpenumbra of a sunspot with developed penumbra. A special attention is paid to the light bridge, which is the brightest feature in the pore seen in the Ca II line centre and shows an enhanced power of chromospheric oscillations at 3–5 mHz. Although the acoustic power flux in the light bridge is five times higher than in the ”quiet” chromosphere, it cannot explain the observed brightness.' address: - '$^1$ Astronomical Institute, Academy of Sciences of the Czech Republic (v.v.i.), Fričova 298, CZ-25165 Ondřejov, Czech Republic' - '$^2$ Charles University in Prague, Faculty of Mathematics and Physics, Astronomical Institute, V Holešovičkách 2, CZ-18000 Prague 8, Czech Republic' - '$^3$ Department of Physics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy' author: - 'M Sobotka$^1$, M Švanda$^{1,2}$, J Jurčák$^1$, P Heinzel$^1$ and D Del Moro$^3$' title: Atmosphere above a large solar pore --- Introduction ============ Pores are small sunspots without penumbra. The absence of a filamentary penumbra in the photosphere has been interpreted as the indication of a simple magnetic structure with mostly vertical field, e.g. \[1, 2\]. Magnetic field lines are observed to be nearly vertical in centres of pores and inclined by about 40$^\circ$ to 60$^\circ$ at their edges \[3, 4\]. Pores contain a large variety of fine bright features, such as umbral dots and light bridges that may be signs of a convective energy transportation mechanism. Light bridges (hereafter LBs) are bright structures in sunspots and pores that separate umbral cores or are embedded in the umbra. Their structure depends on the inclination of local magnetic field and can be granular, filamentary, or a combination of both \[5\]. Many observations confirm that magnetic field in LBs is generally weaker and more inclined with respect to the local vertical. It was shown in \[6\] that the field strength increases and the inclination decreases with increasing height. This indicates the presence of a magnetic canopy above a deeply located field-free region that intrudes into the umbra and forms the LB. Above a LB, \[7\] found a persistent brightening in the TRACE 160 nm bandpass formed in the chromosphere. It was interpreted as a steady-state heating possibly due to constant small-scale reconnections in the inclined magnetic field. In the chromosphere, large isolated sunspots are often surrounded by a pattern of dark, nearly radial fibrils. This pattern, called superpenumbra, is visually similar to the white-light penumbra but extends to a much larger distance from the sunspot. Dark superpenumbral fibrils are the locations of the strongest inverse Evershed flow – an inflow and downflow in the chromosphere toward the sunspot. Time-averaged Doppler measurements indicate the maximum speed of this flow equal to 2–3 km s$^{-1}$ near the outer penumbral border \[8\]. Observations and data processing ================================ A large solar pore NOAA 11005 was observed with the Interferometric Bidimensional Spectrometer IBIS \[9\] attached to the Dunn Solar Telescope (DST) on 15 October 2008 from 16:34 to 17:43 UT, using the DST adaptive optics system. The slowly decaying pore was located at 25.2 N and 10.0 W (heliocentric angle $\theta = 23^\circ$) during our observation. According to \[10\], the maximum photospheric field strength was 2000 G, the inclination of magnetic field at the edge of the pore was 40$^\circ$ and the whole field was inclined by 10$^\circ$ to the west. The IBIS dataset consists of 80 sequences, each containing a full Stokes ($I, Q, U, V$) 21-point spectral scan of the Fe I 617.33 nm line (see \[10\]) and a 21-point $I$-scan of the Ca II 854.2 nm line. The wavelength distance between the spectral points of the Ca II line is 6.0 pm and the time needed to scan the 0.126 nm wide central part of the line profile is 6.4 s. The exposure time for each image was set to 80 ms and each sequence took 52 seconds to complete, thus setting the time resolution. The pixel scale of these images was 0$''$.167. Due to the spectropolarimetric setup of IBIS, the working field of view (FOV) was $228 \times 428$ pixels, i.e., $38''\times 71''.5$. The detailed description of the observations and calibration procedures can be found in \[10,11\]. Complementary observations were obtained with the HINODE/SOT Spectropolarimeter \[12,13\]. The satellite observed the pore on 15 October 2008 at 13:20 UT, i.e., about 3 hours before the start of our observations. From one spatial scan in the full-Stokes profiles of the lines Fe I 630.15 and 630.25 nm we used a part covering the pore umbra with a granular LB. According to \[14\], the inner wings ($\pm 60$ pm) of the infrared Ca II 854.2 nm line sample the middle photosphere at the typical height $h \simeq 250$ km above the $\tau_{500} = 1$ level, while the centre of this line is formed in the middle chromosphere at $h \simeq 1200$–1400 km. This provides a good tool to study the pore and its surroundings at different heights in the atmosphere and to look for relations between the photospheric and chromospheric structures. The observations in the Ca II line are strongly influenced by oscillations and waves present in the chromosphere and upper photosphere. The observed intensity fluctuations in time are caused by real changes of intensity as well as by Doppler shifts of the line profile. To separate the two effects, Doppler shifts of the line profile were measured using the double-slit method \[15\], consisting in the minimisation of difference between intensities of light passing through two fixed slits in the opposite wings of the line. An algorithm based on this principle was applied to the time sequence of Ca II profiles. The distance of the two wavelength points (“slits”) was 36 pm, so that the “slits” were located in the inner wings near the line core, where the intensity gradient of the profile is at maximum and the effective formation height in the atmosphere is approximately 1000 km. The wavelength sampling was increased by the factor of 40 using linear interpolation, thus obtaining the sensitivity of the Doppler velocity measurement 53 m s$^{-1}$. The reference zero of Doppler velocity was defined as a time- and space-average of all measurements. This way we obtained a series of 80 Doppler velocity maps. This method does not take into account the asymmetry of the line profile, e.g., the changes of Doppler velocity with height in the atmosphere. Using the information about the Doppler shifts, all Ca II profiles were shifted to a uniform position with a subpixel accuracy. This way we obtained an intensity data cube ($x, y, \lambda$, scan), where, if we neglect line asymmetries and inaccuracies of the method, the observed intensity fluctuations correspond to the real intensity changes. The oscillations and waves were separated from the slowly evolving intensity and Doppler structures by means of the 3D $k-\omega$ subsonic filter with the phase-velocity cutoff at 6 km s$^{-1}$. Results ======= Examples of intensity maps in the continuum 621 nm, blue wing and centre of the line Ca II 854.2 nm and a Doppler map are shown in Fig. 1. Oscillations and waves are filtered out from these images. A filamentary structure around the pore, composed of radially oriented bright and dark fibrils, is clearly seen in the line-centre intensity and Doppler maps. The area and shape of the filamentary structure are identical in all pairs of the line-centre intensity and Doppler images in the time series. The fibrils begin immediately at the umbral border and in many cases continue till the border of the filamentary structure. Their lengths are identical in the Doppler and intensity maps. However, the fibrils seen in the line centre are spatially uncorrelated with those in the Doppler maps. ![\[fig2\]Time-averaged Doppler map with contours of zero, $-1$ and 1 km s$^{-1}$.](Fig1.eps){width="11cm"} ![\[fig2\]Time-averaged Doppler map with contours of zero, $-1$ and 1 km s$^{-1}$.](Fig2.eps){width="3.3cm"} Concentric running waves originating in the centre of the pore are observed in the unfiltered time series of the line-centre intensity and Doppler images. They propagate through the filamentary structure to the distance 3000–5000 km from the visible border of the pore with a typical speed of 10 km s$^{-1}$. These waves are very similar to running penumbral waves observed in the penumbral chromosphere and in superpenumbrae of developed sunspots \[16\]. The subsonic-filtered time series of Doppler maps was averaged in time to obtain a spatial distribution of mean line-of-sight (LOS) velocities around the pore. The result is shown in Fig. 2 together with contours of zero, $-1$ and 1 km s$^{-1}$. We can see from the figure that the inner part of the filamentary structure contains a positive LOS velocity (away from us, a downflow), while the outer part, located mostly on the limb side, shows a negative LOS velocity (toward us, an upflow). Taking into account the heliocentric angle $\theta = 23^\circ$ and the fact that the plasma moves along magnetic field lines forming a funnel, the negative LOS velocity is only partly caused by real upflows but mostly by horizontal inflows into the pore. The picture of plasma moving toward the pore and flowing down in its vicinity is consistent with the inverse Evershed effect observed in the sunspots’ superpenumbra. All these facts are leading to the conclusion that the filamentary structure observed in the chromosphere above the pore is equivalent to a superpenumbra of a developed sunspot. The missing correlation between the fibrils seen in the line centre and those in the Doppler maps further supports this conclusion, because, according to \[17\], flow channels of the inverse Evershed effect are not identical with superpenumbral filaments.\ A strong granular LB separates the two umbral cores of the pore. It is the brightest feature inside the pore in the photosphere as well as in the chromosphere, where it is by factor of 1.3 brighter than the average brightness in the FOV. The granular structure of the LB is preserved at all heights, from the photospheric continuum level to the formation height of the Ca II line centre. It is interesting that while a typical pattern of reverse granulation, observed in the Ca II wings, appears outside the pore in the middle photosphere ($h \simeq 250$ km), the LB is always composed of small bright granules separated by dark intergranular lanes (Fig. 1). A feature-tracking technique was applied to correlate the LB granules in position and time at different heights in the atmosphere. A correlation was found between the photospheric LB granules in the continuum and Ca II wings (correlation coefficient 0.46). On the other hand, there is no correlation between the chromospheric LB “granules” observed in the Ca II line centre and the photospheric ones in the wings and continuum. The feature-tracking technique has shown that the mean size of the LB granules increases with height from 0$''$.45 in the continuum to 0$''$.50 in the Ca II wings and 0$''$.54 in the Ca II line centre. Similarly, the average width of the LB increases with height from 2$''$ (continuum) to 2$''$.5 (Ca II wings) and 3$''$ (Ca II line centre). On the other hand, it can be expected that the width of the LB magnetic structure decreases with height due to the presence of magnetic canopy. The complementary HINODE observations in two spectral lines Fe I 630.15 and 630.25 nm made it possible to obtain vertical stratification of temperature and magnetic field strength in the LB photosphere, using the inversion code SIR \[18\]. The results, summarised in Table 1, show that the width of LB in temperature maps really increases with height, while the width of the LB magnetic structure decreases and the magnetic field strength increases, confirming the magnetic canopy configuration. [rrrr]{} Height (km) & Width in $T$ & Width in $B$ & $B_{\rm min}$ (G)\ 0 & 2$''$.7 & 1$''$.7 & 0\ 90 & 2$''$.7 & 1$''$.5 & 100\ 180 & 2$''$.7 & 1$''$.3 & 300\ 270 & 2$''$.9 & 1$''$.1 & 500\ 350 & 3$''$.3 & 0$''$.8 & 700\ Power spectra of chromospheric oscillations were calculated using the unfiltered time series of the Ca II line-centre intensity and Doppler images. Power maps derived from the Doppler velocities at frequencies 3–6 mHz are shown in Fig. 3. At the LB position, we can see a strongly enhanced power around 3–5 mHz, comparable with that in a plage near the eastern border of the pore A similar power enhancement was reported in \[19\]. Usually, low-frequency oscillations do not propagate through the temperature minimum from the photosphere to the chromosphere due to the acoustic cut-off at 5.3 mHz \[20\]. To explain our observations, we assume that the low-frequency oscillations leak into the chromosphere along an inclined magnetic field \[19,21\], which is present in the LB and in the plage. The inclined magnetic field in the LB is verified by inversions of the full-Stokes Fe I 617.33 nm profiles \[10\]. The inclination angle, extrapolated to the height of the temperature minimum, is 40$^\circ$–50$^\circ$ to the west thanks to inclined magnetic field lines at the periphery of the larger (eastern) umbral core. These field lines pass above the magnetic canopy of the LB. ![\[fig3\]Power maps of Doppler velocities at frequencies 3–6 mHz. The contours outline the boundaries of the pore and light bridge, observed in the continuum.](Fig3.eps){width="16cm"} Following \[22\], we estimate the acoustic energy flux in the LB chromosphere and compare it with the flux in the “quiet” region. The observed Doppler velocities are used for this purpose. With the estimated magnetic field inclination of 50$^\circ$, the effective acoustic cut-off frequency decreases to the value of 3 mHz in the LB. The total calculated acoustic power flux is 550 W m$^{-2}$ in the LB, while only 110 W m$^{-2}$ in the “quiet” chromosphere. These results seem to be lower than 1840 W m$^{-2}$ presented in \[22\] but one has to bear in mind that this value was obtained for the photosphere, where the power of the acoustic oscillations must be higher than in the chromosphere. Discussion and conclusions ========================== We studied the photosphere and chromosphere above a large solar pore with a granular LB using spectroscopic observations with spatial resolution of 0$''$.3–0$''$.4 in the line Ca II 854.2 nm and photospheric Fe I lines. We have shown that in the chromospheric filamentary structure around the pore, observed in the Ca II line core and Doppler maps, the inverse Evershed effect and running waves are present. Chromospheric fibrils seen in the intensity and Doppler maps are spatially uncorrelated. From these characteristics and from the morphological similarity of the filamentary structure to superpenumbrae of developed sunspots we conclude that the observed pore has a kind of a superpenumbra, in spite of a missing penumbra in the photosphere. A special attention was paid to the granular LB that separated the pore into two umbral cores. The magnetic canopy structure \[6\] is confirmed in this LB. In the middle photosphere ($h \simeq 250$ km, Ca II wings), the reverse granulation is seen around the pore but not in the LB. The reverse granulation is explained by adiabatic cooling of expanding gas in granules, which is only partially cancelled by radiative heating \[23\]. In the LB, hot (magneto)convective plumes at the bottom of the photosphere cannot expand adiabatically in higher photospheric layers due to the presence of magnetic field and the radiative heating dominates, forming small bright granules separated by dark lanes. The positive correlation between the LB structures in the continuum and Ca II line wings indicates that the middle-photosphere structures are heated by radiation from the low photosphere. Since the mean free photon path in the photosphere is larger than 1$''$ for $h > 120$ km, the LB observed in the line wings is broader and its granules are larger than in the continuum due to the diffusion of radiation. In the middle chromosphere ($h \simeq 1300$ km, Ca II centre), the LB is the brightest feature in the pore and it is brighter by factor of 1.3 than the average intensity in the FOV. Since the height in the atmosphere is well above the temperature minimum, the radiative heating cannot be expected. The heating by acoustic waves seems to be a candidate, because the acoustic power flux in the LB is five times higher than in the “quiet” chromosphere. To check this possibility, we have to compare the acoustic power flux with the total radiative cooling. An average profile of the Ca II line in the LB was used to derive a simple semi-empirical model based on the VAL3C chromosphere \[24\], with the temperature increased by 3000 K in the upper chromospheric layers ($h > 900$ km). The net radiative cooling rates were calculated using this model. The resulting height-integrated radiative cooling is approximately 6700 W m$^{-2}$ in the LB chromosphere and 3000 W m$^{-2}$ in the “quiet” chromosphere. The acoustic power fluxes (550 W m$^{-2}$ and 110 W m$^{-2}$, respectively) are by an order of magnitude lower than the estimated total radiative cooling, so that the acoustic power flux does not seem to provide enough energy to reach the observed LB brightness. References {#references .unnumbered} ========== [24]{} Simon G W and Weiss N O 1970 [*Solar Phys.*]{} [**13**]{} 85 Rucklidge A M, Schmidt H U and Weiss N O 1995 [*MNRAS*]{} [**273**]{}, 491 Keppens R and Martínez Pillet V 1996 [*Astron. Astrophys.*]{} [**316**]{} 229 Sütterlin P 1998 [*Astron. Astrophys.*]{} [**333**]{} 305 Sobotka M 1997 [*1st Advances in Solar Physics Euroconference, Advances in the Physics of Sunspots*]{}, ed B Schmieder, J C del Toro Iniesta and M Vázquez, ASP Conference series Vol 118 p 155 Jurčák J, Martínez Pillet V and Sobotka M 2006 [*Astron. Astrophys.*]{} [**453**]{} 1079 Berger T E and Berdyugina S V 2003 [*Astrophys. J.*]{} [**589**]{} L117 Alissandrakis C E, Dialetis D, Mein P et al. 1988 [*Astron. Astrophys.*]{} [**201**]{} 339 Cavallini F 2006 [*Solar Phys.*]{} [**236**]{} 415 Sobotka M, Del Moro D, Jurčák J and Berrilli F 2012 [*Astron. Astrophys.*]{} [**537**]{} A85 Viticchié B, Del Moro D, Criscuoli S and Berrilli F 2010 [*Astrophys. J.*]{} [**723**]{} 787 Kosugi T, Matsuzaki K, Sakao T et al. 2007 [*Solar Phys.*]{} [**243**]{} 3 Tsuneta S, Ichimoto K, Katsukawa Y et al. 2008 [*Solar Phys.*]{} [**249**]{} 167 Cauzzi G, Reardon K P, Uitenbroek H et al. 2008 [*Astron. Astrophys.*]{} [**480**]{} 515 Garcia A, Klvaňa M and Sobotka M 2010 [*Cent. Eur. Astrophys. Bull.*]{} [**34**]{} 47 Christopoulou E B, Georgakilas A A and Koutchmy S 2000 [*Astron. Astrophys.*]{} [**354**]{} 305 Tsiropoula G, Alissandrakis C E, Dialetis D and Mein P 1996 [*Solar Phys.*]{} [**167**]{} 79 Ruiz Cobo B and del Toro Iniesta J C 1992 [*Astrophys. J.*]{} [**398**]{} 375 Stangalini M, Del Moro D, Berrilli F amd Jefferies S M 2011 [*Astron. Astrophys.*]{} [**534**]{} A65 Fossat E, Regulo C, Roca Cortes T et al. 1992 [*Astron. Astrophys.*]{} [**266**]{} 532 Jefferies S M, McIntosh S W, Armstrong J D et al. 2006 [*Astrophys. J.*]{} [**648**]{}, L151 Bello González N, Flores Soriano M, Kneer F and Okunev O 2009 [*Astron. Astrophys.*]{} [**508**]{} 941 Cheung M C M, Schüssler M and Moreno Insertis F 2007 [*Astron. Astrophys.*]{} [**461**]{} 1163 Vernazza J E, Avrett E H and Loeser R 1981 [*Astrophys. J. Suppl. Ser.*]{} [**45**]{} 635

ArXiv

--- abstract: | We present effective pre-training strategies for neural machine translation (NMT) using parallel corpora involving a pivot language, i.e., source-pivot and pivot-target, leading to a significant improvement in source$\rightarrow$target translation. We propose three methods to increase the relation among source, pivot, and target languages in the pre-training: 1) step-wise training of a single model for different language pairs, 2) additional adapter component to smoothly connect pre-trained encoder and decoder, and 3) cross-lingual encoder training via autoencoding of the pivot language. Our methods greatly outperform multilingual models up to +2.6% <span style="font-variant:small-caps;">Bleu</span> in WMT 2019 French$\rightarrow$German and German$\rightarrow$Czech tasks. We show that our improvements are valid also in zero-shot/zero-resource scenarios.\ author: - | Yunsu Kim$^{1\hspace{-0.1em}}\Thanks{\hspace{0.5em}Equal contribution.}$ Petre Petrov$^{1,2*}$ Pavel Petrushkov$^{2}$ Shahram Khadivi$^{2}$ Hermann Ney$^{1}$\ $^{1}$RWTH Aachen University, Aachen, Germany\ [{surname}@cs.rwth-aachen.de]{}\ $^{2}$eBay, Inc., Aachen, Germany\ [{petrpetrov,ppetrushkov,skhadivi}@ebay.com]{}\ bibliography: - 'references.bib' title: | Pivot-based Transfer Learning for Neural Machine Translation\ between Non-English Languages --- Introduction ============ Machine translation (MT) research is biased towards language pairs including English due to the ease of collecting parallel corpora. Translation between non-English languages, e.g., French$\rightarrow$German, is usually done with pivoting through English, i.e., translating French (*source*) input to English (*pivot*) first with a French$\rightarrow$English model which is later translated to German (*target*) with a English$\rightarrow$German model [@de2006catalan; @utiyama2007comparison; @wu2007pivot]. However, pivoting requires doubled decoding time and the translation errors are propagated or expanded via the two-step process. Therefore, it is more beneficial to build a single source$\rightarrow$target model directly for both efficiency and adequacy. Since non-English language pairs often have little or no parallel text, common choices to avoid pivoting in NMT are generating pivot-based synthetic data [@bertoldi2008phrase; @chen2017teacher] or training multilingual systems [@firat2016zero; @johnson2017google]. In this work, we present novel transfer learning techniques to effectively train a single, direct NMT model for a non-English language pair. We pre-train NMT models for source$\rightarrow$pivot and pivot$\rightarrow$target, which are transferred to a source$\rightarrow$target model. To optimize the usage of given source-pivot and pivot-target parallel data for the source$\rightarrow$target direction, we devise the following techniques to smooth the discrepancy between the pre-trained and final models: - Step-wise pre-training with careful parameter freezing. - Additional adapter component to familiarize the pre-trained decoder with the outputs of the pre-trained encoder. - Cross-lingual encoder pre-training with autoencoding of the pivot language. Our methods are evaluated in two non-English language pairs of WMT 2019 news translation tasks: high-resource (French$\rightarrow$German) and low-resource (German$\rightarrow$Czech). We show that NMT models pre-trained with our methods are highly effective in various data conditions, when fine-tuned for source$\rightarrow$target with: - Real parallel corpus - Pivot-based synthetic parallel corpus (*zero-resource*) - None (*zero-shot*) For each data condition, we consistently outperform strong baselines, e.g., multilingual, pivoting, or teacher-student, showing the universal effectiveness of our transfer learning schemes. The rest of the paper is organized as follows. We first review important previous works on pivot-based MT in Section \[sec:related\]. Our three pre-training techniques are presented in Section \[sec:methods\]. Section \[sec:results\] shows main results of our methods with a detailed description of the experimental setups. Section \[sec:analysis\] studies variants of our methods and reports the results without source-target parallel resources or with large synthetic parallel data. Section 6 draws conclusion of this work with future research directions. Related Work {#sec:related} ============ In this section, we first review existing approaches to leverage a pivot language in low-resource/zero-resource MT. They can be divided into three categories: 1. \[sec:pivoting\] **Pivot translation (pivoting).** The most naive approach is reusing (already trained) source$\rightarrow$pivot and pivot$\rightarrow$target models directly, decoding twice via the pivot language [@kauers2002interlingua; @de2006catalan]. One can keep $N$-best hypotheses in the pivot language to reduce the prediction bias [@utiyama2007comparison] and improve the final translation by system combination [@costa2011enhancing], which however increases the translation time even more. In multilingual NMT, modify the second translation step (pivot$\rightarrow$target) to use source and pivot language sentences together as the input. 2. \[sec:pivot-synth\] **Pivot-based synthetic parallel data.** We may translate the pivot side of given pivot-target parallel data using a pivot$\rightarrow$source model [@bertoldi2008phrase], or the other way around translating source-pivot data using a pivot$\rightarrow$target model [@de2006catalan]. For NMT, the former is extended by to compute the expectation over synthetic source sentences. The latter is also called teacher-student approach [@chen2017teacher], where the pivot$\rightarrow$target model (teacher) produces target hypotheses for training the source$\rightarrow$target model (student). 3. **Pivot-based model training.** In phrase-based MT, there have been many efforts to combine phrase/word level features of source-pivot and pivot-target into a source$\rightarrow$target system [@utiyama2007comparison; @wu2007pivot; @bakhshaei2010farsi; @zahabi2013using; @zhu2014improving; @miura2015improving]. In NMT, jointly train for three translation directions of source-pivot-target by sharing network components, where use the expectation-maximization algorithm with the target sentence as a latent variable. deploy intermediate recurrent layers which are common for multiple encoders and decoders, while share all components of a single multilingual model. Both methods train the model for language pairs involving English but enable zero-shot translation for unseen non-English language pairs. For this, encode the target language as an additional embedding and filter out non-target tokens in the output. combine the multilingual training with synthetic data generation to improve the zero-shot performance iteratively, where applies the NMT prediction score and a language model score to each synthetic example as gradient weights. Our work is based on transfer learning [@zoph2016transfer] and belongs to the third category: model training. On the contrary to the multilingual joint training, we suggest two distinct steps: pre-training (with source-pivot and pivot-target data) and fine-tuning (with source-target data). With our proposed methods, we prevent the model from losing its capacity to other languages while utilizing the information from related language pairs well, as shown in the experiments (Section \[sec:results\]). Our pivot adapter (Section \[sec:adapter\]) shares the same motivation with the interlingua component of , but is much compact, independent of variable input length, and easy to train offline. The adapter training algorithm is adopted from bilingual word embedding mapping [@xing2015normalized]. Our cross-lingual encoder (Section \[sec:cross-enc\]) is inspired by cross-lingual sentence embedding algorithms using NMT [@schwenk2017learning; @schwenk2018filtering]. Transfer learning was first introduced to NMT by , where only the source language is switched before/after the transfer. and use shared subword vocabularies to work with more languages and help target language switches. propose additional techniques to enable NMT transfer even without shared vocabularies. To the best of our knowledge, we are the first to propose transfer learning strategies specialized in utilizing a pivot language, transferring a source encoder and a target decoder at the same time. Also, for the first time, we present successful zero-shot translation results only with pivot-based NMT pre-training. Pivot-based Transfer Learning {#sec:methods} ============================= ![Plain transfer learning.[]{data-label="fig:plain"}](plain-transfer.pdf){width="\linewidth"} Our methods are based on a simple transfer learning principle for NMT, adjusted to a usual data condition for non-English language pairs: lots of source-pivot and pivot-target parallel data, little (low-resource) or no (zero-resource) source-target parallel data. Here are the core steps of the plain transfer (Figure \[fig:plain\]): 1. Pre-train a source$\rightarrow$pivot model with a source-pivot parallel corpus and a pivot$\rightarrow$target model with a pivot-target parallel corpus. 2. Initialize the source$\rightarrow$target model with the source encoder from the pre-trained source$\rightarrow$pivot model and the target decoder from the pre-trained pivot$\rightarrow$target model. 3. Continue the training with a source-target parallel corpus. If we skip the last step (for zero-resource cases) and perform the source$\rightarrow$target translation directly, it corresponds to zero-shot translation. Thanks to the pivot language, we can pre-train a source encoder and a target decoder without changing the model architecture or training objective for NMT. On the contrary to other NMT transfer scenarios [@zoph2016transfer; @nguyen2017transfer; @kocmi2018trivial], this principle has no language mismatch between transferor and transferee on each source/target side. Experimental results (Section \[sec:results\]) also show its competitiveness despite its simplicity. Nonetheless, the main caveat of this basic pre-training is that the source encoder is trained to be used by an English decoder, while the target decoder is trained to use the outputs of an English encoder — not of a source encoder. In the following, we propose three techniques to mitigate the inconsistency of source$\rightarrow$pivot and pivot$\rightarrow$target pre-training stages. Note that these techniques are not exclusive and some of them can complement others for a better performance of the final model. Step-wise Pre-training {#sec:step-wise} ---------------------- ![Step-wise pre-training.[]{data-label="fig:step-wise"}](incremental-pre-training.pdf){width="\linewidth"} A simple remedy to make the pre-trained encoder and decoder refer to each other is to train a single NMT model for source$\rightarrow$pivot and pivot$\rightarrow$target in consecutive steps (Figure \[fig:step-wise\]): 1. Train a source$\rightarrow$pivot model with a source-pivot parallel corpus. 2. Continue the training with a pivot-target parallel corpus, while freezing the encoder parameters of 1. In the second step, a target decoder is trained to use the outputs of the pre-trained source encoder as its input. Freezing the pre-trained encoder ensures that, even after the second step, the encoder is still modeling the source language although we train the NMT model for pivot$\rightarrow$target. Without the freezing, the encoder completely adapts to the pivot language input and is likely to forget source language sentences. We build a joint vocabulary of the source and pivot languages so that the encoder effectively represents both languages. The frozen encoder is pre-trained for the source language in the first step, but also able to encode a pivot language sentence in a similar representation space. It is more effective for linguistically similar languages where many tokens are common for both languages in the joint vocabulary. Pivot Adapter {#sec:adapter} ------------- ![Pivot adapter.[]{data-label="fig:adapter"}](pivot-adapter.pdf){width="\linewidth"} Instead of the step-wise pre-training, we can also postprocess the network to enhance the connection between the source encoder and the target decoder which are pre-trained individually. Our idea is that, after the pre-training steps, we adapt the source encoder outputs to the pivot encoder outputs to which the target decoder is more familiar (Figure \[fig:adapter\]). We learn a linear mapping between the two representation spaces with a small source-pivot parallel corpus: 1. Encode the source sentences with the source encoder of the pre-trained source$\rightarrow$pivot model. 2. Encode the pivot sentences with the pivot encoder of the pre-trained pivot$\rightarrow$target model. 3. Apply a pooling to each sentence of 1 and 2, extracting representation vectors for each sentence pair: ($\mathbf{s}$, $\mathbf{p}$). 4. Train a mapping $\mathbf{M}\in\mathbb{R}^{d \times d}$ to minimize the distance between the pooled representations $\mathbf{s}\in\mathbb{R}^{d \times 1}$ and $\mathbf{p}\in\mathbb{R}^{d \times 1}$, where the source representation is first fed to the mapping: $$\begin{aligned} \hat{\mathbf{M}} = \operatorname*{argmin}_{\mathbf{M}} \sum_{\mathbf{s},\mathbf{p}} \|\mathbf{M}\mathbf{s}-\mathbf{p}\|^2 \label{eq:min-vec} \end{aligned}$$ where $d$ is the hidden layer size of the encoders. Introducing matrix notations $\mathbf{S}\in\mathbb{R}^{d \times n}$ and $\mathbf{P}\in\mathbb{R}^{d \times n}$, which concatenate the pooled representations of all $n$ sentences for each side in the source-pivot corpus, we rewrite Equation \[eq:min-vec\] as: $$\begin{aligned} \mathbf{\hat{M}} = \operatorname*{argmin}_{\mathbf{M}} \|\mathbf{M}\mathbf{S}-\mathbf{P}\|^2 \label{eq:min-mtx}\end{aligned}$$ which can be easily computed by the singular value decomposition (SVD) for a closed-form solution, if we put an orthogonality constraint on $\mathbf{M}$ [@xing2015normalized]. The resulting optimization is also called Procrustes problem. The learned mapping is multiplied to encoder outputs of all positions in the final source$\rightarrow$target tuning step. With this mapping, the source encoder emits sentence representations that lie in a similar space of the pivot encoder. Since the target decoder is pre-trained for pivot$\rightarrow$target and accustomed to receive the pivot encoder outputs, it should process the mapped encoder outputs better than the original source encoder outputs. Cross-lingual Encoder {#sec:cross-enc} --------------------- ![Cross-lingual encoder.[]{data-label="fig:cross-lingual"}](cross-lingual-encoder.pdf){width="\linewidth"} As a third technique, we modify the source$\rightarrow$pivot pre-training procedure to force the encoder to have cross-linguality over source and pivot languages; modeling source and pivot sentences in the same mathematical space. We achieve this by an additional autoencoding objective from a pivot sentence to the same pivot sentence (Figure \[fig:cross-lingual\]). The encoder is fed with sentences of both source and pivot languages, which are processed by a shared decoder that outputs only the pivot language. In this way, the encoder is learned to produce representations in a shared space regardless of the input language, since they are used in the same decoder. This cross-lingual space facilitates smoother learning of the final source$\rightarrow$target model, because the decoder is pre-trained to translate the pivot language. The same input/output in autoencoding encourages, however, merely copying the input; it is said to be not proper for learning complex structure of the data domain [@vincent2008extracting]. Denoising autoencoder addresses this by corrupting the input sentences by artificial noises [@hill2016learning]. Learning to reconstruct clean sentences, it encodes linguistic structures of natural language sentences, e.g., word order, better than copying. Here are the noise types we use [@edunov2018understanding]: - Drop tokens randomly with a probability $p_\mathrm{del}$ - Replace tokens with a `<BLANK>` token randomly with a probability $p_\mathrm{rep}$ - Permute the token positions randomly so that the difference between an original index and its new index is less than or equal to $d_\mathrm{per}$ We set $p_\mathrm{del}=0.1$, $p_\mathrm{rep}=0.1$, and $d_\mathrm{per}=3$ in our experiments. The key idea of all three methods is to build a closer connection between the pre-trained encoder and decoder via a pivot language. The difference is in when we do this job: Cross-lingual encoder (Section \[sec:cross-enc\]) changes the encoder pre-training stage (source$\rightarrow$pivot), while step-wise pre-training (Section \[sec:step-wise\]) modifies decoder pre-training stage (pivot$\rightarrow$target). Pivot adapter (Section \[sec:adapter\]) is applied after all pre-training steps. Main Results {#sec:results} ============ We evaluate the proposed transfer learning techniques in two non-English language pairs of WMT 2019 news translation tasks[^1]: French$\rightarrow$German and German$\rightarrow$Czech. **Data** We used the News Commentary v14 parallel corpus and newstest2008-2010 test sets as the source-target training data for both tasks. The newstest sets were oversampled four times. The German$\rightarrow$Czech task was originally limited to unsupervised learning (using only monolingual corpora) in WMT 2019, but we relaxed this constraint by the available parallel data. We used newstest2011 as a validation set and newstest2012/newstest2013 as the test sets. Both language pairs have much abundant parallel data in source-pivot and pivot-target with English as the pivot language. Detailed corpus statistics are given in Table \[tab:corpus-stat\]. **Preprocessing** We used the Moses[^2] tokenizer and applied true-casing on all corpora. For all transfer learning setups, we learned byte pair encoding (BPE) [@sennrich2016neural] for each language individually with 32k merge operations, except for cross-lingual encoder training with joint BPE only over source and pivot languages. This is for modularity of pre-trained models: for example, a French$\rightarrow$English model trained with joint French/English/German BPE could be transferred smoothly to a French$\rightarrow$German model, but would not be optimal for a transfer to e.g., a French$\rightarrow$Korean model. Once we pre-train an NMT model with separate BPE vocabularies, we can reuse it for various final language pairs without wasting unused portion of subword vocabularies (e.g., German-specific tokens in building a French$\rightarrow$Korean model). On the contrary, baselines used joint BPE over all languages with also 32k merges. ----------- ------- ----------- ---------- Words Usage Data Sentences (Source) fr-en 35M 950M en-de 9.1M 170M Fine-tune fr-de 270k 6.9M de-en 9.1M 181M en-cs 49M 658M Fine-tune de-cs 230k 5.1M ----------- ------- ----------- ---------- : Parallel training data statistics.[]{data-label="tab:corpus-stat"} **Model and Training** The 6-layer base Transformer architecture [@vaswani2017attention] was used for all of our experiments. Batch size was set to 4,096 tokens. Each checkpoint amounts to 10k updates for pre-training and 20k updates for fine-tuning. Each model was optimized with Adam [@kingma2014adam] with an initial learning rate of 0.0001, which was multiplied by 0.7 whenever perplexity on the validation set was not improved for three checkpoints. When it was not improved for eight checkpoints, we stopped the training. The NMT model training and transfer were done with the <span style="font-variant:small-caps;">OpenNMT</span> toolkit [@klein-etal-2017-opennmt]. Pivot adapter was trained using the <span style="font-variant:small-caps;">Muse</span> toolkit [@conneau2018word], which was originally developed for bilingual word embeddings but we adjusted for matching sentence representations. ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Direct source$\rightarrow$target 14.8 75.1 16.0 75.1 11.1 81.1 12.8 77.7 Multilingual many-to-many 18.7 71.9 19.5 72.6 14.9 76.6 16.5 73.2 Multilingual many-to-one 18.3 71.7 19.2 71.5 13.1 79.6 14.6 75.8 Plain transfer 17.5 72.3 18.7 71.8 15.4 75.4 18.0 70.9 + Pivot adapter 18.0 71.9 19.1 71.1 15.9 75.0 18.7 70.3 + Cross-lingual encoder 17.4 72.1 18.9 71.8 15.0 75.9 17.6 71.4 + Pivot adapter 17.8 72.3 19.1 71.5 15.6 75.3 18.1 70.8 Step-wise pre-training 18.6 70.7 19.9 70.4 15.6 75.0 18.1 70.9 + Cross-lingual encoder **19.5** **69.8** **20.7** **69.4** **16.2** **74.6** **19.1** **69.9** ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- **Baselines** We thoroughly compare our approaches to the following baselines: 1. *Direct source$\rightarrow$target*: A standard NMT model trained on given source$\rightarrow$target parallel data. 2. *Multilingual*: A single, shared NMT model for multiple translation directions [@johnson2017google]. - *Many-to-many*: Trained for all possible directions among source, target, and pivot languages. - *Many-to-one*: Trained for only the directions *to* target language, i.e., source$\rightarrow$target and pivot$\rightarrow$target, which tends to work better than many-to-many systems [@aharoni2019massively]. In Table \[tab:main\], we report principal results after fine-tuning the pre-trained models using source-target parallel data. As for baselines, multilingual models are better than a direct NMT model. The many-to-many models surpass the many-to-one models; since both tasks are in a low-resource setup, the model gains a lot from related language pairs even if the target languages do not match. Plain transfer of pre-trained encoder/decoder without additional techniques (Figure \[fig:plain\]) shows a nice improvement over the direct baseline: up to +2.7% <span style="font-variant:small-caps;">Bleu</span> for French$\rightarrow$German and +5.2% <span style="font-variant:small-caps;">Bleu</span> for German$\rightarrow$Czech. Pivot adapter provides an additional boost of maximum +0.7% <span style="font-variant:small-caps;">Bleu</span> or -0.7% <span style="font-variant:small-caps;">Ter</span>. Cross-lingual encoder pre-training is proved to be not effective in the plain transfer setup. It shows no improvements over plain transfer in French$\rightarrow$German, and 0.4% <span style="font-variant:small-caps;">Bleu</span> worse performance in German$\rightarrow$Czech. We conjecture that the cross-lingual encoder needs a lot more data to be fine-tuned for another decoder, where the encoder capacity is basically divided into two languages at the beginning of the fine-tuning. On the other hand, the pivot adapter directly improves the connection to an individually pre-trained decoder, which works nicely with small fine-tuning data. Pivot adapter gives an additional improvement on top of the cross-lingual encoder; up to +0.4% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and +0.6% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech. In this case, we extract source and pivot sentence representations from the same shared encoder for training the adapter. Step-wise pre-training gives a big improvement up to +1.2% <span style="font-variant:small-caps;">Bleu</span> or -1.6% <span style="font-variant:small-caps;">Ter</span> against plain transfer in French$\rightarrow$German. It shows the best performance in both tasks when combined with the cross-lingual encoder: up to +1.2% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and +2.6% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech, compared to the multilingual baseline. Step-wise pre-training prevents the cross-lingual encoder from degeneration, since the pivot$\rightarrow$target pre-training (Step 2 in Section \[sec:step-wise\]) also learns the encoder-decoder connection with a large amount of data — in addition to the source$\rightarrow$target tuning step afterwards. Note that the pivot adapter, which inserts an extra layer between the encoder and decoder, is not appropriate after the step-wise pre-training; the decoder is already trained to correlate well with the pre-trained encoder. We experimented with the pivot adapter on top of step-wise pre-trained models — with or without cross-lingual encoder — but obtained detrimental results. Compared to pivot translation (Table \[tab:zero\]), our best results are also clearly better in French $\rightarrow$German and comparable in German$\rightarrow$Czech. Analysis {#sec:analysis} ======== In this section, we conduct ablation studies on the variants of our methods and see how they perform in different data conditions. Pivot Adapter {#pivot-adapter} ------------- ------------------ ---------- ---------- Adapter Training None 18.2 70.7 Max-pooled 18.4 70.5 Average-pooled **18.7** **70.3** Plain transfer 18.0 70.9 ------------------ ---------- ---------- : Pivot adapter variations (German$\rightarrow$Czech). All results are tuned with source-target parallel data.[]{data-label="tab:adapter"} Firstly, we compare variants of the pivot adapter (Section \[sec:adapter\]) in Table \[tab:adapter\]. The row “None” shows that a randomly initialized linear layer already guides the pre-trained encoder/decoder to harmonize with each other. Of course, when we train the adapter to map source encoder outputs to pivot encoder outputs, the performance gets better. For compressing encoder outputs over positions, average-pooling is better than max-pooling. We observed the same trend in the other test set and in French$\rightarrow$German. We also tested nonlinear pivot adapter, e.g., a 2-layer feedforward network with ReLU activations, but the performance was not better than just a linear adapter. Cross-lingual Encoder {#cross-lingual-encoder} --------------------- ------------ ------- ---------- ---------- Trained on Input Clean 15.7 77.7 Noisy 17.5 73.6 Clean 15.9 77.3 Noisy **18.0** **72.7** ------------ ------- ---------- ---------- : Cross-lingual encoder variations (French$\rightarrow$ German). All results are in the zero-shot setting with step-wise pre-training.[]{data-label="tab:cross-enc"} Table \[tab:cross-enc\] verifies that the noisy input in autoencoding is indeed beneficial to our cross-lingual encoder. It improves the final translation performance by maximum +2.1% <span style="font-variant:small-caps;">Bleu</span>, compared to using the copying autoencoding objective. As the training data for autoencoding, we also compare between purely monolingual data and the pivot side of the source-pivot parallel data. By the latter, one can expect a stronger signal for a joint encoder representation space, since two different inputs (in source/pivot languages) are used to produce the exactly same output sentence (in pivot language). The results also tell that there are slight but consistent improvements by using the pivot part of the parallel data. Again, we performed these comparisons in the other test set and German$\rightarrow$Czech, observing the same tendency in results. --------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Multilingual many-to-many 14.1 79.1 14.6 79.1 5.9 - 6.3 99.8 Pivot translation 16.6 72.4 17.9 72.5 16.4 74.5 **19.5** **70.1** Teacher-student 18.7 70.3 20.7 69.5 16.0 75.0 18.5 70.9 Plain transfer 0.1 - 0.2 - 0.1 - 0.1 - Step-wise pre-training 11.0 81.6 11.5 82.5 6.0 92.1 6.5 87.8 + Cross-lingual encoder 17.3 72.1 18.0 72.7 14.1 76.8 16.5 73.5 + Teacher-student **19.3** **69.7** **20.9** **69.3** **16.5** **74.6** 19.1 70.2 --------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Zero-resource/Zero-shot Scenarios --------------------------------- If we do not have an access to any source-target parallel data (*zero-resource*), non-English language pairs have two options for still building a working NMT system, given source-English and target-English parallel data: - *Zero-shot*: Perform source$\rightarrow$target translation using models which have not seen any source-target parallel sentences, e.g., multilingual models or pivoting (Section \[sec:related\].\[sec:pivoting\]). - *Pivot-based synthetic data*: Generate synthetic source-target parallel data using source$\leftrightarrow$English and target$\leftrightarrow$English models (Section \[sec:related\].\[sec:pivot-synth\]). Use this data to train a model for source$\rightarrow$target. Table \[tab:zero\] shows how our pre-trained models perform in zero-resource scenarios with the two options. Note that, unlike Table \[tab:main\], the multilingual baselines exclude source$\rightarrow$target and target$\rightarrow$source directions. First of all, plain transfer, where the encoder and the decoder are pre-trained separately, is poor in zero-shot scenarios. It simply fails to connect different representation spaces of the pre-trained encoder and decoder. In our experiments, neither pivot adapter nor cross-lingual encoder could enhance the zero-shot translation of plain transfer. Step-wise pre-training solves this problem by changing the decoder pre-training to familiarize itself with representations from an already pre-trained encoder. It achieves zero-shot performance of 11.5% <span style="font-variant:small-caps;">Bleu</span> in French$\rightarrow$German and 6.5% <span style="font-variant:small-caps;">Bleu</span> in German$\rightarrow$Czech (newstest2013), while showing comparable or better fine-tuned performance against plain transfer (see also Table \[tab:main\]). With the pre-trained cross-lingual encoder, the zero-shot performance of step-wise pre-training is superior to that of pivot translation in French$\rightarrow$German with only a single model. It is worse than pivot translation in German$\rightarrow$Czech. We think that the data size of pivot-target is critical in pivot translation; relatively huge data for English$\rightarrow$Czech make the pivot translation stronger. Note again that, nevertheless, pivoting (second row) is very poor in efficiency since it performs decoding twice with the individual models. For the second option (pivot-based synthetic data), we compare our methods against the sentence-level beam search version of the teacher-student framework [@chen2017teacher], with which we generated 10M synthetic parallel sentence pairs. We also tried other variants of , e.g., $N$-best hypotheses with weights, but there were no consistent improvements. Due to enormous bilingual signals, the model trained with the teacher-student synthetic data outperforms pivot translation. If tuned with the same synthetic data, our pre-trained model performs even better (last row), achieving the best zero-resource results on three of the four test sets. We also evaluate our best German$\rightarrow$Czech zero-resource model on newstest2019 and compare it with the participants of the WMT 2019 unsupervised news translation task. Ours yield 17.2% <span style="font-variant:small-caps;">Bleu</span>, which is much better than the best single unsupervised system of the winner of the task (15.5%) [@marie-etal-2019-nicts]. We argue that, if one has enough source-English and English-target parallel data for a non-English language pair, it is more encouraged to adopt pivot-based transfer learning than unsupervised MT — even if there is no source-target parallel data. In this case, unsupervised MT unnecessarily restricts the data condition to using only monolingual data and its high computational cost does not pay off; simple pivot-based pre-training steps are more efficient and effective. ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Direct source$\rightarrow$target 20.1 69.8 22.3 68.7 11.1 81.1 12.8 77.7 + Synthetic data 21.1 68.2 22.6 68.1 15.7 76.5 18.5 72.0 Plain transfer 21.8 67.6 23.1 67.5 17.6 73.2 20.3 68.7 + Pivot adapter 21.8 67.6 23.1 67.6 **17.6** **73.0** **20.9** **68.3** + Cross-lingual encoder 21.9 67.7 23.4 67.4 17.5 73.5 20.3 68.7 + Pivot adapter **22.1** **67.5** 23.3 67.5 17.5 73.2 20.6 68.5 Step-wise pre-training 21.8 67.8 23.0 67.8 17.3 73.6 20.0 69.2 + Cross-lingual encoder 21.9 67.6 **23.4** **67.4** 17.5 73.1 20.5 68.6 ---------------------------------- -- ---------- ---------- ---------- ---------- -- ---------- ---------- ---------- ---------- Large-scale Results ------------------- We also study the effect of pivot-based transfer learning in more data-rich scenarios: 1) with large synthetic source-target data (German$\rightarrow$Czech), and 2) with larger real source-target data in combination with the synthetic data (French$\rightarrow$German). We generated synthetic parallel data using pivot-based back-translation [@bertoldi2008phrase]: 5M sentence pairs for German$\rightarrow$Czech and 9.1M sentence pairs for French$\rightarrow$German. For the second scenario, we also prepared 2.3M more lines of French$\rightarrow$German real parallel data from Europarl v7 and Common Crawl corpora. Table \[tab:large\] shows our transfer learning results fine-tuned with a combination of given parallel data and generated synthetic parallel data. The real source-target parallel data are oversampled to make the ratio of real and synthetic data to be 1:2. As expected, the direct source$\rightarrow$target model can be improved considerably by training with large synthetic data. Plain pivot-based transfer outperforms the synthetic data baseline by up to +1.9% <span style="font-variant:small-caps;">Bleu</span> or -3.3% <span style="font-variant:small-caps;">Ter</span>. However, the pivot adapter or cross-lingual encoder gives marginal or inconsistent improvements over the plain transfer. We suppose that the entire model can be tuned sufficiently well without additional adapter layers or a well-curated training process, once we have a large source-target parallel corpus for fine-tuning. Conclusion ========== In this paper, we propose three effective techniques for transfer learning using pivot-based parallel data. The principle is to pre-train NMT models with source-pivot and pivot-target parallel data and transfer the source encoder and the target decoder. To resolve the input/output discrepancy of the pre-trained encoder and decoder, we 1) consecutively pre-train the model for source$\rightarrow$pivot and pivot$\rightarrow$target, 2) append an additional layer after the source encoder which adapts the encoder output to the pivot language space, or 3) train a cross-lingual encoder over source and pivot languages. Our methods are suitable for most of the non-English language pairs with lots of parallel data involving English. Experiments in WMT 2019 French$\rightarrow$German and German$\rightarrow$Czech tasks show that our methods significantly improve the final source$\rightarrow$target translation performance, outperforming multilingual models by up to +2.6% <span style="font-variant:small-caps;">Bleu</span>. The methods are applicable also to zero-resource language pairs, showing a strong performance in the zero-shot setting or with pivot-based synthetic data. We claim that our methods expand the advances in NMT to many more non-English language pairs that are not yet studied well. Future work will be zero-shot translation without step-wise pre-training, i.e., combining individually pre-trained encoders and decoders freely for a fast development of NMT systems for a new non-English language pair. Acknowledgments {#acknowledgments .unnumbered} =============== {width="25.00000%"} {width="20.00000%"} This work has received funding from the European Research Council (ERC) (under the European Union’s Horizon 2020 research and innovation programme, grant agreement No 694537, project “SEQCLAS”) and eBay Inc. The work reflects only the authors’ views and none of the funding agencies is responsible for any use that may be made of the information it contains. [^1]: [^2]:

ArXiv

--- abstract: 'This paper investigates the stochastic permanence of malaria and the existence of a stationary distribution for the stochastic process describing the disease dynamics over sufficiently longtime. The malaria system is highly random with fluctuations from the disease transmission and natural deathrates, which are expressed as independent white noise processes in a family of stochastic differential equation epidemic models. Other sources of variability in the malaria dynamics are the random incubation and naturally acquired immunity periods of malaria. Improved analytical techniques and local martingale characterizations are applied to describe the character of the sample paths of the solution process of the system in the neighborhood of an endemic equilibrium. Emphasis of this study is laid on examination of the impacts of (1) the sources of variability- disease transmission and natural death rates, and (2) the intensities of the white noise processes in the system on the stochastic permanence of malaria, and also on the existence of the stationary distribution for the solution process over sufficiently long time. Numerical simulation examples are presented to illuminate the persistence and stochastic permanence of malaria, and also to numerically approximate the stationary distribution of the states of the solution process.' address: 'Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3042, Statesboro, Georgia, 30460, U.S.A. E-mail:[email protected];[email protected][^1] ' author: - Divine Wanduku title: 'The stochastic permanence of malaria, and the existence of a stationary distribution for a class of malaria models' --- Potential endemic steady state ,permanence in the mean ,Basic reproduction number,Lyapunov functional technique,intensity of white noise process Introduction\[ch1.sec0\] ======================== According to the WHO estimates released in December $2016$, about 212 million cases of malaria occurred in $2015$ resulting in about 429 thousand deaths. In addition, the highest mortality rates were recorded for the sub-Saharan African countries, where about $90\%$ of the global malaria cases occurred, and resulted to about $75\%$ of the global malaria related deaths. Moreover, more than two third of these global malaria related deaths were children younger than or exactly five years old. Furthermore, in spite of the fact that malaria is a curable and preventable disease, and despite all technological advances to control and contain the disease, malaria imposes serious menace to human health and the welfare of many economies in the world. In fact, WHO has reported in $2015$ that nearly half of the world’s population was at risk to malaria, and the disease was actively and continuously transmitted in about $91$ countries in the world. Moreover, the most severely affected economies are the sub-Saharan countries, and the most vulnerable sub-human populations include children younger than five years old, pregnant women, people suffering from HIV/AIDS, and travellers from regions with low malaria transmission to malaria endemic zones[@WHO; @CDC]. These facts serve as motivation to foster research about malaria and understand all aspects of the disease that lead to its containment, and amelioration of the burdens of the disease. Mathematical modeling is one special way of understanding malaria, and malaria models go as far back as 1911 with Ross[@ross] who studied mosquito control. Several other authors such as [@wanduku-biomath; @macdonald; @ngwa-shu; @hyun; @may; @kazeem; @gungala; @anita; @tabo] have also made strides in the understanding of malaria mathematically. Malaria is a vector-borne disease caused by protozoa (a micro-parasitic organism) of the genus *Plasmodium*. There are several different species of the parasite that cause disease in humans namely: *P. falciparum, P. viviax, P. ovale* and *P. malariae*. However, the species that causes the most severe and fatal disease is the *P. falciparum*. Malaria is transmitted between humans by the infectious bite of a female mosquito of the genus *Anopheles*. The complete life cycle of the malaria plasmodium entails two-hosts: (1) the female anopheles mosquito vector, and (2) the susceptible or infectious human being[@malaria; @WHO; @CDC]. The stages of maturation of the plasmodium within the human host is called the *exo-erythrocytic cycle*. Moreover, the total duration of the *exo-erythrocytic cycle* is estimated between 7-30 days depending on the species of plasmodium, with the exceptions of the plasmodia- *P. vivax* and *P. ovale* that may be delayed for as long as 1 to 2 years. See for example [@malaria; @WHO; @CDC]. Also, the stages of development of the plasmodium within the mosquito host is called the *sporogonic cycle*. It is estimated that the duration of the *sporogonic cycle* is over 2 to 3 weeks[@malaria; @WHO; @CDC]. The delay between infection of the mosquito and maturation of the parasite inside the mosquito suggests that the mosquito must survive a minimum of the 2 to 3 weeks to be able to transmit malaria[@malaria]. These facts are important in deriving a mathematical model to represent the dynamics of malaria. More details about the mosquito biting habit, the life cycle of malaria and the key issues related to the mathematical model for malaria in this study are located in Wanduku[@wanduku-biomath], and also in [@malaria; @WHO; @CDC]. It is also important to note that malaria confers natural immunity[@CDC; @lars; @denise] after recovery from the disease. The strength and effectiveness of the natural immunity against the disease depends primarily on the frequency of exposure to the parasites and other biological factors such as age, pregnancy, and genetic structure of red blood cells of people with malaria. The natural immunity against malaria has been studied mathematically by several different authors, for example, [@wanduku-biomath; @hyun]. The duration of the naturally acquired immunity period is random with a range of possible values from zero for individuals with almost no history of the disease (for instance, young babies etc.), to sufficiently long time for people with genetic resistance against the disease( for instance, people with sickle cell anemia, and duffy negative blood type conditions etc.). All of these facts related to the naturally acquired immunity against malaria, and development of the acquired immunity into a mathematical expression are discussed in Wanduku[@wanduku-biomath], and [@CDC; @lars; @denise; @hyun]. As with every other infectious disease dynamics, there is inevitable presence of noise in the dynamics of malaria. Mathematically, the noise in an infectious system over continuous time can be expressed in one way as a Wierner or Brownian motion process obtained as an approximation of a random walk process over an infinitesimally small time interval. Moreover, the central limit theorem is applied to obtain this approximation. There are several different ways to introduce white noise into the infectious system, for example, by considering the variation of the driving parameters of the infectious system, or considering the random perturbation of the density of the system etc. Regardless of the method of introducing the noise into the system, the mathematical systems obtained from the approximation process above are called stochastic differential equation systems. Stochastic systems offer a better representation of reality, and a better fit for most dynamic processes that occur in real life. This is because of the inevitable occurrence of random fluctuations in the dynamic real life systems. Whilst several deterministic systems for malaria dynamics have been studied [@wanduku-biomath; @macdonald; @ngwa-shu; @hyun; @may; @kazeem; @gungala; @anita; @tabo], to the best of the author’s knowledge, little or no mathematical studies authored by other experts exist about malaria in the framework of Ito-Doob type stochastic differential equations. The study by Wanduku[@wanduku-comparative] would appear to lead as the first attempt to understand the impacts of white noise on various aspects of malaria dynamics. And the mode of adding white noise into the malaria dynamics in this study is similar to the earlier studies [@Wanduku-2017; @wanduku-fundamental; @wanduku-delay]. An important investigation in the study of infectious population dynamic systems influenced by white noise is the permanence of the disease, and the existence of a stationary distribution for the infectious system. Several papers in the literature[@aadil; @yanli; @yongli; @yongli-2; @yzhang; @mao-2] have addressed these topics. Investigations about the permanence of the disease in the population seek to find conditions that negatively favor the survival of the endemic population classes (such as the exposed, infectious and removal classes) in the far future time. Moreover, in a white noise influenced infectious system, the permanence of the disease requires the existence of a nonzero average population size for the infectious classes over sufficiently long time. The existence of a stationary distribution for a stochastic infectious system implies that in the far future the statistical properties of the different states of the system can be determined accurately by knowing the distribution of a single random variable, which is the limit of convergence in distribution of the random process describing the dynamics of the disease. Since most realistic stochastic models formulated in terms of stochastic differential equations are nonlinear, and explicit solutions are nontrivial, numerical methods can be used to approximate the stationary distribution for the random process. See for example [@aadil; @yongli-2; @mao-2] Along with the stationary behavior of the stochastic infectious system over sufficiently long time, another topic of investigation concerns the ergodic character of the sample paths of the disease system. The ergodicity of the stochastic disease system ensures that the statistical properties of the disease in the system in the far future time can be understood, and estimated by the sample realizations of disease over sufficiently long time. That is, while insights about the ensemble nature of the disease are difficult to obtain directly from the explicit solutions of the stochastic system because of the nonlinear structure of the system, the stationary and ergodic properties of the stochastic system ensure that sufficient information about the disease is obtained from the sample paths of the disease over sufficiently long time. See for example [@aadil; @yongli-2; @mao-2] Several different studies suggest that the strength or the intensity of the white noise in the infectious system plays a major role on the permanence of the disease, and also on the existence of a stationary distribution for the stochastic system[@aadil; @yanli; @yongli; @yongli-2; @yzhang]. In most of these studies, one can deduce that low intensity of the white noise is associated with the permanence of the disease in the far future time, and consequently lead to the existence of a stationary distribution for the stochastic infectious system. The primary objective of this study is to characterize the role of the intensities of the white noises from different sources in a malaria dynamics on the overall behavior of the disease, and in particular on the permanence of the disease. Furthermore, another objective is to also understand the existence of an endemic stationary distribution, which numerically can be approximated for a given set of parameters corresponding to a malaria scenario. Recently, Wanduku[@wanduku-biomath] presented a class of deterministic models for malaria, where the class type is determined by a generalized nonlinear incidence rate of the disease. The class of epidemic dynamic models incorporates the three delays in the dynamics of malaria from the incubation of the disease inside the mosquito (*sporogonic cycle*), the incubation of the plasmodium inside the human being (*exo-erythrocytic cycle*), and also the period of effective naturally acquired immunity against malaria. Moreover, the delay periods are all random and arbitrarily distributed. Some special cases of the generalized nonlinear incidence rate include (1) a malaria scenario where the response rate of the disease transmission from infectives to susceptible individuals increases initially for small number of infectious individuals, and then saturates with a horizontal asymptote for large and larger number of infectious individuals, (2) a malaria scenario where the response rate of the disease transmission from infectives to susceptible individuals initially decreases, and saturates at a lower horizontal asymptote as the infected population increases, and (3) a malaria scenario where the response rate of the disease transmission from infectives to susceptible individuals initially increases, attains a maximum level and decreases as the number of infected individuals increases etc. Some extensions of Wanduku[@wanduku-biomath] will appear in the context of a general class of vector-borne disease models such as dengue fever and malaria in Wanduku[@wanduku-theorBio], where the role of the different sources of variability on vector-borne diseases are investigated, and their intensities are classified. The focus of [@wanduku-theorBio] is on disease eradication in the steady state population. In the extension Wanduku[@wanduku-comparative], a general class of malaria stochastic models is investigated, and the emphasis is to examine the extend to which the different sources of noise in the system deviate the stochastic dynamics of malaria from its ideal dynamics in the absence of noise. Note that Wanduku[@wanduku-comparative] is a comparative study. The current study extends Wanduku[@wanduku-biomath] by introducing the sources of variability in Wanduku[@wanduku-comparative] with a more detailed formulation of the white noise processes in the malaria dynamics, and provides detailed analytical techniques, biological interpretation and numerical simulation results to comprehend (1) the behavior of the stochastic system in the neighborhood of a potential endemic equilibrium, (2) the stochastic permanence of the disease, and the fundamental role of the intensities of the noises in the system in determining the persistence of the disease, and (3) the existence of a stationary endemic distribution to fully characterize the statistical properties of the states in the system in the long-term. This work is presented as follows:- in Section \[ch1.sec0\], the fundamentals in the derivation of the class of deterministic models for malaria in Wanduku [@wanduku-biomath] are discussed, and essential information to this study is presented. In Section \[ch1.sec0.sec1\], the new class of stochastic models is extensively formulated. In Section \[ch1.sec1\], the model validation results are presented for the stochastic system. In Section \[ch1.sec3\], the persistence of the disease in the human population is exhibited. The permanence of malaria in the mean in the human population is also exhibited in Section \[ch1.sec5\]. Furthermore, the existence of a stationary distribution for the class of stochastic models is presented in Section \[ch1.sec3.sec1\]. Moreover, the ergodicity of the stochastic system is also exhibited in this section. Finally, numerical results are given to test the permanence of malaria, and approximate the stationary distribution of malaria in Section \[ch1.sec4\]. The derivation of the model and some preliminary results {#ch1.sec0} ======================================================== In the recent study by Wanduku [@wanduku-biomath], a class of SEIRS epidemic dynamic models for malaria with three random delays is presented. The delays represent the incubation periods of the infectious agent (plasmodium) inside the vector(mosquito) denoted $T_{1}$, and inside the human host denoted $T_{2}$. The third delay represents the naturally acquired immunity period of the disease $T_{3}$, where the delays are random variables with density functions $f_{T_{1}}, t_{0}\leq T_{1}\leq h_{1}, h_{1}>0$, and $f_{T_{2}}, t_{0}\leq T_{2}\leq h_{2}, h_{2}>0$ and $f_{T_{3}}, t_{0}\leq T_{3}<\infty$. Furthermore, the joint density of $T_{1}$ and $T_{2}$ given by $f_{T_{1},T_{2}}, t_{0}\leq T_{1}\leq h_{1} , t_{0}\leq T_{2}\leq h_{2}$, is also expressed as $f_{T_{1},T_{2}}=f_{T_{1}}.f_{T_{2}}, t_{0}\leq T_{1}\leq h_{1} , t_{0}\leq T_{2}\leq h_{2}$, since it is assumed that the random variables $T_{1}$ and $T_{2}$ are independent (see [@wanduku-biomath]). The independence between $T_{1}$ and $T_{2}$ is justified from the understanding that the incubation of the infectious agent for the vector-borne disease depends on the suitable biological environmental requirements for incubation inside the vector and the human body which are unrelated. Furthermore, the independence between $T_{1}$ and $T_{3}$ follows from the lack of any real biological evidence to justify the connection between the incubation of the infectious agent inside the vector and the acquired natural immunity conferred to the human being. But $T_{2}$ and $T_{3}$ may be dependent as biological evidence suggests that the naturally acquired immunity is induced by exposure to the infectious agent. By employing similar reasoning in [@cooke; @qun; @capasso; @huo], the expected incidence rate of the disease or force of infection of the disease at time $t$ due to the disease transmission process between the infectious vectors and susceptible humans, $S(t)$, is given by the expression $\beta \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}S(t)G(I(t-s))ds$, where $\mu$ is the natural death rate of individuals in the population, and it is assumed for simplicity that the natural death rate for the vectors and human beings are the same. Assuming exponential lifetimes for the people and vectors in the population, the term $0<e^{-\mu s}\leq 1, s\in [t_{0}, h_{1}], h_{1}>0$ represents the survival probability rate of exposed vectors over the incubation period, $T_{1}$, of the infectious agent inside the vectors with the length of the period given as $T_{1}=s, \forall s \in [t_{0}, h_{1}]$, where the vectors acquired infection at the earlier time $t-s$ from an infectious human via a successful infected blood meal, and become infectious at time $t$. Furthermore, it is assumed that the survival of the vectors over the incubation period of length $s\in [t_{0}, h_{1}]$ is independent of the age of the vectors. In addition, $I(t-s)$, is the infectious human population at earlier time $t-s$, $G$ is a nonlinear incidence function of the disease dynamics, and $\beta$ is the average number of effective contacts per infectious individual per unit time. Indeed, the force of infection, $\beta \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}S(t)G(I(t-s))ds$ signifies the expected rate of new infections at time $t$ between the infectious vectors and the susceptible human population $S(t)$ at time $t$, where the infectious agent is transmitted per infectious vector per unit time at the rate $\beta$. Furthermore, it is assumed that the number of infectious vectors at time $t$ is proportional to the infectious human population at earlier time $t-s$. Moreover, it is further assumed that the interaction between the infectious vectors and susceptible humans exhibits nonlinear behavior, for instance, psychological and overcrowding effects, which is characterized by the nonlinear incidence function $G$. Therefore, the force of infection given by $$\label{ch1.sec0.eqn0} \beta \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}S(t)G(I(t-s))ds,$$ represents the expected rate at which infected individuals leave the susceptible state and become exposed at time $t$. The susceptible individuals who have acquired infection from infectious vectors but are non infectious form the exposed class $E$. The population of exposed individuals at time $t$ is denoted $E(t)$. After the incubation period, $T_{2}=u\in [t_{0}, h_{2}]$, of the infectious agent in the exposed human host, the individual becomes infectious, $I(t)$, at time $t$. Applying similar reasoning in [@cooke-driessche], the exposed population, $E(t)$, at time $t$ can be written as follows $$\label{ch1.sec0.eqn1a} E(t)=E(t_{0})e^{-\mu (t-t_{0})}p_{1}(t-t_{0})+\int^{t}_{t_{0}}\beta S(\xi)e^{-\mu T_{1}}G(I(\xi-T_{1}))e^{-\mu(t-\xi)}p_{1}(t-\xi)d \xi,$$ where $$\label{ch1.seco.eqn1b} p_{1}(t-\xi)=\left\{\begin{array}{l}0,t-\xi\geq T_{2},\\ 1, t-\xi< T_{2} \end{array}\right.$$ represents the probability that an individual remains exposed over the time interval $[\xi, t]$. It is easy to see from (\[ch1.sec0.eqn1a\]) that under the assumption that the disease has been in the population for at least a time $t>\max_{t_{0}\leq T_{1}\leq h_{1}, t_{0}\leq T_{2}\leq h_{2}} {( T_{1}+ T_{2})}$, in fact, $t>h_{1}+h_{2}$, so that all initial perturbations have died out, the expected number of exposed individuals at time $t$ is given by $$\label{ch1.sec0.eqn1} E(t)=\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)\int_{t-u}^{t}\beta \int^{h_{1}}_{t_{0}} f_{T_{1}}(s) e^{-\mu s}S(v)G(I(v-s))e^{-\mu(t-u)}dsdvdu.$$ Similarly, for the removal population, $R(t)$, at time $t$, individuals recover from the infectious state $I(t)$ at the per capita rate $\alpha$ and acquire natural immunity. The natural immunity wanes after the varying immunity period $T_{3}=r\in [ t_{0},\infty]$, and removed individuals become susceptible again to the disease. Therefore, at time $t$, individuals leave the infectious state at the rate $\alpha I(t)$ and become part of the removal population $R(t)$. Thus, at time $t$ the removed population is given by the following equation $$\label{ch1.sec0.eqn2a} R(t)=R(t_{0})e^{-\mu (t-t_{0})}p_{2}(t-t_{0})+\int^{t}_{t_{0}}\alpha I(\xi)e^{-\mu(t-\xi)}p_{2}(t-\xi)d \xi,$$ where $$\label{ch1.sec0.eqn2b} p_{2}(t-\xi)=\left\{\begin{array}{l}0,t-\xi\geq T_{3},\\ 1, t-\xi< T_{3} \end{array}\right.$$ represents the probability that an individual remains naturally immune to the disease over the time interval $[\xi, t]$. But it follows from (\[ch1.sec0.eqn2a\]) that under the assumption that the disease has been in the population for at least a time $t> \max_{t_{0}\leq T_{1}\leq h_{1}, t_{0}\leq T_{2}\leq h_{2}, T_{3}\geq t_{0}}{(T_{1}+ T_{2}, T_{3})}\geq \max_{t_{0}\leq T_{3}}{(T_{3})}$, in fact, the disease has been in the population for sufficiently large amount of time so that all initial perturbations have died out, then the expected number of removal individuals at time $t$ can be written as $$\label{ch1.sec0.eqn2} R(t)=\int_{t_{0}}^{\infty}f_{T_{3}}(r)\int_{t-r}^{t}\alpha I(v)e^{-\mu (t-v)}dvdr.$$ There is also constant birth rate $B$ of susceptible individuals in the population. Furthermore, individuals die additionally due to disease related causes at the rate $d$. Moreover, $B>$, $d>0$, and all the other parameters are nonnegative. A compartmental framework illustrating the transition rates between the different states in the system and also showing the delays in the disease dynamics is given in Figure \[ch1.sec4.figure 1\]. ![The compartmental framework illustrates the transition rates between the states $S,E,I,R$ of the system. It also shows the incubation delay $T_{2}$ and the naturally acquired immunity $T_{3}$ periods. \[ch1.sec4.figure 1\]](SEIRS-compartmental-framework-edited-april-5-2017.eps){height="8cm"} It follows from (\[ch1.sec0.eqn0\]), (\[ch1.sec0.eqn1\]), (\[ch1.sec0.eqn2\]) and the transition rates illustrated in the compartmental framework in Figure \[ch1.sec4.figure 1\] above that the family of SEIRS epidemic dynamic models for a malaria and vector-borne diseases in general in the absence of any random environmental fluctuations in the disease dynamics can be written as follows: $$\begin{aligned} dS(t)&=&\left[ B-\beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu S(t)+ \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu r}dr \right]dt,\nonumber\\ &&\label{ch1.sec0.eq3}\\ dE(t)&=& \left[ \beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu E(t)\right.\nonumber\\ &&\left.-\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu \right]dt,\label{ch1.sec0.eq4}\\ &&\nonumber\\ dI(t)&=& \left[\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu- (\mu +d+ \alpha) I(t) \right]dt,\nonumber\\ &&\label{ch1.sec0.eq5}\\ dR(t)&=&\left[ \alpha I(t) - \mu R(t)- \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu s}dr \right]dt,\label{ch1.sec0.eq6}\end{aligned}$$ where the initial conditions are given in the following: Let $h= h_{1}+ h_{2}$ and define $$\begin{aligned} &&\left(S(t),E(t), I(t), R(t)\right) =\left(\varphi_{1}(t),\varphi_{2}(t), \varphi_{3}(t),\varphi_{4}(t)\right), t\in (-\infty,t_{0}],\nonumber\\% t\in [t_{0}-h,t_{0}],\quad and\quad= &&\varphi_{k}\in \mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+}),\forall k=1,2,3,4, \nonumber\\ &&\varphi_{k}(t_{0})>0,\forall k=1,2,3,4,\nonumber\\ \label{ch1.sec0.eq06a}\end{aligned}$$ where $\mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+})$ is the space of continuous functions with the supremum norm $$\label{ch1.sec0.eq06b} ||\varphi||_{\infty}=\sup_{ t\leq t_{0}}{|\varphi(t)|}.$$ The following general properties of the incidence function $G$ studied in [@wanduku-biomath] are given as follows: \[ch1.sec0.assum1\] 1. $G(0)=0$. 2. $G(I)$ is strictly monotonic on $[0,\infty)$. 3. $G''(I)<0$ $\Leftrightarrow$ $G(I)$ is differentiable concave on $[0,\infty)$. 4. $\lim_{I\rightarrow \infty}G(I)=C, 0\leq C<\infty$ $\Leftrightarrow$ $G(I)$ has a horizontal asymptote $0\leq C<\infty$. 5. $G(I)\leq I, \forall I>0$ $\Leftrightarrow$ $G(I)$ is at most as large as the identity function $f:I\mapsto I$ over the positive all $I\in (0,\infty)$. Note that an incidence function $G$ that satisfies Assumption \[ch1.sec0.assum1\] $A1$-$A5$ can be used to describe the disease transmission rate of a vector-borne disease scenario, where the disease dynamics is represented by the system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]), and the disease transmission rate between the vectors and the human beings initially increases or decreases for small values of the infectious population size, and is bounded or steady for sufficiently large size of the infectious individuals in the population. It is noted that Assumption \[ch1.sec0.assum1\] is a generalization of some subcases of the assumptions $A1$-$A5$ investigated in [@gumel; @zhica; @kyrychko; @qun]. Some examples of frequently used incidence functions in the literature that satisfy Assumption \[ch1.sec0.assum1\]$A1$-$A5$ include: $G(I(t))=\frac{I(t)}{1+\alpha I(t)}, \alpha>0$, $G(I(t))=\frac{I(t)}{1+\alpha I^{2}(t)}, \alpha>0$, $G(I(t))=I^{p}(t),0<p<1$ and $G(I)=1-e^{-aI}, a>0$. In the analysis of the deterministic malaria model (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) with initial conditions in (\[ch1.sec0.eq06a\])-(\[ch1.sec0.eq06b\]) in Wanduku[@wanduku-biomath], the threshold values for disease eradication such as the basic reproduction number for the disease when the system is in steady state are obtained in both cases where the delays in the system $T_{1}, T_{2}$ and $T_{3}$ are constant and also arbitrarily distributed. It should be noted that the assumption of constant delay times representing the incubation period of the disease in the vector, $T_{1}$, incubation period of the disease in the host, $T_{2}$, and immunity period of the disease in the human population, $T_{3}$ in (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) is equivalent to the special case of letting the probability density functions $f_{T_{i}}, i=1,2,3$ of the random variables $T_{1}, T_{2}$ and $T_{3}$ be the dirac-delta function. That is, $$\label{ch1.sec2.eq4} f_{T_{i}}(s)=\delta(s-T_{i})=\left\{\begin{array}{l}+\infty, s=T_{i},\\ 0, otherwise, \end{array}\right. , i=1, 2, 3.$$ Moreover, under the assumption that $T_{1}\geq 0, T_{2}\geq 0$ and $T_{3}\geq 0$ are constant, the following expectations can be written as $E(e^{-2\mu (T_{1}+T_{2})})=e^{-2\mu (T_{1}+T_{2})} $, $E(e^{-2\mu T_{1}})=e^{-2\mu T_{1}} $ and $E(e^{-2\mu T_{3}})=e^{-2\mu T_{3}} $. When the delays in the system are all constant, the basic reproduction number of the disease is given by $$\label{ch1.sec2.lemma2a.corrolary1.eq4} \hat{R}^{*}_{0}=\frac{\beta S^{*}_{0} }{(\mu+d+\alpha)}.$$ This threshold value $\hat{R}^{*}_{0}=\frac{\beta S^{*}_{0} }{(\mu+d+\alpha)}$ from (\[ch1.sec2.lemma2a.corrolary1.eq4\]), represents the total number of infectious cases that result from one malaria infectious individual present in a completely disease free population with state given by $S^{*}_{0}=\frac{B}{\mu}$, over the average lifetime given by $\frac{1 }{(\mu+d+\alpha)}$ of a person who has survived from disease related death $d$, natural death $\mu$ and recovered at rate $\alpha$ from infection. Hence, $\hat{R}^{*}_{0}$ is also the noise-free basic reproduction number of the disease, whenever the incubation periods of the malaria parasite inside the human and mosquito hosts given by $T_{i}, i=1,2$, and also the period of effective naturally acquired immunity against malaria given by $T_{3}$, are all positive constants. Furthermore, the threshold condition $\hat{R}^{*}_{0}<1$ is required for the disease to be eradicated from the noise free human population, whenever the constant delays in the system also satisfy the following: $$\label{ch1.sec2.lemma2a.corrolary1.eq7} T_{max}\geq \frac{1}{2\mu}\log{\frac{\hat{R}^{*}_{1}}{1-\hat{R}^{*}_{0}}},$$ where $$\label{ch1.sec2.lemma2a.corrolary1.eq8} T_{max}=\max{(T_{1}+T_{2}, T_{3})},$$ and $$\label{ch1.sec2.lemma2a.corrolary1.eq3} \hat{R}^{*}_{1}=\frac{\beta S^{*}_{0} \hat{K}^{*}_{0}+\alpha}{(\mu+d+\alpha)},$$ with some constant $\hat{K}^{*}_{0}>0$ (in fact, $\hat{K}^{*}_{0}=4 $). On the other hand, when the delays in the system $T_{i}, i=1,2$ are random, and arbitrarily distributed, the basic reproduction number is given by $$\label{ch1.sec2.theorem1.corollary1.eq3} R_{0}=\frac{\beta S^{*}_{0} \hat{K}_{0}}{(\mu+d+\alpha)}+\frac{\alpha}{(\mu+d+\alpha)},$$ where, $\hat{K}_{0}>0$ is a constant that depends only on $S^{*}_{0}$ (in fact, $\hat{K}_{0}=4+ S^{*}_{0} $). In addition, malaria is eradicated from the system in the steady state, whenever $R_{0}\leq 1$, $\hat{U}_{0}\leq 1$ and $\hat{V}_{0}\leq 1$, where $$\label{ch1.sec2.theorem1.corollary1.eq4} \hat{U}_{0}=\frac{2\beta S^{*}_{0}+\beta +\alpha + 2\frac{\mu}{\tilde{K}(\mu)^{2}}}{2\mu},$$ and $$\label{ch1.sec2.theorem1.corollary1.eq5} \hat{V}_{0}=\frac{(2\mu \tilde{K}(\mu)^{2} + \alpha + \beta (2S^{*}_{0}+1 ) )}{2\mu},$$ are other threshold values for the stability of the disease-free steady state $E_{0}=(S^{*}_{0},0,0),S^{*}_{0}=\frac{B}{\mu}$. Note that the threshold value $R_{0}$ in (\[ch1.sec2.theorem1.corollary1.eq3\]) is a modification of the basic reproduction number $\hat{R}^{*}_{0}$ defined in (\[ch1.sec2.lemma2a.corrolary1.eq4\]), and it is therefore the corresponding noise-free basic reproduction number for the disease dynamics described by deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]), whenever the delays $T_{i}, i=1,2,3$ in the system are random variables. See Wanduku[@wanduku-biomath] for more conceptual and biological interpretation of the threshold values for disease eradication. The stochastic extension of the deterministic model (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) is derived and studied in the following section. The stochastic model {#ch1.sec0.sec1} ==================== Stochastic models are more realistic because of the inevitable occurrence of random fluctuations in the dynamics of diseases, and in addition, stochastic models provide a better fit for these disease scenarios than their deterministic counterparts. There are several different techniques to add gaussian noise processes into a dynamic system. One method involves adding the noise into the system as direct influence to the state of the system, where the random fluctuations in the system are, for instance, (1) proportional to the state of the system, or (2) proportional to the deviation of the state of the system from a nonzero steady state etc. Another approach to adding white noise into a dynamic system involves (3) incorporating the random fluctuations in the driving parameters of the system such as the birth, death, recovery and disease transmission rates etc. of an infectious system. See for example [@imf]. In this study, the third approach is utilized to model the random environmental fluctuations in the disease transmission rate $\beta$, and also in the natural death rates $\mu$ of the different states $S(t)$, $E(t)$ , $I(t)$ and $R(t)$ of the human population. This approach entails the construction of a random walk process for the rates $\beta$, and $\mu$ over an infinitesimally small interval $[t, t+dt]$ and applying the central limit theorem. See for example [@wanduku-bookchapter] For $t\geq t_{0}$, let $(\Omega, \mathfrak{F}, P)$ be a complete probability space, and $\mathfrak{F}_{t}$ be a filtration (that is, sub $\sigma$- algebra $\mathfrak{F}_{t}$ that satisfies the following: given $t_{1}\leq t_{2} \Rightarrow \mathfrak{F}_{t_{1}}\subset \mathfrak{F}_{t_{2}}; E\in \mathfrak{F}_{t}$ and $P(E)=0 \Rightarrow E\in \mathfrak{F}_{0} $ ). The variability in the disease transmission and natural death rates are represented by independent white noise or Wiener processes with drifts, and the rates are expressed as follows: $$\label{ch1.sec0.eq7} \mu \rightarrow \mu + \sigma_{i}\xi_{i}(t),\quad \xi_{i}(t) dt= dw_{i}(t),i=S,E,I,R, \quad \beta \rightarrow \beta + \sigma_{\beta}\xi_{\beta}(t),\quad \xi_{\beta}(t)dt=dw_{\beta}(t),$$ where $\xi_{i}(t)$ and $w_{i}(t)$ represent the standard white noise and normalized wiener processes for the $i^{th}$ state at time $t$, with the following properties: $w(0)=0, E(w(t))=0, var(w(t))=t$. Furthermore, $\sigma_{i},i=S,E,I,R $, represents the intensity of the white noise process due to the random natural death rate of the $i^{th}$ state, and $\sigma_{\beta}$ is the intensity of the white noise process due to the random disease transmission rate. Moreover, the $ w_{i}(t),i=S,E,I,R,\beta,\forall t\geq t_{0}$, are all independent. The detailed formulation of the expressions in (\[ch1.sec0.eq7\]) will appear in the book chapter by Wanduku[@wanduku-bookchapter]. The ideas behind the formulation of the expressions in (\[ch1.sec0.eq7\]) are given in the following. The constant parameters $\mu$ and $\beta$ represent the natural death and disease transmission rates per unit time, respectively. In reality, random environmental fluctuations impact these rates turning them into random variables $\tilde{\mu}$ and $\tilde{\beta}$. Thus, the natural death and disease transmission rates over an infinitesimally small interval of time $[t, t+dt]$ with length $dt$ is given by the expressions $\tilde{\mu}(t)=\tilde{\mu}dt$ and $\tilde{\beta}(t)=\tilde{\beta}dt$, respectively. It is assumed that there are independent and identical random impacts acting upon these rates at times $t_{j+1}$ over $n$ subintervals $[t_{j}, t_{j+1}]$ of length $\triangle t=\frac{dt}{n}$, where $t_{j}=t_{0}+j\triangle t, j=0,1,\cdots,n$, and $t_{0}=t$. Furthermore, it is assumed that $\tilde{\mu}(t_{0})=\tilde{\mu}(t)=\mu dt$ is constant or deterministic, and $\tilde{\beta}(t_{0})=\tilde{\beta}(t)=\beta dt$ is also a constant. It follows that by letting the independent identically distributed random variables $Z_{i},i=1,\cdots,n $ represent the random effects acting on the natural death rate, then it follows further that the rate at time $t_{n}=t+dt$, that is, $$\label{ch1.sec0.eq7.eq1} \tilde{\mu}(t+dt)=\tilde{\mu}(t)+\sum_{j=1}^{n}Z_{j},$$ where $E(Z_{j})=0$, and $Var(Z_{j})=\sigma^{2}_{i}\triangle t, i\in \{S, E, I, R\}$. Note that $\tilde{\beta}(t+dt)$ can similarly be expressed as (\[ch1.sec0.eq7.eq1\]). And for sufficient large value of $n$, the summation in (\[ch1.sec0.eq7.eq1\]) converges in distribution by the central limit theorem to a random variable which is identically distributed as the wiener process $\sigma_{i}(w_{i}(t+dt)-w_{i}(t))=\sigma_{i}dw_{i}(t)$, with mean $0$ and variance $\sigma^{2}_{i}dt, i\in \{S, E, I, R\}$. It follows easily from (\[ch1.sec0.eq7.eq1\]) that $$\label{ch1.sec0.eq7.eq2} \tilde{\mu}dt =\mu dt+ \sigma_{i}dw_{i}(t), i\in \{S, E, I, R\}.$$ Similarly, it can be easily seen that $$\label{ch1.sec0.eq7.eq2} \tilde{\beta}dt =\beta dt+ \sigma_{\beta}dw_{\beta}(t).$$ Note that the intensities $\sigma^{2}_{i},i=S,E,I,R, \beta $ of the independent white noise processes in the expressions $\tilde{\mu}(t)=\mu dt + \sigma_{i}\xi_{i}(t)$ and $\tilde{\beta} (t)=\beta dt + \sigma_{\beta}\xi_{\beta}(t)$ that represent the natural death rate, $\tilde{\mu}(t)$, and disease transmission rate, $\tilde{\beta} (t)$, at time $t$, measure the average deviation of the random variable disease transmission rate, $\tilde{\beta}$, and natural death rate, $\tilde{\mu}$, about their constant mean values $\beta$ and $\mu$, respectively, over the infinitesimally small time interval $[t, t+dt]$. These measures reflect the force of the random fluctuations that occur during the disease outbreak at anytime, and which lead to oscillations in the natural death and disease transmission rates overtime, and consequently lead to oscillations of the susceptible, exposed, infectious and removal states of the system over time during the disease outbreak. Thus, in this study the words “strength” and “intensity” of the white noise are used synonymously. Also, the constructions “strong noise” and “weak noise” are used to refer to white noise with high and low intensities, respectively. It is easy to see from (\[ch1.sec0.eq7\]) that the random natural death rate per unit time denoted $\tilde{\mu}$ is given by $\tilde{\mu}=\mu + \sigma_{i}\xi_{i}(t),\quad \xi_{i}(t) dt= dw_{i}(t),i=S,E,I,R,$. It follows further that under the assumption of independent deaths in the human population, so that the number of natural deaths $N(t)$ over an interval $[t_{0}, t_{0}+t]$ of length $t$ follows a Poisson process $\{N(t),t\geq t_{0}\}$ with intensity of the process $E(\tilde{\mu})=\mu$, and mean $E(N(t))=E(\tilde{\mu}t)=\mu t$, then the time until death is exponentially distributed with mean $\frac{1}{\mu}$. Moreover, the survival function is given by $$\label{ch1.sec0.eq7.eq3} \mathfrak{S}(t)=e^{-\mu t},t>0.$$ Substituting (\[ch1.sec0.eq7\])-(\[ch1.sec0.eq7.eq3\]) into the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) leads to the following generalized system of Ito-Doob stochastic differential equations describing the dynamics of vector-borne diseases in the human population. $$\begin{aligned} dS(t)&=&\left[ B-\beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu S(t)+ \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu r}dr \right]dt\nonumber\\ &&-\sigma_{S}S(t)dw_{S}(t)-\sigma_{\beta} S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))dsdw_{\beta}(t) \label{ch1.sec0.eq8}\\ dE(t)&=& \left[ \beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu E(t)\right.\nonumber\\ &&\left.-\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu \right]dt\nonumber\\ &&-\sigma_{E}E(t)dw_{E}(t)+\sigma_{\beta} S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))dsdw_{\beta}(t)\nonumber\\ &&-\sigma_{\beta} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdudw_{\beta}(t)\label{ch1.sec0.eq9}\\ dI(t)&=& \left[\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu- (\mu +d+ \alpha) I(t) \right]dt\nonumber\\ &&-\sigma_{I}I(t)dw_{I}(t)+\sigma_{\beta} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdudw_{\beta}(t)\nonumber\\ &&\label{ch1.sec0.eq10}\\ dR(t)&=&\left[ \alpha I(t) - \mu R(t)- \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu s}dr \right]dt-\sigma_{R}R(t)dw_{R}(t),\label{ch1.sec0.eq11}\end{aligned}$$ where the initial conditions are given in the following: Let $h= h_{1}+ h_{2}$ and define $$\begin{aligned} &&\left(S(t),E(t), I(t), R(t)\right) =\left(\varphi_{1}(t),\varphi_{2}(t), \varphi_{3}(t),\varphi_{4}(t)\right), t\in (-\infty,t_{0}],\nonumber\\% t\in [t_{0}-h,t_{0}],\quad and\quad= &&\varphi_{k}\in \mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+}),\forall k=1,2,3,4, \nonumber\\ &&\varphi_{k}(t_{0})>0,\forall k=1,2,3,4,\nonumber\\ \label{ch1.sec0.eq12}\end{aligned}$$ where $\mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+})$ is the space of continuous functions with the supremum norm $$\label{ch1.sec0.eq13} ||\varphi||_{\infty}=\sup_{ t\leq t_{0}}{|\varphi(t)|}.$$ Furthermore, the random continuous functions $\varphi_{k},k=1,2,3,4$ are $\mathfrak{F}_{0}-measurable$, or independent of $w(t)$ for all $t\geq t_{0}$. In a similar structure to the study [@cooke-driessche], one major advantage of the family of vector-borne disease models (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is the sufficiency in its simplistic nature to provide insights about the vector-borne disease in the human population with limited characterization or limited knowledge of the life cycle of the disease vector. This model provides a suitable platform to study control strategies against the disease with primary focus directed to the human population, whenever there is limited information about the vector life cycle, for instance, in an emergency situation where there is a sudden deadly vector-borne disease outbreak, and there is limited time to investigate the biting habits and life cycles of the vectors. Observe that (\[ch1.sec0.eq9\]) and (\[ch1.sec0.eq11\]) decouple from the other two equations in the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). Nevertheless, for mathematical convenience the results in this paper will be shown for the vector $X(t)=(S(t), E(t), I(t))^{T}$. The following notations are utilized: $$\label{ch1.sec0.eq13b} \left\{ \begin{array}{lll} Y(t)&=&(S(t), E(t), I(t), R(t))^{T} \\ X(t)&=&(S(t), E(t), I(t))^{T} \\ N(t)&=&S(t)+ E(t)+ I(t)+ R(t). \end{array} \right.$$ Model validation \[ch1.sec1\] ============================= The existence and uniqueness of solution of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is exhibited in the following theorem. Moreover, the feasibility region of the the solution process $\{X(t), t\geq t_{0}\}$ of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is defined. The standard methods utilized in the earlier studies[@wanduku-determ; @Wanduku-2017; @wanduku-delay; @divine5] are applied to establish the results. It should be noted that the existence and qualitative behavior of the positive solution process of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) depend on the sources (natural death or disease transmission rates) of variability in the system. As it is shown below, certain sources of variability lead to very complex uncontrolled behavior of the sample paths of the system. The following lemma describes the behavior of the positive local solution process for the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). This result will be useful in establishing the existence and uniqueness results for the global solutions of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). \[ch1.sec1.lemma1\] Suppose for some $\tau_{e}>t_{0}\geq 0$ the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) with initial condition in (\[ch1.sec0.eq12\]) has a unique positive solution process $Y(t)\in \mathbb{R}^{4}_{+}$, for all $t\in (-\infty, \tau_{e}]$, then it follows that if $N(t_{0})\leq \frac{B}{\mu}$, and the intensities of the independent white noise processes in the system satisfy $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, then $N(t)\leq \frac{B}{\mu}$, and in addition, the set denoted by $$\label{ch1.sec1.lemma1.eq1} D(\tau_{e})=\left\{Y(t)\in \mathbb{R}^{4}_{+}: N(t)=S(t)+ E(t)+ I(t)+ R(t)\leq \frac{B}{\mu}, \forall t\in (-\infty, \tau_{e}] \right\}=\bar{B}^{(-\infty, \tau_{e}]}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\right),$$ is locally self-invariant with respect to the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), where $\bar{B}^{(-\infty, \tau_{e}]}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\right)$ is the closed ball in $\mathbb{R}^{4}_{+}$ centered at the origin with radius $\frac{B}{\mu}$ containing the local positive solutions defined over $(-\infty, \tau_{e}]$. If the intensities of the independent white noise processes in the system satisfy $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, then $Y(t)\in \mathbb{R}^{4}_{+}$ and $N(t)\geq 0$, for all $t\in (-\infty, \tau_{e}]$. Proof:\ It follows directly from (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) that when $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, then $$\label{ch1.sec1.lemma1.eq2} dN(t)=[B-\mu N(t)-dI(t)]dt$$ The result in (a.) follows easily by observing that for $Y(t)\in \mathbb{R}^{4}_{+}$, the equation (\[ch1.sec1.lemma1.eq2\]) leads to $N(t)\leq \frac{B}{\mu}-\frac{B}{\mu}e^{-\mu(t-t_{0})}+N(t_{0})e^{-\mu(t-t_{0})}$. And under the assumption that $N(t_{0})\leq \frac{B}{\mu}$, the result follows immediately. The result in (b.) follows directly from Theorem \[ch1.sec1.thm1\].\ The following theorem presents the existence and uniqueness results for the global solutions of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). \[ch1.sec1.thm1\] Given the initial conditions (\[ch1.sec0.eq12\]) and (\[ch1.sec0.eq13\]), there exists a unique solution process $X(t,w)=(S(t,w),E(t,w), I(t,w))^{T}$ satisfying (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), for all $t\geq t_{0}$. Moreover, the solution process is positive for all $t\geq t_{0}$ a.s. and lies in $D(\infty)$, whenever the intensities of the independent white noise processes in the system satisfy $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. That is, $S(t,w)>0,E(t,w)>0, I(t,w)>0, \forall t\geq t_{0}$ a.s. and $X(t,w)\in D(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\right)$, where $D(\infty)$ is defined in Lemma \[ch1.sec1.lemma1\], (\[ch1.sec1.lemma1.eq1\]). Also, the solution process is positive for all $t\geq t_{0}$ a.s. and lies in $\mathbb{R}^{4}_{+}$, whenever the intensities of the independent white noise processes in the system satisfy $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. That is, $S(t,w)>0,E(t,w)>0, I(t,w)>0, \forall t\geq t_{0}$ a.s. and $X(t,w)\in \mathbb{R}^{4}_{+}$. Proof:\ A similar proof of this result appears in a more general study of vector-borne diseases in Wanduku[@wanduku-theorBio], nevertheless it is added here for completion. It is easy to see that the coefficients of (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) satisfy the local Lipschitz condition for the given initial data (\[ch1.sec0.eq12\]). Therefore there exist a unique maximal local solution $X(t,w)=(S(t,w), E(t,w), I(t,w))$ on $t\in (-\infty,\tau_{e}(w)]$, where $\tau_{e}(w)$ is the first hitting time or the explosion time of the process[@mao]. The following shows that $X(t,w)\in D(\tau_{e})$ almost surely, whenever $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, where $D(\tau_{e}(w))$ is defined in Lemma \[ch1.sec1.lemma1\] (\[ch1.sec1.lemma1.eq1\]), and also that $X(t,w)\in \mathbb{R}^{4}_{+}$, whenever $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. Define the following stopping time $$\label{ch1.sec1.thm1.eq1} \left\{ \begin{array}{lll} \tau_{+}&=&sup\{t\in (t_{0},\tau_{e}(w)): S|_{[t_{0},t]}>0,\quad E|_{[t_{0},t]}>0,\quad and\quad I|_{[t_{0},t]}>0 \},\\ \tau_{+}(t)&=&\min(t,\tau_{+}),\quad for\quad t\geq t_{0}.\\ \end{array} \right.$$ and lets show that $\tau_{+}(t)=\tau_{e}(w)$ a.s. Suppose on the contrary that $P(\tau_{+}(t)<\tau_{e}(w))>0$. Let $w\in \{\tau_{+}(t)<\tau_{e}(w)\}$, and $t\in [t_{0},\tau_{+}(t))$. Define $$\label{ch1.sec1.thm1.eq2} \left\{ \begin{array}{ll} V(X(t))=V_{1}(X(t))+V_{2}(X(t))+V_{3}(X(t)),\\ V_{1}(X(t))=\ln(S(t)),\quad V_{2}(X(t))=\ln(E(t)),\quad V_{3}(X(t))=\ln(I(t)),\forall t\leq\tau_{+}(t). \end{array} \right.$$ It follows from (\[ch1.sec1.thm1.eq2\]) that $$\label{ch1.sec1.thm1.eq3} dV(X(t))=dV_{1}(X(t))+dV_{2}(X(t))+dV_{3}(X(t)),$$ where $$\begin{aligned} %% \nonumber % Remove numbering (before each equation) dV_{1}(X(t)) &=& \frac{1}{S(t)}dS(t)-\frac{1}{2}\frac{1}{S^{2}(t)}(dS(t))^{2}\nonumber \\ &=&\left[ \frac{B}{S(t)}-\beta \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu + \frac{\alpha}{S(t)} \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu r}dr \right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{S}-\frac{1}{2}\sigma^{2}_{\beta}\left(\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds\right)^{2}\right]dt\nonumber\\ &&-\sigma_{S}dw_{S}(t)-\sigma_{\beta} \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))dsdw_{\beta}(t), \label{ch1.sec1.thm1.eq4}\end{aligned}$$ $$\begin{aligned} % \nonumber % Remove numbering (before each equation) dV_{2}(X(t)) &=& \frac{1}{E(t)}dE(t)-\frac{1}{2}\frac{1}{E^{2}(t)}(dE(t))^{2} \nonumber\\ &=& \left[ \beta \frac{S(t)}{E(t)}\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds - \mu \right.\nonumber\\ &&\left.-\beta\frac{1}{E(t)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu \right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{E}-\frac{1}{2}\sigma^{2}_{\beta}\frac{S^{2}(t)}{E^{2}(t)}\left(\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))ds\right)^{2}\right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{\beta}\frac{1}{E^{2}(t)}\left(\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu \right)^{2}\right]dt\nonumber\\ &&-\sigma_{E}dw_{E}(t)+\sigma_{\beta} \frac{S(t)}{E(t)}\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(t-s))dsdw_{\beta}(t)\nonumber\\ &&-\sigma_{\beta}\frac{1}{E(t)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdudw_{\beta}(t),\nonumber\\ \label{ch1.sec1.thm1.eq5}\end{aligned}$$ and $$\begin{aligned} % \nonumber % Remove numbering (before each equation) dV_{3}(X(t)) &=& \frac{1}{I(t)}dI(t)-\frac{1}{2}\frac{1}{I^{2}(t)}(dI(t))^{2}\nonumber \\ &=& \left[\beta \frac{1}{I(t)}\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu- (\mu +d+ \alpha)\right. \nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{I}-\frac{1}{2}\sigma^{2}_{\beta}\left(\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdu \right)^{2}\right]dt\nonumber\\ &&-\sigma_{I}dw_{I}(t)+\sigma_{\beta}\frac{1}{I(t)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(t-s-u))dsdudw_{\beta}(t)\nonumber\\ &&\label{ch1.sec1.thm1.eq6}\end{aligned}$$ It follows from (\[ch1.sec1.thm1.eq3\])-(\[ch1.sec1.thm1.eq6\]) that for $t<\tau_{+}(t)$, $$\begin{aligned} % \nonumber % Remove numbering (before each equation) V(X(t))-V(X(t_{0})) &\geq& \int^{t}_{t_{0}}\left[-\beta \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(\xi-s))ds-\frac{1}{2}\sigma^{2}_{S}\right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{\beta}\left(\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(\xi-s))ds\right)^{2}\right]d\xi\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%V_2 %%%%%%%%%%%%%%%%%%%%% &&+ \int_{t}^{t_{0}}\left[-\beta\frac{1}{E(\xi)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(\xi-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(\xi-s-u))dsdu \right.\nonumber\\ %&&\left. \right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{E}-\frac{1}{2}\sigma^{2}_{\beta}\frac{S^{2}(\xi)}{E^{2}(\xi)}\left(\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(\xi-s))ds\right)^{2}\right.\nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{\beta}\frac{1}{E^{2}(\xi)}\left(\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(\xi-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(\xi-s-u))dsdu \right)^{2}\right]d\xi\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V_3 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &&+ \int_{t}^{t_{0}}\left[- (3\mu +d+ \alpha)-\frac{1}{2}\sigma^{2}_{I}\right. \nonumber\\ &&\left.-\frac{1}{2}\sigma^{2}_{\beta}\left(\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(\xi-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(\xi-s-u))dsdu \right)^{2}\right]d\xi\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%Diffussion v_1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &&+\int_{t}^{t_{0}}\left[-\sigma_{S}dw_{S}(\xi)-\sigma_{\beta} \int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(\xi-s))dsdw_{\beta}(\xi)\right]\nonumber \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%V_2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &&+\int_{t}^{t_{0}}\left[-\sigma_{E}dw_{E}(\xi)+\sigma_{\beta} \frac{S(\xi)}{E(\xi)}\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s}G(I(\xi-s))dsdw_{\beta}(\xi)\right]\nonumber\\ &&-\int_{t}^{t_{0}}\left[\sigma_{\beta}\frac{1}{E(\xi)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(\xi-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(\xi-s-u))dsdudw_{\beta}(\xi)\right]\nonumber\\ %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%V_3 &&+\int_{t_{0}}^{t}\left[-\sigma_{I}dw_{I}(\xi)\right.\nonumber\\ &&\left.+\sigma_{\beta}\frac{1}{I(\xi)} \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(\xi-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu s-\mu u}G(I(\xi-s-u))dsdudw_{\beta}(\xi)\right].\nonumber\\ &&\label{ch1.sec1.thm1.eq7} %%%%%%%%%%%%%%%%%%%%%\end{aligned}$$ Taking the limit on (\[ch1.sec1.thm1.eq7\]) as $t\rightarrow \tau_{+}(t)$, it follows from (\[ch1.sec1.thm1.eq1\])-(\[ch1.sec1.thm1.eq2\]) that the left-hand side $V(X(t))-V(X(t_{0}))\leq -\infty$. This contradicts the finiteness of the right-handside of the inequality (\[ch1.sec1.thm1.eq7\]). Hence $\tau_{+}(t)=\tau_{e}(w)$ a.s., that is, $X(t,w)\in D(\tau_{e})$, whenever $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, and $X(t,w)\in \mathbb{R}^{4}_{+}$, whenever $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. The following shows that $\tau_{e}(w)=\infty$. Let $k>0$ be a positive integer such that $||\vec{\varphi}||_{1}\leq k$, where $\vec{\varphi}=\left(\varphi_{1}(t),\varphi_{2}(t), \varphi_{3}(t)\right), t\in (-\infty,t_{0}]$ defined in (\[ch1.sec0.eq12\]), and $||.||_{1}$ is the $p-sum$ norm defined on $\mathbb{R}^{3}$, when $p=1$. Define the stopping time $$\label{ch1.sec1.thm1.eq8} \left\{ \begin{array}{ll} \tau_{k}=sup\{t\in [t_{0},\tau_{e}): ||X(s)||_{1}=S(s)+E(s)+I(s)\leq k, s\in[t_{0},t] \}\\ \tau_{k}(t)=\min(t,\tau_{k}). \end{array} \right.$$ It is easy to see that as $k\rightarrow \infty$, $\tau_{k}$ increases. Set $\lim_{k\rightarrow \infty}\tau_{k}(t)=\tau_{\infty}$. Then it follows that $\tau_{\infty}\leq \tau_{e}$ a.s. We show in the following that: (1.) $\tau_{e}=\tau_{\infty}\quad a.s.\Leftrightarrow P(\tau_{e}\neq \tau_{\infty})=0$, (2.) $\tau_{\infty}=\infty\quad a.s.\Leftrightarrow P(\tau_{\infty}=\infty)=1$. Suppose on the contrary that $P(\tau_{\infty}<\tau_{e})>0$. Let $w\in \{\tau_{\infty}<\tau_{e}\}$ and $t\leq \tau_{\infty}$. Define $$\label{ch1.sec1.thm1.eq9} \left\{ \begin{array}{ll} \hat{V}_{1}(X(t))=e^{\mu t}(S(t)+E(t)+I(t)),\\ \forall t\leq\tau_{k}(t). \end{array} \right.$$ The Ito-Doob differential $d\hat{V}_{1}$ of (\[ch1.sec1.thm1.eq9\]) with respect to the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is given as follows: $$\begin{aligned} % \nonumber % Remove numbering (before each equation) d\hat{V}_{1} &=& \mu e^{\mu t}(S(t)+E(t)+I(t)) dt + e^{\mu t}(dS(t)+dE(t)+dI(t)) \\ &=& e^{\mu t}\left[B+\alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu r}dr-(\alpha + d)I(t)\right]dt\nonumber\\ &&-\sigma_{S}e^{\mu t}S(t)dw_{S}(t)-\sigma_{E}e^{\mu t}E(t)dw_{E}(t)-\sigma_{I}e^{\mu t}I(t)dw_{I}(t)\label{ch1.sec1.thm1.eq10}\end{aligned}$$ Integrating (\[ch1.sec1.thm1.eq9\]) over the interval $[t_{0}, \tau]$, and applying some algebraic manipulations and simplifications it follows that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) V_{1}(X(\tau)) &=& V_{1}(X(t_{0}))+\frac{B}{\mu}\left(e^{\mu \tau}-e^{\mu t_{0}}\right)\nonumber\\ &&+\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}\left(\int_{t_{0}-r}^{t_{0}}\alpha I(\xi)d\xi-\int_{\tau-r}^{\tau}\alpha I(\xi)d\xi\right)dr-\int_{t_{0}}^{\tau}d I(\xi)d\xi \nonumber\\ &&+\int^{\tau}_{t_{0}}\left[-\sigma_{S}e^{\mu \xi}S(\xi)dw_{S}(\xi)-\sigma_{E}e^{\mu \xi}E(\xi)dw_{E}(\xi)-\sigma_{I}e^{\mu \xi}I(\xi)dw_{I}(\xi)\right]\label{ch1.sec1.thm1.eq11}\end{aligned}$$ Removing negative terms from (\[ch1.sec1.thm1.eq11\]), it implies from (\[ch1.sec0.eq12\]) that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) V_{1}(X(\tau)) &\leq& V_{1}(X(t_{0}))+\frac{B}{\mu}e^{\mu \tau}\nonumber\\ &&+\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}\left(\int_{t_{0}-r}^{t_{0}}\alpha \varphi_{3}(\xi)d\xi\right)dr \nonumber\\ &&+\int^{\tau}_{t_{0}}\left[-\sigma_{S}e^{\mu \xi}S(\xi)dw_{S}(\xi)-\sigma_{E}e^{\mu \xi}E(\xi)dw_{E}(\xi)-\sigma_{I}e^{\mu \xi}I(\xi)dw_{I}(\xi)\right]\label{ch1.sec1.thm1.eq12}\end{aligned}$$ But from (\[ch1.sec1.thm1.eq9\]) it is easy to see that for $\forall t\leq\tau_{k}(t)$, $$\label{ch1.sec1.thm1.eq12a} ||X(t)||_{1}=S(t)+E(t)+I(t)\leq V(X(t)).$$ Thus setting $\tau=\tau_{k}(t)$, then it follows from (\[ch1.sec1.thm1.eq8\]), (\[ch1.sec1.thm1.eq12\]) and (\[ch1.sec1.thm1.eq12a\]) that $$\label{ch1.sec1.thm1.eq13} k=||X(\tau_{k}(t))||_{1}\leq V_{1}(X(\tau_{k}(t)))$$ Taking the limit on (\[ch1.sec1.thm1.eq13\]) as $k\rightarrow \infty$ leads to a contradiction because the left-hand-side of the inequality (\[ch1.sec1.thm1.eq13\]) is infinite, but following the right-hand-side from (\[ch1.sec1.thm1.eq12\]) leads to a finite value. Hence $\tau_{e}=\tau_{\infty}$ a.s. The following shows that $\tau_{e}=\tau_{\infty}=\infty$ a.s. Let $\ w\in \{\tau_{e}<\infty\}$. It follows from (\[ch1.sec1.thm1.eq11\])-(\[ch1.sec1.thm1.eq12\]) that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) I_{\{\tau_{e}<\infty\}}V_{1}(X(\tau)) &\leq& I_{\{\tau_{e}<\infty\}}V_{1}(X(t_{0}))+I_{\{\tau_{e}<\infty\}}\frac{B}{\mu}e^{\mu \tau}\nonumber\\ &&+I_{\{\tau_{e}<\infty\}}\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}\left(\int_{t_{0}-r}^{t_{0}}\alpha \varphi_{3}(\xi)d\xi\right)dr\nonumber\\ &&+I_{\{\tau_{e}<\infty\}}\int^{\tau}_{t_{0}}\left[-\sigma_{S}e^{\mu \xi}S(\xi)dw_{S}(\xi)-\sigma_{E}e^{\mu \xi}E(\xi)dw_{E}(\xi)-\sigma_{I}e^{\mu \xi}I(\xi)dw_{I}(\xi)\right]. \nonumber\\ \label{ch1.sec1.thm1.eq14}\end{aligned}$$ Suppose $\tau=\tau_{k}(t)\wedge T$, where $ T>0$ is arbitrary, then taking the expected value of (\[ch1.sec1.thm1.eq14\]) follows that $$\label{ch1.sec1.thm1.eq14a} E(I_{\{\tau_{e}<\infty\}}V_{1}(X(\tau_{k}(t)\wedge T))) \leq V_{1}(X(t_{0}))+\frac{B}{\mu}e^{\mu T}$$ But from (\[ch1.sec1.thm1.eq12a\]) it is easy to see that $$\label{ch1.sec1.thm1.eq15} I_{\{\tau_{e}<\infty,\tau_{k}(t)\leq T\}}||X(\tau_{k}(t))||_{1}\leq I_{\{\tau_{e}<\infty\}}V_{1}(X(\tau_{k}(t)\wedge T))$$ It follows from (\[ch1.sec1.thm1.eq14\])-(\[ch1.sec1.thm1.eq15\]) and (\[ch1.sec1.thm1.eq8\]) that $$\begin{aligned} P(\{\tau_{e}<\infty,\tau_{k}(t)\leq T\})k&=&E\left[I_{\{\tau_{e}<\infty,\tau_{k}(t)\leq T\}}||X(\tau_{k}(t))||_{1}\right]\nonumber\\ &\leq& E\left[I_{\{\tau_{e}<\infty\}}V_{1}(X(\tau_{k}(t)\wedge T))\right]\nonumber\\ &\leq& V_{1}(X(t_{0}))+\frac{B}{\mu}e^{\mu T}. %&&+\sum_{r=1}^{M}\sum_{i=1}^{n_{r}}\sum_{q=1}^{M}\sum_{l=1}^{n_{q}}\int_{0}^{\infty}f^{rq}_{il}(t)\left[\varrho^{q}_{l}\int^{t_{0}}_{-t}\varphi^{rq}_{il2}(s) %e^{\delta^{q}_{l}s}ds\right]dt\nonumber\\ \label{ch1.sec1.thm1.eq16} \end{aligned}$$ It follows immediately from (\[ch1.sec1.thm1.eq16\]) that $P(\{\tau_{e}<\infty,\tau_{\infty}\leq T\})\rightarrow 0$ as $k\rightarrow \infty$. Furthermore, since $T<\infty$ is arbitrary, we conclude that $P(\{\tau_{e}<\infty,\tau_{\infty}< \infty\})= 0$. Finally, by the total probability principle, $$\begin{aligned} P(\{\tau_{e}<\infty\})&=&P(\{\tau_{e}<\infty,\tau_{\infty}=\infty\})+P(\{\tau_{e}<\infty,\tau_{\infty}<\infty\})\nonumber\\ &\leq&P(\{\tau_{e}\neq\tau_{\infty}\})+P(\{\tau_{e}<\infty,\tau_{\infty}<\infty\})\nonumber\\ &=&0.\label{ch1.sec1.thm1.eq17} \end{aligned}$$ Thus from (\[ch1.sec1.thm1.eq17\]), $\tau_{e}=\tau_{\infty}=\infty$ a.s.. In addition, $X(t)\in D(\infty)$, whenever $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, and $X(t,w)\in \mathbb{R}^{4}_{+}$, whenever $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. \[ch1.sec0.remark1\]Theorem \[ch1.sec1.thm1\] and Lemma \[ch1.sec1.lemma1\] signify that the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) has a unique positive solution process $Y(t)\in \mathbb{R}^{4}_{+}$ globally for all $t\in (-\infty, \infty)$. Furthermore, it follows that every positive sample path for the stochastic system that starts in the closed ball centered at the origin with a radius of $\frac{B}{\mu}$, $D(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\right)$, will continue to oscillate and remain bounded in the closed ball for all time $t\geq t_{0}$, whenever the intensities of the independent white noise processes in the system satisfy $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. Hence, the set $D(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\right)$ is a positive self-invariant set, or the feasibility region for the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), whenever $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. In the case where the intensities of the independent white noise processes in the system satisfy $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, the sample path solutions are positive and unique, and continue to oscillate in the unbounded space of positive real numbers $\mathbb{R}^{4}_{+}$. In other words, all positive sample path solutions of the system that start in the bounded region $D(\infty)$, remain bounded for all time, whenever $\sigma_{i}=0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$, and the positive sample paths may become unbounded, whenever $\sigma_{i}>0$, $i\in \{S, E, I\}$ and $\sigma_{\beta}\geq 0$. The implication of this result to the disease dynamics represented by (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is that the occurrence of noise exclusively from the disease transmission rate allows a controlled situation for the disease dynamics, since the positive solutions exist within a well-defined positive self invariant space. The additional source of variability from the natural death rate of any of the different disease classes (susceptible, exposed, infectious or removed), may lead to more complex and uncontrolled situations for the disease dynamics, since it is obvious that the intensities of the white noise processes from the natural death rates of the different states in the system may drive the positive sample path solutions of the system unbounded. Some examples of uncontrolled disease situations that can occur when the positive solutions are unbounded include:- (1) extinction of the population, (2) failure of existence of an infection-free steady population state, wherein the disease can be controlled into the state, and (3) a sudden significant random rise or drop of a given state, such as the infectious state, from a low to high value, or vice versa over a short time period etc. It is shown in the subsequent sections that the stronger noise in the system tends to enhance the persistence of the disease, and possible eventual extinction of the human population. Stochasticity of the endemic equilibrium and persistence of disease\[ch1.sec3\] =============================================================================== From a probabilistic perspective, the stochastic asymptotic stability (in the sense of Lyapunov) of an endemic equilibrium $E_{1}$, whenever it exists, ensures that every sample path for the stochastic system that starts in the neighborhood of the steady state $E_{1}$, has a high probability of oscillating in the neighborhood of the state, and eventually converges to that steady state, almost surely. The biological significance of the stochastic stability of the endemic equilibrium $E_{1}$, whenever it exists is that, there exists a steady state for all disease related states in the population (exposed, infectious and removed), denoted $E_{1}$, where all sample paths for the disease related states that start in the neighborhood of the state $E_{1}$, have a high probability of oscillating in the neighborhood of the state $E_{1}$, and eventually converge to that steady state in the definite sense. In other words, in the long term, there is certainty of an endemic population, which persists the disease. Epidemiologists require the basic reproduction numbers $R^{*}_{0}$ or $R_{0}$ defined in (\[ch1.sec2.lemma2a.corrolary1.eq4\]) and (\[ch1.sec2.theorem1.corollary1.eq3\]), respectively, to satisfy the conditions $R^{*}_{0}>1$ or $R_{0}\geq 1$ for the disease to persist.These facts are discussed in this section, and examples provided to substantiate the results. It is easy to see that the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) does not have a nontrivial or endemic steady state, whenever at least one of the intensities of the independent white noise processes in the system $\sigma_{i}>0, i=S, E, I, R, \beta$. Nevertheless, when the intensities of the noises of the system are infinitesimally small, that is, $\sigma_{i}=0, i=S, E, I, R, \beta$, the resulting system behaves approximately in the same manner as the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]), which has an endemic equilibrium $E_{1}$ studied in [@wanduku-biomath]. Thus, in this section, the asymptotic behavior of the sample paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) in the neighborhood of the potential endemic steady state, denoted $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$, is exhibited. The following results are quoted from [@wanduku-biomath] about sufficient conditions for the existence of the endemic equilibrium of the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]). \[ch1.sec3.thm1\] Suppose the threshold condition $R_{0}>1$ is satisfied, where $R_{0}$ is defined in (\[ch1.sec2.theorem1.corollary1.eq3\]). It follows that when the delays in the system namely $T_{i}, i=1, 2, 3$ are random, and arbitrarily distributed, then the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) has a unique positive equilibrium state denoted by $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$, whenever $$\label{ch1.sec3.thm1.eq1} E(e^{-\mu(T_{1}+T_{2})})\geq \frac{\hat{K}_{0}+\frac{\alpha}{\beta \frac{B}{\mu}}}{G'(0)},$$ where $\hat{K}_{0}$ is also defined in (\[ch1.sec2.theorem1.corollary1.eq3\]). Proof:\ See[@wanduku-biomath]. \[ch1.sec3.thm1.corrolary1\] Suppose the incubation periods of the malaria plasmodium inside the mosquito and human hosts $T_{1}$ and $T_{2}$, and also the period of effective natural immunity against malaria inside the human being $T_{3}$ are constant. Let the threshold condition $R^{*}_{0}>1$ be satisfied, where $R^{*}_{0}$ is defined in (\[ch1.sec2.lemma2a.corrolary1.eq4\]). It follows that the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) has a unique positive equilibrium state denoted by $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$, whenever $$\label{ch1.sec3.thm1.corrolary1.eq1} T_{1}+T_{2}\leq\frac{1}{\mu}\log{(G^{'}(0))}.%E(e^{-\mu(T_{1}+T_{2})})\geq \frac{\hat{K}_{0}+\frac{\alpha}{\beta \frac{B}{\mu}}}{G'(0)},$$ Proof:\ See[@wanduku-biomath]. It should be noted from Assumption \[ch1.sec0.assum1\] that the nonlinear function $G$ is bounded. Therefore, suppose $$\label{ch1.sec3.rem1.eqn1} G^{*}=\sup_{x>0}{G(x)},$$ then it is easy to see that $0\leq G(x)\leq G^{*},\forall x>0$. It follows further from Assumption \[ch1.sec0.assum1\] that given $\lim _{I\rightarrow \infty}{G(I)}=C$, if $G$ is strictly monotonic increasing then $G^{*}\leq C$. Also, if $G$ is strictly monotonic decreasing then $G^{*}\geq C$. It easy to see that when the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) is perturbed by the noise in the system from at least one of the different sources- natural death or disease transmission rates, that is, whenever at least one of $\sigma_{i}> 0,i=S,E,I,R,\beta$, then the nontrivial steady state $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$ ceases to exist for the resulting perturbed system from (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]). It is important to understand the extend to which the sample paths are deviated from the endemic steady state $E_{1}$, under the influence of the noises in the system. The following lemma will be utilized to prove the results that characterize the asymptotic behavior of the sample paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) in the neighborhood of the nontrivial steady state $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$, whenever at least one of $\sigma_{i} \neq 0,i=S,E,I,R,\beta$. \[ch1.sec3.lemma1\]Let the hypothesis of Theorem \[ch1.sec3.thm1\] be satisfied and define the $\mathcal{C}^{2,1}-$ function $V:\mathbb{R}^{3}_{+}\times \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ where $$\label{ch1.sec3.lemma1.eq1} V(t)=V_{1}(t)+V_{2}(t)+V_{3}(t) +V_{4}(t),$$ where, $$\label{ch1.sec3.lemma1.eq2} V_{1}(t)=\frac{1}{2}\left(S(t)-S^{*}_{1}+E(t)-E^{*}_{1}+I(t)-I^{*}_{1}\right)^{2},$$ $$\label{ch1.sec3.lemma1.eq3} V_{2}(t)=\frac{1}{2}\left(S(t)-S^{*}_{1}\right)^{2}$$ $$\label{ch1.sec3.lemma1.eq4} V_{3}(t)=\frac{1}{2}\left(S(t)-S^{*}_{1}+E(t)-E^{*}_{1}\right)^{2}.$$ and $$\begin{aligned} V_{4}(t)&=&\frac{3}{2}\frac{\alpha}{\lambda(\mu)}\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-2\mu r}dr(I(\theta)-I^{*})^{2}d\theta dr\nonumber\\ &&+\frac{\beta S^{*}_{1}}{\lambda(\mu)}(G'(I^{*}_{1}))^{2} \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}\int^{t}_{t-(s+u)}(I(\theta)-I^{*}_{1})^{2}d\theta dsdu\nonumber\\ %%%%% %&&+\frac{\beta }{\lambda(\mu)}(G'(I^{*}_{1}))^{2} \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}\int^{t}_{t-(s+u)}(I(\theta)-I^{*}_{1})^{2}d\theta %dsdu\nonumber\\ %%%%% &&+[\frac{\beta \lambda(\mu)}{2} \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}\int^{t}_{t-(s+u)}G^{2}(I(\theta))(S(\theta)-S^{*}_{1})^{2}d\theta dsdu\nonumber\\ %%%%% %%%%% &&+\sigma^{2}_{\beta} \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}\int^{t}_{t-(s+u)}G^{2}(I(\theta))(S(\theta)-S^{*}_{1})^{2}d\theta dsdu,\nonumber\\\label{ch1.sec3.lemma1.eq4b}\end{aligned}$$ where $\lambda(\mu)>0$ is a real valued function of $\mu$. Suppose $\tilde{\phi}_{1}$, $\tilde{\psi}_{1}$ and $\tilde{\varphi}_{1}$ are defined as follows $$\begin{aligned} % \nonumber % Remove numbering (before each equation) \tilde{\phi}_{1}&=&3\mu-\left[2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+3\sigma ^{2}_{S}+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)E(e^{-2\mu T_{1}})\right.\nonumber\\ &&\left.+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}+\sigma^{2}_{\beta}(G^{*})^{2}\right)E(e^{-2\mu (T_{1}+T_{2})})\right]\label{ch1.sec3.lemma1.eq5a}\\ \tilde{\psi}_{1} &=& 2\mu-\left[\frac{\beta }{2\lambda{(\mu)}}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+ \frac{2\mu}{\lambda{(\mu)}}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha \lambda{(\mu)}+ 2 \sigma^{2}_{E} \right]\label{ch1.sec3.lemma1.eq5b}\\ \tilde{\varphi}_{1}&=& (\mu + d+\alpha)-\left[(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} + \sigma^{2}_{I}+\frac{3\alpha}{2\lambda{(\mu)}}E(e^{-2\mu T_{3}})\right.\nonumber\\ &&\left. +\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)E(e^{-2\mu (T_{1}+T_{2})})\right].\label{ch1.sec3.lemma1.eq5c} \end{aligned}$$ The differential operator $dV$ applied to $V(t)$ with respect to the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) can be written as follows: $$\label{ch1.sec3.lemma1.eq5} dV=LV(t)dt + \overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)},$$ where for $\overrightarrow{w(t)}=(w_{S},w_{E}, w_{I}, w_{\beta})^{T}$ and the function $(S(t), E(t), I(t))\mapsto g(S(t), E(t), I(t))$, is defined as follows: $$\begin{aligned} &&\overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)}= -\sigma_{S}(3(S(t)-S^{*}_{1})+2(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})S(t)dw_{S}(t)\nonumber\\ &&-\sigma_{E}(2(S(t)-S^{*}_{1})+2(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})E(t)dw_{E}(t)\nonumber\\ &&-\sigma_{I}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})I(t)dw_{I}(t)\nonumber\\ &&-\sigma_{\beta}(S(t)-S^{*}_{1})S(t)\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))dsdw_{\beta}(t)\nonumber\\ &&-\sigma_{\beta}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}S(t-u)G(I(t-s-u))dsdudw_{\beta}(t),\nonumber\\\label{ch1.sec3.lemma1.eq6} \end{aligned}$$ and $LV$ satisfies the following inequality $$\begin{aligned} LV(t)&\leq& -\left\{\tilde{\phi}_{1}(S(t)-S^{*}_{1})^{2}+\tilde{\psi}_{1}(E(t)-E^{*}_{1})^{2}+\tilde{\varphi}_{1}(I(t)-I^{*}_{1})^{2}\right\}\nonumber\\ &&+3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}}).\nonumber\\\label{ch1.sec3.lemma1.eq7} \end{aligned}$$ Proof\ From (\[ch1.sec3.lemma1.eq2\])-(\[ch1.sec3.lemma1.eq4\]) the derivative of $V_{1}$, $V_{2}$ and $V_{3}$ with respect to the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) can be written in the form: $$\begin{aligned} dV_{1}&=&LV_{1}dt -\sigma_{S}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})S(t)dw_{S}(t)\nonumber\\ &&-\sigma_{E}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})E(t)dw_{E}(t)\nonumber\\ &&-\sigma_{I}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1})+I(t)-I^{*}_{1})I(t)dw_{I}(t),\nonumber\\\label{ch1.sec3.lemma1.proof.eq1}\end{aligned}$$ $$\begin{aligned} dV_{2}&=&LV_{2}dt -\sigma_{S}((S(t)-S^{*}_{1}))S(t)dw_{S}(t)\nonumber\\ &&-\sigma_{\beta}(S(t)-S^{*}_{1})S(t)\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))dsdw_{\beta}(t)\nonumber\\\label{ch1.sec3.lemma1.proof.eq2}\end{aligned}$$ and $$\begin{aligned} dV_{3}&=&LV_{3}dt -\sigma_{S}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))S(t)dw_{S}(t)-\sigma_{S}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))E(t)dw_{E}(t)\nonumber\\ &&-\sigma_{\beta}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}S(t-u)G(I(t-s-u))dsdudw_{\beta}(t),\nonumber\\\label{ch1.sec3.lemma1.proof.eq3}\end{aligned}$$where utilizing (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]), $LV_{1}$, $LV_{2}$ and $LV_{3}$ can be written as follows: $$\begin{aligned} LV_{1}(t)&=&-\mu (S(t)-S^{*}_{1})^{2}-\mu (E(t)-E^{*}_{1})^{2}-(\mu + d+ \alpha) (I(t)-I^{*}_{1})^{2}\nonumber\\ &&-2\mu (S(t)-S^{*}_{1})(E(t)-E^{*}_{1})-(2\mu + d+ \alpha) (S(t)-S^{*}_{1})(I(t)-I^{*}_{1})\nonumber\\ &&-(2\mu + d+ \alpha) (E(t)-E^{*}_{1})(I(t)-I^{*}_{1})\nonumber\\ &&+\alpha((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1})+(I(t)-I^{*}_{1}))\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}(I(t-r)-I^{*}_{1})dr\nonumber\\ &&+\frac{1}{2}\sigma^{2}_{S}S^{2}(t)+\frac{1}{2}\sigma^{2}_{E}E^{2}(t)+\frac{1}{2}\sigma^{2}_{I}I^{2}(t),\label{ch1.sec3.lemma1.proof.eq4}\end{aligned}$$ $$\begin{aligned} LV_{2}(t)&=&-\mu (S(t)-S^{*}_{1})^{2}+\alpha(S(t)-S^{*}_{1})\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}(I(t-r)-I^{*}_{1})dr\nonumber\\ &&-\beta(S(t)-S^{*}_{1})^{2}\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))ds\nonumber\\ &&-\beta S^{*}_{1}(S(t)-S^{*}_{1})\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}(G(I(t-s))-G(I^{*}_{1}))ds\nonumber\\ &&+\frac{1}{2}\sigma^{2}_{S}S^{2}(t)+\frac{1}{2}\sigma^{2}_{\beta}S^{2}(t)\left(\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))ds\right)^{2},\label{ch1.sec3.lemma1.proof.eq5}\end{aligned}$$ and $$\begin{aligned} LV_{3}(t)&=&-\mu (S(t)-S^{*}_{1})^{2}-\mu (E(t)-E^{*}_{1})^{2}-2\mu (S(t)-S^{*}_{1})(E(t)-E^{*}_{1})\nonumber\\ &&+\alpha((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}(I(t-r)-I^{*}_{1})dr\nonumber\\ &&-\beta(S(t)-S^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(S(t-u)-S^{*}_{1})G(I(t-s-u))dsdu\nonumber\\ &&-\beta(E(t)-E^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(S(t-u)-S^{*}_{1})G(I(t-s-u))dsdu\nonumber\\ &&-\beta S^{*}_{1}(S(t)-S^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(G(I(t-s-u))-G(I^{*}_{1}))dsdu\nonumber\\ &&-\beta S^{*}_{1}(E(t)-E^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(G(I(t-s-u))-G(I^{*}_{1}))dsdu\nonumber\\ %%%%%%%%%&& &&+\frac{1}{2}\sigma^{2}_{S}S^{2}(t)+\frac{1}{2}\sigma^{2}_{E}E^{2}(t)\nonumber\\ %%%%%%%%%&& &&+\frac{1}{2}\sigma^{2}_{\beta}\left(\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}S(t-u)G(I(t-s-u))dsdu\right)^{2}.\label{ch1.sec3.lemma1.proof.eq6}\end{aligned}$$ From (\[ch1.sec3.lemma1.proof.eq4\])-(\[ch1.sec3.lemma1.proof.eq6\]), the set of inequalities that follow will be used to estimate the sum $LV_{1}(t)+LV_{2}(t)+LV_{3}(t)$. That is, applying $Cauchy-Swartz$ and $H\ddot{o}lder$ inequalities, and also applying the algebraic inequality $$\label{ch2.sec2.thm2.proof.eq2*} 2ab\leq \frac{a^{2}}{g(c)}+b^{2}g(c)$$ where $a,b,c\in \mathbb{R}$, and the function $g$ is such that $g(c)> 0$, the terms associated with the integral term (sign) $\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}(I(t-r)-I^{*}_{1})dr$ are estimated as follows: $$\label{ch1.sec3.lemma1.proof.eq7} (a(t)-a^{*})\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-\mu r}(I(t-r)-I^{*}_{1})dr\leq \frac{\lambda(\mu)}{2}(a-a^{*})^{2} + \frac{1}{2\lambda(\mu)}\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-2\mu r}(I(t-r)-I^{*}_{1})^{2}dr,$$ where $a(t)\in \{S(t), E(t), I(t)\}$ and $a^{*}\in \{S^{*}_{1}, E^{*}_{1}, I^{*}_{1}\}$. Furthermore, the terms with the integral sign that depend on $G(I(t-s))$ and $G(I(t-s-u))$ are estimated as follows: $$\begin{aligned} &&-\beta(S(t)-S^{*}_{1})^{2}\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))ds \leq \frac{\beta\lambda(\mu)}{2}(S(t)-S^{*}_{1})^{2}\nonumber\\ &&+\frac{\beta}{2\lambda(\mu)}(S(t)-S^{*}_{1})^{2}\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-2\mu s}G^{2}(I(t-s))ds.\nonumber\\ && -\beta(E(t)-E^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(S(t-u)-S^{*}_{1})G(I(t-s-u))dsdu\leq \frac{\beta}{2\lambda(\mu)}(E(t)-E^{*}_{1})^{2} \nonumber\\ &&+\frac{\beta\lambda(\mu)}{2}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}(S(t-u)-S^{*}_{1})^{2}G^{2}(I(t-s-u))dsdu. \label{ch1.sec3.lemma1.proof.eq8} \end{aligned}$$ The terms with the integral sign that depend on $G(I(t-s))-G(I^{*}_{1})$ and $G(I(t-s-u))-G(I^{*}_{1})$ are estimated as follows: $$\begin{aligned} &&-\beta S^{*}_{1}(S(t)-S^{*}_{1})\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}(G(I(t-s))-G(I^{*}_{1}))ds\leq \frac{\beta S^{*}_{1}\lambda(\mu)}{2}(S(t)-S^{*}_{1})^{2}\nonumber\\ && +\frac{\beta S^{*}_{1}}{2\lambda(\mu)}\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-2\mu s}(I(t-s)-I^{*}_{1})^{2}\left(\frac{G(I(t-s))-G(I^{*}_{1})}{I(t-s)-I^{*}_{1}}\right)^{2}ds\nonumber\\ &&\leq \frac{\beta S^{*}_{1}\lambda(\mu)}{2}(S(t)-S^{*}_{1})^{2}\nonumber\\ && +\frac{\beta S^{*}_{1}}{2\lambda(\mu)}\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-2\mu s}(I(t-s)-I^{*}_{1})^{2}\left(G'(I^{*}_{1})\right)^{2}ds.\nonumber\\ &&-\beta S^{*}_{1}(E(t)-E^{*}_{1})\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}(G(I(t-s-u))-G(I^{*}_{1}))dsdu\leq \frac{\beta S^{*}_{1}\lambda(\mu)}{2}(E(t)-E^{*}_{1})^{2}\nonumber\\ && +\frac{\beta S^{*}_{1}}{2\lambda(\mu)}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu s}(I(t-s-u)-I^{*}_{1})^{2}\left(\frac{G(I(t-s-u))-G(I^{*}_{1})}{I(t-s-u)-I^{*}_{1}}\right)^{2}ds\nonumber\\ &&\leq \frac{\beta S^{*}_{1}\lambda(\mu)}{2}(E(t)-E^{*}_{1})^{2}\nonumber\\ && +\frac{\beta S^{*}_{1}}{2\lambda(\mu)}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu s}(I(t-s-u)-I^{*}_{1})^{2}\left(G'(I^{*}_{1})\right)^{2}ds,\nonumber\\\label{ch1.sec3.lemma1.proof.eq9}\end{aligned}$$ where the inequality in (\[ch1.sec3.lemma1.proof.eq9\]) follows from Assumption \[ch1.sec0.assum1\]. That is, $G$ is a differentiable monotonic function with $G''(I)<0$, and consequently, $0< \frac{G(I)-G(I^{*}_{1})}{(I-I^{*}_{1})}\leq G'(I^{*}_{1}), \forall I>0$. By employing the $Cauchy-Swartz$ and $H\ddot{o}lder$ inequalities, and also applying the following algebraic inequality $(a+b)^{2}\leq 2a^{2}+ 2b^{2}$, the last set of terms with integral signs on (\[ch1.sec3.lemma1.proof.eq5\])-(\[ch1.sec3.lemma1.proof.eq6\]) are estimated as follows: $$\begin{aligned} &&\frac{1}{2}\sigma^{2}_{\beta}S^{2}(t)\left(\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))ds\right)^{2}\leq \sigma^{2}_{\beta}\left((S(t)-S^{*}_{1})^{2}+(S^{*}_{1})^{2}\right)\times\nonumber\\ &&\times\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-2\mu s}G^{2}(I(t-s))ds.\nonumber\\ &&\frac{1}{2}\sigma^{2}_{\beta}\left(\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu (s+u)}S(t-u)G(I(t-s-u))dsdu\right)^{2}\leq \sigma^{2}_{\beta}\times\nonumber\\ &&\times\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}\left((S(t-u)-S^{*}_{1})^{2}+(S^{*}_{1})^{2}\right)G^{2}(I(t-s-u))dsdu.\nonumber\\\label{ch1.sec3.lemma1.proof.eq10}\end{aligned}$$ By further applying the algebraic inequality (\[ch2.sec2.thm2.proof.eq2\*\]) and the inequalities (\[ch1.sec3.lemma1.proof.eq7\])-(\[ch1.sec3.lemma1.proof.eq10\]) on the sum $LV_{1}(t)+LV_{2}(t)+LV_{3}(t)$, it is easy to see from (\[ch1.sec3.lemma1.proof.eq4\])-(\[ch1.sec3.lemma1.proof.eq6\]) that $$\begin{aligned} &&LV_{1}(t)+LV_{2}(t)+LV_{3}(t)\leq (S(t)-S^{*}_{1})^{2}\left[-3\mu +2\mu \lambda(\mu) +(2\mu+ d+\alpha)\frac{\lambda(\mu)}{2}+\alpha\lambda(\mu)\right.\nonumber\\ &&\left.+ \frac{\beta\lambda(\mu)}{2}+\frac{\beta S^{*}_{1}\lambda(\mu)}{2}+ \frac{\beta}{2\lambda(\mu)}(G^{*})^{2}E(e^{-2\mu T_{1}})+ 3\sigma^{2}_{S}+\sigma^{2}_{S} (G^{*})^{2}E(e^{-2\mu T_{1}})\right]\nonumber\\ &&(E(t)-E^{*}_{1})^{2}\left[-2\mu +\frac{2\mu }{\lambda(\mu)} +(2\mu+ d+\alpha)\frac{\lambda(\mu)}{2}+\alpha\lambda(\mu)+ \frac{\beta}{2\lambda(\mu)}+\frac{\beta S^{*}_{1}\lambda(\mu)}{2}+ 2\sigma^{2}_{E}\right]\nonumber\\ %%%%%\right.\nonumber\\&&\left. &&+(I(t)-I^{*}_{1})^{2}\left[-(\mu+d+\alpha) +(2\mu+ d+\alpha)\frac{1}{\lambda(\mu)}+\frac{\alpha\lambda(\mu)}{2}+ \frac{\beta}{2\lambda(\mu)}+\frac{\beta S^{*}_{1}\lambda(\mu)}{2}+ \sigma^{2}_{I}\right]\nonumber\\ %\right.\nonumber\\ %&&\left. &&+\frac{3\alpha}{2\lambda(\mu)}\int_{t_{0}}^{\infty}f_{T_{3}}(r)e^{-2\mu r}(I(t-r)-I^{*}_{1})^{2}dr\nonumber\\ &&+\frac{\beta S^{*}_{1}}{\lambda(\mu)}(G'(I^{*}_{1}))^{2}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}(I(t-s-u)-I^{*}_{1})^{2}dsdu\nonumber\\ &&+\frac{\beta \lambda(\mu)}{2}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}G^{2}(I(t-s-u))(S(t-s)-S^{*}_{1})^{2}dsdu\nonumber\\ &&+3\sigma^{2}_{S}(S^{*}_{1})^{2}+2\sigma^{2}_{E}(E^{*}_{1})^{2}+ \sigma^{2}_{I}(I^{*}_{1})^{2}\nonumber\\ &&+\sigma^{2}_{\beta}(S^{*}_{1})^{2}\int_{t_{0}}^{h_{1}}f_{T_{1}}(r)e^{-2\mu }f_{T_{1}}(s)e^{-2\mu s}G^{2}(I(t-s))ds\nonumber\\ &&+\sigma^{2}_{\beta}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-2\mu (s+u)}G^{2}(I(t-s-u))(S(t-s-u)-S^{*}_{1})^{2}dsdu\nonumber\\\label{ch1.sec3.lemma1.proof.eq11}\end{aligned}$$ But $V(t)=V_{1}(t)+V_{2}(t)+V_{3}(t)+V_{4}(t)$, therefore from (\[ch1.sec3.lemma1.proof.eq11\]), (\[ch1.sec3.lemma1.eq4b\]) and (\[ch1.sec3.lemma1.proof.eq4\])-(\[ch1.sec3.lemma1.proof.eq6\]), the results in (\[ch1.sec3.lemma1.eq5\])-(\[ch1.sec3.lemma1.eq7\]) follow directly. Theorems \[\[ch1.sec3.thm1\], \[ch1.sec3.thm1.corrolary1\]\] assert that the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) has an endemic equilibrium denoted $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$, whenever the basic reproduction numbers $R_{0}$ and $R^{*}_{0}$ for the disease in the absence of noise in the system satisfy $R_{0}>1$ and $R^{*}_{0}>1$, respectively. One common technique to obtain insight about the asymptotic behavior of the sample paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) near the potential endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$ for the stochastic system, is to characterize the long-term average distance of the paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) from the endemic equilibrium $E_{1}$. Indeed, justification for this technique is the fact that for the second order stochastic solution process $\{X(t),t\geq t_{0}\}$ of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) defined in Theorem \[ch1.sec1.thm1\], the long-term average distance of the sample paths from the endemic equilibrium $E_{1}$, denoted $\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}||X(s)-E_{1}||ds}$ estimates the long-term ensemble mean denoted\ $\limsup_{t\rightarrow \infty}{E||X(t)-E_{1}||}$, almost surely. Moreover, if the solution process $\{X(t),t\geq t_{0}\}$ of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is stationary and ergodic, then the long-term ensemble mean $\limsup_{t\rightarrow \infty}{E||X(t)-E_{1}||}=E||X-E_{1}||$, where $X$ is the limit of convergence in distribution of the solution process $\{X(t),t\geq t_{0}\}$. That is, $\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}||X(s)-E_{1}||ds}=E||X-E_{1}||$, almost surely. These stationary and ergodic properties of the solution process $\{X(t),t\geq t_{0}\}$ are discussed in details in Section \[ch1.sec3.sec1\]. For convenience, the following notations are introduced and used in the rest of the results that follow in the subsequent sections. Let $a_{1}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{S}, \sigma^{2}_{\beta})$, $a_{2}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{I})$, $a_{3}(\mu, d, \alpha, \beta, B)$, and $a_{3}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{E})$ represent the following set of parameters $$\label{ch1.sec3.lemma1.proof.eq13a} \left\{ \begin{array}{lll} a_{1}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{S}, \sigma^{2}_{\beta})&=&2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+3\sigma ^{2}_{S}\\ &&+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}+\sigma^{2}_{\beta}(G^{*})^{2}\right)\left(\frac{1}{G'(0)}\right)^{2}\\%\label{ch1.sec3.lemma1.proof.eq13a}\\ a_{1}(\mu, d, \alpha, \beta, B)&=&2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}\\%\nonumber\\ &&+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}\right)\left(\frac{1}{G'(0)}\right)^{2}\\%\label{ch1.sec3.lemma1.proof.eq13a1}\\ %\hat{K}_{0}+\frac{\alpha}{\beta \frac{B}{\mu}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a_{2}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{I})&=&(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} + \sigma^{2}_{I}\\%\nonumber\\ &&+\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)\left(\frac{1}{G'(0)}\right)^{2}\\%\label{ch1.sec3.lemma1.proof.eq13b}\\ a_{2}(\mu, d, \alpha, \beta, B)&=&(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2}\\% \nonumber\\ &&+\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)\left(\frac{1}{G'(0)}\right)^{2}\\%\label{ch1.sec3.lemma1.proof.eq13b1}\\ %\hat{K}_{0}+\frac{\alpha}{\beta \frac{B}{\mu}}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% a_{3}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{E})&=&\frac{\beta }{2\lambda{(\mu)}}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+ \frac{2\mu}{\lambda{(\mu)}}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha \lambda{(\mu)}+ 2 \sigma^{2}_{E}\\%\nonumber\\\label{ch1.sec3.lemma1.proof.eq13c} a_{3}(\mu, d, \alpha, \beta, B)&=&\frac{\beta }{2\lambda{(\mu)}}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+ \frac{2\mu}{\lambda{(\mu)}}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha \lambda{(\mu)}%\nonumber\\\label{ch1.sec3.lemma1.proof.eq13c1} \end{array} \right.$$ Also let $\tilde{a}_{1}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{S}, \sigma^{2}_{\beta})$, $\tilde{a}_{1}(\mu, d, \alpha, \beta, B)$, $\tilde{a}_{2}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{I})$, $\tilde{a}_{2}(\mu, d, \alpha, \beta, B)$ represent the following set of parameters $$\label{ch1.sec3.thm2.proof.eq4a} \left\{ \begin{array}{lll} \tilde{a}_{1}(\mu, d, \alpha, \beta, B,\sigma ^{2}_{S}, \sigma^{2}_{\beta})&=&2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+3\sigma ^{2}_{S}\\%\nonumber\\ &&+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}+\sigma^{2}_{\beta}(G^{*})^{2}\right)+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)\\%\label{ch1.sec3.thm2.proof.eq4a}\\ \tilde{a}_{1}(\mu, d, \alpha, \beta, B)&=&2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}\\%\nonumber\\ &&+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}\right)+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}\right)\\%\label{ch1.sec3.thm2.proof.eq4a1}\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \tilde{a}_{2}(\mu, d, \alpha, \beta, B, \sigma^{2}_{I})&=&(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} + \sigma^{2}_{I}\\%\nonumber\\ &&+\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)+\frac{3\alpha}{2\lambda(\mu)},\\%\label{ch1.sec3.thm2.proof.eq4b}\\ \tilde{a}_{2}(\mu, d, \alpha, \beta, B)&=&(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} +\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)+\frac{3\alpha}{2\lambda(\mu)}. \end{array} \right.$$ The result in Theorem \[ch1.sec3.thm2\] characterizes the behavior of the sample paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) in the neighborhood of the nontrivial steady states $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$ defined in Theorem \[ch1.sec3.thm1.corrolary1\], whenever the incubation and natural immunity delay periods of the disease denoted by $T_{1}$, $T_{2}$, and $T_{3}$ are constant for all individuals in the population, and Theorem \[ch1.sec1.thm1\]\[a.\] holds. The following partial result from \[[@maobook], Theorem 3.4\] called the strong law of large number for local martingales will be used to establish the result. \[ch1.sec3.thm2.lemma1.lemma1\] Let $M=\{M_{t}\}_{t\geq 0}$ be a real valued continuous local martingale vanishing at $t=0$. Then $$\lim_{t\rightarrow \infty}{<M(t),M(t)>}=\infty, \quad a.s.\quad\Rightarrow\quad \lim_{t\rightarrow\infty} \frac{M(t)}{<M(t), M(t)>}=0,\quad a.s.$$ and also $$\limsup_{t\rightarrow \infty}{\frac{<M(t),M(t)>}{t}}<\infty, \quad a.s.\quad\Rightarrow\quad \lim_{t\rightarrow\infty} \frac{M(t)}{t}=0,\quad a.s.$$ The notation $<M(t),M(t)>$ is used to denote the quadratic variation of the local martingale $M=\{M(t),\forall t\geq t_{0}\}$. Recall, the assumptions that $T_{1}, T_{2}$ and $T_{3}$ are constant, is also equivalent to the special case of letting the probability density functions of $T_{1}, T_{2}$ and $T_{3}$ to be the dirac-delta function defined in (\[ch1.sec2.eq4\]). Moreover, under the assumption that $T_{1}\geq 0, T_{2}\geq 0$ and $T_{3}\geq 0$ are constant, it follows from (\[ch1.sec3.lemma1.eq5a\])-(\[ch1.sec3.lemma1.eq5c\]), that $E(e^{-2\mu (T_{1}+T_{2})})=e^{-2\mu (T_{1}+T_{2})} $, $E(e^{-2\mu T_{1}})=e^{-2\mu T_{1}} $ and $E(e^{-2\mu T_{3}})=e^{-2\mu T_{3}} $. \[ch1.sec3.thm2\] Let the hypotheses of Theorem \[ch1.sec1.thm1\]\[a.\], Theorem \[ch1.sec3.thm1.corrolary1\] and Lemma \[ch1.sec3.lemma1\] be satisfied and let $$\begin{aligned} \mu>\max{ \left(\frac{1}{3}a_{1}(\mu, d, \alpha, \beta, B, \sigma^{2}_{S}=0,\sigma^{2}_{\beta}),\frac{1}{2}a_{3}(\mu, d, \alpha, \beta, B)\right)},\quad and\nonumber\\ (\mu+d+\alpha)>a_{2}(\mu, d, \alpha, \beta, B ).\label{ch1.sec3.thm2.eq1} %\quad and\quad 2\mu> a_{3}(\mu, d, \alpha, \beta, B). \end{aligned}$$ Also let the delay times $T_{1}, T_{2}$ and $T_{3}$ be constant, that is, the probability density functions of $T_{1}, T_{2}$ and $T_{3}$ respectively denoted by $f_{T_{i}}, i=1, 2, 3$ are the dirac-delta functions defined in (\[ch1.sec2.eq4\]). Furthermore, let the constants $T_{1}, T_{2}$ and $T_{3}$ satisfy the following set of inequalities: $$\label{ch1.sec3.thm2.eq2} T_{1}>\frac{1}{2\mu}\log{\left(\frac{\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)}{(3\mu-a_{1}(\mu, d, \alpha, \beta, B, \sigma^{2}_{S}=0,\sigma^{2}_{\beta}))}\right)},$$ $$\label{ch1.sec3.thm2.eq3} T_{2}<\frac{1}{2\mu}\log{\left(\frac{(3\mu-a_{1}(\mu, d, \alpha, \beta, B, \sigma^{2}_{S}=0,\sigma^{2}_{\beta}))}{\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)\left(\frac{1}{G'(0)}\right)^{2}}\right)},$$and $$\label{ch1.sec3.thm2.eq3b} T_{3}>\frac{1}{2\mu}\log{\left(\frac{\frac{3\alpha}{2\lambda(\mu)}}{(\mu+d+\alpha)-a_{2}(\mu, d, \alpha, \beta, B)}\right)}.$$ There exists a positive real number $\mathfrak{m}_{1}>0$, such that $$\begin{aligned} &&\limsup_{t\rightarrow \infty}\frac{1}{t}\int^{t}_{0}\left[ ||X(v)-E_{1}||_{2}\right]^{2}dv\nonumber\\ &&\leq \frac{\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-2\mu (T_{1}+T_{2})}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-\mu T_{1}}}{\mathfrak{m}_{1}},\nonumber\\ \label{ch2.sec3.thm2.eq4}\end{aligned}$$almost surely, where $X(t)$ is defined in (\[ch1.sec0.eq13b\]), and $||.||_{2}$ is the natural Euclidean norm in $\mathbb{R}^{2}$. Proof:\ From Lemma \[ch1.sec3.lemma1\], (\[ch1.sec3.lemma1.eq5a\])-(\[ch1.sec3.lemma1.eq5c\]), it is easy to see that under the assumptions in (\[ch1.sec3.thm1.corrolary1.eq1\]) and (\[ch1.sec3.thm2.eq1\])-(\[ch1.sec3.thm2.eq3b\]), then $\tilde{\phi}_{1}>0$, $\tilde{\psi}_{1}>0$ and $\tilde{\varphi}_{1}>0$. Therefore, from (\[ch1.sec3.lemma1.eq5\])-(\[ch1.sec3.lemma1.eq7\]) it is also easy to see that $$\begin{aligned} dV&=&LV(t)dt + \overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)},\nonumber\\ &\leq&-\min\{\tilde{\phi}_{1}, \tilde{\psi}_{1}, \tilde{\varphi}_{1}\}\left[ (S(t)-S^{*}_{1})^{2}+ (E(t)-E^{*}_{1})^{2}+ (I(t)-I^{*}_{1})^{2}\right]\nonumber\\ && +3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-2\mu (T_{1}+T_{2})}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-\mu T_{1}}\nonumber\\ &&+ \overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)},\label{ch1.sec3.thm2.proof.eq1} \end{aligned}$$ Integrating both sides of (\[ch1.sec3.thm2.proof.eq1\]) from 0 to $t$, it follows that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) &&(V(t)-V(0))\leq -\mathfrak{m_{1}}\int^{t}_{0}\left[ (S(v)-S^{*}_{1})^{2}+ (E(v)-E^{*}_{1})^{2}+ (I(v)-I^{*}_{1})^{2}\right]dv\nonumber\\ && +\left(3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-2\mu (T_{1}+T_{2})}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}e^{-\mu T_{1}}\right)t,\nonumber\\ &&+\int^{t}_{0}\overrightarrow{g}(S(v), E(v), I(v))d\overrightarrow{w(v)},\label{ch2.sec3.thm2.proof.eq2}\end{aligned}$$ where $V(0)\geq 0$ and $$\label{ch2.sec3.thm2.proof.eq3} \mathfrak{m}_{1}=min(\tilde{\phi},\tilde{ \psi},\tilde{\varphi})>0.$$ are constants and $$\begin{aligned} &&\overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)}= -\sigma_{\beta}(S(t)-S^{*}_{1})S(t)e^{-\mu T_{1}}G(I(t-T_{1}))dw_{\beta}(t)\nonumber\\ &&-\sigma_{\beta}((S(t)-S^{*}_{1})+(E(t)-E^{*}_{1}))e^{-\mu (T_{1}+T_{2})}S(t-T_{2})G(I(t-T_{1}-T_{2}))dw_{\beta}(t),\nonumber\\\label{ch2.sec3.thm2.proof.eq4} \end{aligned}$$ since Theorem \[ch1.sec1.thm1\]\[a.\] holds and $T_{1}$ and $T_{2}$ are constants. Now, define $$\begin{aligned} N_{1}(t)=-\int^{t}_{0}\sigma_{\beta}(S(v)-S^{*}_{1})S(v)e^{-\mu T_{1}}G(I(v-T_{1}))dw_{\beta}(v),\quad and\quad \nonumber\\ N_{2}(t)=-\int^{t}_{0}\sigma_{\beta}((S(v)-S^{*}_{1})+(E(v)-E^{*}_{1}))e^{-\mu (T_{1}+T_{2})}S(v-T_{2})G(I(v-T_{1}-T_{2}))dw_{\beta}(v).\nonumber\\ \label{ch2.sec3.thm2.proof.eq5} \end{aligned}$$ Also, the quadratic variations of $N_{1}(t)$ and $N_{2}(t)$ in (\[ch2.sec3.thm2.proof.eq5\]) are given by $$\begin{aligned} <N(t)_{1}(t), N_{1}(t)>&=&\int^{t}_{0}\sigma^{2}_{\beta}(S(v)-S^{*}_{1})^{2}S^{2}(v)e^{-2\mu T_{1}}G^{2}(I(v-T_{1}))dv,\nonumber\\ <N(t)_{2}(t), N_{2}(t)>&=&\int^{t}_{0}\sigma^{2}_{\beta}((S(v)-S^{*}_{1})+(E(v)-E^{*}_{1}))^{2}e^{-2\mu (T_{1}+T_{2})}S^{2}(v-T_{2})G^{2}(I(v-T_{1}-T_{2}))dv.\nonumber\\\label{ch2.sec3.thm2.proof.eq6} \end{aligned}$$ It follows that when Theorem \[ch1.sec1.thm1\]\[a.\] holds, then from Assumption \[ch1.sec0.assum1\] and (\[ch2.sec3.thm2.proof.eq6\]), $$\begin{aligned} <N(t)_{1}, N_{1}(t)>&\leq& \int^{t}_{0}\sigma^{2}_{\beta}\left(\frac{\beta}{\mu}+S^{*}_{1}\right)^{2}\left(\frac{\beta}{\mu}\right)^{2}e^{-2\mu T_{1}}\left(\frac{\beta}{\mu}\right)^{2}dv\nonumber\\ &=&\sigma^{2}_{\beta}\left(\frac{\beta}{\mu}+S^{*}_{1}\right)^{2}\left(\frac{\beta}{\mu}\right)^{4}e^{-2\mu T_{1}}t.\label{ch2.sec3.thm2.proof.eq7}\end{aligned}$$ From (\[ch2.sec3.thm2.proof.eq7\]), it is easy to see that $\limsup_{t\rightarrow \infty }{\frac{1}{t}<N(t)_{1}, N_{1}(t)>}<\infty$, a.s.. Therefore by the strong law of large numbers for local martingales in Lemma \[ch1.sec3.thm2.lemma1.lemma1\], it follows that $\limsup_{t\rightarrow \infty }\frac{1}{t}N_{1}(t)=0$, a.s. By the same reasoning, it can be shown that $\limsup_{t\rightarrow \infty }\frac{1}{t}N_{2}(t)=0$, a.s. Moreover, from (\[ch2.sec3.thm2.proof.eq4\]), it can be seen that $\limsup_{t\rightarrow \infty }\int^{t}_{0}\overrightarrow{g}(S(v), E(v), I(v))d\overrightarrow{w(v)}=0$, a.s. Hence, diving both sides of (\[ch2.sec3.thm2.proof.eq2\]) by $t$ and $\mathfrak{m}_{1}$, and taking the $\limsup_{t\rightarrow \infty}$, then (\[ch2.sec3.thm2.eq4\]) follows directly. \[ch2.sec3.thm2.rem1\] Theorem \[ch1.sec3.thm2\] asserts that when the basic reproduction number $R^{*}_{0}$ defined in (\[ch1.sec2.lemma2a.corrolary1.eq4\]) satisfies $R^{*}_{0}>1$, and the disease dynamics is perturbed by random fluctuations exclusively in the disease transmission rate, that is, the intensity $\sigma_{\beta}>0$, it is noted earlier that the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) does not have an endemic equilibrium state. Nevertheless, the conditions in Theorem \[ch1.sec3.thm2\] provide estimates for the constant delay times $T_{1}, T_{2}$ and $T_{3}$ in (\[ch1.sec3.thm2.eq2\])-(\[ch1.sec3.thm2.eq3b\]) in addition to other parametric restrictions in (\[ch1.sec3.thm2.eq1\]) that are sufficient for the solution paths of the perturbed stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) to oscillate near the nontrivial steady state, $E_{1}$, of the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) found in Theorem \[ch1.sec3.thm1.corrolary1\]. The result in (\[ch2.sec3.thm2.eq4\]) estimates the average distance between the sample paths of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), and the nontrivial steady state $E_{1}$. Moreover, (\[ch2.sec3.thm2.eq4\]) depicts the size of the oscillations of the paths of the stochastic system relative to $E_{1}$, where smaller values for the intensity ( $\sigma_{\beta}> 0$) lead to oscillations of the paths in close proximity to the steady state $E_{1}$, and vice versa. In a biological context, the result of this theorem signifies that when the basic reproduction number exceeds one, and the other parametric restrictions in (\[ch1.sec3.thm2.eq1\])-(\[ch1.sec3.thm2.eq3b\]) are satisfied, then the disease related classes ($E, I, R$), and consequently the disease in a whole will persist in state near the endemic equilibrium state $E_{1}$. Moreover, stronger noise in the system from the disease transmission rate of the disease tends to persist the disease in state further away from the endemic equilibrium state $E_{1}$, and vice versa. Nevertheless, in this case of variability exclusively from the disease transmission rate, the numerical simulation results in Example \[ch1.sec4.subsec1.1\] suggest that continuous decrease in size of some of the subpopulation classes- susceptible, exposed, infectious and removal may occur, as the intensity of the noise from the disease transmission rate increases, but there is no definite indication of extinction of the entire human population over time. Note that, comparing to the simulation results in Example \[ch1.sec4.subsec1.2\], there is some evidence that the strength of the noise from the disease transmission rate persists the disease, but not to the point of extinction of the entire population (or at least not at the same rate as the case of the strength of the noises from the natural deathrates). These observations in Example \[ch1.sec4.subsec1.1\] also support the remark for Theorem \[ch1.sec1.thm1\]\[a\] that the sample paths for the stochastic system exhibit non-complex behaviors such as extinction of the entire human population, when the disease dynamics is perturbed exclusively by noise from the disease transmission rate, compared to the complex behavior observed when the disease dynamics is perturbed by noise from the natural death rates. These facts suggest that malaria control policies in the event where the disease is persistent, should focus on reducing the intensity of the fluctuations in the disease transmission rate, perhaps through vector control and better care of the people in the population to keep the transmission rate constant, in order to reduce the number of malaria cases which lead to the persistence of the disease. The subsequent result provides more general conditions irrespective of the probability distribution of the random variables $T_{1}, T_{2}$ and $T_{3}$, that are sufficient for the trajectories of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) to oscillate near the nontrivial steady state $E_{1}$ of the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]), whenever the intensities of the gaussian noises in the system are positive, that is whenever $\sigma_{i}>0, i=S, E, I, R, \beta$. The following result from \[Lemma 2.2 [@maobook]\] and \[lemma 15,[@Li2008]\] will be used to achieve this result. \[ch1.sec3.thm3.lemma1\] Let $M(t); t\geq 0$ be a continuous local martingale with initial value $M(0)=0$. Let $<M(t),M(t)>$ be its quadratic variation. Let $\delta >1$ be a number, and let $\nu_{k}$ and $\tau_{k}$ be two sequences of positive numbers. Then, for almost all $w\in\Omega$, there exists a random integer $k_{0}=k_{0}(w)$ such that, for all $k\geq k_{0}$, $$M(t)\leq \frac{1}{2}\nu_{k}<M(t), M(t)>+\frac{\delta ln k}{\nu_{k}}, 0\leq t\leq \tau_{k}.$$ Proof:\ See [@maobook; @Li2008]. \[ch2.sec3.thm3\] Suppose the hypotheses of Theorem \[ch1.sec1.thm1\]\[b.\], Theorem \[ch1.sec3.thm1\] and Lemma \[ch1.sec3.lemma1\] are satisfied, and let $$\label{ch1.sec3.thm3.eq1} \mu> \max\left\{\frac{1}{3}\tilde{a}_{1}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{S}, \sigma^{2}_{\beta}),\frac{1}{2}a_{3}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{E})\right\}\quad and\quad(\mu+d+\alpha)>\tilde{a}_{2}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{I}).$$ It follows that for any arbitrary probability distribution of the delay times: $T_{1}, T_{2}$ and $T_{3}$, there exists a positive real number $\mathfrak{m}_{2}>0$ such that $$\begin{aligned} &&\limsup_{t\rightarrow \infty}\frac{1}{t}\int^{t}_{0}\left[ (S(v)-S^{*}_{1})^{2}+ (E(v)-E^{*}_{1})^{2}+ (I(v)-I^{*}_{1})^{2}\right]dv\nonumber\\ &&\leq \frac{3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}})}{\mathfrak{m}_{2}},\nonumber\\ \label{ch2.sec3.thm3.eq2}\end{aligned}$$almost surely. Proof:\ From Lemma \[ch1.sec3.lemma1\], (\[ch1.sec3.lemma1.eq5a\])-(\[ch1.sec3.lemma1.eq5c\]), $$\begin{aligned} % \nonumber % Remove numbering (before each equation) -\tilde{\phi}_{1}&=&-3\mu+\left[2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+3\sigma ^{2}_{S}+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)E(e^{-2\mu T_{1}})\right.\nonumber\\ &&\left.+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}+\sigma^{2}_{\beta}(G^{*})^{2}\right)E(e^{-2\mu (T_{1}+T_{2})})\right]\nonumber\\ &&\leq -3\mu+\left[2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+3\sigma ^{2}_{S}+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}+\sigma^{2}_{\beta}(G^{*})^{2}\right)\right.\nonumber\\ &&\left.+\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}+\sigma^{2}_{\beta}(G^{*})^{2}\right)\right]\nonumber\\ &&=-\left(3\mu-\tilde{a}_{1}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{S}, \sigma^{2}_{\beta})\right)\label{ch1.sec3.thm3.proof.eq1}\\ %%%%% %%%%%% -\tilde{\psi}_{1} &=& -2\mu+\left[\frac{\beta }{2\lambda{(\mu)}}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+ \frac{2\mu}{\lambda{(\mu)}}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha \lambda{(\mu)}+ 2 \sigma^{2}_{E} \right]\nonumber\\ &&=-\left(2\mu-a_{3}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{E})\right)\label{ch1.sec3.thm3.proof.eq2}\\ -\tilde{\varphi}_{1}&=& -(\mu + d+\alpha)+\left[(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} + \sigma^{2}_{I}+\frac{3\alpha}{2\lambda{(\mu)}}E(e^{-2\mu T_{3}})\right.\nonumber\\ &&\left. +\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)E(e^{-2\mu (T_{1}+T_{2})})\right]\nonumber\\ &\leq& -(\mu + d+\alpha)+\left[(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2} + \sigma^{2}_{I}+\frac{3\alpha}{2\lambda{(\mu)}}\right.\nonumber\\ &&\left. +\left(\frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}} \right)\right]\nonumber\\ &&=-\left((\mu + d+\alpha)-\tilde{a}_{2}(\mu, d, \alpha, \beta, B, \sigma ^{2}_{I})\right),\label{ch1.sec3.thm3.proof.eq3} \end{aligned}$$since $0<E(e^{-2\mu (T_{i})})\leq 1, i=1, 2,3$. It follows from (\[ch1.sec3.lemma1.eq5\])-(\[ch1.sec3.lemma1.eq7\]) and (\[ch1.sec3.thm3.proof.eq1\])-(\[ch1.sec3.thm3.proof.eq3\]) that $$\begin{aligned} dV&=&LV(t)dt + \overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)},\nonumber\\ &\leq&-\min\{\left(3\mu-\tilde{a}_{1}(\mu, d, \alpha, \beta, B)\right), \left(2\mu-a_{3}(\mu, d, \alpha, \beta, B)\right), \left((\mu + d+\alpha)-\tilde{a}_{2}(\mu, d, \alpha, \beta, B)\right)\}\times\nonumber\\ &&\times \left[ (S(t)-S^{*}_{1})^{2}+ (E(t)-E^{*}_{1})^{2}+ (I(t)-I^{*}_{1})^{2}\right]\nonumber\\ && +3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}})\nonumber\\ &&+ \overrightarrow{g}(S(t), E(t), I(t))d\overrightarrow{w(t)},\label{ch1.sec3.thm3.proof.eq4} \end{aligned}$$ where under the assumptions (\[ch1.sec3.thm3.eq1\]) in the hypothesis, $$\label{ch2.sec3.thm3.proof.eq5} \mathfrak{m}_{2}=\min\{\left(3\mu-\tilde{a}_{1}(\mu, d, \alpha, \beta, B)\right), \left(2\mu-a_{3}(\mu, d, \alpha, \beta, B)\right), \left((\mu + d+\alpha)-\tilde{a}_{2}(\mu, d, \alpha, \beta, B)\right)\}>0.$$ Integrating both sides of (\[ch1.sec3.thm3.proof.eq4\]) from 0 to $t$, it follows that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) &&(V(t)-V(0))\leq -\mathfrak{m_{2}}\int^{t}_{0}\left[ (S(v)-S^{*}_{1})^{2}+ (E(v)-E^{*}_{1})^{2}+ (I(v)-I^{*}_{1})^{2}\right]dv\nonumber\\ && +\left(3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}})\right)t\nonumber\\ &&+M(t), \label{ch2.sec3.thm3.proof.eq6}\end{aligned}$$ where $V(0)$ is constant and from (\[ch1.sec3.lemma1.eq6\]), the function $M(t)$ is defined as follows:- $$\begin{aligned} M(t)&=&\int^{t}_{0}\overrightarrow{g}(S(v), E(v), I(v))d\overrightarrow{w(v)}\nonumber\\ &=&\sum^{3}_{i=1}M^{S}_{i}(t)+\sum^{3}_{i=1}M^{E}_{i}(t)+\sum^{3}_{i=1}M^{I}_{i}(t)+\sum^{3}_{i=1}M^{\beta}_{i}(t),\nonumber\\ \label{ch2.sec3.thm3.proof.eq7} \end{aligned}$$ and $$\begin{aligned} M^{S}_{1}(t)=-\int^{t}_{0}3\sigma_{S}(S(v)-S^{*}_{1})S(v)dw_{S}(v),\nonumber\\ M^{S}_{2}(t)=-\int^{t}_{0}2\sigma_{S}(E(v)-E^{*}_{1})S(v)dw_{S}(v),\nonumber\\ M^{S}_{3}(t)=-\int^{t}_{0}\sigma_{S}(I(v)-I^{*}_{1})S(v)dw_{S}(v).\nonumber\\ \label{ch2.sec3.thm3.proof.eq8} \end{aligned}$$ Also, $$\begin{aligned} M^{E}_{1}(t)=-\int^{t}_{0}2\sigma_{E}(S(v)-S^{*}_{1})E(v)dw_{E}(v),\nonumber\\ M^{E}_{2}(t)=-\int^{t}_{0}2\sigma_{E}(E(v)-E^{*}_{1})E(v)dw_{E}(v),\nonumber\\ M^{E}_{3}(t)=-\int^{t}_{0}\sigma_{E}(I(v)-I^{*}_{1})E(v)dw_{E}(v),\nonumber\\ \label{ch2.sec3.thm3.proof.eq9} \end{aligned}$$ and $$\begin{aligned} M^{I}_{1}(t)=-\int^{t}_{0}2\sigma_{I}(S(v)-S^{*}_{1})I(v)dw_{I}(v),\nonumber\\ M^{I}_{2}(t)=-\int^{t}_{0}2\sigma_{I}(E(v)-E^{*}_{1})I(v)dw_{I}(v),\nonumber\\ M^{I}_{3}(t)=-\int^{t}_{0}\sigma_{I}(I(v)-I^{*}_{1})I(v)dw_{I}(v).\nonumber\\ \label{ch2.sec3.thm3.proof.eq10} \end{aligned}$$ Furthermore, $$\begin{aligned} M^{\beta}_{1}(t)=-\int^{t}_{0}\sigma_{\beta}(S(v)-S^{*}_{1})S(v)E(e^{-\mu T_{1}}G(I(v-T_{1})))dw_{\beta}(v),\nonumber\\ M^{\beta}_{2}(t)=-\int^{t}_{0}\sigma_{\beta}(S(v)-S^{*}_{1})E(S(v-T_{2})e^{-\mu (T_{1}+T_{2})}G(I(v-T_{1}-T_{2})))dw_{\beta}(v),\nonumber\\ M^{\beta}_{3}(t)=-\int^{t}_{0}\sigma_{\beta}(E(v)-E^{*}_{1})E(S(v-T_{2})e^{-\mu (T_{1}+T_{2})}G(I(v-T_{1}-T_{2})))dw_{\beta}(v).\nonumber\\ \label{ch2.sec3.thm3.proof.eq11} \end{aligned}$$ Applying Lemma \[ch1.sec3.thm3.lemma1\], choose $\delta=\frac{2}{12}, \nu_{k}=\nu$, and $\tau_{k}=k$, then there exists a random number $k_{j}(w)>0, w\in \Omega$, and $j=1,2,\ldots 12$ such that $$\label{ch2.sec3.thm3.proof.eq12} M^{S}_{i}(t)\leq \frac{1}{2}\nu <M^{S}_{i}(t), M^{S}_{i}(t)>+ \frac{\frac{2}{12}}{\nu}\ln{k}, \forall t\in [0,k], i=1,2,3,$$ $$\label{ch2.sec3.thm3.proof.eq13} M^{E}_{i}(t)\leq \frac{1}{2}\nu <M^{E}_{i}(t), M^{E}_{i}(t)>+ \frac{\frac{2}{12}}{\nu}\ln{k}, \forall t\in [0,k], i=1,2,3,$$ $$\label{ch2.sec3.thm3.proof.eq14} M^{I}_{i}(t)\leq \frac{1}{2}\nu <M^{I}_{i}(t), M^{S}_{i}(t)>+ \frac{\frac{2}{12}}{\nu}\ln{k}, \forall t\in [0,k], i=1,2,3,$$ $$\label{ch2.sec3.thm3.proof.eq15} M^{\beta}_{i}(t)\leq \frac{1}{2}\nu <M^{\beta}_{i}(t), M^{\beta}_{i}(t)>+ \frac{\frac{2}{12}}{\nu}\ln{k}, \forall t\in [0,k], i=1,2,3,$$ where the quadratic variations $<M^{a}_{i}(t), M^{a}_{i}(t)>, \forall i=1,2,3$ and $a\in \{S, E, I, \beta\}$ are computed in the same manner as (\[ch2.sec3.thm2.proof.eq6\]). It follows from (\[ch2.sec3.thm3.proof.eq7\]) that for all $ k>\max_{j=1,2,\ldots,12}{k_{j}(w)}>0$, $$\begin{aligned} M(t)&\leq& \frac{1}{2}\nu \sum^{3}_{i=1}<M^{S}_{i}(t), M^{S}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{E}_{i}(t), M^{E}_{i}(t)>\nonumber\\ &&+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{I}_{i}(t), M^{I}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{\beta}_{i}(t), M^{\beta}_{i}(t)> \nonumber\\ &&+\frac{2}{\nu}\ln{k}, \forall t\in [0,k].\label{ch2.sec3.thm3.proof.eq16} \end{aligned}$$ Now, rearranging and diving both sides (\[ch2.sec3.thm3.proof.eq6\]) of by $t$ and $\mathfrak{m}_{2}$, it follows that for all $t\in [k-1,k]$, $$\begin{aligned} &&\frac{1}{t}\int^{t}_{0}\left[ (S(v)-S^{*}_{1})^{2}+ (E(v)-E^{*}_{1})^{2}+ (I(v)-I^{*}_{1})^{2}\right]dv\nonumber\\ &&\leq \frac{3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}})}{\mathfrak{m}_{2}}\nonumber\\ &&+\frac{1}{t}\left(\frac{1}{\mathfrak{m}_{2}}\right)\left(\frac{1}{2}\nu \sum^{3}_{i=1}<M^{S}_{i}(t), M^{S}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{E}_{i}(t), M^{E}_{i}(t)>\right)\nonumber\\ &&+\frac{1}{t}\left(\frac{1}{\mathfrak{m}_{2}}\right)\left(\frac{1}{2}\nu \sum^{3}_{i=1}<M^{I}_{i}(t), M^{I}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{\beta}_{i}(t), M^{\beta}_{i}(t)> \right)\nonumber\\ &&+\frac{2}{\nu (k-1)}\ln{k}\left(\frac{1}{\mathfrak{m}_{2}}\right).% ,\nonumber\\ \label{ch2.sec3.thm3.proof.eq17} \end{aligned}$$ Thus letting $k\rightarrow \infty$, then $t\rightarrow \infty$. It follows from (\[ch2.sec3.thm3.proof.eq17\]) that $$\begin{aligned} &&\limsup_{t\rightarrow \infty}\frac{1}{t}\int^{t}_{0}\left[ (S(v)-S^{*}_{1})^{2}+ (E(v)-E^{*}_{1})^{2}+ (I(v)-I^{*}_{1})^{2}\right]dv\nonumber\\ &&\leq \frac{3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}})}{\mathfrak{m}_{2}}\nonumber\\ &&+\limsup_{t\rightarrow \infty}\frac{1}{t}\left(\frac{1}{\mathfrak{m}_{2}}\right)\left(\frac{1}{2}\nu \sum^{3}_{i=1}<M^{S}_{i}(t), M^{S}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{E}_{j}(t), M^{E}_{i}(t)>\right)\nonumber\\ &&+\limsup_{t\rightarrow \infty}\frac{1}{t}\left(\frac{1}{\mathfrak{m}_{2}}\right)\left(\frac{1}{2}\nu \sum^{3}_{i=1}<M^{I}_{i}(t), M^{I}_{i}(t)>+\frac{1}{2}\nu \sum^{3}_{i=1}<M^{\beta}_{i}(t), M^{\beta}_{i}(t)> \right)\nonumber\\.% ,\nonumber\\ \label{ch2.sec3.thm3.proof.eq18} \end{aligned}$$ Finally, by sending $\nu\rightarrow 0$, the result in (\[ch2.sec3.thm3.eq2\]) follows immediately from (\[ch2.sec3.thm3.proof.eq18\]). \[ch1.sec3.rem2\] When the disease dynamics is perturbed by random fluctuations in the disease transmission or natural death rates, that is, when at least one of the intensities $\sigma^{2}_{i}> 0, i= S, E, I, \beta$, it has been noted earlier that the nontrivial steady state, $E_{1}$, of the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) found in Theorem \[ch1.sec3.thm1\] no longer exists for the perturbed stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). Nevertheless, the conditions in Theorem \[ch1.sec3.thm2\] provide restrictions for the constant delays $T_{1}$, $T_{2}$ and $T_{3}$ in (\[ch1.sec3.thm2.eq2\])-(\[ch1.sec3.thm2.eq3b\]) and parametric restrictions in (\[ch1.sec3.thm2.eq1\]) which are sufficient for the sample path solutions of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) to oscillate in the neighborhood of the potential endemic equilibrium $E_{1}$. Also, the conditions in Theorem \[ch2.sec3.thm3\] provide general restrictions in (\[ch1.sec3.thm3.eq1\]) irrespective of the probability distribution of the random variable delay times $T_{1}, T_{2}$ and $T_{3}$ that are sufficient for the solutions of the perturbed stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) to oscillate near the nontrivial steady state, $E_{1}$, of the deterministic system (\[ch1.sec0.eq3\])-(\[ch1.sec0.eq6\]) found in Theorem \[ch1.sec3.thm1\]. The results in (\[ch2.sec3.thm2.eq4\]) and (\[ch2.sec3.thm3.eq2\]) characterize the average distance between the trajectories of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), and the nontrivial steady state $E_{1}$. Moreover, the results also signify that the size of the oscillations of the trajectories of the stochastic system relative to the state $E_{1}$ depends on the intensities of the noises in the system, that is the sizes of $\sigma^{2}_{i}> 0, i= S, E, I, \beta$. Indeed, it can be seen easily that the trajectories oscillate much closer to $E_{1}$, whenever the intensities ($\sigma^{2}_{i}> 0, i= S, E, I, \beta$) are small, and vice versa. Permanence of malaria in the stochastic system\[ch1.sec5\] ========================================================== In this section, the permanence of malaria in the human population is investigated, whenever the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is subject to the influence of random environmental fluctuations from the disease transmission and natural death rates. Permanence in the mean of the disease seeks to determine whether there is always a positive significant average number of individuals of the disease related classes namely- exposed $(E)$, infectious $(I)$, and removal $(R)$ subclasses in the population over sufficiently long time. That is, seeking to determine whether $\lim_{t\rightarrow\infty}E(||E(t)||)>0$, $\lim_{t\rightarrow\infty}E(||I(t)||)>0$ and $\lim_{t\rightarrow\infty}E(||R(t)||)>0$. In the absence of explicit solutions for the nonlinear stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), this information about the asymptotic average number of individuals in the disease related classes is obtained via Lyapunov techniques from the examination of the statistical properties of the paths of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), in particular, determining the size of all average sample path estimates for the disease related subclasses over sufficiently long time. That is, to determine whether the following hold: $\liminf_{t\rightarrow\infty} \frac{1}{t}\int^{t}_{0}||E(s)||ds>0$, $\liminf_{t\rightarrow\infty} \frac{1}{t}\int^{t}_{0}||I(s)||ds>0$ and $\liminf_{t\rightarrow\infty} \frac{1}{t}\int^{t}_{0}||R(s)||ds>0$. The following definition is given for the stochastic version of strong permanence in the mean of a disease. \[ch1.sec5.definition1\] The system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is said to be almost surely permanent in the mean[@chen-biodyn] (in the strong sense), if $$\begin{aligned} && \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}S(s)ds}>0, a.s., \quad \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}E(s)ds}>0, a.s., \nonumber\\ && \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}I(s)ds}>0, a.s., \quad \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}R(s)ds}>0, a.s.\label{ch1.sec5.definition1.eq1} \end{aligned}$$ where $(S(t), E(t), I(t), R(t))$ is any positive solution of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). The method applied to show the permanence in the mean of the disease in the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is similar in the cases where (1) all the delays in the systems $T_{1}, T_{2}$ and $T_{3}$ are constant and finite, and (2) in the case where the delays $T_{1}, T_{2}$ and $T_{3}$ are random variables. Thus, without loss of generality, the results for the permanence in the mean of the disease in the stochastic system is presented only in the case of random delays in the system. Some ideas in [@yanli] can be used to prove this result. \[ch1.sec5.thm1\] Assume that the conditions of Theorem \[ch1.sec3.thm1\] and Theorem \[ch2.sec3.thm3\] are satisfied. Define the following $$\begin{aligned} \label{ch1.sec5.thm1.eq1} \hat{h}\equiv\hat{h}(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})&=&3(S^{*}_{1})^{2}+ 2(E^{*}_{1})^{2}+(I^{*}_{1})^{2}+(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}}),\nonumber\\ \sigma^{2}_{max}&=& \max{(\sigma^{2}_{S}, \sigma^{2}_{E}, \sigma^{2}_{I}, \sigma^{2}_{\beta})}. \end{aligned}$$ Assume further that the following relationship is satisfied $$\label{ch1.sec5.thm1.eq2} \sigma^{2}_{max}<\frac{\mathfrak{m}_{2}}{\hat{h}}\min{((S^{*}_{1})^{2}, (E^{*}_{1})^{2}, (I^{*}_{1})^{2})}\equiv \hat{\tau},$$ where $\mathfrak{m}_{2}$ is defined in(\[ch2.sec3.thm3.proof.eq5\]). Then it follows that $$\begin{aligned} && \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}S(v)dv}>0,a.s., \quad \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}E(v)dv}>0,a.s. \nonumber\\ && and\quad \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}I(v)dv}>0,a.s.\label{ch1.sec5.thm1.eq3} \end{aligned}$$ In other words, the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is permanent in the mean. Proof:\ It is easy to see that when Theorem \[ch1.sec3.thm1\] and Theorem \[ch2.sec3.thm3\] are satisfied, then it follows from (\[ch2.sec3.thm3.eq2\]) that $$\begin{aligned} &&\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0} (S(v)-S^{*}_{1})^{2}dv}\leq \sigma^{2}_{max}\frac{\hat{h}}{\mathfrak{m}_{2}},\nonumber\\ &&\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0} (E(v)-E^{*}_{1})^{2}dv}\leq \sigma^{2}_{max}\frac{\hat{h}}{\mathfrak{m}_{2}},\nonumber\\ &&\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0} (I(v)-I^{*}_{1})^{2}dv}\leq \sigma^{2}_{max}\frac{\hat{h}}{\mathfrak{m}_{2}}.\quad a.s. \label{ch1.sec5.thm1.proof.eq1}\end{aligned}$$For each $a(t)\in \left\{S(t), E(t), I(t)\right\}$ it is easy to see that $$\label{ch1.sec5.thm1.proof.eq2} 2(a^{*}_{1})^{2}-2a^{*}_{1}a(t)\leq (a^{*}_{1})^{2}+ (a(t)-a^{*}_{1})^{2}.$$ It follows from (\[ch1.sec5.thm1.proof.eq2\]) that $$\begin{aligned} \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}a(v)dv}&\geq& \frac{a^{*}_{1}}{2}-\limsup_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0} \frac{(a(v)-a^{*}_{1})^{2}}{2a^{*}_{1}}dv}\nonumber\\ &\geq&\frac{a^{*}_{1}}{2}-\sigma^{2}_{max}\frac{\hat{h}}{\mathfrak{m_{2}}}\frac{1}{2a^{*}_{1}}\label{ch1.sec5.thm1.proof.eq3}\end{aligned}$$ For each $a(t)\in \left\{S(t), E(t), I(t)\right\}$, the inequalities in (\[ch1.sec5.thm1.eq3\]) follow immediately from (\[ch1.sec5.thm1.proof.eq3\]), whenever (\[ch1.sec5.thm1.eq2\]) is satisfied.\ The following result about the permanence of the disease in the system in the case where the delays $T_{1}, T_{2}$ and $T_{3}$ are all constant is stated. \[ch1.sec5.thm2\] Assume that the conditions of Theorem \[ch1.sec3.thm2\] and Theorem \[ch1.sec3.thm1.corrolary1\] are satisfied. Define the following $$\begin{aligned} \hat{h}\equiv\hat{h}(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})&=&(S^{*}_{1})^{2}(G^{*})^{2}e^{-2\mu (T_{1}+T_{2})}+(S^{*}_{1})^{2}(G^{*})^{2}e^{-\mu T_{1}},\label{ch1.sec5.thm2.eq1} %\sigma^{2}_{\beta}&=& \max{(\sigma^{2}_{S}, \sigma^{2}_{E}, \sigma^{2}_{I}, \sigma^{2}_{\beta})}. \end{aligned}$$ Assume further that the following relationship is satisfied $$\label{ch1.sec5.thm2.eq2} \sigma^{2}_{\beta}<\frac{\mathfrak{m}_{1}}{\hat{h}}\min{((S^{*}_{1})^{2}, (E^{*}_{1})^{2}, (I^{*}_{1})^{2})}\equiv \hat{\tau},$$ where $\mathfrak{m}_{1}$ is defined in(\[ch2.sec3.thm2.proof.eq3\]). Then it follows that $$\begin{aligned} && \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}S(v)dv}>0,a.s. \quad \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}E(v)dv}>0,a.s. \nonumber\\ && \liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}I(v)dv}>0,a.s.\label{ch1.sec5.thm2.eq3} \end{aligned}$$ In other words, the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is permanent in the mean. Proof:\ The proof is similar to the proof of Theorem \[ch1.sec5.thm1\] above. \[ch1.sec5.rem1\] Theorems \[\[ch1.sec5.thm2\] &\[ch1.sec5.thm1\]\] provide sufficient conditions for the permanence in the mean of the vector-borne disease in the population, where the conditions depend on the intensities of the white noise processes in the system. Indeed, for Theorem \[ch1.sec5.thm1\], the condition in (\[ch1.sec5.thm1.eq2\]) suggests that there is a bound for the intensities of the white noise processes in the system that allows the disease to persist in the population permanently, provided that the noise free basic reproduction number of the disease in (\[ch1.sec2.theorem1.corollary1.eq3\]) satisfies $R_{0}>1$. Moreover, if one defines the term $$\label{ch1.sec5.rem1.eq1} l_{a}(\sigma^{2}_{max})=\frac{a^{*}_{1}}{2}-\sigma^{2}_{max}\frac{\hat{h}}{\mathfrak{m_{2}}}\frac{1}{2a^{*}_{1}}, \forall a(t)\in \left\{S(t), E(t), I(t)\right\},$$ then it is easy to see from (\[ch1.sec5.thm1.proof.eq3\]) that $ l_{a}(\sigma^{2}_{max})$ is the lower bound for all average sample path estimates for the ensemble mean of the different states $a(t)\in \left\{S(t), E(t), I(t)\right\}$ of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) over sufficiently large amount of time, where $a^{*}_{1}\in \left\{S^{*}_{1}, E^{*}_{1}, I^{*}_{1}\right\}$. It can be observed from (\[ch1.sec5.rem1.eq1\]) and (\[ch1.sec5.thm1.eq2\]) that $$\label{ch1.sec5.rem1.eq2} \lim_{\sigma^{2}_{max}\rightarrow 0^{+}}{l_{a}(\sigma^{2}_{max})}=\frac{1}{2}a^{*}_{1},\quad and \quad \lim_{\sigma^{2}_{max}\rightarrow \hat{\tau}^{-}}{l_{a}(\sigma^{2}_{max})}=0.$$ That is, the presence of noise in the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) with significant continuously increasing intensity values (i.e. $\sigma^{2}_{max}\rightarrow \hat{\tau}^{-}$ ) allow for smaller values of the asymptotic average limiting value of all sample paths ($\liminf_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{0}a(v)dv}$) of the states $a(t)\in \left\{S(t), E(t), I(t)\right\}$ of the system, and vice versa. This observation suggests that the occurrence of “stronger” noise (noise with higher intensity) in the system suppresses the permanence of the disease in the population by allowing smaller average values for the sample paths of the disease related states $E, I, R$ in the population over sufficiently large amount of time. However, it should be noted that the smaller average values for the disease related classes $E, I, R$ over sufficiently long time are also matched with smaller average values for the susceptible class $S$ as the intensities of the noises in the system rise. This observation suggests that there is a general decrease in the population size over sufficiently long time as the intensities of the noises in the system increase in magnitude. Therefore, one can conclude that for small to moderate values for the intensities of the noises in the system, the disease persists permanently with higher average values for the disease related classes. Furthermore, as the magnitude of the intensities of the noises increase to higher values, the human population may become extinct over sufficiently long time. This observation is illustrated in the Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\] and Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\], and much more in Figures \[ch1.sec4.figure 1\]-\[ch1.sec4.figure 3\]. The statistical significance of this result is noting that if all the sample paths for a given state $a(t)\in \left\{S(t), E(t), I(t)\right\}$ are bounded from below on average by the same significantly larger positive value $ l_{a}(\sigma^{2}_{max})$ asymptotically, then the ensemble means corresponding to the given states are also bounded by the same value asymptotically. This fact is more apparent as the stationary and ergodic properties of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) are shown in the next section. Stationary distribution and ergodic property {#ch1.sec3.sec1} ============================================ The knowledge of the distribution of the solutions of a system of stochastic differential equations holds the key to fully understand the statistical and probabilistic properties of the solutions at any time, and overall to characterize the uncertainties of the states of the stochastic system over time. Furthermore, the ergodicity of the solutions of a stochastic system ensure that one obtains insights about the long-term behavior of the system, that is the statistical properties of the solutions of the system via knowledge of the average behavior of the sample paths or sample realizations of the system over finite or sufficiently large time interval. In this section, the long term distribution and the ergodicity of the positive solutions of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) are characterized in the neighborhood of a potential endemic steady state $E_{1}$ for the system obtained in Theorem \[ch1.sec3.thm1\]. It is shown below that the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) has a stationary endemic distribution for the solutions of the system. First, the definition of a stationary distribution for a continuous-time and continuous-state Markov process or for the solution of a system of stochastic differential equations is presented in the following: \[ch1.sec3.sec1.defn1\] (see [@yongli]) Denote $\mathbb{P}_{\gamma}$ the corresponding probability distribution of an initial distribution $\gamma$, which describes the initial state of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) at time $t=0$. Suppose that the distribution of $Y(t)=(S(t), E(t), I(t), R(t))$ with initial distribution $\gamma$ converges in some sense to a distribution $\pi=\pi_{\gamma}$ ( a priori $\pi$ may depend on the initial distribution $\gamma$), that is, $$\label{ch1.sec3.sec1.defn1.eq1} \lim_{t\rightarrow \infty}\mathbb{P}_{\gamma}\{X(t)\in F\}=\pi(F)$$ for all measurable sets $F$, then we say that the stochastic system of differential equations has a stationary distribution $\pi(.)$. The standard method utilized in [@yongli; @yongli-2; @yzhang] is applied to establish this result. The following assumptions are made:- let $Y(t)$ be a regular temporary homogeneous Markov process in the $d$-dimensional space $E_{d}\subseteq \mathbb{R}^{d}_{+}$ described by the stochastic differential equation $$\label{ch1.sec3.sec1.eq1} dY(t)=b(Y,t)dt+\sum_{r=1}^{k}g_{r}(Y, t)dB_{r}(t).$$ Then the diffusion matrix can be defined as follows $$\label{ch1.sec3.sec1.eq2} A(Y)=(a_{ij}(y)), a_{ij}(y)=\sum_{r=1}^{k}g^{i}_{r}(Y, t)g^{j}_{r}(Y, t).$$ The following lemma describes the existence of a stationary solution for (\[ch1.sec3.sec1.eq1\]). \[ch1.sec3.sec1.lemma1\] The Markov process $Y(t)$ has a unique ergodic stationary distribution $\pi(.)$ if a bounded domain $D\subset E_{d}$ with regular boundary $\Gamma$ exists and there exists a constant $M>0$ satisfying $\sum^{d}_{ij}a_{ij}(x)\xi_{i}\xi_{j}\geq M|\xi|^{2}$, $x\in D, \xi\in \mathbb{R}^{d}$. there is a $\mathcal{C}^{1,2}$-function $V\geq 0$ such that $LV$ is negative for any $E_{d}\backslash D$. Then $$\label{ch1.sec3.sec1.lemma1.eq1} \mathbb{P}\left\{ \lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}f(Y(t))dt=\int_{E_{d}}f(y)\pi(dy)\right\}=1,$$ for all $y\in E_{d}$, where $f(.)$ is an integrable function with respect to the measure $\pi$. The result that follows characterizes the existence of stationary distribution for the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) in the general case where the delay times $T_{1}, T_{2}$ and $T_{3}$ in the system are distributed. \[ch1.sec3.sec1.thm1\] Let the assumptions in Theorem \[ch2.sec3.thm3\] be satisfied, and let $$\label{ch1.sec3.sec1.thm1.eq0} R=3\min_{(S, E, I)\in \mathbb{R}^{3}_{+}}{\left(\frac{1}{\left(\frac{1}{3}\Phi_{1}- \sigma^{2}_{\beta}\left(\frac{2}{3}(G^{*})^{2}+2\theta^{2}_{1}\right)\right)},\frac{2}{\Phi_{2}}\left(1+\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2}\right), \frac{1}{\Phi_{3}}\left(1+2\sigma^{2}_{\beta}\theta^{2}_{2}\right) \right)}<\infty.$$ Also define the following parameters:- $$\label{ch1.sec3.sec1.thm1.eq1} \Phi=3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}}),$$ and $$\label{ch1.sec3.sec1.thm1.eq1-1} R_{min}=3\min{\left(\frac{1}{\left(\frac{1}{3}\Phi_{1}- \sigma^{2}_{\beta}\left(\frac{2}{3}(G^{*})^{2}+2(I^{*}_{1})^{2}\right)\right)},\frac{2}{\Phi_{2}}\left(1+4\sigma^{2}_{\beta}(S^{*}_{1}I^{*}_{1})^{2}\right), \frac{1}{\Phi_{3}}\left(1+2\sigma^{2}_{\beta}(S^{*}_{1}I^{*}_{1})^{2}\right) \right)}<\infty,$$ where, $$\begin{aligned} % \nonumber % Remove numbering (before each equation) \theta_{1} &=&\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(I(t-s))ds,\label{ch1.sec3.sec1.thm1.proof.eq1} \\ \theta_{2} &=&\int_{t_{0}}^{h_{2}}f_{T_{2}}S(t-u)\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu( s+u)}G(I(t-s-u))dsdu,\label{ch1.sec3.sec1.thm1.proof.eq2}\\ \Phi_{1}&=&3\mu-\left[2\mu\lambda{(\mu)}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha\lambda{(\mu)}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}\right.\nonumber\\ &&+\left.\left(\frac{\beta \lambda{(\mu)}(G^{*})^{2}}{2}\right)+\left(\frac{\beta (G^{*})^{2}}{2\lambda{(\mu)}}\right)\right],\label{ch1.sec3.sec1.thm1.proof.eq2a}\\ \Phi_{2}&=&2\mu-\left[\frac{\beta }{2\lambda{(\mu)}}+\frac{\beta S^{*}_{1}\lambda{(\mu)}}{2}+ \frac{2\mu}{\lambda{(\mu)}}+(2\mu+d+\alpha)\frac{\lambda{(\mu)}}{2}+\alpha \lambda{(\mu)}\right],\label{ch1.sec3.sec1.thm1.proof.eq2b}\\ \Phi_{3}&=&(\mu+d+\alpha)-\left[(2\mu+d+\alpha)\frac{1}{\lambda{(\mu)}}+ \frac{\alpha\lambda{(\mu)}}{2}+\frac{3\alpha}{2\lambda(\mu)} \right.\nonumber\\ &&\left.+ \frac{\beta S^{*}_{1}(G'(I^{*}_{1}))^{2}}{\lambda{(\mu)}}\right].\label{ch1.sec3.sec1.thm1.proof.eq2c} \end{aligned}$$ It follows that there is a unique stationary distribution $\pi(.)$ for the solutions of (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), whenever $$\label{ch1.sec3.sec1.thm1.eq2} \Phi<\min{[\Phi_{1}(S^{*}_{1})^{2}, \Phi_{2}(E^{*}_{1})^{2}, \Phi_{3}(I^{*}_{1})^{2}]},$$ and $$\label{ch1.sec3.sec1.thm1.eq2-1} R<R_{min},\quad and\quad R_{min}+\Phi<||E_{1}-0||^{2}.$$ Moreover, the solution of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) is ergodic. Proof:\ The results will be shown for the vector $X=(S, E, I)$ corresponding to the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]). Furthermore, the hypothesis $\mathbf{H1}$ in Lemma \[ch1.sec3.sec1.lemma1\] is verified in the following. It is easy to see that when the conditions in (\[ch1.sec3.thm3.eq1\]) given in Theorem \[ch2.sec3.thm3\] are satisfied, it follows from (\[ch1.sec3.sec1.thm1.proof.eq2a\])-(\[ch1.sec3.sec1.thm1.proof.eq2c\]) that $$\label{ch1.sec3.sec1.thm1.proof.eq2d} 2(G^{*})^{2}\sigma^{2}_{\beta} +3\sigma^{2}_{S}<\Phi_{1},\quad 2\sigma^{2}_{E}<\Phi_{2}\quad and\quad \sigma^{2}_{I}<\Phi_{3}.$$ In addition, the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) can be rewritten in the form (\[ch1.sec3.sec1.eq1\]), where $d=3$, and the diffusion matrix $ A(X)=(a_{ij}(x))$ is defined as follows: $$\label{ch1.sec3.sec1.thm1.proof.eq3} \left\{ \begin{array}{ccc} a_{11} & = &\sigma^{2}_{S}S^{2}+\sigma^{2}_{\beta}S^{2}\theta^{2}_{1}, \\ a_{12} & = &\sigma^{2}_{\beta}S\theta_{1}\theta_{2}-\sigma^{2}_{\beta}S^{2}\theta^{2}_{1}, \\ a_{13} & = &-\sigma^{2}_{\beta}S\theta_{1}\theta_{2}, \end{array} \right.$$ $$\label{ch1.sec3.sec1.thm1.proof.eq4} \left\{ \begin{array}{lll} a_{21} & = &a_{12},\\%\sigma^{2}_{S}S^{2}+\sigma^{2}_{\beta}S^{2}\theta^{2} \\ a_{22} & = &\sigma^{2}_{E}E^{2}+\sigma^{2}_{\beta}S^{2}\theta^{1}-2\sigma^{2}_{\beta}S\theta_{1}\theta_{2}+\sigma^{2}_{\beta}\theta^{2}_{2}, \\ a_{23} & = &\sigma^{2}_{\beta}S\theta_{1}\theta_{2}-\sigma^{2}_{\beta}\theta^{2}_{2}, \end{array} \right.$$ and $$\label{ch1.sec3.sec1.thm1.proof.eq5} a_{31}=a_{13}, a_{32}=a_{32}, a_{33}=\sigma^{2}_{I}I^{2}+\sigma^{2}_{\beta}\theta^{2}_{2}.$$ Also, define the sets $U_{1}$ and $U_{2}$ as follows $$\label{ch1.sec3.sec1.thm1.proof.eq6} U_{1}=\left\{(S, E, I)\in \mathbb{R}^{3}_{+}| \mathfrak{m}_{2}(S-S^{*}_{1})^{2}+\mathfrak{m}_{2}(E-E^{*}_{1})^{2}+\mathfrak{m}_{2}(I-I^{*}_{1})^{2}\leq \Phi\right\}$$ and $$\begin{aligned} %<\frac{1}{6}\left[\frac{\Phi_{1}}{\sigma^{2}_{\beta}}-2(G^{*})^{2}\right] U_{2}&=&\left\{(S, E, I)\in \mathbb{R}^{3}_{+}|S^{2}(\sigma^{2}_{S}-2\sigma^{2}_{\beta}\theta^{2}_{1})\geq 1,E^{2}\geq \frac{1}{\sigma^{2}_{E}}\left[1+\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2}\right], I^{2}\geq \frac{1}{\sigma^{2}_{I}}\left(1+2\sigma^{2}_{\beta}\theta^{2}_{2}\right) \right\}.\nonumber\\ \label{ch1.sec3.sec1.thm1.proof.eq7} \end{aligned}$$ One can see from (\[ch1.sec3.sec1.thm1.proof.eq2d\]) that $$\begin{aligned} U_{2} &\subset&\left\{(S, E, I)\in \mathbb{R}^{3}_{+}| S^{2}>\frac{1}{\left(\frac{1}{3}\Phi_{1}- \sigma^{2}_{\beta}\left(\frac{2}{3}(G^{*})^{2}+2\theta^{2}_{1}\right)\right)},\quad E^{2}>\frac{2}{\Phi_{2}}\left(1+\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2}\right),\quad\right.\nonumber\\ &&\left. I^{2}>\frac{1}{\Phi_{3}}\left(1+2\sigma^{2}_{\beta}\theta^{2}_{2}\right) \right\},\nonumber\\ &\subset& \left(\bar{B}_{\mathbb{R}^{3}_{+}}(0; R)\right)^{c},\label{ch1.sec3.sec1.thm1.proof.eq7-1} \end{aligned}$$ where the set $\left(\bar{B}_{\mathbb{R}^{3}_{+}}(0; R)\right)^{c}$ is the complement of the closed ball or sphere in $\mathbb{R}^{3}_{+}$ centered at the origin $(S, E, I)=(0,0,0)$ with radius given by $$\label{ch1.sec3.sec1.thm1.proof.eq7-2} R=3\min_{(S, E, I)\in \mathbb{R}^{3}_{+}}{\left(\frac{1}{\left(\frac{1}{3}\Phi_{1}- \sigma^{2}_{\beta}\left(\frac{2}{3}(G^{*})^{2}+2\theta^{2}_{1}\right)\right)},\frac{2}{\Phi_{2}}\left(1+\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2}\right), \frac{1}{\Phi_{3}}\left(1+2\sigma^{2}_{\beta}\theta^{2}_{2}\right) \right)}<\infty.$$ In addition, the symbol “$\subset$” signifies the set operation of proper subset, and $\Phi$ and $\mathfrak{m}_{2}$ are defined in (\[ch1.sec3.sec1.thm1.eq1\]), Theorem \[ch2.sec3.thm3\] and (\[ch2.sec3.thm3.proof.eq5\]). Moreover, $ \mathfrak{m}_{2}$ is given as follows: $$\begin{aligned} % %\nonumber % Remove numbering (before each equation) % \Phi=3\sigma^{2}_{S}(S^{*}_{1})^{2}+ 2\sigma^{2}_{E}(E^{*}_{1})^{2}+\sigma^{2}_{I}(I^{*}_{1})^{2}+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-2\mu (T_{1}+T_{2})})+\sigma^{2}_{\beta}(S^{*}_{1})^{2}(G^{*})^{2}E(e^{-\mu T_{1}}) \nonumber\\ % \label{ch1.sec3.sec1.thm1.proof.eq7a}\\ \mathfrak{m}_{2}=\min\{\left(3\mu-\tilde{a}_{1}(\mu, d, \alpha, \beta, B)\right), \left(2\mu-a_{3}(\mu, d, \alpha, \beta, B)\right), \left((\mu + d+\alpha)-\tilde{a}_{2}(\mu, d, \alpha, \beta, B)\right)\}.\nonumber\\ \label{ch1.sec3.sec1.thm1.proof.eq7b} \end{aligned}$$ It is easy to see that when the conditions in (\[ch1.sec3.thm3.eq1\]) which are presented in Theorem \[ch2.sec3.thm3\] are satisfied, then $\mathfrak{m}_{2}>0$. Also, observe that $U_{1}$ defined in (\[ch1.sec3.sec1.thm1.proof.eq6\]) represents the interior and boundary of a sphere in $\mathbb{R}^{3}_{+}$ with radius $\Phi$, centered at the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1}, I^{*}_{1})$. Furthermore, when (\[ch1.sec3.sec1.thm1.eq2\]) holds, it is easy to see that $$\label{ch1.sec3.sec1.thm1.proof.eq7c} \Phi\ll \Phi_{1}(S^{*}_{1})^{2}+ \Phi_{2}(E^{*}_{1})^{2}+ \Phi_{3}(I^{*}_{1})^{2}<\max{(\Phi_{1}, \Phi_{2}, \Phi_{3})}||E_{1}-0||^{2}.$$ Thus, from (\[ch1.sec3.sec1.thm1.proof.eq7c\]), the set $U_{1}$ is non-empty and totally enclosed in $\mathbb{R}^{3}_{+}$. Furthermore, the endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})\in U_{2}$. Indeed, it can be seen from the Assumption \[ch1.sec0.assum1\] that $\theta_{1}\leq I^{*}_{1}$ and $\theta_{2}\leq S^{*}_{1}I^{*}_{1}$, whenever $X(t)=E_{1}$. Therefore, since $R<R_{min}$ from (\[ch1.sec3.sec1.thm1.eq2-1\]), it implies that $$\label{ch1.sec3.sec1.thm1.proof.eq7c-1} E_{1}\in \left(\bar{B}_{\mathbb{R}^{3}_{+}}(0; R_{min})\right)^{c}\subset \left(\bar{B}_{\mathbb{R}^{3}_{+}}(0; R)\right)^{c}.$$ One also notes that the set $U_{1}$ does not overlap or intersect with the closed ball $\bar{B}_{\mathbb{R}^{3}_{+}}(0; R)$, since $R_{min}+\Phi<||E_{1}-0||^{2}$ from (\[ch1.sec3.sec1.thm1.eq2-1\]). That is, $U_{1}\subset \left(\bar{B}_{\mathbb{R}^{3}_{+}}(0; R)\right)^{c}$. Now, define the new set $$\label{ch1.sec3.sec1.thm1.proof.eq7d} U=U_{1}\cap U_{2}.$$ Clearly, $U\neq \emptyset$ since $E_{1}\in U$. Also, from (\[ch1.sec3.sec1.thm1.proof.eq7d\]) the set $U\subset U_{1}\subset \mathbb{R}^{3}_{+}$ and $U\subset U_{2}\subset \mathbb{R}^{3}_{+}$, thus, the set $U$ is totally enclosed in $\mathbb{R}^{3}_{+}$, and hence, the set $U$ is well-defined. Now, let $(S, E, I)\in U=U_{1}\cap U_{2}$. It follows from (\[ch1.sec3.sec1.eq2\]) and (\[ch1.sec3.sec1.thm1.proof.eq3\])-(\[ch1.sec3.sec1.thm1.proof.eq5\]), that $$\begin{aligned} % \nonumber % Remove numbering (before each equation) \sum^{3}_{ij}a_{ij}(x)\xi_{i}\xi_{j}=&& \sigma^{2}_{I}S^{2}\xi^{2}_{1} +\sigma^{2}_{E}E^{2}\xi^{2}_{2} +\sigma^{2}_{I}I^{2}\xi^{2}_{3}\nonumber\\ &&+\sigma^{2}_{\beta}S^{2}\theta^{2}_{1}(\xi_{1}-\xi_{2})^{2}+\sigma^{2}_{\beta}\theta^{2}_{2}(\xi_{2}-\xi_{3})^{2}-2\sigma^{2}_{\beta}S\theta_{1}\theta_{2}\xi^{2}_{2}\nonumber\\ &&+2\sigma^{2}_{\beta}S\theta_{1}\theta_{2}\xi_{1}\xi_{2}+2\sigma^{2}_{\beta}S\theta_{1}\theta_{2}\xi_{2}\xi_{3}-2\sigma^{2}_{\beta}S\theta_{1}\theta_{2}\xi_{1}\xi_{3}.\label{ch1.sec3.sec1.thm1.proof.eq8} \end{aligned}$$ By applying the following algebraic inequalities $\min_{a,b}{(-2ab,2ab)}>-a^{2}-b^{2}$, to $\sum^{3}_{ij}a_{ij}(x)\xi_{i}\xi_{j}, \forall \xi=(\xi_{1},\xi_{2})$, it is easy to see that (\[ch1.sec3.sec1.thm1.proof.eq8\]) becomes the following: $$\begin{aligned} % \nonumber % Remove numbering (before each equation) \sum^{3}_{ij}a_{ij}(x)\xi_{i}\xi_{j}\geq&& (\sigma^{2}_{S}S^{2}-2\sigma^{2}_{\beta}S^{2}\theta^{2}_{1})\xi^{2}_{1} +(\sigma^{2}_{E}E^{2}-\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2})\xi^{2}_{2}\nonumber\\ &&+(\sigma^{2}_{I}I^{2}-2\sigma^{2}_{\beta}\theta^{2}_{2})\xi^{2}_{3}+\sigma^{2}_{\beta}S^{2}\theta^{2}_{1}(\xi_{1}-\xi_{2})^{2}\nonumber\\ &&+\sigma^{2}_{\beta}\theta^{2}_{2}(\xi_{2}-\xi_{3})^{2}.\label{ch1.sec3.sec1.thm1.proof.eq9} \end{aligned}$$ Define the constant $M$ as follows $$\label{ch1.sec3.sec1.thm1.proof.eq10} M=\min{[(\sigma^{2}_{S}S^{2}-2\sigma^{2}_{\beta}S^{2}\theta^{2}_{1}),(\sigma^{2}_{E}E^{2}-\sigma^{2}_{\beta}(S\theta_{1}+\theta_{2})^{2}),(\sigma^{2}_{I}I^{2}-2\sigma^{2}_{\beta}\theta^{2}_{2}) ]}.$$ Clearly, from (\[ch1.sec3.sec1.thm1.proof.eq7\]), one can see that $M\geq 1>0, \forall (S,E,I)\in U$. It is also seen that by taking $D$ to be a neighborhood of $U$, then for $\forall (S, E, I)\in \bar{D}$, it follows from (\[ch1.sec3.sec1.thm1.proof.eq9\])-(\[ch1.sec3.sec1.thm1.proof.eq10\]) that $$\label{ch1.sec3.sec1.thm1.proof.eq10a} \sum^{3}_{ij}a_{ij}(x)\xi_{i}\xi_{j}\geq M|\xi|^{2},$$ where $\bar{D}$ is the closure of the set $D$. The hypothesis $\mathbf{H2}$ in Lemma \[ch1.sec3.sec1.lemma1\] follows from (\[ch1.sec3.lemma1.eq7\]), where it can be seen that $$\label{ch1.sec3.sec1.thm1.proof.eq10b} LV(t)<-\mathfrak{m}_{2}(S-S^{*}_{1})^{2}-\mathfrak{m}_{2}(E-E^{*}_{1})^{2}-\mathfrak{m}_{2}(I-I^{*}_{1})^{2}+ \Phi.$$ Furthermore, from (\[ch1.sec3.sec1.thm1.proof.eq6\])-(\[ch1.sec3.sec1.thm1.proof.eq7d\]), it follows that $LV(t)<0$, for all $X=(S, E, I)\in \mathbb{R}^{3}_{+}\backslash D$. Since the hypotheses $\mathbf{H1}$ and $\mathbf{H2}$ hold, it implies from Lemma \[ch1.sec3.sec1.lemma1\] that the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) has stationary Ergodic solutions. \[ch1.sec3.sec1.thm1.rem1\] Theorem \[ch1.sec3.sec1.thm1\] signifies that the positive solution process $Y(t)=(S(t), E(t), I(t), R(t))\in R_{+}^{4}\times [t_{0}, \infty)$ of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) converges in distribution to a unique random variable (for instance denoted $Y_{s}=(S_{s}, E_{s}, I_{s}, R_{s})$) that has the stationary distribution $\pi(.)$ in the neighborhood of the endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1}, R^{*}_{1})$, whenever the conditions of Theorem \[ch1.sec3.thm1\], Theorem \[ch2.sec3.thm3\] and (\[ch1.sec3.sec1.thm1.eq2\])-(\[ch1.sec3.sec1.thm1.eq2-1\]) are satisfied. The stationary feature of the solutions process $\{Y(t),t\geq t_{0}\}$ ensures that all the statistical properties such as the mean, variance, and moments etc. remain the same over time for every random vector $Y(t)$, whenever $t\geq t_{0}$ is sufficiently large and fixed. In other words, the ensemble mean, variance and moments etc. for the solution process $\{Y(t),t\geq t_{0}\}$ exist for sufficiently large time, and they are constant, and also correspond to the mean, variance and moments etc. of the stationary distribution $\pi(.)$. The ergodic feature of the solution process $\{Y(t),t\geq t_{0}\}$ also allows for insights about the statistical properties of the entire process via knowledge of the sample paths over sufficiently large amount of time. For example, over sufficiently large time, the average value of the sample path given by $\hat{\mu}_{f}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}f(Y(t))dt$ accurately estimates the corresponding ensemble mean given by $\mu_{f}=E(f(Y(t)))=\int_{E_{d}}f(y)\pi(dy)$, where $f(.)$ is an integrable function with respect to the measure $\pi$. In particular, $\hat{\mu}_{S}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}S(t)dt$ estimates the ensemble mean $\mu_{S}=E(S(t))=\int_{E_{d}}S\pi(dS)$, and $\hat{\mu}_{I}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}I(t)dt$ estimates the ensemble mean $\mu_{S}=E(I(t))=\int_{E_{d}}I\pi(dI)$ etc. over sufficiently large values of $t\geq t_{0}$. Note that in the absence of explicit sample path solutions of the nonlinear stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), the stationary distribution $\pi(.)$ is numerically approximated in Section \[ch1.sec4\] for specified sets of system parameters of the stochastic system. Example {#ch1.sec4} ======= In this study, the examples exhibited in this section are used to facilitate understanding about the influence of the intensity or “strength” of the noise in the system on the persistence of the disease in the population, and also to illustrate the existence of a stationary distribution $\pi(.)$ for the states of the system. These objectives are achieved in a simplistic manner by examining the behavior of the sample paths of the different states ($S, E, I, R$) of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) in the neighborhood of the potential endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1}, R^{*}_{1})$ of the system, and also by generating graphical representations for the distributions of samples of observations of the states ($S, E, I, R$) of the system at a specified time $t\in [t_{0},\infty)$. Recall Theorem \[ch1.sec3.thm1.corrolary1\] asserts that potential endemic equilibrium $E_{1}$ exists, whenever the basic reproduction number $R^{*}_{0}>1$, where $R^{*}_{0}$ is defined in (\[ch1.sec2.lemma2a.corrolary1.eq4\]). It follows that when the conditions of Theorem \[ch1.sec3.thm1.corrolary1\] are satisfied, then the endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1}, R^{*}_{1})$ satisfies the following system $$\label{ch1.sec4.eq1} \left\{ \begin{array}{lll} &&B-\beta Se^{-\mu T_{1}}G(I)-\mu S+\alpha I e^{-\mu T_{3}}=0\\ &&\beta Se^{-\mu T_{1}}G(I)-\mu E -\beta Se^{-\mu (T_{1}+T_{2})}G(I)=0\\ &&\beta Se^{-\mu (T_{1}+T_{2})}G(I)-(\mu+d+\alpha)I=0\\ &&\alpha I-\mu R-\alpha I e^{-\mu T_{3}}=0 \end{array} \right.$$ Example 1: Effect of the intensity of white noise on the persistence of the disease {#ch1.sec4.subsec1} ------------------------------------------------------------------------------------ The following convenient list of parameter values in Table \[ch1.sec4.table2\] are used to generate and examine the paths of the different states of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), whenever $R^{*}_{0}>1$, and the intensities of the white noise processes in the system continuously increase. It is easily seen that for this set of parameter values, $R^{*}_{0}=80.7854>1$. Furthermore, the endemic equilibrium for the system $E_{1}$ is given as follows:- $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Disease transmission rate $\beta$ 0.6277 --------------------------------- ---------- ----------------------- Constant Birth rate $B$ $ \frac{22.39}{1000}$ Recovery rate $\alpha$ 0.05067 Disease death rate $d$ 0.01838 Natural death rate $\mu$ $0.002433696$ Incubation delay time in vector $T_{1}$ 2 units Incubation delay time in host $T_{2}$ 1 unit Immunity delay time $T_{3}$ 4 units : A list of specific values chosen for the system parameters for the examples in subsection \[ch1.sec4.subsec1\][]{data-label="ch1.sec4.table2"} The Euler-Maruyama stochastic approximation scheme[^2] is used to generate trajectories for the different states $S(t), E(t), I(t), R(t)$ over the time interval $[0,T]$, where $T=\max(T_{1}+T_{2}, T_{3})=4$. The special nonlinear incidence functions $G(I)=\frac{aI}{1+I}, a=0.05$ in [@gumel] is utilized to generate the numeric results. Furthermore, the following initial conditions are used $$\label{ch1.sec4.eq1} \left\{ \begin{array}{l l} S(t)= 10,\\ E(t)= 5,\\ I(t)= 6,\\ R(t)= 2, \end{array} \right. \forall t\in [-T,0], T=\max(T_{1}+T_{2}, T_{3})=4.$$ ### Effect of the intensity of white noise $\sigma_{\beta}$ on the persistence and permanence of the disease {#ch1.sec4.subsec1.1} ![(a-i), (b-i), (c-i) and (d-i) show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever the only source of the noise in the system is from the disease transmission rate, and the strength of the noise in the system is relatively small, that is, $\sigma_{i}=0.5, \forall i\in \{S, E, I, R, \beta\} $. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}=9.168214$, $\min{(E(t))}=4.946437$, $\min{(I(t))}=5.43688$ and $\min{(R(t))}=1.93282$. []{data-label="ch1.sec4.subsec1.1.figure 1"}](newpersistence-sigmas-beta-is-0_5-and-R_0-bigger-than-1.eps){height="6cm"} ![$(a-ii)$, $(b-ii)$, $(c-ii)$ and $(d-ii)$ show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever the only source of the noise in the system is from the disease transmission rate, and the strength of the noise in the system is relatively high, that is, $\sigma_{i}=5, \forall i\in \{S, E, I, R, \beta\} $. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}=3.614151$, $\min{(E(t))}=4.611832$, $\min{(I(t))}=5.43688$ and $\min{(R(t))}=1.93282$. []{data-label="ch1.sec4.subsec1.1.figure 2"}](newpersistence-sigmas-beta-is-5-and-R_0-bigger-than-1.eps){height="6cm"} ![$(a-iii)$, $(b-iii)$, $(c-iii)$ and $(d-iii)$ show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever the only source of the noise in the system is from the disease transmission rate, and the strength of the noise in the system is relatively higher, that is, $\sigma_{i}=10, \forall i\in \{S, E, I, R, \beta\} $. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}=1.217144$, $\min{(E(t))}=4.465998$, $\min{(I(t))}=5.43688$ and $\min{(R(t))}=1.93282$. []{data-label="ch1.sec4.subsec1.1.figure 3"}](newpersistence-sigmas-beta-is-10-and-R_0-bigger-than-1.eps){height="6cm"} The Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\] and Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\] can be used to examine the persistence and permanence of the disease in the human population exhibited in Theorem \[ch1.sec3.thm2\] and Theorem \[ch1.sec5.thm2\], respectively. The following observations are made:- (1) the occurrence of noise in the disease transmission rate, that is, $\sigma_{\beta}>0$, results in random fluctuations mainly in the susceptible and exposed states depicted in Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\](a-i)-(a-iii), (b-i)-(b-iii). No major oscillations are observed in the infectious and removed states $I(t)$ and $R(t)$. This observation is also more significant in the approximate uniform distributions observed for the $I(t)$ and $R(t)$ states at the time $t=900$ depicted in Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\](c-iv)-(c-vi), (d-iv)-(d-vi), based on 1000 sample points for $I(t)$ and $R(t)$ at the fixed time $t=900$ (that is, 1000 sample observations of $I(900)$ and $R(900)$). \(2) Increasing the intensity of the noise from the disease transmission rate, that is, as $\sigma_{\beta}$ increases from $0.5$ to $10$, it results in a rise in infection with many more people becoming exposed to the disease. This fact is depicted in Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\](a-i)-(a-iii), (b-i)-(b-iii), where a new higher maximum value for the trajectories of the exposed state $E(t)$ , and a new lower minimum value for the trajectory of the susceptible state $S(t)$ are attained over the time interval $[0,1000]$, respectively, across the figures as $\sigma_{\beta}$ increases from $0.5$ to $10$. Therefore, stronger noise in the disease dynamics from the disease transmission rate leads to more persistence of the disease. This observation about the persistence of disease is also significant in the approximate distributions for the susceptible and exposed states, $I(t)$ and $R(t)$, respectively, at the time $t=900$ depicted in Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\](a-iv)-(a-vi), (b-iv)-(b-vi), based on 1000 sample points for $I(t)$ and $R(t)$ at the fixed time $t=900$. Indeed, it can be seen from Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\](a-iv)-(a-vi), (b-iv)-(b-vi) that when the intensity of the noise in the system is $\sigma_{\beta}=0.5$, the distributions of $S(900)$ and $E(900)$ are closely symmetric with one peak, with the center for $S(900)$ approximately between $(9.5, 10.5)$, and about $97.3\%$ of the values between $(9.0,11.0)$. The center for $E(900)$ approximately between $(4.5, 5.5)$, and about $97.3\%$ of the values between $(4.0,6.0)$. Now, when the intensity increases to $\sigma_{\beta}=5$, and to $\sigma_{\beta}=10$, the distributions of $S(900)$ and $E(900)$ continuously become more sharply skewed, with $S(900)$ skewed the right with center ( utilizing the mode of $S(900)$) shifting to the left with majority of the possible values for $S(900)$ continuously decreasing in magnitude from approximately under the interval $(0, 30)$, to the interval $(0,20)$. These changes of the shape of the distribution, and decrease of the range of values in the support for the distribution of $S(900)$ as the the intensity rises from $\sigma_{\beta}=5$, and to $\sigma_{\beta}=10$, indicates that more susceptible tend to get infected at the time $t=900$ as the intensity of the noise rises. Similarly, when the intensity increases from $\sigma_{\beta}=0.5$ to $\sigma_{\beta}=5$, and also from $\sigma_{\beta}=5$ to $\sigma_{\beta}=10$, the distribution of $E(900)$ is skewed to the left with center ( utilizing the mode of $E(900)$) shifting to the right with majority of the possible values in the support for $E(900)$ continuously increasing in magnitude from under the interval $(0, 10)$, to the interval $(0,20)$. These changes of the shape of the distribution, and increase in the magnitude of the range of values in the the support for $E(900)$ as the the intensity rises from $\sigma_{\beta}=5$, to $\sigma_{\beta}=10$, indicates that more susceptible people tend to get infected and become exposed at the time $t=900$. (3)Finally, the remark about the influence of the strength of the noise on the stochastic permanence in the mean of the disease in Remark \[ch1.sec5.rem1\] can be examined using Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\], and Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\]. Recall, (\[ch1.sec5.rem1.eq2\]) in Remark \[ch1.sec5.rem1\] corresponding to Theorem \[ch1.sec5.thm2\] asserts that when the intensity of the noise $\sigma_{\beta}$ is infinitesimally small, a larger asymptotic lower bound $l_{a}(\sigma^{2}_{max})=l_{a}(\sigma^{2}_{\beta}),a\in \{S, E, I\}$ is attained for the average in time of the sample path of each state heavily influenced by the random fluctuations in the system, which in this scenario is the $S(t)$ and $E(t)$ states. Across the figures, as the intensity rises from $\sigma_{\beta}=0.5$, to $\sigma_{\beta}=10$ in Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\](a-i)-(a-iii), (b-i)-(b-iii), and Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 6\](a-iv)-(a-vi), (b-iv)-(b-vi), smaller minimum values for the paths of $S(t)$ and $E(t)$ are observed in Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 3\](a-i)-(a-iii), (b-i)-(b-iii). Also, the centers (utilizing the mean) for $S(900)$ and $E(100)$ continuously decrease in magnitude as the intensity rises from $\sigma_{\beta}=0.5$, and to $\sigma_{\beta}=10$. Indeed, this is evident since for the figures that are symmetric with a single peak, that is, Figures \[ch1.sec4.subsec1.1.figure 6\](a-iv)-(b-iv) corresponding to $\sigma_{\beta}=0.5$, the measures of the center (mean, mode, and median) are all approximately equal and higher in magnitude, while for the figures that are skewed ( to the left, or to the right), that is, Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 5\] ((b-iv) and (b-v) skewed to the left, and (a-iv), (a-v) skewed to the right), the position of the mean is also pulled further to the direction of the skew, which is relatively lower in magnitude compared to the observation in Figures \[ch1.sec4.subsec1.1.figure 6\]. And note that despite the fact that the approximate distribution of $S(900)$ is skewed to the right in Figures \[ch1.sec4.subsec1.1.figure 4\]-\[ch1.sec4.subsec1.1.figure 5\](a-iv)-(a-v), the mean continuously becomes smaller in magnitude as the intensity rises from $\sigma_{\beta}=0.5$ to $\sigma_{\beta}=10$. ![(a-iv), (b-iv), (c-iv) and (d-iv) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=900$, whenever the only source of noise in the system is from the disease transmission rate, and the intensity of the noise in the system is relatively higher, that is, $\sigma_{\beta}=10$. []{data-label="ch1.sec4.subsec1.1.figure 4"}](newhistogram-persistence-sigmas-beta-is-10-and-R_0-bigger-than-1.eps){height="6cm"} ![(a-v), (b-v), (c-v) and (d-v) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=900$, whenever the only source of noise in the system is from the disease transmission rate, and the intensity of the noise in the system is relatively high, that is, $\sigma_{\beta}=5$. []{data-label="ch1.sec4.subsec1.1.figure 5"}](newhistogram-persistence-sigmas-beta-is-5-and-R_0-bigger-than-1.eps){height="6cm"} ![(a-vi), (b-vi), (c-vi) and (d-vi) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=900$, whenever the only source of noise in the system is from the disease transmission rate, and the intensity of the noise in the system is relatively low, that is, $\sigma_{\beta}=0.5$. []{data-label="ch1.sec4.subsec1.1.figure 6"}](newhistogram-persistence-sigmas-beta-is-0_5-and-R_0-bigger-than-1.eps){height="6cm"} ### Joint effect of the intensities of white noise $\sigma_{i},i=S, E, I, \beta$ on the persistence of the disease {#ch1.sec4.subsec1.2} ![(a1), (b1), (c1) and (d1) show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever various sources of the noise in the system (i.e. natural death and disease transmission rates) are assumed to have the same strength or intensity (i.e. $\sigma_{i}=\sigma_{\beta}, i\in \{S, E, I, R\}$), and the strength of the noises in the system are relatively small, that is, $\sigma_{i}=0.5, \forall i\in \{S, E, I, R, \beta\} $. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}= 6.941153$, $\min{(E(t))}=3.693744$, $\min{(I(t))}=4.004954$ and $\min{(R(t))}= 1.384626$. []{data-label="ch1.sec4.figure 1"}](persistence-sigmas-is-0_5-and-R_0-bigger-than-1.eps){height="6cm"} ![(a2), (b2), (c2) and (d2) show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever the various sources of the noise in the system (i.e. natural death and disease transmission rates) are assumed to have the same strength or intensity (i.e. $\sigma_{i}=\sigma_{\beta}, i\in \{S, E, I, R\}$), and the strength of the noises in the system are relatively moderate, that is, $\sigma_{i}=1.5, \forall i\in \{S, E, I, R, \beta\} $. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}= 0.6116444$, $\min{(E(t))}=0.3496159$, $\min{(I(t))}=0.4439692$ and $\min{(R(t))}= -1.499425$. []{data-label="ch1.sec4.figure 2"}](persistence-sigmas-is-1_5-and-R_0-bigger-than-1.eps){height="6cm"} ![(a3), (b3), (c3) and (d3) show the trajectories of the disease states $(S,E,I,R)$ respectively, whenever the various sources of the noise in the system (i.e. natural death and disease transmission rates) are assumed to have the same strength or intensity (i.e. $\sigma_{i}=\sigma_{\beta}, i\in \{S, E, I, R\}$), and the strength of the noises in the system are relatively high, that is, $\sigma_{i}=2.5, \forall i\in \{S, E, I, R, \beta\}$. The broken lines represent the endemic equilibrium $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(0.2286216,0.07157075,0.9929282)$. Furthermore, $\min{(S(t))}=0.03717095$, $\min{(E(t))}=-2.204265$, $\min{(I(t))}=0.03397191$ and $\min{(R(t))}= -2.645903$. []{data-label="ch1.sec4.figure 3"}](persistence-sigmas-is-2_5-and-R_0-bigger-than-1.eps){height="6cm"} The Figures \[ch1.sec4.figure 1\]-\[ch1.sec4.figure 3\] can be used to examine the persistence of the disease in the human population as the intensities of the white noises in the system equally and continuously rise in value between $0.5$ to $2.5$, that is, for $\sigma_{i}=0.5, 1.5, 2.5, \forall i\in \{S, E, I, R, \beta\}$. It can be observed from Figure \[ch1.sec4.figure 1\] that when the intensity of noise is relatively small, that is, $\sigma_{S}=\sigma_{E}=\sigma_{\beta}=\sigma_{I}=\sigma_{R}=0.5$, the disease persists in the population with a significantly higher lower bounds for the disease classes $E, I$, and $R$. The lower bound for these disease classes are seen to continuously decrease in value as the magnitude of the intensities rise from $\sigma_{i}= 0.5$ to $\sigma_{i}= 1.5, \forall i\in \{S, E, I, R, \beta\}$, and also increase to $\sigma_{i}= 2.5, \forall i\in \{S, E, I, R, \beta\}$. This observation confirms the result in Theorems \[\[ch1.sec5.thm1\]&\[ch1.sec5.thm2\]\] and Remark \[ch1.sec5.rem1\], which asserts that an increase in the intensities of the noises in the system tends to lead to persistence of the disease with smaller lower bounds for the paths of the disease related classes, while a decrease in the strength of the noise in the system allows the disease to persist with a relatively higher lower margin for the paths of the disease related classes. It is important to note that the continuous decrease in the lower bounds for the paths of the disease related states $E, I, R$ are also matched with continuous decrease in the lower margin for the susceptible class exhibited in Figures \[ch1.sec4.figure 1\]-\[ch1.sec4.figure 3\] \[(a1), (a2), (a3)\]. This observation suggests that the population gets extinct overtime with continuous rise in the intensity of the noises in the system. ![(a4), (b4), (c4) and (d4) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=600$, whenever the intensities of the noises in the system are relatively small, that is, $\sigma_{E}=\sigma_{I}=\sigma_{R}=0.5$. []{data-label="ch1.sec4.figure 4"}](Histogram-persistence-sigmas-is-0_5-and-R_0-bigger-than-1.eps){height="6cm"} ![(a5), (b5), (c5) and (d5) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=600$, whenever the intensities of the noises in the system are relatively moderate, that is, $\sigma_{E}=\sigma_{I}=\sigma_{R}=1.5$. []{data-label="ch1.sec4.figure 5"}](Histogram-persistence-sigmas-is-1_5-and-R_0-bigger-than-1.eps){height="6cm"} ![(a6), (b6), (c6) and (d6) show the approximate distribution of the different disease states $(S,E,I,R)$ respectively, at the specified time $t=600$, whenever the intensities of the noises in the system are relatively high, that is, $\sigma_{E}=\sigma_{I}=\sigma_{R}=2.5$. []{data-label="ch1.sec4.figure 6"}](Histogram-persistence-sigmas-is-2_5-and-R_0-bigger-than-1.eps){height="6cm"} Figure \[ch1.sec4.figure 4\]-Figure \[ch1.sec4.figure 6\] provides a clearer picture about the effect of the rise in the intensity of the noises in the system on the persistence of the disease at any given time, for example, when the time is $t=600$. The statistical graphs in Figure \[ch1.sec4.figure 4\]-Figure \[ch1.sec4.figure 6\] are based on samples of 1000 simulation observations for the different states in the system $S, E, I$ and $R$ at the time $t=600$. For the susceptible population in Figure \[ch1.sec4.figure 4\]-Figure \[ch1.sec4.figure 6\]\[(a4)-a(6)\], it can be seen that the majority of possible values in the support for $S(600)$ occur in $(0,60)$. However, the frequency of these values dwindle with the rise in the intensity of the noises in the system from $\sigma_{i}= 0.5, \forall i\in \{S, E, I, R, \beta\}$ to $\sigma_{i}= 2.5, \forall i\in \{S, E, I, R, \beta\}$. Moreover, the much smaller values in the support for $S(600)$ tend to occur more frequently as is depicted in Figure \[ch1.sec4.figure 5\] and Figure \[ch1.sec4.figure 6\]. This observation suggests that the rise in the intensity of the noise in the system increases the probability of occurrence for smaller values in the support of the susceptible population state $S$ at time $t=200$, and this further suggests that more susceptible people tend to converted out of the susceptible state, either as a result of infection or natural death. Similar observations can be made for the disease related classes:- exposed($E$), infectious ($I$) and removal($R$) populations in Figure \[ch1.sec4.figure 4\]-Figure \[ch1.sec4.figure 6\]\[(b4)-b(6)\], \[(c4)-c(6)\] and \[(d4)-d(6)\] respectively. It can be seen that the majority of possible values in the support for $E(600)\leq 50$, $0\leq I(600)\leq 30$, and $R(600)\leq 20$. However, the frequency of these values also dwindle with the rise in the intensity of the noises in the system from $\sigma_{i}= 0.5, \forall i\in \{S, E, I, R, \beta\}$ to $\sigma_{i}= 2.5, \forall i\in \{S, E, I, R, \beta\}$. Moreover, much smaller values in the support for $I(600)$ tend to occur more frequently as is seen in Figure \[ch1.sec4.figure 5\] and Figure \[ch1.sec4.figure 6\], while negative values for $E(600)$ and $R(600)$ tend to occur the most for these disease related classes. This observation suggests that the rise in the intensity of the noise in the system increases the probability for smaller values in the support of the disease related states $E, I, R$ in the population at time $t=200$ to occur. This observation further suggests that more exposed people tend to be converted from the exposed state either as a result of more exposed people turning into full blown infectious individuals or natural death. For the infectious population, this observation suggests that more infectious people tend to be converted from the infectious state either because more infectious people tend to recover from the disease or die naturally, or die from disease related causes. Also, for the recovery class, the high probability of occurrence of smaller values in the support for $R(200)$ suggests that more removed individuals tend to be converted out of the state either because more naturally immune persons tend to lose their naturally acquired immunity and become susceptible again to the disease, or because they tend to die naturally. It is important to note that the occurrence of the negative values in the support for $E(600)$ and $R(600)$ with high probability values signifies that in many occasions, the exposed and removal populations become extinct. The numerical simulation results in the subsections \[\[ch1.sec4.subsec1.1\]-\[ch1.sec4.subsec1.2\]\] suggest and highlight an important fact that the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) exhibits higher sensitivity (such as high persistence of disease, tendency for the population to become extinct etc.) to changes in the magnitude of the intensities of the noises from the natural deathrates (i.e. $\sigma_{i}=0.5, 1.5, 2.5, \forall i\{S, E, I, R\})$ observed in Figures \[ch1.sec4.figure 1\]-\[ch1.sec4.figure 6\], compared to changes in the magnitude of the intensity of the noise from the disease transmission rate (i.e. $\sigma_{\beta}=0.5, 5, 10$) exhibited in Figures \[ch1.sec4.subsec1.1.figure 1\]-\[ch1.sec4.subsec1.1.figure 6\]. This fact can be easily seen in the occurrence and size of the oscillations in the sample paths of the different states $S, E, I, R$ represented in the figures, as the intensities continuously change values from small to high values. For example, the continuous changes in the intensity $\sigma_{\beta}=0.5, 5, 10$ result to a possibility for extinction of the human population only for very high intensity values, while small changes in the intensities $\sigma_{i}=0.5, 1.5, 2.5, \forall i\{S, E, I, R\})$ result to the possibility of extinction of the human population at disproportionately smaller magnitudes of the intensity values. This suggests that the growth rates of the intensities in the system influence the qualitative behavior of the trajectories of the system, and consequently impact the disease dynamics over time. This fact about growth rates of the intensities of the white noise processes in the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), and various classifications of the growth rates and the qualitative behavior of the disease dynamics is the central theme of the study Wanduku[@wanduku-theorBio]. Evidence of stationary distribution {#ch1.sec4.subsec2} ----------------------------------- The following new convenient set of parameter values in Table \[ch1.sec4.table3\] are used to examine the results of Theorem \[ch1.sec3.sec1.thm1\] about the existence of a stationary distribution for the different states of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), whenever $R^{*}_{0}>1$, and the delay times $T_{1}, T_{2}$ and $T_{3}$ are constant. It is easily seen that for this set of parameter values, $R^{*}_{0}=10.59709>1$. Furthermore, the endemic equilibrium for the system $E_{1}$ is given as follows:- $E_{1}=(S^{*}_{1},E^{*}_{1},I^{*}_{1})=(2.83334,0.2755608,2.521028)$. And, the parameter $\lambda(\mu)$ is taken to be $\lambda(\mu)=\mu e^{-\mu(T_{1}+T_{2})}$. Also, the intensities of the white noise processes in the system are $\sigma_{\beta}=2$ and $\sigma_{i}=0.2, \forall i\{S, E, I, R\})$. In addition, for the condition in (\[ch1.sec3.sec1.thm1.eq2\]), it is also easy to see that $\Phi=0.01223636$, and $$\min{[\Phi_{1}(S^{*}_{1})^{2}, \Phi_{2}(E^{*}_{1})^{2}, \Phi_{3}(I^{*}_{1})^{2}]}=\min{(2.697389, 0.03695982, 1.546866) }.$$ Thus, (\[ch1.sec3.sec1.thm1.eq2\]) is satisfied. The approximate distributions of the different disease states $S, E, I, R$ based on samples of 5000 simulation realizations, at the different times $t=600$, $t=700$ and $t=900$ are given in Figure \[ch1.sec4.figure 7\]-Figure \[ch1.sec4.figure 9\], respectively. Disease transmission rate $\beta$ $0.6277e^{-\mu(T_{1}+T_{2})}$ --------------------------------- ---------- -------------------------------------------- Constant Birth rate $B$ $ 1$ Recovery rate $\alpha$ $5.067\times 10^{-8}e^{-\mu(T_{1}+T_{2})}$ Disease death rate $d$ $1.838\times 10^{-8}e^{-\mu(T_{1}+T_{2})}$ Natural death rate $\mu$ $0.2433696$ Incubation delay time in vector $T_{1}$ 2 units Incubation delay time in host $T_{2}$ 1 unit Immunity delay time $T_{3}$ 4 units : A list of specific values chosen for the system parameters for the example in subsection \[ch1.sec4.subsec2\].[]{data-label="ch1.sec4.table3"} ![(a7), (b7), (c7) and (d7) depict an approximate distribution for the different disease states $(S,E,I,R)$ respectively, at the time when $t=600$. []{data-label="ch1.sec4.figure 7"}](statstionary-distribution-at-t-600.eps){height="6cm"} ![ (a8), (b8), (c8) and (d8) depict an approximate distribution for the different disease states $(S,E,I,R)$ respectively, at the time when $t=700$. []{data-label="ch1.sec4.figure 8"}](statstionary-distribution-at-t-700.eps){height="6cm"} ![ (a9), (b9), (c9) and (d9) depict an approximate distribution for the different disease states $(S,E,I,R)$ respectively, at the time when $t=900$. []{data-label="ch1.sec4.figure 9"}](statstionary-distribution-at-t-900.eps){height="6cm"} It is easy to see from Figure \[ch1.sec4.figure 7\]-Figure \[ch1.sec4.figure 9\] that the approximate distributions for the different states $S, E, I$ and $R$ have approximately common location, scale and shape parameters. These same conclusions can be reached about the location, scale and shape parameters of the corresponding distributions for the different states $S, E, I$ and $R$ obtained for all bigger times than $t=600$, that is, $\forall t\geq 600$. This clearly proves the existence of a stationary distribution for the states of the stochastic system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]), which is the limit of convergence (in distribution) of the sequence (collection) of distributions of the different states of the system, that is, $S(t), E(t), I(t), R(t)$, indexed by the time $t>0$, whenever the conditions in Theorem \[ch1.sec3.sec1.thm1\] are satisfied, and all possible simulation realizations are generated for the different states in the system at each time $t$. Furthermore, one can infer from the descriptive statistics of the approximated distributions presented in Figure \[ch1.sec4.figure 7\]-Figure \[ch1.sec4.figure 9\] that the location or centering parameters (e.g. ensemble mean) of the true stationary distribution for the susceptible population lies approximately in the interval \[6.5,8.0\]. Moreover, the shape of the distribution is approximately symmetric with a single peak. Similarly, for the disease related classes namely:- exposed, infectious and removal classes, one can infer that the shapes of the stationary distributions for each of these classes are approximately symmetric with a single peak, and the location or centering parameters lie approximately in the intervals $[2.45,2.85]$, $[2.8,3.6]$ and $[0.95,1.90]$, respectively. It is also important to note that whilst the statistical parameters such as the measures of center, variation and moments of the stationary distribution for the states of the system (\[ch1.sec0.eq8\])-(\[ch1.sec0.eq11\]) may change for each given set of values selected for the parameters of the system in Table \[ch1.sec4.table3\], the stationary distribution obtained is unique for that set of system parameter values. Conclusion ========== The presented family of stochastic malaria models with nonlinear incidence rates, random delays and environmental perturbations characterizes the general dynamics of malaria in a highly random environment with variability originating from (1.) the disease transmission rate between mosquitoes and humans, and also from (2.) the natural death rates of the susceptible, exposed, infectious and removal individuals of the human population. The random environmental fluctuations are formulated as independent white noise processes, and the malaria dynamics is expressed as a family of Ito-Doob type stochastic differential equations. Moreover, the family type is determined by a general nonlinear incidence function $G$ with various mathematical properties. The nonlinear incidence function $G$ can be used to describe the nonlinear character of malaria transmission rates in various disease scenarios where the rate of malaria transmission may initially increase or decrease, and become steady or bounded as the number of malaria cases increase in the population. This work furthers investigation about the malaria project initiated in Wanduku[@wanduku-biomath; @wanduku-comparative] and focuses on (1) the stochastic permanence of malaria, and (2) the existence of a stationary distribution for the random process describing the states of the disease over time. The investigation about these two aspects above centers on analyzing the behavior of the sample paths of the different states of the stochastic process in the neighborhood of the potential endemic equilibrium state of the dynamic system. Lyapunov functional techniques, and other local martingale characterizations are applied to characterize the trajectories of the solution process of the stochastic dynamic system. In addition, much emphasis is laid on analyzing the impacts of (a) the sources of the noises-disease transmission or natural death rates, and (b) the intensities of the noises on the permanence of malaria and the existence of a stationary distribution for the disease. Expansive and exhaustive discussions are presented to elucidate the qualitative character of the permanence of malaria and stationary distribution for the disease under the influence of the different sources, and different intensities of the noises in the system. The results of this study are illuminated by detailed numerical simulation examples that examine the trajectories of the states of the stochastic process in the neighborhood of the endemic equilibrium, and also under the influence of the different sources, and different intensities of the noises in the system. The numerical simulation results suggest that higher intensities of the white processes in the system drive the sample paths of the stochastic system further away from the potential endemic steady state. Moreover, the intensities of the noises from the natural death rates seem to have stronger consequences on the evolution of the disease in the system compared to the intensity of the noise from the disease transmission rate. Also, there is some evidence of a high chance of the population becoming extinct over time, whenever the intensities of the white noise processes become large. Furthermore, in the absence of explicit solution process for the nonlinear stochastic system of differential equations, the stationary distribution is numerically approximated for a given set of parameter values for the stochastic dynamic system of equations. References ========== [300]{} Y. Cai, Y. kang et al., a stochastic epidemic model incorporatinmg media coverage, commun. math sci, vol. 14, n0.4,(2016) 893-910 A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A Stochastic Differential Equation SIS Epidemic Model, SIAM J. Appl. Math., 71(3), (2011) 876–902 A. lahrouz, L. Omari, extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics & Probability Letters 83(4):(2013)960–968 D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, International Journal of Biomathematics Vol. 11, No. 6 (2018) 1850085 (46 pages), DOI: 10.1142/S1793524518500857 Li, Y., On the almost surely asymptotic bounds of a class of ornstein-Uhlenbeck Process in finifte dimensions, Journal of Systems Science and Complexity, 21 (2008), 416-426. X. Mao, stochastic differential equations and application, 2nd ed., WP, 2007 A. G. Ladde, G.S.Ladde, An introduction to differential equations: stochastic modelling, methods and analysis, vol 2, world scientific publishing, 2013 D. Wanduku, Modeling highly random dynamical infectious systems (book chapter), in press, 2017 D. Wanduku, Analyzing the qualitative properties of white noise on a family of infectious disease models in a highly random environment, available at arXiv:1808.09842 \[q-bio.PE\] D. Wanduku, A comparative stochastic and deterministic study of a class of epidemic dynamic models for malaria: exploring the impacts of noise on eradication and persistence of disease, in press, 2017 Y. Cai, Y. Kang, W. Wang, a stochastic SIRS epidemic model with nonlinear incidence, applied mathematics and computation 305 (2017)221-240 Y. zhang, k. Fan, S. Gao, S. Chen, A remark on stationary distributionof a stochastic SIR epidemic model with double saturated rates, applied mathamtics letters 76 (2018) 46-52 Cooke, Kenneth L. Stability analysis for a vector disease model. Rocky Mountain Journal of Mathematics 9 (1979), no. 1, 31-42 Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal. 42 (2000), 931-947. E. Beretta, V. Kolmanovskii, L. Shaikhet, Stability of epidemic model with time delay influenced by stochastic perturbations, Mathematics and Computers in Simulation 45 (1998) 269-277 Vincenzo Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics,Volume 97 1993 Liu WM, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J.Math Biol (1987); 25:359 Capasso V, Serio G.A. A generalization of the Kermack-Mckendrick deterministic epidemic model. Math Biosc 1978; 42:43 D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate. Math Biosci. 208(2):(2007) 419-29 Huo, Hai-Feng; Ma, Zhan-Ping, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Communications in Nonlinear Science and Numerical Simulation, Volume 15, Issue 2,(2010) p. 459-468. S.M. Moghadas, A.B. Gumel, Global Statbility of a two-stage epidemic model with generalized nonlinear incidence, Mathematics and computers in simulation 60 (2002), 107-118 Yuliya N. Kyrychko, Konstantin B. Blyussb, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Analysis: Real World Applications Volume 6, Issue 3, July 2005, Pages 495-507 Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi, Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence, Communications in Nonlinear Science and Numerical Simulation Volume 40, November 2016, Pages 89-99. Q. Liu, Q. Chen Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence Physica A, 428 (2015), pp. 140–153 Cooke KL, van den Driessche P., Analysis of an SEIRS epidemic model with two delays, J Math Biol. 1996 Dec;35(2):240-60. Nguyen Huu Du, Nguyen Ngoc Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters 64 (2017) 223-230 Joaquim P. Mateusa, , César M. Silvab, Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Analysis: Real World Applications Volume 34, April 2017, Pages 379-402 M. De la Sena, S. Alonso-Quesadaa, A. Ibeasb, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Applied Mathematics and Computation Volume 270, 1 November 2015, Pages 953-976 Zhichao Jianga, b, Wanbiao Mab, Junjie Wei, Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Mathematics and Computers in Simulation Volume 122, April 2016, Pages 35–54 Joaquim P. Mateus, César M. Silva, A non-autonomous SEIRS model with general incidence rate , Applied Mathematics and Computation Volume 247, 15 November 2014, Pages 169-189 M. De la Sen, S. Alonso-Quesada, , A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Applied Mathematics and Computation Volume 270, 1 November 2015, Pages 953-976 Shujing Gao, Zhidong Teng, Dehui Xie, The effects of pulse vaccination on SEIR model with two time delays, Applied Mathematics and Computation Volume 201, Issues 1-2, 15 July 2008, Pages 282-292 B. G. Sampath Aruna Pradeep , Wanbiao Ma, Global Stability Analysis for Vector Transmission Disease Dynamic Model with Non-linear Incidence and Two Time Delays, Journal of Interdisciplinary Mathematics, Volume 18, 2015 - Issue 4 Zhenguo Bai, Yicang Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Analysis: Real World Applications Volume 13, Issue 3, June 2012, Pages 1060-1068 Yanli Zhou, Weiguo Zhang, Sanling Yuan, Hongxiao Hu, Persistence And Extinction In Stochastic Sirs Models With General Nonlinear Incidence Rate, Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 42, pp. 1-17. Eric Avila, ValesBruno Buonomo, Analysis of a mosquito-borne disease transmission model with vector stages and nonlinear forces of infection, Ricerche di Matematica November 2015, Volume 64, Issue 2, pp 377-390 S. Syafruddin, Mohd Salmi Md. Noorani, Lyapunov function of SIR and SEIR model for transmission of dengue fever disease, International Journal of Simulation and Process Modelling (IJSPM), Vol. 8, No. 2-3, 2013 Liuyong Pang, Shigui Ruan, Sanhong Liu , Zhong Zhao , Xinan Zhang, Transmission dynamics and optimal control of measles epidemics, Applied Mathematics and Computation 256 (2015) 131–147 Ling ZhuHongxiao Hu, A stochastic SIR epidemic model with density dependent birth rate, Advances in Difference Equations December 2015, 2015:330 G.S. Ladde, V. Lakshmikantham, Random Differential Inequalities, Academic press, New York, 1980 R. Ross, The Prevention of Malaria, John Murray, London, 1911. Macdonald, G., The analysis of infection rates in diseases in which superinfection occurs. Trop. Dis. Bull. 47, (1950) 907-915 G. A. Ngwa, W. Shu, A mathematical model for endemic malaria with variable human and mosquito population, Math. computer. model. 32, (2000)747-763 Hyun, M.Y., Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector). Rev. Saude Publica 2000, 34 (3), 223–231 Anderson, R.M., May, R.M., Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford, 1991. G.A. Ngwa, A.M. Niger, A.B. Gumel, Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector, Appl. Math. Comput. 217 (2010) 3286. Ngonghala CN, Ngwa GA, Teboh-Ewungkem MI, Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission. Math Biosci, 240(1):(2012)45-62 N. Chitnis, J.M. Hyman, J.M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol. 70 (2008) 1272. M.I. Teboh-Ewungkem, T. Yuster, A within-vector mathematical model of plasmodium falciparum and implications of incomplete fertilization on optimal gametocyte sex ratio, J. Theory Biol. 264 (2010) 273. D. Wanduku, Complete Global Analysis of a Two-Scale Network SIRS Epidemic Dynamic Model with Distributed Delay and Random Perturbation, Applied Mathematics and Computation Vol. 294 (2017) p. 49 - 76 D. Wanduku, G.S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model, nonlinear Analysis: Real World Applications 13(2012)794-816 D. Wanduku, G.S. Ladde, The global analysis of a stochastic two-scale Network Human Epidemic Dynamic Model With Varying Immunity Period, Journal of Applied Mathematics and Physics, 2017, 5, 1150-1173 D. Wanduku and G.S. Ladde Global Stability of Two-Scale Network Human Epidemic Dynamic Model, Neural, Parallel, and Scientific Computations 19 (2011) 65-90 D. Wanduku, G.S. Ladde , Fundamental Properties of a Two-scale Network stochastic human epidemic Dynamic model, Neural, Parallel, and Scientific Computations 19(2011) 229-270 M. Xuerong, Stochastic differential equations and applications Horwood Publishing Ltd.(2008), 2nd ed. G.S., Ladde, Cellular Systems-II. Stability of Campartmental Systems. Math. Biosci. 30(1976), 1-21 James M. Crutcher, Stephen L. Hoffman, Malaria, Chapter 83-malaria, Medical Microbiology, 4th edition, Galveston (TX): University of Texas Medical Branch at Galveston; 1996. http://www.who.int/denguecontrol/human/en/ https://www.cdc.gov/malaria/about/disease.html L. Hviid, Naturally acquired immunity to Plasmodium falciparum malaria ,Acta Tropica 95(3): October 2005, 270-5 Denise L. Doolan, Carlota Dobano,J. Kevin Baird, Acquired Immunity to Malaria, clinical microbiology reviews,Vol. 22, No. 1, Jan. 2009, p. 13–36 Yakui Xue And Xiafeng Duan, Dynamic Analysis Of An Sir Epidemic Model With Nonlinear Incidence Rate And Double Delays, International Journal Of Information And Systems SciencesVolume 7, Number 1, (2011) Pages 92–102 Yoshiaki Muroya , Yoichi Enatsu, Yukihiko Nakata,Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate, Journal of Mathematical Analysis and Applications Volume 377, Issue 1, 1 May 2011, Pages 1–14 W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (4) (1987) 359–380 A. Korobeinikov, P.K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (1) (2004) 57–60. Chen, L, Chen, J., Nonlinear Biologiical Dynamical System, beijing, Science Press, 1993. Y. Kuang, delay Differential Equations with Applications in population Dynamics, Academic Press, boston. 1993 D. Wanduku, G.S. Ladde, Global stability of a two-scale network SIR delayed epidemic dynamic model Proceedings of Dynamic Systems and Applications 6 (2012) 437–441 D. Wanduku, Two-Scale Network Epidemic Dynamic Model for Vector Borne Diseases, Proceedings of Dynamic Systems and Applications 6 (2016) 228–232 [^1]: Corresponding author. Tel: +14073009605. [^2]: A seed is set on the random number generator to reproduce the same sequence of random numbers for the Brownian motion in order to generate reliable graphs for the trajectories of the system under different intensity values for the white noise processes, so that comparison can be made to identify differences that reflect the effect of intensity values.

ArXiv

--- author: - 'Michael Shell,  John Doe,  and Jane Doe, [^1]' title: | Bare Advanced Demo of IEEEtran.cls for\ IEEE Computer Society Journals --- [Shell : Bare Advanced Demo of IEEEtran.cls for IEEE Computer Society Journals]{} Introduction {#sec:introduction} ============ demo file is intended to serve as a “starter file” for IEEE Computer Society journal papers produced under LaTeX using IEEEtran.cls version 1.8b and later. I wish you the best of success. mds August 26, 2015 Subsection Heading Here ----------------------- Subsection text here. ### Subsubsection Heading Here Subsubsection text here. Conclusion ========== The conclusion goes here. Proof of the First Zonklar Equation =================================== Appendix one text goes here. Appendix two text goes here. Acknowledgments {#acknowledgments .unnumbered} =============== Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank... [1]{} H. Kopka and P. W. Daly, *A Guide to [LaTeX]{}*, 3rd ed.1em plus 0.5em minus 0.4emHarlow, England: Addison-Wesley, 1999. [Michael Shell]{} Biography text here. [John Doe]{} Biography text here. [Jane Doe]{} Biography text here. [^1]: Manuscript received April 19, 2005; revised August 26, 2015.

ArXiv

--- abstract: 'The quantum Zakharov system in three-spatial dimensions and an associated Lagrangian description, as well as its basic conservation laws are derived. In the adiabatic and semiclassical case, the quantum Zakharov system reduces to a quantum modified vector nonlinear Schrödinger (NLS) equation for the envelope electric field. The Lagrangian structure for the resulting vector NLS equation is used to investigate the time-dependence of the Gaussian shaped localized solutions, via the Rayleigh-Ritz variational method. The formal classical limit is considered in detail. The quantum corrections are shown to prevent the collapse of localized Langmuir envelope fields, in both two and three-spatial dimensions. Moreover, the quantum terms can produce an oscillatory behavior of the width of the approximate Gaussian solutions. The variational method is shown to preserve the essential conservation laws of the quantum modified vector NLS equation.' author: - 'F. Haas' - 'P. K. Shukla' title: Quantum and classical dynamics of Langmuir wave packets --- Introduction ============ The Zakharov system [@Zakharov], describing the coupling between Langmuir and ion-acoustic waves, is one of the basic plasma models, see Ref. [@Goldman; @Thornhill] for reviews. Recently [@Garcia], a quantum modified Zakharov system was derived, by means of the quantum plasma hydrodynamic model [@Haas]–[@HaasQMHD]. In this context, enhancement of the quantum effects was then shown [*e. g.*]{} to suppress the four-wave decay instability. Subsequently [@Marklund], a kinetic treatment of the quantum Zakharov system has shown that the modulational instability growth rate can be increased in comparison to the classical case, for partially coherent Langmuir wave electric fields. Also [@Haasvar], a variational formalism was obtained and used to study the radiation of localized structures described by the quantum Zakharov system. Bell shaped electric field envelopes of electron plasma oscillations in dense quantum plasmas obeying Fermi statistics were analyzed in Ref. [@Shukla]. More mathematically-oriented works on the quantum Zakharov equations concern its Lie symmetry group [@Tang] and the derivation of exact solutions [@Abdou]–[@Yang]. Finally, there is evidence of hyperchaos in the reduced temporal dynamics arising from the quantum Zakharov equations [@Misra]. All these paper refer to quantum Zakharov equations in one-spatial-dimension only. In the present work, we extend the quantum Zakharov system to fully three-dimensional space, allowing also for the magnetic field perturbation. In the classical case, both heuristic arguments and numerical simulations indicate that the ponderomotive force can produce finite-time collapse of Langmuir wave packets in two- or three-dimensions [@Goldman], [@Zakharov2; @Zakharov3]. This is in contrast to the one-dimensional case, whose solutions are smooth for all time. A dynamic rescaling method was used for the time-evolution of electrostatic self-similar and asymptotically self-similar solutions in two- and three-dimensions, respectively [@Landman]. Allowing for transverse fields shows that singular solutions of the resulting vector Zakharov equations are weakly anisotropic, for a large class of initial conditions [@Papanicolaou]. The electrostatic nonlinear collapse of Langmuir wave packets in the ionospheric and laboratory plasmas has been observed [@Dubois; @Robinson]. Also, the collapse of Langmuir wave packets in beam plasma experiments verifies the basic concepts of strong Langmuir turbulence, as introduced by Zakharov [@Cheung]. The analysis of the coupled longitudinal and transverse modes in the classical strong Langmuir turbulence has been less studied [@Alinejad]–[@Li], as well as the intrinsically magnetized case [@Pelletier], which can lead to upper-hybrid wave collapse [@Stenflo]. Finally, Zakharov-like equations have been proposed for the electromagnetic wave collapse in a radiation background [@Marklund2]. It is expected that the ponderomotive force causing the collapse of localized solutions in two- or three-space dimensions could be weakened by the inclusion of quantum effects, making the dynamics less violent. This conjecture is checked after establishing the quantum Zakharov system in higher-dimensional space and using its variational structure in association with a (Rayleigh-Ritz) trial function method. The manuscript is organized in the following fashion. In Section 2, the quantum Zakharov system in three-spatial-dimensions is derived by means of the usual two-time scale method applied to the fully 3D quantum hydrodynamic model. In Section 3, the 3D quantum Zakharov system is shown to be described by a Lagrangian formalism. The basic conservation laws are then also derived. When the density fluctuations are so slow in time so that an adiabatic approximation is possible, and treating the quantum term of the low-frequency equation as a perturbation, a quantum modified vector nonlinear Schrödinger equation for the envelope electric field is obtained. In Section 4, the variational structure is used to analyze the temporal dynamics of localized (Gaussian) solutions of this quantum NLS equation, through the Rayleigh-Ritz method, in two-spatial-dimensions. Section 5 follows the same strategy, extended to fully 3D space. Special attention is paid to the comparison between the classical and quantum cases, with considerable qualitative and quantitative differences. Section 6 contains conclusions. Quantum Zakharov equations in $3+1$ dimensions ============================================== The starting point for the derivation of the electromagnetic quantum Zakharov equations is the quantum hydrodynamic model for an electron-ion plasma, Equations (20)-(28) of Ref. [@HaasQMHD]. For the electron fluid pressure $p_e$, consider the equation of state for spin $1/2$ particles at zero temperature, $$\label{e1} p_e = \frac{3}{5}\,\frac{m_{e}v_{Fe}^2 \,n_{e}^{5/3}}{n_{0}^{2/3}} \,,$$ where $m_e$ is the electron mass, $v_{Fe}$ is the Fermi electron thermal speed, $n_e$ is the electron number density and $n_0$ is the equilibrium particle number density both for electron and ions. The pressure and quantum effects (due to their larger mass) are neglected for the ions. Also due to the larger ion mass, it is possible to introduce a two-time scale decomposition, $n_e = n_0 + \delta n_s + \delta n_f$, $n_i = n_0 + \delta n_s$, ${\bf u}_e = \delta{\bf u}_s + \delta{\bf u}_f$, ${\bf u}_i = \delta{\bf u}_s$, ${\bf E} = \delta{\bf E}_s + \delta{\bf E}_f$, ${\bf B} = \delta{\bf B}_f$, where the subscripts $s$ and $f$ refer to slowly and rapidly changing quantities, respectively. Also, ${\bf u}_e$ is the electron fluid velocity, $n_i$ the ion number density, ${\bf u}_i$ the ion fluid velocity, ${\bf E}$ the electric field, and ${\bf B}$ the magnetic field. Notice that it is assumed that there is no slow contribution to the magnetic field, a restriction which allows to get ${\bf B} = (m_{e}/e)\,\nabla\times\delta{\bf u}_f$ (see Equation (2.21) of Ref. [@Thornhill]), where $-e$ is the electron charge. Including a slow contribution to the magnetic field could be an important improvement, but this is outside the scope of the present work. Following the usual approximations [@Thornhill; @Garcia], the quantum corrected 3D Zakharov equations read $$\begin{aligned} \label{e2} 2i\omega_{pe}\frac{\partial{\bf\tilde{E}}}{\partial t} &-& c^2\, \nabla\times(\nabla\times{\bf\tilde{E}}) + v_{Fe}^2 \nabla(\nabla\cdot{\bf\tilde{E}}) = \nonumber \\ &=& \frac{\delta n_s}{n_0} \,\omega_{pe}^2 \,{\bf\tilde{E}} + \frac{\hbar^2}{4m_{e}^2}\nabla\left[\nabla^2 (\nabla\cdot{\bf\tilde{E}})\right] \,, \\ \label{e3} \frac{\partial^2 \delta n_s}{\partial t^2} &-& c_{s}^2 \,\nabla^2 \delta n_s - \frac{\varepsilon_0}{4m_i}\nabla^2 (|{\bf\tilde{E}}|^2) + \frac{\hbar^2}{4m_e m_i} \,\nabla^4 \delta n_s = 0 \,.\end{aligned}$$ Here ${\bf\tilde{E}}$ is the slowly varying envelope electric field defined via $${\bf E}_f = \frac{1}{2}\,({\bf\tilde{E}} \, e^{-i\omega_{pe}t} + {\bf\tilde{E}}^{*} \, e^{i\omega_{pe}t}) \,,$$ where $\omega_{pe}$ is the electron plasma frequency. Also, in Eqs. (\[e2\]–\[e3\]) $c$ is the speed of light in vacuum, $\hbar$ the scaled Planck constant, $\varepsilon_0$ the vacuum permittivity and $m_i$ the ion mass. In addition, $c_{s}^2 = \kappa_B T_{Fe}/m_i \,,$ where $\kappa_B T_{Fe} = m_e v_{Fe}^2$. Therefore, $c_s$ is a Fermi ion-acoustic speed, with the Fermi temperature replacing the thermal temperature for the electrons. In comparison to the classical Zakharov system (see Eqs. (2.48a)–(2.48b) of Ref. [@Thornhill]), there is the inclusion of the extra dispersive terms proportional to $\hbar^2$ in Eqs. (\[e2\])–(\[e3\]). Other quantum difference is the presence of the Fermi speed instead of the thermal speed in the last term at the left hand side of Eq. (\[e2\]). From the qualitative point of view, the terms proportional to $\hbar^2$ are responsible for extra dispersion which can avoid collapsing of Langmuir envelopes, at least in principle. This possibility is investigated in Sections 4 and 5. Finally, notice the non trivial form of the fourth order derivative term in Eq. (\[e2\]). It is not simply proportional to $\nabla^{4} {\bf\tilde{E}}$ as could be wrongly guessed from the quantum Zakharov equations in $1+1$ dimensions, where there is a $\sim \partial^{4}{\bf\tilde{E}}/\partial x^4$ contribution [@Garcia]. It is useful to consider the rescaling $$\begin{aligned} \label{e4} \bar{\bf r} &=& \frac{2\sqrt{\mu}\,\omega_{pe}\,{\bf r}}{v_{Fe}} \,, \quad \bar{t} = 2\,\mu\,\omega_{pe}t \,, \\ n &=& \frac{\delta n_s}{4\mu n_0} \,, \quad {\bf\cal E} = \frac{e\,\tilde{\bf E}}{4\,\sqrt{\mu}\,m_{e}\omega_{pe}v_{Fe}} \,, \nonumber\end{aligned}$$ where $\mu = m_{e}/m_{i}$. Then, dropping the bars in ${\bf r}, t$, we obtain $$\begin{aligned} \label{e5} i\frac{\partial{\bf\cal E}}{\partial t} &-& \frac{c^2}{v_{Fe}^2} \nabla\times(\nabla\times{\bf\cal E}) + \nabla(\nabla\cdot{\bf\cal E}) = \nonumber \\ &=& n \, {\bf\cal E} + \Gamma\,\nabla\left[\nabla^2 (\nabla\cdot{\bf\cal E})\right] \,, \\ \label{e6} \frac{\partial^2 n}{\partial t^2} &-& \nabla^2 n - \nabla^2 (|{\bf\cal E}|^2) + \Gamma\, \nabla^4 n = 0 \,, \end{aligned}$$ where $$\label{e7} \Gamma = \frac{m_e}{m_i}\left(\frac{\hbar\,\omega_{pe}}{\kappa_{B}T_{Fe}}\right)^2$$ is a non-dimensional parameter associated with the quantum effects. Usually, it is an extremely small quantity, but it is nevertheless interesting to retain the $\sim \Gamma$ terms, specially for the collapse scenarios. The reason is not only due to a general theoretical motivation, but also because from some simple estimates one concludes that these terms become of the same order as some of other terms in Eqs. (\[e2\])–(\[e3\]) provided that the characteristic length $l$ for the spatial derivatives becomes as small as the mean inter-particle distance, $l \sim n_{0}^{-1/3}$. Of course, the Zakharov equations are not able to describe the late stages of the collapse, since they do not include dissipation, which is unavoidable for short scales. But even Landau damping would be irrelevant for a zero-temperature Fermi plasma, where the main influence comes from the Pauli pressure. In the left-hand side of Eq. (\[e5\]), the $\nabla(\nabla\cdot{\bf\cal E})$ term is retained because the $\sim c^{2}/v_{Fe}^2$ transverse term disappears in the electrostatic approximation. In the adiabatic limit, neglecting $\partial^{2} n/\partial t^2$ in Eq. (\[e6\]) and under appropriated boundary conditions, it follows that $$\label{n} n = - |{\bf\cal E}|^2 + \Gamma\, \nabla^{2} n \,,$$ When $\Gamma \neq 0$, it is not easy to directly express $n$ as a function of $|{\bf\cal E}|$ as in the classical case. Therefore, the adiabatic limit is not enough to derive a vector nonlinear Schrödinger equation, due to the coupling in Eq. (\[n\]). Lagrangian structure and conservation $\!\!\!$ laws =================================================== The quantum Zakharov equations (\[e5\])–(\[e6\]) can be described by the Lagrangian density $$\begin{aligned} {\cal L} &=& \frac{i}{2}\,\Bigl(\,{\bf\cal E}^{*}\cdot\frac{\partial{\bf\cal E}}{\partial t} - {\bf\cal E}\cdot\frac{\partial{\bf\cal E}^{*}}{\partial t}\,\Bigr) - \frac{c^2}{v_{Fe}^2} |\nabla\times {\bf\cal E}|^2 - |\nabla\cdot{\bf\cal E}|^2 - \Gamma \,|\nabla(\nabla\cdot{\bf\cal E})|^2 \nonumber \\ \label{e8} &+& n\,\Bigl(\,\frac{\partial\alpha}{\partial t} - |{\bf\cal E}|^2\,\Bigr) - \frac{1}{2}\,\Bigl(n^2 + \Gamma |\nabla n|^2 + |\nabla\alpha|^2\Bigr) \,,\end{aligned}$$ where $n$, the auxiliary function $\alpha$ and the components of ${\bf\cal E}, {\bf\cal E}^{*}$ are regarded as independent fields. Remark: for the particular form (\[e8\]) and for a generic field $\psi$, one computes the functional derivative as $$\label{e9} \frac{\delta{\cal L}}{\delta\psi} = \frac{\partial{\cal L}}{\partial\psi} - \frac{\partial}{\partial r_i}\,\frac{\partial{\cal L}}{\partial\psi/\partial r_i} - \frac{\partial}{\partial t}\,\frac{\partial{\cal L}}{\partial\psi/\partial t} + \frac{\partial^2}{\partial r_{i}\,\partial r_j}\, \frac{\partial{\cal L}}{\partial^{2}\psi/\partial r_i \partial r_j} \,,$$ using the summation convention and where $r_i$ are cartesian components. Taking the functional derivatives with respect to $n$ and $\alpha$, we have $$\label{e10} \frac{\partial\alpha}{\partial t} = n + |{\bf\cal E}|^2 - \Gamma\nabla^2 n ,$$ and $$\label{e11} \frac{\partial n}{\partial t} = \nabla^2 \alpha \,,$$ respectively. Eliminating $\alpha$ from Eqs. (\[e10\]) and (\[e11\]) we obtain the low frequency equation. In addition, the functional derivatives with respect to ${\bf\cal E}^{*}$ and ${\bf\cal E}$ produce the high-frequency equation and its complex conjugate. The present formalism is inspired by the Lagrangian formulation of the classical Zakharov equations [@Gibbons]. The quantum Zakharov equations admit as exact conserved quantities the “number of plasmons" of the Langmuir field, $$\label{e12} N = \int |{\bf\cal E}|^2 \,d{\bf r} \,,$$ the linear momentum (with components $P_i, \, i = x, y, z$), $$\label{e13} P_i = \int \Bigl[\frac{i}{2} \left({\cal E}_j \,\frac{\partial{\cal E}^{*}_j}{\partial r_i} - {\cal E}^{*}_j \,\frac{\partial{\cal E}_j}{\partial r_i}\right) - n \,\frac{\partial\alpha}{\partial r_i} \Bigr] \,d{\bf r}$$ and the Hamiltonian, $$\begin{aligned} {\cal H} &=& \int \Bigl[n|{\bf\cal E}|^2 + \frac{c^2}{v_{Fe}^2}\, |\nabla\times{\bf\cal E}|^2 + |\nabla\cdot{\bf\cal E}|^2 + \Gamma\,|\nabla(\nabla\cdot{\bf\cal E})|^2 \nonumber \\ \label{e14} &+& \frac{1}{2}\,\Bigl(n^2 + \Gamma |\nabla n|^2 + |\nabla\alpha|^2\Bigr) \Bigr] \,d{\bf r} \,.\end{aligned}$$ Furthermore, there is also a preserved angular momenta functional, but it is not relevant in the present work. These four conserved quantities can be associated, through Noether’s theorem, to the invariance of the action under gauge transformation, time translation, space translation and rotations, respectively. The conservation laws can be used [*e. g.*]{} to test the accuracy of numerical procedures. Also, observe that equations (\[e6\]) and (\[n\]) for the adiabatic limit are described by the same Lagrangian density (\[e8\]). In this approximation, it suffices to set $\alpha \equiv 0$. In addition to the adiabatic limit, Eq. (\[n\]) can be further approximated to $$\label{e15} n = - |{\bf\cal{E}}|^2 - \Gamma \nabla^{2}(|{\bf\cal{E}}|^{2}) \,,$$ assuming that the quantum term is a perturbation. In this way and using Eq. (\[e5\]), a quantum modified vector nonlinear Schrödinger equation is derived $$\begin{aligned} i\frac{\partial{\bf\cal E}}{\partial t} &+& \nabla(\nabla\cdot{\bf\cal E}) - \frac{c^2}{v_{Fe}^2} \nabla\times(\nabla\times{\bf\cal E}) + |{\bf\cal{E}}|^2 {\bf\cal{E}} = \nonumber \\ \label{e16} &=& \Gamma\nabla\left[\nabla^2 (\nabla\cdot{\bf\cal E})\right] -\Gamma \, {\bf\cal E} \nabla^{2}(|{\bf\cal{E}}|^{2}) \,. \end{aligned}$$ The appropriate Lagrangian density ${\cal L}_{ad,sc}$ for the semiclassical equation (\[e16\]) is given by $$\begin{aligned} {\cal L}_{ad,sc} &=& \frac{i}{2}\,\Bigl(\,{\bf\cal{E}}^{*}\cdot\frac{\partial{\bf\cal{E}}}{\partial t} - {\bf\cal{E}}\cdot\frac{\partial{\bf\cal{E}}^{*}}{\partial t}\,\Bigr) - \frac{c^2}{v_{Fe}^2}|\nabla\times{\bf\cal{E}}|^2 - |\nabla\cdot{\bf\cal{E}}|^2 \nonumber \\ \label{e17} &-& \Gamma \,|\nabla(\nabla\cdot{\bf\cal{E}})|^2 + \frac{1}{2}\,|{\bf\cal{E}}|^4 - \frac{\Gamma}{2}\,\Bigl|\nabla[\,|{\bf\cal{E}}|^2] \Bigr|^2 \,,\end{aligned}$$ where the independent fields are taken as ${\bf\cal{E}}$ and ${\bf\cal{E}}^{*}$ components. The expression $N$ for the number of plasmons in Eq. (\[e12\]) remains valid as a constant of motion in the joint adiabatic and semiclassical limit, as well as the momentum ${\bf P}$ in Eq. (\[e13\]) with $\alpha \equiv 0$. Finally, the Hamiltonian $$\begin{aligned} {\cal H}_{ad,sc} = \int \Bigl[\,\frac{c^2}{v_{Fe}^2}\,|\nabla&\times&{\bf\cal{E}}|^2 + |\nabla\cdot{\bf\cal{E}}|^2 + \Gamma \,|\nabla(\nabla\cdot{\bf\cal{E}})|^2 \nonumber \\ \label{e18} &-& \frac{1}{2}\,|{\bf\cal{E}}|^4 + \frac{\Gamma}{2}\,\Bigl|\nabla[\,|{\bf\cal{E}}|^2\,] \Bigr|^2 \,\,\Bigr] \,d{\bf r} \end{aligned}$$ is also a conserved quantity. In the following, the influence of the quantum terms in the right-hand side of Eq. (\[e16\]) are investigated, assuming adiabatic conditions for collapsing quantum Langmuir envelopes. Other scenarios for collapse, like the supersonic one [@Landman; @Papanicolaou], could also be relevant and shall be investigated in the future. Variational solution in two dimensions ====================================== Consider the adiabatic semiclassical system defined by Eq. (\[e16\]). We refer to localized solution for this vector NLS equation as (quantum) “Langmuir wave packets", or envelopes. As discussed in detail in [@Gibbons] in the purely classical case, Langmuir wave packets will become singular in a finite time, provided the energy is not bounded from below. Of course, explicit analytic Langmuir envelopes are difficult to derive. A fruitful approach is to make use of the Lagrangian structure for deriving approximate solutions. This approach has been pursued in [@Malomed] for the classical and in [@Haasvar] for the quantum Zakharov system. Both studies considered the internal vibrations of Langmuir envelopes in one-spatial-dimension. Presently, we shall apply the time-dependent Rayleigh-Ritz method for the higher-dimensional cases. A priori, it is expected that the quantum corrections would inhibit the collapse of localized solutions, in view of wave-packet spreading. To check this conjecture, and to have more definite information on the influence of the quantum terms, first we consider the following [*Ansatz*]{}, $$\label{e19} {\bf\cal{E}} = \left(\frac{N}{\pi}\right)^{1/2}\,\frac{1}{\sigma}\,\exp\left(-\frac{\rho^2}{2\sigma^2}\right)\, \exp\left(i(\Theta + k\rho^2)\right)\,\,(\cos\phi, \sin\phi, 0) \,,$$ which is appropriate for two-spatial-dimensions. Here $\sigma, k, \Theta$ and $\phi$ are real functions of time, and $\rho = \sqrt{x^2+y^2}$. The normalization condition (\[e12\]) is automatically satisfied (in 2D the spatial integrations reduce to integrations on the plane). Other localized forms, involving [*e. g.*]{} a [*sech*]{} type dependence, could have been also proposed. Here a Gaussian form was suggested mainly for the sake of simplicity [@Fedele]. Notice that the envelope electric field (\[e19\]) is not necessarily electrostatic: it can carry a transverse ($\nabla\times{\bf\cal{E}} \neq 0$) component. The free functions in Eq. (\[e19\]) should be determined by extremization of the action functional associated with the Lagrangian density (\[e17\]). A straightforward calculation gives $$\begin{aligned} L_2 \equiv \int\,{\cal L}_{ad,sc}\,dx\,dy &=& - N \,\Bigl[\dot\Theta + \sigma^2 \dot{k} + \frac{2c^2}{v_{Fe}^2}\,k^2 \sigma^2 + \frac{1}{2}\,\left( \frac{c^2}{v_{Fe}^2} - \frac{N}{2\pi}\right)\,\frac{1}{\sigma^2} \nonumber \\ \label{e20} &+& 8\Gamma k^2 + 16\Gamma k^4 \sigma^4 + \left(1+\frac{N}{2\pi}\right)\,\frac{\Gamma}{\sigma^4}\,\Bigr] \,,\end{aligned}$$ where only the main quantum contributions are retained. Now $L_2$ is the Lagrangian for a mechanical system, after the spatial form of the envelope electric field was defined in advance via Eq. (\[e19\]). Of special interest is the behavior of the dispersion $\sigma$. For a collapsing solution one could expect that $\sigma$ goes to zero in a finite time. The phase $\Theta$ and the chirp function $k$ should be regarded as auxiliary fields. Notice that $L_2$ is not dependent on the angle $\phi$, which remains arbitrary as far as the variational method is concerned. Applying the functional derivative of $L_2$ with respect to $\Theta$, we obtain $$\label{e21} \frac{\delta L_2}{\delta\Theta} = 0 \quad \rightarrow \quad \dot{N} = 0 \,,$$ so that the variational solution preserves the number of plasmons, as expected. The remaining Euler-Lagrange equations are $$\begin{aligned} \label{e22} \frac{\delta L_2}{\delta k} = 0 \quad \rightarrow \quad \sigma\dot\sigma &=& \frac{2 c^2}{v_{Fe}^2}\,\sigma^2 k + 8\Gamma k + 32\Gamma\sigma^4 k^3 \,,\\ \frac{\delta L_2}{\delta\sigma} = 0 \quad \rightarrow \quad \sigma\dot{k} &=& - \frac{2c^2}{v_{Fe}^2}\,k^2 \sigma + \frac{1}{2}\,\left( \frac{c^2}{v_{Fe}^2} - \frac{N}{2\pi}\right)\,\frac{1}{\sigma^3} - 32\Gamma k^4 \sigma^3 \nonumber \\ \label{e23} &+& \left(1+\frac{N}{2\pi}\right)\,\frac{2\Gamma}{\sigma^5} \,.\end{aligned}$$ The exact solution of the nonlinear system (\[e22\]–\[e23\]) is difficult to obtain, but at least the dynamics was reduced to ordinary differential equations. It is instructive to analyze the purely classical ($\Gamma \equiv 0$) case first. This is specially true, since to our knowledge the Rayleigh-Ritz method was not applied to the vector NLS equation (\[e16\]), even for classical systems. The reason can be due to the calculational complexity induced by the transverse term. When $\Gamma = 0$, Eq. (\[e22\]) gives $k = v_{Fe}^2 \dot\sigma/2c^2 \sigma$. Inserting this in Eq. (\[e23\]) we have $$\label{e24} \ddot\sigma = - \frac{\partial V_{2c}}{\partial\sigma} \,,$$ where the pseudo-potential $V_{2c}$ is $$\label{e25} V_{2c} = \frac{c^2}{2v_{Fe}^2}\,\left( \frac{c^2}{v_{Fe}^2} - \frac{N}{2\pi}\right)\,\frac{1}{\sigma^2} \,.$$ From Eq. (\[e25\]) it is evident that the repulsive character of the pseudo-potential will be converted into an attractive one, whenever the number of plasmons exceeds a threshold, $$\label{e26} N > \frac{2\pi c^2}{v_{Fe}^2} \,,$$ a condition for Langmuir wave packet collapse in the classical two-dimensional case. The interpretation of the result is as follows. When the number of plasmons satisfy Eq. (\[e26\]), the refractive $\sim |{\bf\cal{E}}|^4$ term dominates over the dispersive terms in the Lagrangian density (\[e17\]), producing a singularity in a finite time. Finally, notice the ballistic motion when $N = 2\pi c^{2}/v_{Fe}^2$, which can also lead to singularity. Further insight follows after evaluating the energy integral (\[e18\]) with the [*Ansatz*]{} (\[e19\]), which gives, after eliminating $k$, $$\label{e27} {\cal H}_{ad,sc,2c} = \frac{N v_{Fe}^2}{c^2}\,\left[\frac{\dot\sigma^2}{2} + V_{2c}\right] \quad (\Gamma \equiv 0) \,.$$ Of course, this energy first integral could be obtained directly from Eq. (\[e24\]). However, the plausibility of the variational solution is reinforced, since Eq. (\[e27\]) shows that it preserves the exact constant of motion ${\cal H}_{ad,sc}$. In addition, in the attractive (collapsing) case the energy (\[e27\]) is not bounded from bellow. In the quantum ($\Gamma \neq 0$) case, Eq. (\[e22\]) becomes a cubic equation in $k$, whose exact solution is too cumbersome to be of practical use. It is better to proceed by successive approximations, taking into account that the quantum and electromagnetic terms are small. In this way, one arrives at $$\label{e28} \ddot\sigma = - \frac{\partial V_{2}}{\partial\sigma} \,,$$ where the pseudo-potential $V_{2}$ is $$\label{e29} V_{2} = \frac{c^2}{2v_{Fe}^2}\,\left( \frac{c^2}{v_{Fe}^2} - \frac{N}{2\pi}\right)\,\frac{1}{\sigma^2} + \frac{\Gamma c^2}{v_{Fe}^2}\,\left(1+\frac{N}{2\pi}\right)\,\frac{1}{\sigma^4}\,.$$ Now, even if the threshold (\[e26\]) is exceeded, the repulsive $\sim\sigma^{-4}$ quantum term in $V_2$ will prevent singularities. This adds quantum diffraction as another physical mechanism, besides dissipation and Landau damping, so that collapsing Langmuir wave packets are avoided in vector NLS equation. Also, similar to Eq. (\[e27\]), it can be shown that the approximate dynamics preserves the energy integral, even in the quantum case. Indeed, calculating from Eq. (\[e18\]) and the variational solution gives ${\cal H}_{ad,sc}$ as $${\cal H}_{ad,sc,2} = \frac{N v_{Fe}^2}{c^2}\,\left[\frac{\dot\sigma^2}{2} + V_{2}\right] \quad (\Gamma \geq 0) \,.$$ From Eq. (\[e28\]), obviously $\dot{\cal H}_{ad,sc,2} = 0$. It should be noticed that oscillations of purely quantum nature are obtained when the number of plasmons exceeds the threshold (\[e26\]). Indeed, in this case the pseudo-potential $V_2$ in Eq. (\[e29\]) assumes a potential well form as shown in Figure 1, which clearly admits oscillations around a minimum $\sigma = \sigma_{m}$. Here, $$\label{e30} \sigma_{m} = 2\,\left[\frac{\Gamma (1 + N/2\pi)}{N/2\pi - c^2/v_{Fe}^2}\right]^{1/2} \,.$$ Also, the minimum value of $V_2$ is $$V_{2}(\sigma_{m}) = - \frac{c^2}{16\Gamma\,v_{Fe}^2}\,\frac{(N/2\pi - c^2/v_{Fe}^2)^2}{1+N/2\pi} > - \frac{1}{16\Gamma}\,\left(\frac{N}{2\pi}-\frac{c^2}{v_{Fe}^2}\right)^2 \,,$$ the last inequality follows since Eq. (\[e26\]) is assumed. Therefore, a deepest potential well is obtained when $N$ is increasing. Also, for too large quantum effects the trapping of the localized electric field in this potential well would be difficult, since $V_{2}(\sigma_{m}) \rightarrow 0_{-}$ as $\Gamma$ increases. This is due to the dispersive nature of the quantum corrections. The frequency $\omega$ of the small amplitude oscillations is derived linearizing Eq. (\[e28\]) around the equilibrium point (\[e30\]). Restoring physical coordinates via Eq. (\[e4\]) this frequency is calculated as $$\begin{aligned} \omega &=& \frac{c}{\sqrt{2}\,v_{Fe}}\,\left(\frac{\kappa_{B}T_{Fe}}{\hbar\,\omega_{pe}}\right)^2\, \frac{(N/2\pi-c^{2}/v_{Fe}^{2})^{3/2}}{1+N/2\pi}\,\,\omega_{pe} \nonumber \\ \label{e31} &<& \frac{v_{Fe}}{\sqrt{2}\,c}\,\left(\frac{\kappa_{B}T_{Fe}}{\hbar\,\omega_{pe}}\right)^2\, \left(\frac{N}{2\pi}-\frac{c^2}{v_{Fe}^2}\right)^{3/2}\,\omega_{pe} \,.\end{aligned}$$  To conclude, the variational solution suggests that the extra dispersion arising from the quantum terms would inhibit the collapse of Langmuir wave packets in two-spatial-dimensions. Moreover, for sufficient electric field energy (which is proportional to $N$), instead of collapse there will be oscillations of the width of the localized solution, due to the competition between the classical refraction and the quantum diffraction. The frequency of linear oscillations is then given by Eq. (\[e31\]). The emergence of a pulsating Langmuir envelope is a qualitatively new phenomena, which could be tested quantitatively in experiments. Variational solution in three-dimensions ======================================== It is worth to study the dynamics of localized solutions for the vector NLS equation (\[e16\]) in fully three-dimensional space. For this purpose, we consider the Gaussian form $$\label{e32} {\bf\cal{E}} = \left(\frac{N}{(\sqrt{\pi}\,\sigma)^3}\right)^{1/2}\!\!\!\!\!\exp\left[-\frac{r^2}{2\sigma^2}\! +\!i(\Theta\!+\!k\,r^2)\right](\cos\phi\sin\theta, \sin\phi\sin\theta, \cos\theta) \,,$$ where $\sigma, k, \Theta, \theta$ and $\phi$ are real functions of time and $r = \sqrt{x^2+y^2+z^2}$, applying the Rayleigh-Ritz method just like in the last Section. The normalization condition (\[e12\]) is automatically satisfied with Eq. (\[e32\]), which, occasionally, can also support a transverse ($\nabla\times{\bf\cal{E}} \neq 0$) part. Proceeding as before, the Lagrangian $$\begin{aligned} L_3 \equiv \int\,{\cal L}_{ad,sc}\,d{\bf r} &=& - N \,\Bigl[\dot\Theta + \frac{3}{2}\,\sigma^2 \dot{k} + \frac{4\,c^2}{v_{Fe}^2}\,k^2 \sigma^2 + \frac{c^2}{v_{Fe}^2\,\sigma^2} - \frac{N}{4\sqrt{2}\,\pi^{3/2}\,\sigma^3} \nonumber \\ \label{e33} &+& 10\,\Gamma k^2 + 20\,\Gamma k^4 \sigma^4 + \frac{5\,\Gamma}{4\,\sigma^4} +\frac{3\,\Gamma N}{4\sqrt{2}\,\pi^{3/2}\,\sigma^5}\,\Bigr] \end{aligned}$$ is derived. In comparison to the reduced 2D-Lagrangian in Eq. (\[e20\]), there are different numerical factors as well as qualitative changes due to higher-order nonlinearities. Also, the angular variables $\theta$ and $\phi$ don’t appear in $L_3$. The main remaining task is to analyze the dynamics of the width $\sigma$ as a function of time. This is achieved from the Euler-Lagrange equations for the action functional associated to $L_3$. As before, $\delta L_{3}/\delta\Theta = 0$ gives $\dot{N}=0$, a consistency test satisfied by the variational solution. The other functional derivatives yield $$\begin{aligned} \label{e34} \frac{\delta L_3}{\delta k} = 0 \rightarrow \sigma\dot\sigma &=& \frac{4k}{3}\, \left[\frac{2\, c^2}{v_{Fe}^2}\,\sigma^2 + 5\Gamma\,(1+4k^2 \sigma^4)\right] \,,\\ \frac{\delta L_3}{\delta\sigma} = 0 \rightarrow \sigma\dot{k} &=& \frac{1}{3}\, \Bigl[- \frac{8\,c^2}{v_{Fe}^2}\,k^2 \sigma + \frac{2\,c^2}{v_{Fe}^2 \sigma^3} - \frac{3\,N}{4\,\sqrt{2}\,\pi^{3/2}\,\sigma^4} \nonumber \\ \label{e35} &-& 80\, \Gamma k^4 \sigma^3 + \frac{5\,\Gamma}{\sigma^5} + \frac{15\,\Gamma N}{4\,\sqrt{2}\,\pi^{3/2}\,\sigma^6}\Bigr] \,.\end{aligned}$$ In the formal classical limit ($\Gamma \equiv 0$), and using Eq. (\[e34\]) to eliminate $k$, we obtain $$\label{e36} \ddot\sigma = - \frac{\partial V_{3c}}{\partial\sigma} \,,$$ where now the pseudo-potential $V_{3c}$ is $$\label{e37} V_{3c} = \frac{c^2}{v_{Fe}^2}\,\left( \frac{8\,c^2}{9\,v_{Fe}^2\,\sigma^2} - \frac{2\,N}{9\,\sqrt{2}\,\pi^{3/2}\,\sigma^3}\right) \,.$$ The form (\[e37\]) shows a generic singular behavior, since the attractive $\sim \sigma^{-3}$ term will dominate for sufficiently small $\sigma$, irrespective of the value of $N$. Hence, in fully three-dimensional space there is more “room" for a collapsing dynamics. Figure 2 shows the qualitative form of $V_{3c}$, attaining a maximum at $\sigma = \sigma_M$, where $$\label{e38} \sigma_M = \frac{3\, v_{F}^2\,N}{8\,\sqrt{2}\,\pi^{3/2}\,c^2} \,.$$  By Eq. (\[e35\]) and using successive approximations in the parameter $\Gamma$ to eliminate $k$ via Eq. (\[e34\]), we obtain $$\label{ee} \ddot\sigma = - \frac{\partial V_{3}}{\partial\sigma} \,,$$ where $$\label{e39} V_3 = \frac{8\,c^2}{3\,v_{Fe}^2}\,\left[\frac{c^2}{3\,v_{Fe}^2\,\sigma^2} - \frac{N}{12\,\sqrt{2}\,\pi^{3/2}\,\sigma^3} + \frac{5\,\Gamma}{12\,\sigma^4} + \frac{\Gamma\,N}{4\,\sqrt{2}\,\pi^{3/2}\,\sigma^5}\right] \,.$$ The quantum terms are repulsive and prevent collapse, since they dominate for sufficiently small $\sigma$. Moreover, when $\Gamma \neq 0$ an oscillatory behavior is possible, provided a certain condition, to be explained in the following, is meet. To examine the possibility of oscillations, consider $V_{3}'(\sigma) = 0$, the equation for the critical points of $V_3$. Under the rescaling $s = \sigma/\sigma_{M}$, where $\sigma_M$ (defined in Eq. (\[e38\])) is the maximum of the purely classical pseudo-potential, the equation for the critical points read $$\label{e40} V_{3}' = 0 \quad \rightarrow \quad s^3 - s^2 + \frac{4\,g}{27} = 0 \,,$$ where $$\label{e41} g = \frac{480\,\pi^3\,\Gamma\,c^4}{N^2\,v_{Fe}^4}$$ is a new dimensionless parameter. In deriving Eq. (\[e40\]), it was omitted a term negligible except if $s \sim c^2/v_{Fe}^2$, which is unlikely. The quantity $g$ plays a decisive rôle on the shape of $V_3$. Indeed, calculating the discriminant shows that the solutions to the cubic in Eq. (\[e40\]) are as follows: (a) $g < 1 \rightarrow$ three distinct real roots (one negative and two positive); (b) $g = 1 \rightarrow$ one negative root, one (positive) double root; (c) $g > 1 \rightarrow$ one (negative) real root, two complex conjugate roots. Therefore, $g < 1$ is the condition for the existence of a potential well, which can support oscillations. This is shown in Figure 3. The analytic formulae for the solutions of the cubic in Eq. (\[e40\]) are cumbersome and will be omitted.  Restoring physical coordinates, the necessary condition for oscillations is rewritten as $$\label{e42} g < 1 \quad \rightarrow \quad \frac{\varepsilon_0}{2}\,\int\,|\tilde{\bf E}|^2\,d{\bf r} > \frac{\sqrt{30\pi}}{\gamma}\,\,m_e\,v_{Fe}\,c \,,$$ where $\gamma = e^2/4\,\pi\varepsilon_{0}\,\hbar\,c \simeq 1/137$ is the fine structure constant. From Eq. (\[e42\]) it is seen that for sufficient electrostatic energy the width $\sigma$ of the localized envelope field can show oscillations, supported by the competition between classical refraction and quantum diffraction. Also, due to the Fermi pressure, for large particle densities the inequality (\[e42\]) becomes more difficult to be met, since $v_{Fe} \sim n_{0}^{1/3}$. For example, when $n_0 \sim 10^{36}\,m^{-3}$ (white dwarf), the right-hand-side of Eq. (\[e42\]) is $0.6\,$ GeV. For $n_0 \sim 10^{33}\,m^{-3}$ (the next generation intense laser-solid density plasma experiments), it is $57.5$ MeV. Finally, notice that ${\cal H}_{ad,sc}$ from Eq. (\[e18\]), evaluated with the variational solution (\[e32\]), is proportional to $\dot{\sigma}^2/2 + V_3$, which is a constant of motion for Eq. (\[ee\]). Therefore, the approximate solution preserves one of the basic first integrals of the vector NLS equation (\[e16\]), as it should be. Conclusion ========== In this paper, the quantum Zakharov system in fully three-dimensional space has been derived. An associated Lagrangian structure was found, as well as the pertinent conservation laws. From the Lagrangian formalism, many possibilities are opened. Here, the variational description was used to analyze the behavior of localized envelope electric fields of Gaussian shape, in both two- and three-space dimensions. It was shown that the quantum corrections induce qualitative and quantitative changes, inhibiting singularities and allowing for oscillations of the width of the Langmuir envelope field. This new dynamics can be tested in experiments. In particular, the rôle of the parameter $g$ and the inequality in Eq. (\[e42\]) should be investigated. However, the variational method was applied only for the adiabatic and semiclassical case, which allows to derive the quantum modified vector NLS equation (\[e16\]). Other, more general, scenarios for the solutions of the fully three-dimensional quantum Zakharov system are also worth to study, with numerical and real experiments. .5cm [**Acknowledgments**]{} .5cm This work was partially supported by the Alexander von Humboldt Foundation. Fernando Haas also thanks Professors Mattias Marklund and Gert Brodin for their warm hospitality at the Department of Physics of Umeå University, where part of this work was produced. [99]{} V. E. Zakharov, Zh. Eksp. Teor. Fiz. [**62**]{}, 1745 (1972) \[Sov. Phys. JETP [**35**]{}, 908 (1972)\]. M. V. Goldman, Rev. Mod. Phys. [**56**]{}, 709 (1984). S. G. Thornhill and D. ter Haar, Phys. Reports [**43**]{}, 43 (1978). L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Phys. Plasmas [**12**]{}, 012302 (2005). F. Haas, G. Manfredi and M. R. Feix, Phys. Rev. E [**62**]{}, 2763 (2000). G. Manfredi and F. Haas, Phys. Rev. B [**64**]{}, 075316 (2001). F. Haas, Phys. Plasmas [**12**]{}, 062117 (2005). M. Marklund, Phys. Plasmas [**12**]{}, 082110 (2005). F. Haas, Phys. Plasmas [**14**]{}, 042309 (2007). P. K. Shukla and B. Eliasson, Phys. Rev. Lett. [**96**]{}, 245001 (2006); Phys. Lett. A [**372**]{}, 2893 (2008). X. Y. Tang and P. K. Shukla, Phys. Scripta [**76**]{}, 665 (2007). M. A. Abdou and E. M. Abulwafa, Z. Naturforsch. A [**63**]{}, 646 (2008). S. A. El-Wakil and M. A. Abdou, Nonl. Anal. TMA [**68**]{}, 235 (2008). Q. Yang, C. Q. Dai, X. Y. Wang and J. F. Zhang, J. Phys. Soc. Japan [**74**]{}, 2492 (2005). See the comments about this work in Ref. [@Tang]. A. P. Misra, D. Ghosh and A. R. Chowdhury, Phys. Lett. A [**372**]{}, 1469 (2008). V. E. Zakharov, A. F. Mastryukov and V. H. Sinakh, Fiz. Plazmy [**1**]{}, 614 (1975) \[Sov. J. Plasma Phys. [**1**]{}, 339 (1975)\]. V. E. Zakharov, [*Handbook of Plasma Physics*]{}, eds. M. N. Rosenbluth and R. Z. Sagdeev (Elsevier, New York, 1984), vol. 2, p. 81. M. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem and X. P. Wang, Phys. Rev. A [**46**]{}, 7869 (1992). G. C. Papanicolaou, C. Sulem, P. L. Sulem and X. P. Wang, Phys. Fluids B [**3**]{}, 969 (1991). D. F. Dubois, A. Hanssen, H. A. Rose and D. Russel, J. Geophys. Res. [**98**]{}, 17543 (1993). P. A. Robinson and D. H. Newman, Phys. Fluids B [**2**]{}, 3120 (1990). P. Y. Cheung and A. Y. Wong, Phys. Fluids [**18**]{}, 1538 (1985). H. Alinejad, P. A. Robinson, I. H. Cairns, O. Skjaeraasen and C. Sobhanian, Phys. Plasmas [**14**]{}, 082304 (2007). K. Akimoto, H. L. Rowland and K. Papadopoulos, Phys. Fluids [**31**]{}, 2185 (1988). L. H. Li and X. Q. Li, Phys. Fluids B [**5**]{}, 3819 (1993). G. Pelletier, H. Sol and E. Asseo, Phys. Rev. A [**38**]{}, 2552 (1988). L. Stenflo, Phys. Rev. Lett. [**48**]{}, 1441 (1982). M. Marklund, G. Brodin and L. Stenflo, Phys. Rev. Lett. [**91**]{}, 163601 (2003). J. Gibbons, S. G. Thornhill, M. J. Wardrop and D. ter Haar, J. Plasma Phys. [**17**]{}, 153 (1977). B. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro and L. Stenflo, Phys. Rev. E [**55**]{}, 962 (1997). R. Fedele, U. de Angelis and T. Katsouleas, Phys. Rev. A [**33**]{}, 4412 (1986).

ArXiv

--- abstract: 'The formation of charged pion condensate under parallel electromagnetic fields is studied within the two-flavor Nambu–Jona-Lasinio model. The technique of Schwinger proper time method is extended to explore the quantity locating in the off-diagonal flavor space, i.e., charged pion. We obtain the associated effective potential as a function of the strength of the electromagnetic fields and find out that it contains a sextic term which possibly induce weakly first order phase transition. Dependence of pion condensation on model parameters is investigated.' address: - 'Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China' - 'Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China' - 'School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China' - 'Matrosov Institute for System Dynamics and Control Theory , 664033, Irkutsk, Russia' - author: - Jingyi Chao - Mei Huang - Andrey Radzhabov title: Charged pion condensation under parallel electromagnetic fields --- Introduction ============ The phase structure of Quantum Chromodynamics (QCD) at high temperature/density and other extreme conditions has attracted lots of attentions and been a main topic of heavy ion collisions. The perturbative QCD predicts a free gas of quarks and gluons at high temperature limit and a color-flavor-locking phase at very high baryon density but low temperature. However, the QCD vacuum has a rather complicated nonperturbative structure, and the QCD phase diagram is not a simple transition between the hadron phase with non-zero chiral condensate to the weakly coupled quark-gluon plasma as expected long time ago [@Cleymans:1985wb], but instead a rich structure of different phases with corresponding condensates. These phases could include different color superconducting states or inhomogeneous chiral condensates [@Buballa:2003qv; @Buballa:2014tba; @Andersen:2018osr]. Recently, QCD phase structure under strong magnetic fields has drawn great interests  [@Kharzeev:2007jp; @Skokov:2009qp; @Hattori:2016emy; @Andersen:2014xxa; @Miransky:2015ava; @Huang:2015oca]. The strong magnetic fields can be generated with the strength up to $B\sim10^{18\sim 20}$ G in the non-central heavy ion collisions [@Skokov:2009qp; @Deng:2012pc], and is expected to be on the order of $10^{18}$-$10^{20}$ G [@Duncan:1992hi; @Blaschke:2018mqw] in the inner core of magnetars. Lots of interesting phenomena under strong magnetic fields have been discussed, for example, the magnetic catalysis [@Klevansky:1989vi; @Klimenko:1990rh; @Gusynin:1995nb; @Gusynin:1999pq], inverse magnetic catalysis [@Bali:20111213; @Bali:2012zg; @Bali:2013esa] effect, the chiral magnetic effect (CME) [@Kharzeev:2007jp; @Kharzeev:2007tn; @Fukushima:2008xe] and the vacuum superconductivity [@Chernodub:2010qx; @Chernodub:2011mc]. Moreover, it was pointed out that under the parallel electromagnetic fields, the neutral pion condensation can be formed [@Cao:2015cka; @Wang:2018gmj] due to the connection of field with axial anomaly. If only QCD interaction is included, the axial isospin currents is anomaly free. It turns out that anomaly emerges associated with the coupling of quarks to electromagnetism, where the axial isospin currents is given by $${\partial}_{\mu}j_{5}^{\,\mu 3}=-\frac{e^{2}}{16\pi^{2}}{\varepsilon}^{{\alpha}{\beta}\mu\nu}F_{{\alpha}{\beta}}F_{\mu\nu}\cdot{\hbox{tr}}{\left}[\tau^{3}Q^{2}{\right}].$$ Here $Q$ is the matrix of quark electric charges and $F$ is the field strength. The corresponding process is $\pi_{0}\to {\gamma}{\gamma}$. The decay of a neutral pion into two photons, which had been a puzzle for some time in the 1960s, is the most successful proof of chiral anomaly. Above solution led to the discovery of the Adler–Bell–Jackiw anomaly [@Adler:2004qt]. In the asymmetric flavor space, one can introduce a chiral isospin chemical potential $\mu_{I}^5$ corresponding to the current $\bar{\psi}{\gamma}_{0}{\gamma}_{5}\tau_{3}\psi$, which is similar to the isospin chemical potential $\mu_{I}$ with respect to $\bar{\psi}{\gamma}_{0}\tau_{3}\psi$. It has been a long history of investigating the pion condensation under the isospin asymmetric nuclear matter. In the beginning this effect is discussed for case nuclear matter in neutron-star interiors [@Sawyer:1972cq; @Sawyer:1973fv; @Voskresensky:1980nk] or superdense and supercharged nuclei [@Migdal:1978az]. The pion condensation of charged or neutral pion modes in QCD vacuum are also considered in the frameworks of effective models with quark degrees of freedom [@Son:2000xc; @He:2006tn; @Mao:2014hga; @Khunjua:2017khh] or in lattice calculations [@Brandt:2016zdy; @Brandt:2018bwq]. The degeneracy between $\pi_{0}$ and $\pi_{\pm}$ is destroyed because of the axial isospin chemical potential. It is worth to pursuing the detailed behaviors of charged pions in a strict manner. Hence, in this work, we focus on the possibility of charged pion condensation under the parallel electromagnetic fields in the framework of the ${\mathrm}{SU}(2) \times {\mathrm}{SU}(2)$ NJL model [@Nambu:1961tp; @Nambu:1961fr]. For this purpose, we develop a full routine to derive the mean-field thermodynamical potential of the NJL model with nonzero charged pion condensate ${\langle}\bar{\psi} i \gamma_5 \tau_{\pm} \psi {\rangle}$ in the off-diagonal flavor space under the parallel electromagnetic fields. Calculations are performed with Schwinger proper time method [@Schwinger:1951nm] and the proper time regularization in the NJL model is used. Through the paper we only consider the model at zero temperature and chemical potential and restrict ourselves to the case of the electric field anti-parallel to the magnetic field. \[intr\] Lagrangian {#NJL} ========== The Lagrangian of the ${\mathrm}{SU}(2)\times {\mathrm}{SU}(2)$ NJL model is in the form of [@Nambu:1961tp; @Nambu:1961fr; @Volkov:1986zb; @Vogl:1991qt; @Klevansky:1992qe; @Hatsuda:1994pi; @Volkov:2005kw] $$\begin{aligned} \mathcal{L}_{{\mathrm}{NJL}} = \bar{\psi}\left(i \slashed{D} - m_0 \right) \psi + \mathrm{G} \left[ \left( \bar{\psi} \psi \right)^2+ \left( \bar{\psi} i\gamma_5 \tau_i \psi \right)^2 \right],\end{aligned}$$ where $\bar{\psi}(x)=(\bar{u}(x),\bar{d}(x))$ are $u$ and $d$ anti-quark fields. The limit of equal current masses for $u,d$, $m_u=m_d\equiv m_{0}$ is considered. $\gamma_i$, $\tau_{i}$ are conventional Dirac and Pauli matrices and $\tau_{0}$ is the unit matrix. $\slashed{D}$ is the covariant derivative and, in the two flavor space, expressed as $$\begin{aligned} D_{\mu}={\left}({\partial}_{\mu}-{\mathop{}\!i}QA_{\mu}{\right})\tau_{0}-{\mathop{}\!i}qA_{\mu}\tau_{3},\end{aligned}$$ where $Q=\frac{1}{2}{\left}(q_{u}+q_{d}{\right})$ and $q=\frac{1}{2}{\left}(q_{u}-q_{d}{\right})$. Introducing auxiliary bosonic fields $\pi$, $\sigma$, with the help of Hubbard-Stratonovich transformation one can integrate over the quark fields, then obtains the following effective Lagrangian: $$\begin{aligned} \mathcal{L} = \frac{\sigma^2+\vec{\pi}^2}{4\mathrm{G}}-{\mathop{}\!i}{\hbox{Tr}}\ln S^{-1}, \label{eLagrangian}\end{aligned}$$ where $S^{-1}$ is the inverse quark propagator and $$\begin{aligned} S^{-1}= {\mathop{}\!i}\slashed{D} - M, \quad M=m_{0}\tau_{0}-\sigma\tau_{0}-{\mathop{}\!i}\gamma_5\pi_{i} \tau_i. \label{Propagator}\end{aligned}$$ The auxiliary bosonic fields could have a nonzero vacuum expectation values and therefore it is necessary to shift them as $\sigma = \sigma^\prime-{\langle}\sigma{\rangle}$, $\pi_{i}=\pi_{i}^\prime- {\langle}\pi_{i}{\rangle}$. Equations of motion for mean-fields ${\langle}\sigma{\rangle}$, ${\langle}\pi_{i}{\rangle}$ are obtained from the Lagrangian (\[eLagrangian\]) after elimination from its linear terms, i.e. $$\begin{aligned} \frac{\delta \mathcal{L} }{\delta {\langle}\sigma{\rangle}}\biggl|_{\substack{ \sigma^\prime =0 \\ \pi_{i}^\prime=0 }}, \quad \frac{\delta \mathcal{L} }{\delta {\langle}\pi_{i}{\rangle}}\biggl|_{\substack{ \sigma^\prime =0 \\ \pi_{i}^\prime=0 }} =0.\end{aligned}$$ As a result, under different conditions the ${\langle}{\sigma}{\rangle}$, ${\langle}\pi_{i}{\rangle}$ condensates have non-zero values and the non-zero value of scalar condensate leads to a formation of constituent quarks with dynamical quark mass $m=m_{0}-{\langle}{\sigma}{\rangle}$. Let us denote the second term of effective Lagrangian (\[eLagrangian\]) as $\mathcal{S}_{eff}=-{\mathop{}\!i}{\hbox{Tr}}\ln S^{-1}$. Then the gap equations for ${\langle}{\sigma}{\rangle}$ and ${\langle}\pi_{i}{\rangle}$ takes the form $$\begin{aligned} m=m_{0}-2\mathrm{G}\frac{\partial\mathcal{S}_{eff}}{\partial {\langle}\sigma{\rangle}},\quad {\langle}\pi_{i}{\rangle}=-2\mathrm{G}\frac{\partial \mathcal{S}_{eff}}{\partial {\langle}\pi_{i}{\rangle}}.\end{aligned}$$ The calculation of $\mathcal{S}_{eff}$ is presented in the following section. The effective potential {#main} ======================= Without loss of generality, one can choose ${\langle}\pi_{i}{\rangle}={\left}(\pi_{1},0,0{\right})$ and therefore “mass” in quark propagator Eq.(\[Propagator\]) is $M=m\tau_{0}+{\mathop{}\!i}\pi_{1}{\gamma}_{5}\tau_{1}$. Since ${\mathrm}{Det}{\left}({\mathop{}\!i}\slashed{D}-M{\right})={\mathrm}{Det}\,{\Gamma}{\left}({\mathop{}\!i}\slashed{D}-M{\right}){\Gamma}$, where ${\Gamma}={\gamma}_{5}\tau_{3}$, the second term of the Lagrangian Eq.(\[eLagrangian\]) is replaced to $$\mathcal{S}_{eff}=-\frac{{\mathop{}\!i}}{2}\ln{\mathrm}{Det}{\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right}),$$ where $\slashed{\mathcal{D}}^{2}=\slashed{D}^{2}-{\gamma}_{5}{\gamma}^{\mu}\pi_{1}{\left}[\tau_{1},D_{\mu}{\right}]$. By using the method of proper time, we represent $\mathcal{S}_{eff}$ as following: $$\label{eqn_seff} \mathcal{S}_{eff}={\hbox{Tr}}\int\limits_{1/{\Lambda}^{2}}^{\infty}{\mathop{}\!i}\,\frac{{\mathop{}\!d}s}{2s}\int{\hbox{tr}}{\left}{\langle}x\big|{\mathop{}\!e}^{-{\mathop{}\!i}{\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right})s}\big|x'{\right}{\rangle}{\mathop{}\!d}^{4}x,$$ where the ultraviolet cutoff $1/{\Lambda}^{2}$ has been explicitly introduced, ${\hbox{tr}}$ and ${\hbox{Tr}}$ means the trace taking in the spinor and flavor space, respectively. From now on, we will work in the Euclidean space. Following notations are introduced: $$\begin{aligned} \label{AlBeLa} & {\alpha}=m^{2}+\pi_{1}^{2}-\frac{1}{2}{\sigma}^{\mu\nu}{\lambda}_{\mu\nu},\quad {\beta}_{\nu}=q\pi_{1}{\gamma}_{5}{\gamma}^{\mu}F_{\mu\nu}\tau_{2},\quad\nonumber \\ & {\lambda}_{\mu\nu}=q_{f}F_{\mu\nu},\end{aligned}$$ where $q_{f}={\mathrm}{Diag}(q_{u},q_{d})$ and ${\sigma}^{\mu\nu}=\frac{{\mathop{}\!i}}{2}{\left}[{\gamma}^{\mu},{\gamma}^{\nu}{\right}]$. In order to obtain $\mathcal{S}_{eff}$, it is then straightforward to look for the solution of $G(x,y;s)$ obeying a second order differential equation ${\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right})G{\left}(x,y;s{\right})={\delta}{\left}(x,y;s{\right})$. The explicit form is $$\begin{aligned} \slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}&={\partial}^{2}_{x}+\alpha(y)+\beta_{\mu}(y){\left}(x-y{\right})^{\mu}+\nonumber\\ & +\frac{1}{4}{\lambda}^{2}_{\mu\nu}{\left}(x-y{\right})^{\mu}{\left}(x-y{\right})^{\nu}.\end{aligned}$$ Performing the Fourier transform, one finds, $$\begin{aligned} \label{eqn_diff_p} {\left}(-p^{2}+\alpha-{\mathop{}\!i}\beta_{\mu}\frac{{\partial}}{{\partial}p_{\mu}}-\frac{1}{4}{\lambda}^{2}_{\mu\nu}\frac{{\partial}^{2}}{{\partial}p_{\mu}{\partial}p_{\nu}}{\right})G(p;s)=1.\end{aligned}$$ As suggested in the the reference [@Brown:1975bc] one can solve the equation in the form $$\begin{aligned} \label{eqn_G_pV2} G(p;s)={\mathop{}\!e}^{-\alpha s}{\mathop{}\!e}^{ p\cdot A(s)\cdot p+B(s) \cdot p+C(s)},\end{aligned}$$ whose associated descriptions of matrix $A$, vector $B$ and scalar $C$ are $$\begin{aligned} \label{eqn_sol_ABC} &A={\lambda}^{-1}\tan{\lambda}s,\quad B=-2{\mathop{}\!i}{\beta}\cdot{\lambda}^{-2}{\left}(1-\sec{\lambda}s{\right}),\\ &C=-\frac{1}{2}\,{\hbox{tr}}\ln\cos{\lambda}s-{\beta}\cdot{\lambda}^{-3}{\left}(\tan{\lambda}s-{\lambda}s{\right})\cdot{\beta}\nonumber.\end{aligned}$$ For simplicity here and below indexes are not shown. Plugging the form of ${\beta}$ in Eq.(\[AlBeLa\]) into vector $B$ and restoring indexes one has $$\begin{aligned} B_{\mu}=-2{\mathop{}\!i}q\pi_{1}\tau_{2}{\gamma}_{5}{\gamma}^{\nu}F_{\nu{\alpha}}{\left}[{\lambda}^{-2}{\left}(1-\sec{\lambda}s{\right}){\right}]^{{\alpha}}_{\mu}.\end{aligned}$$ Vector $B$ contains Dirac matrix, not commuting with ${\sigma}^{\mu\nu}$. Therefore, we emphasize that one should be careful with tracing in spinor space and integrating in momentum space. Introducing notations $P_{1}=\frac{1}{2}{\sigma}{\lambda}s$ and $P_{2}=p\cdot A(s)\cdot p+B(s) \cdot p$, one has ${\left}[{\sigma}{\lambda}s, p\cdot A(s)\cdot p{\right}]=0$ and the part with matrices in exponent Eq.(\[eqn\_G\_pV2\]) can be expanded as $$\begin{aligned} &{\mathop{}\!e}^{P_{1}+P_{2}}\simeq {\mathop{}\!e}^{P_{1}}{\mathop{}\!e}^{P_{2}}{\mathop{}\!e}^{-\frac{1}{2}[P_{1},P_{2}]}=\nonumber \\ &\quad\quad={\mathop{}\!e}^{\frac{1}{2}{\sigma}{\lambda}s}{\mathop{}\!e}^{p\cdot A(s)\cdot p+B(s) \cdot p}{\mathop{}\!e}^{-\frac{1}{4}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]}.\end{aligned}$$ We denote $-\frac{1}{4}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]=\frac{1}{2}q\pi_{1} Os$, where $O$ has a structure of the form $O=Q\tau_{2}O_{1}\mathbb{B}_{1}p+q\tau_{1}O_{2}\mathbb{B}_{2}p$ and $\mathbb{B}$ will render in Eq. (\[Bdefinition\]). Shorthand matrix notation is applied, i.e. $\mathbb{F}=F_{\mu}^{\nu}$. To find the eigenvalue of $O$, we square it and get $$\begin{aligned} O^{2}&=Q^{2}{\left}(\tau_{2}O_{1}\mathbb{B}_{1}p{\right})^{2}+q^{2}{\left}(\tau_{1}O_{2}\mathbb{B}_{2}p{\right})^{2} -\nonumber \\ &-{\mathop{}\!i}qQ\tau_{3}{\left}[O_{1}\mathbb{B}_{1}p,O_{2}\tilde{\mathbb{B}}_{2}p{\right}],\\ &O_{1}={\mathop{}\!i}{\left}[{\sigma}_{\mu\nu},{\gamma}_{5}{\gamma}^{{\alpha}} {\right}]=2{\gamma}_{5}g_{\nu}^{{\alpha}}{\gamma}_{\mu}-2{\gamma}_{5}g_{\mu}^{{\alpha}}{\gamma}_{\nu},\nonumber\\ &O_{2}={\left}\{{\sigma}_{\mu\nu},{\gamma}_{5}{\gamma}^{{\alpha}}{\right}\}=-2{\varepsilon}^{{\alpha}{\beta}}_{\;\mu\nu}{\gamma}_{{\beta}}.\nonumber\end{aligned}$$ With help of relation $\tau_{2}q_{f}\tau_{2}={\mathrm}{Diag}{\left}(q_{d}, q_{u}{\right})=\tilde{q}_{f}$, the $\tilde{\mathbb{B}}{\left}(\mathbb{B}{\right})$ are shown as $$\begin{aligned} \label{Bdefinition} &\tilde{\mathbb{B}}_{1}{\left}(\mathbb{B}_{1}{\right})=\frac{1}{\mathsf{q}^{2}}{\left}[1-\sec \mathsf{q}\mathbb{F}s{\right}],\quad \nonumber \\ &\tilde{\mathbb{B}}_{2}{\left}(\mathbb{B}_{2}{\right})= \frac{\bar{\mathbb{F}}\mathbb{F}}{\mathbb{F}^{2}} \frac{1}{\mathsf{q}^{2}} {\left}[1-\sec\mathsf{q}\mathbb{F}s{\right}],\end{aligned}$$ where $\mathsf{q}=\tilde{q}_{f}$ or $q_{f}$ for $\tilde{\mathbb{B}},\mathbb{B}$ respectively; $\mathbb{F}$ and $\bar{\mathbb{F}}$ are field strength tensor $F^{\mu\nu}$ and dual field strength tensor $\bar{F}^{\mu\nu}=\frac{1}{2}{\varepsilon}^{\mu\nu{\alpha}{\beta}}F_{{\alpha}{\beta}}$, in shorthand notations. Moreover, ${\left}(\tau_{2}O_{1}\mathbb{B}_{1}p{\right})^{2}=-16\mathbb{B}_{1}\tilde{\mathbb{B}}_{1}p^{2}$, ${\left}(\tau_{1}O_{2}\mathbb{B}_{2}p{\right})^{2}=16\mathbb{B}_{2}\tilde{\mathbb{B}}_{2}p^{2}$ and $[O_{1}\mathbb{B}_{1}p,O_{2}\tilde{\mathbb{B}}_{2}p]=-32{\gamma}_{5}\mathbb{B}_{1}\tilde{\mathbb{B}}_{2}p^{2}$. Applying the system that in a Lorentz frame where the electromagnetic field vectors are anti-parallel, e.g., ${\mathbf}{B}=-{\mathbf}{E}=f\hat{z}$, one gets $\mathbb{F}^{2}=f^2\,{\mathrm}{Diag}{\left}(-,+,+,-{\right})$ and $\bar{\mathbb{F}}\mathbb{F}=-f^2{\delta}_{\mu\nu}$ in Euclidean metric $(-,-,-,-)$, hence that $\bar{\mathbb{F}}\mathbb{F}/\,\mathbb{F}^{2}=f^{2}\mathbb{F}^{-2}$. Besides, ${\left}[1-\sec\mathsf{q}\mathbb{F}s{\right}]$ contains even powers of $\mathbb{F}$. It causes $O^{2}=-16Q^{2}\mathsf{p}_{1}^{2}+16q^{2}\mathsf{p}_{2}^{2}+32{\mathop{}\!i}{\gamma}_{5}Qq\mathsf{p}_{1}\cdot\mathsf{p}_{2}$ in a simply manner, where $\mathsf{p}_{1}=p_{\shortparallel}+p_{\perp}$, $\mathsf{p}_{2}=p_{\shortparallel}-p_{\perp}$, $p_{\shortparallel}=b_{\shortparallel}(p_{0},0,0,p_{3})$ and $p_{\perp}=b_{\perp}(0,p_{1},p_{2},0)$. The forms of $b_{\shortparallel}$ and $b_{\perp}$ are taken as $$\label{eqn_b03} b_{\shortparallel}=\frac{(1-\sec q_{f}s)^{\frac{1}{2}}(1-\sec\tilde{q}_{f}s)^{\frac{1}{2}}}{q_{f}\tilde{q}_{f}},$$ $$\label{eqn_b12} b_{\perp}=\frac{(1-\operatorname{sech}q_{f}s)^{\frac{1}{2}}(1-\operatorname{sech}\tilde{q}_{f}s)^{\frac{1}{2}}}{q_{f}\tilde{q}_{f}}.$$ Here and below in we rescale the integration variable as $s=s^\prime/f$ and omit prime. Because ${\gamma}_{5}^{2}=1$ associated with eigenvalue $\pm 1$, it follows that $O$ has four eigenvalues [@Schwinger:1951nm], written as $$\begin{aligned} \label{eqn_O_squar} \mathcal{O}=\pm 4{\left}({\mathop{}\!i}Q\mathsf{p}_{1}\pm q\tau_{3}\mathsf{p}_{2}{\right}).\end{aligned}$$ Let ${\theta}=q\pi_{1} s/f$, one has $$\begin{aligned} {\hbox{tr}}\,{\mathop{}\!e}^{\frac{1}{2}{\theta}O}=\mathsf{T}=\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right}),\end{aligned}$$ which follow the method applied in [@dittrich2000probing]. The full statement is that $$\begin{aligned} \label{eqn_O_exp} \exp{\left}[\frac{1}{2}{\theta}O{\right}]=\mathsf{T}+{\mathop{}\!i}{\gamma}_{5}\mathsf{U}+\frac{O\mathsf{V}}{2K^{2}}+\frac{{\mathop{}\!i}{\gamma}_{5}O\mathsf{W}}{2K^{2}},\end{aligned}$$ where $K^{2}=\mathsf{p}_{1}^{2}=\mathsf{p}_{2}^{2}$. $\mathsf{T}, \mathsf{U}, \mathsf{V}$ and $\mathsf{W}$ are scalars. Similarly, $$\begin{aligned} \label{eqn_F_exp} \exp{\left}[q_{f}\frac{{\sigma}Fs}{2f}{\right}]=\mathsf{P}-{\mathop{}\!i}{\gamma}_{5}\mathsf{Q}+\frac{{\sigma}F}{2f}\,\mathsf{R}-\frac{{\mathop{}\!i}{\gamma}_{5}{\sigma}F}{2f}\,\mathsf{S}.\end{aligned}$$ Since $$\begin{aligned} {\hbox{tr}}{\left}(O^{2}{\mathop{}\!e}^{\frac{1}{2}{\theta}O}{\right})=\frac{{\partial}^{2}}{{\partial}^{2}{\theta}}{\hbox{tr}}{\left}(4{\mathop{}\!e}^{\frac{1}{2}{\theta}O}{\right})=4\frac{{\partial}^{2}\mathsf{T}}{{\partial}^{2}{\theta}},\end{aligned}$$ apply the identity of [Eq. (\[eqn\_O\_squar\])]{}, it derives that $$\begin{aligned} \mathsf{U}=\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right}).\end{aligned}$$ Proceeding with the direct differentiation of the exponential function via our basic trick, we get $$\begin{aligned} &\mathsf{V}=\frac{1}{Q^{2}+q^{2}} \biggl(Q\mathsf{p}_{1}\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})+\nonumber\\ &\quad\quad\quad+\tau_{3}q\mathsf{p}_{2}\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})\biggr), \nonumber\\ &\mathsf{W}=\frac{1}{Q^{2}+q^{2}} \biggl(\tau_{3}q\mathsf{p}_{2}\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})-\nonumber\\ &\quad\quad\quad-Q\mathsf{p}_{1}\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})\biggr).\end{aligned}$$ Known in [@dittrich2000probing], one has $$\begin{aligned} &\mathsf{P}=\cos q_{f}s\cosh q_{f}s,\, \mathsf{Q}=\sin q_{f}s\sinh q_{f}s \nonumber\\ &\mathsf{R}=(\sinh q_{f}s\cos q_{f}s+\cosh q_{f}s\sin q_{f}s)/2,\\ &\mathsf{S}=(\sinh q_{f}s\cos q_{f}s-\cosh q_{f}s\sin q_{f}s)/2.\nonumber\end{aligned}$$ Then, we perform an approximate expansion $$\begin{aligned} \label{eqn_ap2bp} &{\mathop{}\!e}^{ p\cdot A(s)\cdot p+B(s)\cdot p}\simeq {\mathop{}\!e}^{p\cdot A(s)\cdot p}{\mathop{}\!e}^{B(s)\cdot p} \\ &={\mathop{}\!e}^{ p\cdot A(s)\cdot p}{\left}(\cos\varrho +B(s) \cdot p\frac{\sin\varrho}{\varrho}{\right})\nonumber \end{aligned}$$ where $\varrho=2q\pi_{1}k/f$ and $k={\left}(\mathsf{p}_{1}\cdot\mathsf{p}_{2}{\right})^{\frac{1}{2}}$. Now, it is allowed us to integrate with respect to $p$ and take the trace in the spinor space. With help of the [Eq. (\[eqn\_O\_exp\])]{}, [Eq. (\[eqn\_F\_exp\])]{} and [Eq. (\[eqn\_ap2bp\])]{}, one has $$\begin{aligned} L(s)&={\hbox{tr}}\int{\mathop{}\!e}^{\frac{1}{2f}{\sigma}{\lambda}s}{\mathop{}\!e}^{p\cdot A(s)\cdot p+B(s)\cdot p}{\mathop{}\!e}^{-\frac{1}{4f}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]}{\mathop{}\!d}^{4}p \nonumber\\ &=L_{0}(s)+L_{1}(s)+L_{2}(s).\end{aligned}$$ Here ${\langle}X{\rangle}$ denotes integrating in momentum and tracing in spinor space ${\hbox{tr}}\int X{\mathop{}\!e}^{p\cdot A\cdot p}{\mathop{}\!d}^{4}p$. It gives $$\begin{aligned} &L_{0}(s)={\left}{\langle}\cos\varrho\mathsf{T}\mathsf{P}{\right}{\rangle}, \, \, L_{1}(s)={\left}{\langle}\cos\varrho\mathsf{U}\mathsf{Q}{\right}{\rangle}, \, \, L_{2}(s)=\\ &={\left}< \frac{2\tilde{q}_{f}\sin\varrho}{K^{2}k} {\left}[q_{f}K^{2}{\left}(\mathsf{W}\mathsf{S}-\mathsf{V}\mathsf{R}{\right})+\tilde{q}_{f}k^{2}{\left}(\mathsf{V}\mathsf{S}+\mathsf{W}\mathsf{R}{\right}){\right}]{\right}>. \nonumber\end{aligned}$$ The integration with respect to momentum $p$ is in the Gaussian form, which can be taken easily with result $$\begin{aligned} &{\langle}1{\rangle}=\mathcal{N}=\pi^{2}{\mathrm}{Det}A^{-\frac{1}{2}},\,A={\mathrm}{Diag}{\left}(a_{\shortparallel}, a_{\perp}, a_{\perp}, a_{\shortparallel}{\right})\nonumber\\ &{\langle}K^{2}{\rangle}=\frac{\mathcal{N}}{2}\,{\hbox{tr}}{\left}(\frac{D_{+}}{A}{\right}),\, {\langle}k^{2}{\rangle}=\frac{\mathcal{N}}{2}\,{\hbox{tr}}{\left}(\frac{D_{-}}{A}{\right})\end{aligned}$$ The matrices $D_{\pm}={\mathrm}{Diag}{\left}(b^{2}_{\shortparallel}, \pm b^{2}_{\perp}, \pm b^{2}_{\perp}, b^{2}_{\shortparallel}{\right})$, which read from [Eq. (\[eqn\_b03\])]{} and [Eq. (\[eqn\_b12\])]{}. From [Eq. (\[eqn\_sol\_ABC\])]{}, one has $a_{\shortparallel}=\tan q_{f}s/{\left}(q_{f} f{\right})$ and $a_{\perp}=\tanh q_{f}s/{\left}(q_{f} f{\right})$. The higher orders corrections ${\langle}K^{4}{\rangle}$, ${\langle}k^{4}{\rangle}$ and $ {\langle}k^{2}K^{2}{\rangle}$ can be drawn in a similar manner, which are abbreviated here. Since ${\theta}p\sim\pi_{1}ps/f\sim \pi_{1}p/{\Lambda}^{2}\ll 1$ and the integration is exponential suppressed for large $s$, it enables us to approximate $\sin(a{\theta}p)$, $\sinh(a{\theta}p)\sim a{\theta}p$ and $\cos(a{\theta}p)$, $\cosh(a{\theta}p)\sim 1$. Hence, it acquires $\mathsf{T}\sim 1$, $\mathsf{U}\sim k^{2}s$, $\mathsf{V}\sim K^{2}s$ and $\mathsf{W}\sim K^{2}k^2 s$. Finally, take the integration with respect to $s$ to get $$\begin{aligned} \label{eqn_eff_potentioal} &\mathcal{S}_{eff}= {\mathcal}{S}_{eff}^{0}+{\mathcal}{S}_{eff}^{1}+{\mathcal}{S}_{eff}^{2}, \\ &{\mathcal}{S}_{eff}^{i}=\frac{N_c }{4\pi^{2}}{\hbox{Tr}}\int_{f/{\Lambda}^{2}}^{\infty} \frac{{\mathop{}\!d}s}{2s} {\mathop{}\!e}^{-h(s)} S_{eff}^{i}(s),\quad \nonumber \end{aligned}$$ where $-h(s)=-(m^{2}+\pi_{1}^{2})s/f+{C}(s)-\frac{1}{2}\ln{\hbox{tr}}{A}$, and $$\begin{aligned} &{C}(s)-\frac{\ln{\hbox{tr}}{A}}{2}= -\ln\frac{\sin q_{f}s\sinh q_{f}s}{q_{f}^{2}f^{2}}\nonumber\\ &\quad\quad\quad-\frac{2q^{2}{\pi_{1}}^{2}}{\tilde{q}_{f}^{3}f}{\left}(2\tilde{q}_{f}s-\tan \tilde{q}_{f}s-\tanh \tilde{q}_{f}s{\right}).\end{aligned}$$ The detailed integrands $S_{eff}^{i}(s)$ are $$\begin{aligned} &S_{eff}^{0}(s) = \mathsf{P},\quad \nonumber \\ &S_{eff}^{1}(s) =4 \tau_{3} \frac{Q q^3\pi_{1}^2 s^2}{f^2{\mathcal}{N}}\frac{}{}{\left}{\langle}k^2{\right}{\rangle}\mathsf{Q} ,\quad \label{exprS2} \\ &S_{eff}^{2}(s) =\frac{8\tilde{q}_{f}q^2\pi_{1}^2 s}{f^2{\mathcal}{N}} {\left}(-q_{f}{\left}{\langle}K^2{\right}{\rangle}\mathsf{R}+\tilde{q}_{f}{\left}{\langle}k^2{\right}{\rangle}\mathsf{S}{\right}) +\nonumber\\ &\quad\quad+\tau_{3} \frac{32\tilde{q}_{f} Qq^5 \pi_{1}^4 s^3}{3f^4{\mathcal}{N}} {\left}(q_{f}{\left}{\langle}K^2k^2{\right}{\rangle}\mathsf{S}+\tilde{q}_{f}{\left}{\langle}k^4{\right}{\rangle}\mathsf{R}{\right}).\nonumber \end{aligned}$$ Eventually, we have the effective potential which takes the following form: $$\begin{aligned} \Omega = \frac{(m-m_{0})^2+\pi_1^2}{4\mathrm{G}} + \mathcal{S}_{eff}.\end{aligned}$$ Numerical results ================= The NJL model is nonrenormalizable and therefore the UV cut-off should be employed in order to get reasonable results, where a proper time regularization is applied in the work, i.e., the integration with respect to $s$ start from $f/{\Lambda}^{2}$. We perform calculations of integral expression for $\mathcal{S}_{eff}$ in Eq. (\[eqn\_eff\_potentioal\]) numerically. In the limit of zero field $f$ the expression leads to the usual proper-time regularization scheme of NJL model. Therefore, for numerical estimation we use the model parameterization from Ref. [@Inagaki:2015lma]. Namely, in [@Inagaki:2015lma] there are five sets of model parameters for proper-time regularization scheme which are fitted in favor of observable values of pion mass and weak pion decay constant. For convenience we present them in Table 1. In the set 1 the constituent quark mass is $178$ MeV and for set 5 is $372$ MeV. The constituent quark masses for other parameterizations are in between these two cases. Therefore, one can consider set 1 and set 5 as limited cases for the predictions of the NJL model. Set $m_0$\[MeV\] $\Lambda$\[MeV\] G\[GeV$^{-2}$\] $m$\[MeV\] ----- -------------- ------------------ ----------------- ------------ -- -- -- 1 3.0 1464 1.61 178 2 5.0 1097 3.07 204 3 8.0 849 5.85 245 4 10.0 755 8.13 265 5 15.0 645 17.2 372 : Parameters of the NJL model in the proper-time regularization taken from [@Inagaki:2015lma]. \[TableParameters\] The important point of calculation is that integrand of $\mathcal{S}_{eff}$ contain singularities and one should specify how to deal with them. The singularities which are generated by trigonometric functions tangent and cotangent of $q_i s$ for quark flavor i are located at real axis and by hyperbolic functions at imaginary axis. We shift $s$ to the complex plane $s+i\epsilon$, see Fig. \[ContTikz\], since we prefer to running a numerical calculation of integral instead of residues summation like what used in [@Inagaki:2003ac; @Ruggieri:2016xww]. In principle, the effective potential at finite $f$ acquires an imaginary part which correspond to pair-production because of Schwinger mechanism [@Schwinger:1951nm; @Tavares:2018poq; @Cao:2015dya]. We figure out that the imaginary part is smaller than the real part in current work. Plus, the subtle effect of Schwinger mechanism is out of the scope of the present paper and will not discuss here. ![Contour on complex $s$-plane. Singularities for a quark of flavor i which are related to tangent are shown by open circles while filled circles correspond to those of cotangent $\cot(q_i s)$.[]{data-label="ContTikz"}](ContTikz2){width="47.00000%"} In Figs. \[Scan5ff001\], \[Scan5ff010\], \[Scan5ff035\], the behavior of effective potential for Set 5 of model parameters is plotted for field values $f=0.01,0.2,0.450$ GeV$^2$, respectively. We found the following typical behavior for three regions: 1) For small field $f=0.01$ GeV$^2$ as shown in Fig. \[Scan5ff001\], the system is in usual (almost vacuum) chiral symmetry breaking phase with nonzero sigma condensate and zero pion condensate; 2) For moderate field $f=0.2$ GeV$^2$, seen in Fig. \[Scan5ff010\], the additional minima appears in the effective potential and the system takes a chiral rotation in $\sigma -\pi_1$ plane to have a nonzero pion condensate, $\pi_1$; 3) For large field $f=0.450$ GeV$^2$, read from Fig. \[Scan5ff035\], the minimum with $\pi_1=0$ is energetically favorable. There are two sources to break the chiral symmetry: spontaneous chiral symmetry breaking due to presence of quark condensate ${\langle}\bar{\psi} \psi {\rangle}$ and explicit chiral symmetry breaking due to nonzero current quark mass in the Lagrangian. Therefore, we investigate not only the reality situation but also for $m_0\to 0$. To systematically perform this task, we vary $m_{0}$ and recalculate $m$ while $\mathrm{G}$ and $\Lambda$ have the same values, i.e. we consider $m$ as a function of $m_{0}$ [@Bernard:1992mp]. In the following we denote the physical value of current quark mass as $m_{0}^\star$. The behaviors of $m$ and $\pi_1$ as a function of field $f$ are presented in Fig. \[Set1Set5MassDelta\] for different values of ratio $m_{0}/m_{0}^\star=0.01,0.1,0.5,1.0$. The left and right sides are obtained by model parameter sets 1 and 5, respectively. It is straightforward to figure out that for small current quark mass the system is more preferable to chirally rotate from zero to nonzero value $\pi_1$, leaving the total order parameter of chiral symmetry breaking $|M|=\sqrt{m^2+\pi_1^2}$ unchanged. With increasing of $m_{0}$ the situation becomes more complicated. The phase of pion condensation even never show up for $m_{0}=m_{0}^\star$ in the model parameter Set 1. Conclusions {#con} =========== In this paper the charged pion condensation under the parallel electromagnetic fields is calculated in the framework of the NJL model by using Schwinger proper-time method. The configuration of field is chosen, the electric field being anti-parallel to the magnetic one, to have a zero first Lorentz invariant, $I_{1}=\mathbf{E}^{2}-\mathbf{B}^{2}$, and a nonzero second Lorentz invariant, $I_2=\mathbf{E} \cdot \mathbf{B}$. We find that in the chiral limit the system is favorable to form a both nonzero condensation of scalar and charged pion, i.e. rotating in the chiral group. Chiral condensates aligning to pseudo mesons space has been found in [@Cao:2015cka] by the methods of $\chi{\mathrm}{PT}$ and NJL model, where the system is immediately straighten up $\pi_0$ direction in the chiral limit once the second Lorentz invariant $I_2$ turned on. The main difference of charged condensation is that the system will across a weakly first order phase transition to zero pion condensate and then a second order phase transition to chirally symmetric phase as the field strength increasing, while it, characterizing by $\pi_{0}$, is a whole second order phase transition as shown in [@Cao:2015cka]. The underlying mechanism are two folds. One is the obviously coupling between charged pions and electromagnetism. Another reason is that a more complicated influence of anomalous diagrams are implicitly included, not only $\pi_0\rightarrow\gamma\gamma$ but also $\gamma\rightarrow \pi_+\pi_-\pi_0$. Indeed, if assuming condensation in the neutral channel ${\langle}\sigma{\rangle}$ nears a second order phase transition, its effective potential has the form ${\mathcal}{S}_{eff}^{0}\sim -c_{0}M^{2}+c_{1}M^{4}/f$ according to Ginzburg-Landau theory [@Ginzburg:1950sr]. However, if we include $\pi_{\pm}$ as an additional degree of freedom and non-degenerate with $\pi_{0}$, read from [Eq. (\[eqn\_eff\_potentioal\])]{}, the potential arranges as: ${\mathcal}{S}_{eff}^{2}\sim -\tilde{c}_{1}M^{4}/f+c_{2}M^{6}/f^{2}$. As a result, we have a weakly first order phase transition and effective potential in the form of $$\begin{aligned} {\Omega}=\frac{M^{2}}{4 \mathrm{G}}-c_{0}M^{2}+\frac{{\left}(c_{1}-\tilde{c}_{1}{\right})M^{4}}{f}+\frac{c_{2}M^{6}}{f^{2}}.\end{aligned}$$ Our numerical simulations support these arguments, read from Figs. \[Scan5ff001\], \[Scan5ff010\], \[Scan5ff035\]. The mass of current quarks plays an important role and it denies our claim at some regions of the model parameters. It requires a further study via the first principle calculation, such as Dyson-Schwinger equation or functional renormalization group methods. Application of the charged pion condensation to the case heavy-ion collisions or neutron stars interior need an extension to finite temperature and/or chemical potential. We will explore this extension in future. Acknowledgments {#Ackn} =============== We are grateful to Maxim Chernodub, Nikolai Kochelev, Marco Ruggieri and Pengming Zhang for the useful discussions. J.Y.C. is supported by the NSFC under Grant number: 11605254 and Major State Basic Research Development Program in China (No. 2015CB856903). M.H. is supported by the NSFC under Grant No. 11725523, 11735007 and 11261130311(CRC 110 by DFG and NSFC). A.R. is supported by the CAS President’s international fellowship initiative (Grant No. 2017VMA0045), Council for Grants of the President of the Russian Federation (project NSh-8081.2016.9) and numerical calculations are performed on computing cluster “Akademik V.M. Matrosov” (http://hpc.icc.ru). {width="49.00000%"} {width="43.00000%"} {width="49.00000%"} {width="43.00000%"} {width="49.00000%"} {width="43.00000%"} {width="47.00000%"} {width="47.00000%"} [10]{} J. Cleymans, R. V. Gavai, E. Suhonen, Phys. Rept. [**130**]{} (1986) 217. M. Buballa, Phys. Rept. [**407**]{} (2005) 205. M. Buballa, S. Carignano, Prog. Part. Nucl. Phys. [**81**]{} (2015) 39. J. O. Andersen, P. Kneschke, [](http://arxiv.org/abs/1802.01832). D. E. Kharzeev, L. D. McLerran, H. J. Warringa, Nucl. Phys. A [**803**]{} (2008) 227. V. Skokov, A. [Yu]{}. Illarionov, V. Toneev, Int. J. Mod. Phys. A [**24**]{} (2009) 5925. K. Hattori, X.-G. Huang, Nucl. Sci. Tech. [**28**]{} (2) (2017) 26. J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys.  [**88**]{} (2016) 025001. V. A. Miransky and I. A. Shovkovy, Phys. Rept.  [**576**]{} (2015) 1. X. G. Huang, Rept. Prog. Phys.  [**79**]{}, no. 7 (2016) 076302. W. -T. Deng and X. -G. Huang, Phys. Rev. C [**85**]{} (2012) 044907. R. C. Duncan, C. Thompson, Astrophys. J. [**392**]{} (1992) L9. D. Blaschke, N. Chamel, [](http://arxiv.org/abs/1803.01836). S. P. Klevansky and R. H. Lemmer, Phys. Rev. D [**39**]{} (1989) 3478. K. G. Klimenko, Theor. Math. Phys.  [**89**]{} (1992) 1161 \[Teor. Mat. Fiz.  [**89**]{} (1991) 211\]. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**462**]{} (1996) 249. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**563**]{} (1999) 361. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer and K. K. Szabo, JHEP [**1202**]{} (2012) 044. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer, Phys. Rev. D [**86**]{} (2012) 071502. G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber and A. Schaefer, JHEP [**1304**]{} (2013) 130. D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A [**797**]{} (2007) 67. K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev.  D [**78**]{} (2008) 074033 . M. N. Chernodub, Phys. Rev. D [**82**]{} (2010) 085011. M. N. Chernodub, Phys. Rev. Lett.  [**106**]{} (2011) 142003 . G. Cao, X.-G. Huang, Phys. Lett. B [**757**]{} (2016) 1. L. Wang, G. Cao, X. G. Huang and P. Zhuang, Phys. Lett. B [**780**]{} (2018) 273. S. L. Adler, [Anomalies to all orders]{}, in: G. ’t Hooft (Ed.), 50 years of Yang-Mills theory, 2005, pp. 187–228. R. F. Sawyer, Phys. Rev. Lett. [**29**]{} (1972) 382. R. F. Sawyer, D. J. Scalapino, Phys. Rev. D [**7**]{} (1973) 953. D. N. Voskresensky, N. [Yu]{}. Anisimov, Sov. Phys. JETP [**51**]{} (1980) 13 \[Zh. Eksp. Teor. Fiz. [**78**]{} (1980) 28\]. A. B. Migdal, Rev. Mod. Phys. [**50**]{} (1978) 107. D. T. Son and M. A. Stephanov, Phys. Rev. Lett.  [**86**]{} (2001) 592. L. He, M. Jin and P. Zhuang, Phys. Rev. D [**74**]{} (2006) 036005. S. Mao, Phys. Rev. D [**89**]{} (2014) 116006. T. G. Khunjua, K. G. Klimenko, R. N. Zhokhov, V. C. Zhukovsky, Phys. Rev. D [**95**]{} (2017) 105010. B. B. Brandt, G. Endrodi, PoS [**LATTICE2016**]{} (2016) 039. B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert, J. Schaffner-Bielich, S. Schmalzbauer, [](http://arxiv.org/abs/1802.06685). Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**122**]{} (1961) 345. Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**124**]{} (1961) 246. J. S. Schwinger, Phys. Rev. [**82**]{} (1951) 664. M. K. Volkov, Sov. J. Part. Nucl. [**17**]{} (1986) 186, \[Fiz. Elem. Chast. Atom. Yadra [**17**]{} (1986) 433\]. U. Vogl, W. Weise, Prog. Part. Nucl. Phys. [**27**]{} (1991) 195. S. P. Klevansky, Rev. Mod. Phys. [**64**]{} (1992) 649. T. Hatsuda, T. Kunihiro, Phys. Rept. [**247**]{} (1994) 221. M. K. Volkov, A. E. Radzhabov, Phys. Usp. [**49**]{} (2006) 551. M. R. Brown, M. J. Duff, Phys. Rev. D [**11**]{} (1975) 2124. W. Dittrich and H. Gies, [*Probing the quantum vacuum: perturbative effective action approach in quantum electrodynamics and its application*]{} (2000), Springer Science & Business Media. H. Kohyama, D. Kimura, T. Inagaki, Nucl. Phys. B [**896**]{} (2015) 682. T. Inagaki, D. Kimura, T. Murata, Int. J. Mod. Phys. A [**20**]{} (2005) 4995. M. Ruggieri, Z. Y. Lu, G. X. Peng, Phys. Rev. D [**94**]{} (2016) 116003. W. R. Tavares, S. S. Avancini, [](http://arxiv.org/abs/1801.10566). G. Cao, X.-G. Huang, Phys. Rev. D [**93**]{} (2016) 016007. V. Bernard, A. A. Osipov, U. G. Meissner, Phys. Lett. B [**285**]{} (1992) 119. V. L. Ginzburg, L. D. Landau, Zh. Eksp. Teor. Fiz. [**20**]{} (1950) 1064.

ArXiv

--- abstract: 'Regression analysis is an important machine learning task used for predictive analytic in business, sports analysis, etc. In regression analysis, optimization algorithms play a significant role in search the coefficients in the regression model. In this paper, nonlinear regression analysis using a recently developed meta-heuristic Multi-Verse Optimizer (MVO) is proposed. The proposed method is applied to 10 well-known benchmark nonlinear regression problems. A comparative study has been conducted with Particle Swarm Optimizer (PSO). The experimental results demonstrate that the proposed method statistically outperforms PSO algorithm.' author: - Jayri Bagchi - Tapas Si date: 'Received: date / Accepted: date' title: 'Nonlinear Regression Analysis Using Multi-Verse Optimizer' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction {#intro} ============ Regression analysis is a statistical method to explain the relationship between independent and dependent variables or parameters and predict the coefficients of the function. Linear regression analysis involves those functions that are linear combination of the independent parameters whereas nonlinear regression analysis is a type of regression analysis in which given data is modeled with a function that is a nonlinear combination of multiple independent variables. Some examples on nonlinear regression functions are exponential, logarithmic, trigonometric, power functions. Regression is the most important and widely used statistical technique with many applications in business and economics.\ OZSOY et al. [@Ref2] performed an estimation of nonlinear regression model parameters using PSO. This study compares the optimal and estimated parameters and its results hence show that the estimation of the coefficients using PSO yield reliable results. Mohanty [@Ref9] applied PSO to astronomical data analysis. The results show that PSO requires tuning of few parameters compared to GA but is found to be slightly worse than GA. A case study of PSO in regression analysis by Cheng et al. [@Ref6] utilized PSO to solve a regression problem in the dielectric relaxation field. The results show that PSO with ring structure has a good mean solution than PSO with a star structure. Erdogmus and Ekiz [@Ref9] proposed a nonlinear regression analysis using PSO and GA for some test problems shows that GA shows better performance in estimating the values of the coefficients. Their work further shows that such heuristic optimization algorithms can be an alternative to classic optimization methods. Lu et al. [@Ref3] performed a selection of most important descriptors to build QSAR models using modified PSO (PSO-MLR) and compared the results with GA (GA-MLR). The results reveal that PSO-MLR performed better than GA-MLR for the prediction set. Barmaplexis et al. [@Ref4] applied multi-linear regression, PSO and artificial neural networks in the pre-formulation phase of mini-tablet preparation to establish an acceptable processing window and identify product design space. Their results show that DoE-MLR regression equations gave good fitting results for 5 out of 8 responses whereas GP gave the best results for the other 3 responses. PSO-ANNs was only to fit all selected responses simultaneously. Cerny et al. [@Ref5] proposed a new type of genotype for Prefix Gene Expression Programming (PGEP). PGEP, improved from Gene Expression Programming (GEP) is used for Signomial Regression(SR). The method was called Differential Evolution-Prefix Gene Expression Programming (DE-PGEP) which allows for expression and constants to co-exist in the same vector spaced representation and be evolved simultaneously. Park et al. [@Ref7] proposed PSO based Signomial Regression (PSR) to solve non-linear regression problems. Their work attempted to solve the signomial function by estimating the parameters using PSO. Mishra [@Ref12] evaluates the performance of Differential Evolution at nonlinear curve fitting. Results show that DE has been successful to obtain optimum results even if parameter domains were wide but it couldn’t reach near-optimal results for the CPC-X problems which are the challenge problems for any nonlinear least-squares algorithm. Gilli et al. [@Ref13] used DE, PSO and Threshold Accepting methods to estimate the parameters of linear regression. Yang et al. [@Ref10] constructed the linear regression models for the symbolic interval-values data using PSO.\ The objective of this paper is to perform a nonlinear regression analysis using the MVO algorithm [@Ref1]. The proposed method is applied to 10 well-known benchmark nonlinear regression problems. A comparative study is conducted with PSO [@Ref8]. The experimental results with statistical analysis demonstrate that the proposed method outperforms PSO. Organization of this paper -------------------------- The remaining of the paper is organized as follows: the proposed method is discussed in section \[sec:1\]. The experimental setup including the regression models and dataset description is given in section \[sec:2\]. The results and discussion are given in section \[sec:3\]. Finally, the conclusion with future works is given in section \[sec:4\]. Materials & Methods {#sec:1} =================== Regression Analysis ------------------- Regression analysis is a statistical technique to estimate the relationships among the variables of a function. It is a commonly used method for obtaining the prediction function for predicting the values of the response variable using predictor variables [@Ref16]. There are three types of variable in regression such as - The unknown coefficients or parameters, denoted as $\beta$, may be represent a scalar or a vector - The independent variable or predictor variable, i.e., input vector $X=(x_1,x_2,\ldots,x_n)$ - The dependent variable or response variable, i.e., output $y$ The regression model in basic form can be defined as: $$y \approx f(x,\beta)$$ where $\beta=(\beta_0, \beta_1, \beta_2,\ldots, \beta_m)$.\ A linear regression model is a model of which output variable is the linear combination of coefficients and input variables and it is defined as [@Ref15]: $$y=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_nx_n+\xi$$ where $\xi$ is a random variable, a disturbance that perturbs the output $y$. A nonlinear regression model is a model of which output variable is the nonlinear combination of coefficients and input variables. The nonlinear regression model is defined as follows [@Ref15]: $$y=f(x,\beta)+ \xi$$ where $f$ is the nonlinear function. In regression analysis, an optimizer is used to search the coefficients, i.e., parameters so that the model fits well the data. In the current work, the unknown parameters of different nonlinear regression models are searched using MVO algorithm. The MVO algorithm is discussed next. MVO Algorithm ------------- Multi-Verse Optimizer is an optimization algorithm whose design is inspired by the multiverse theory in Physics [@Ref1]. Multiverse theory in Physics states that there exist multiple universes and each universe possesses its own inflation rate which is responsible for the creation of stars, planets, asteroids, meteroids, black holes, white holes, wormholes, physical laws for that universe. For a universe to be stable, it must have a minimum inflation rate. So the goal of the MVO algorithm is to find the best solution by reducing the inflation rate of the universes which is also the fitness value. Now, observations from multiverse theory show that universes with higher inflation rate have more white holes and universes with a low inflation rates have more black holes. So to have a stable situation, objects from white holes have to travel to black holes. Also the objects in each universe may travel randomly to the best universe through wormholes. In MVO, each solution represents a universe and each variable to be an object in the universe. Further, the inflation rate is assigned to each universe which is proportional to the fitness value of each universe. MVO uses the concept of black holes and white holes for exploring search spaces and wormholes to exploit search spaces. When a tunnel is established between two universes, it is assumed that the universe with a higher inflation rate has more white holes and the universe with a lower inflation rate has more black holes. So universes exchange objects from white holes to black holes which improves the average inflation rates of all universes over the iterations. In order to mathematically model the above idea, the roulette Wheel mechanism is used that selects one of the universes with a high inflation rate to contain a white hole and allows objects from that universe to move into the universe containing a black hole and relatively low inflation rate. At every iteration, universes are sorted and one of them is selected by the roulette wheel to have a white hole. Assuming that $U$ is the matrix of universes with $d$ parameters and $n$ candidate solutions. Roulette wheel selection mechanism based on the normalized inflation rate is illustrated as below: $$x_i(j) = \begin{cases} x_k(j) & r_1<NI(U_i) \\ x_i(j) & r_1>=NI(U_i) \end{cases}$$ Here $x_i(j)$ indicates $j$th parameter of $i$th universe, $U_i$ shows $i$th universe, $NI(U_i)$ is the normalized inflation rate of $i$th universe, $r_1$ is a random number in the interval $[0,1]$ and $x_k(j)$ is the $j$th parameter of the $k$th universe selected by roulette wheel mechanism.\ As this is done with the sorted universes, so the universes with low inflation rates have a higher probability if sending objects through white/black holes. Now in order to perform exploitation, it is considered each universe has wormholes to transport its objects randomly through space. In order to improve average inflation rates, it is assumed that wormhole tunnels are established between a universe and the best universe obtained so far. The formulation of the mechanism is: $$x_i^j = \begin{cases} \begin{cases} x_j+TDR*((ub_j-lb_j)*r_4+lb_j) & r_3<0.5\\ x_j-TDR*((ub_j-lb_j)*r_4+lb_j) & r_3>=0.5 \end{cases} & r_2<WEP\\ x_i^j & r_2>=WEP \end{cases}$$ Here $x_j$ indicates the $j$th parameter of best universe formed so far, $TDR$ and $WEP$ are coefficients, $lb_j$ is the lower bound of $j$th variable, $ub_j$ is the upper bound of $j$th variable, $x_i^j$ indicates the $j$th parameter of $i$th universe, and $r_2$, $r3$, $r_4$ are random numbers in $[0,1]$. $WEP$ is the wormhole existence probability and $TDR$ is the traveling distance rate. $WEP$ is for defining the probability of the existence of wormhole. It is to be increased linearly over the iterations for better exploitation results. TDR defines the distance rate that an object can travel through the wormhole to the best universe obtained so far. $TDR$ is decreased over the iterations to increase the accuracy of local search by the following rule: $$TDR=1-\frac{l^\frac{1}{p}}{L^\frac{1}{p}}$$ where $l$ is the current iteration and $L$ is the maximum number of iterations. $p$ is the exploitation accuracy over iterations and generally, it is set to $6$. The update rule of $WEP$ is as follows: $$WEP=WEP_{\min}+l*\left(\frac{WEP_{\max}-WEP_{\min}}{L}\right)$$ where $WEP_{\min}$ and $WEP_{\max}$ indicate the minimum and maximum range of $WEP$. ![Flowchart of MVO.[]{data-label="fig:mvo_flowchart"}](mvo_flowchart.jpg){height="18cm" width="11cm"} Regression Analysis Using MVO ----------------------------- In this work, MVO is used to search the unknown parameters ($\beta$) of the nonlinear regression model. Let assume a model has $m$ number of parameters and then the dimension of the universe in MVO is $m$. The $i$th universe is represented by $X_i=(x_1,x_2,\ldots,x_m)$. The inflation rate of the universe is the objective function value. The mean square error (MSE) is used as an objective function in this work. The MSE is calculated as follows: $$MSE=\frac{1}{N}\sum_{i=1}^{N}(y_{i}- y_{i}{'})^{2}$$ where $y_i$ and $y_{i}{'}$ are the target and predicted output of the $i$th input data respectively. MVO algorithm minimizes MSE to fit the data. $N$ is the number of samples in the dataset. Experimental Setup {#sec:2} ================== Regression Model & Dataset Description -------------------------------------- In this work, 10 regression model is analyzed and the datasets for the models have been collected from [@Ref14]. The description of different regression models and their dataset is given in Table \[tab:model\].\ [p[0.3cm]{}p[1.0cm]{}p[4.4cm]{}p[2.5cm]{}p[1.2cm]{}]{} Sl. No. & Name & Model & No. of coefficients & No. of samples\ 1 & Misra1a & $\beta_1(1-exp(-\beta_2x))$ & 2 & 14\ 2 & Gauss1 & $\beta_1exp(-\beta_2x)+\beta_3\frac{-(x-\beta_4)^2}{\beta_5^2}+\beta_6\frac{-(x-\beta_7)^2}{\beta_8^2}$ & 8 & 250\ 3 & DanWood & $\beta_1x^{\beta_2}$ & 2 & 6\ 4 & Nelson & $exp(\beta_1-\beta_2x_1exp(-\beta_3x_2))$ & 3 & 128\ 5 & Lanczos2 & $\beta_1exp(-\beta_2x)+\beta_3exp(-\beta_4x)+\beta_5exp(-\beta_6x)$ & 6 & 24\ 6 & Roszman1 & $\beta_1-\beta_2x-\frac{arctan\frac{\beta_3}{x-\beta4}}{\pi}$ & 4 & 25\ 7 & ENSO & $\beta_1+\beta_2cos\frac{2\pi x}{12}+\beta_3sin\frac{2\pi x}{12}+\beta_5cos\frac{2\pi x}{\beta_4}+\beta_6sin\frac{2\pi x}{\beta_4}+\beta_8cos\frac{2\pi x}{\beta_7}+\beta_9sin\frac{2\pi x}{\beta_7}$ & 9 & 168\ 8 & MGH09 & $\frac{\beta_1(x^2+x\beta_2)}{x^2+x\beta_3+\beta_4}$ & 4 & 11\ 9 & Thurber & $\frac{\beta_1+\beta_2x+\beta_3x^2+\beta_4x^3}{1+\beta_5x+\beta_6x^2+\beta_7x^3}$ & 7 & 37\ 10 & Rat42 & $\frac{\beta_1}{1+exp(\beta_2-\beta_3x)}$ & 3 & 9\ Parameters Setting ------------------ The parameters of MVO are set as the following: Number of universe = $30$, $WEP_{\max} = 1$, $WEP_{\min} = 0.2$, exploitation accuracy ($p$)= $6$, the maximum number of iterations=$100$.\ The parameters of PSO are set as the following: population size=$30$, $w_{\max}=0.9$, $w_{\min}=0.4$, $c_1=2.05, c_2=2.05$, the maximum number of iterations=$100$. PC Configuration ---------------- - CPU: Intel i3-4005U 1.70GHz - RAM: 4GB - Operating System: Windows 7 - Software Tool: MATLAB R2018a Results & Discussion {#sec:3} ==================== In this work, nonlinear regression analysis has been performed using the MVO algorithm for $10$ regression models. ‘Hold-out’ cross-validation method is used. $80\%$ of the dataset is used in training of the model and the remaining $20\%$ of the dataset is used as test data for the model. The experiment is repeated $31$ times for each model. The same experiment is conducted using PSO for the comparative study. The quality of the results has been measured in terms of training and testing MSE errors over 31 independent runs. The mean and standard deviation of MSE values in training over $31$ runs are given in Table \[tab:training\]. The mean and standard deviation of MSE values in testing over $31$ runs are given in Table \[tab:1\]. [lll]{} Model & PSO & MVO\ Misra1a & 3.389(0.1696) &**0.2638(0.1246)**\ Gauss1 & 80.4833(130.1000) & **5.5966(0.4977)**\ Danwood & 0.0076(0.0288) & **6.05E-04(1.42E-05)**\ Nelson & 0.0499(0.0441) & **0.0271(1.37E-04)**\ Lanczos2 & 5.96E-04(8.37E-04) & **2.76E-06(5.01E-06)**\ Roszman1 & 2.62E-05(2.20E-05) & **1.56E-05(4.42E-07)**\ ENSO & 11.2482(1.48E-06) & 11.2482(1.82E-05)\ MGH09 & **2.58E-05(1.01E-06)** & 2.65E-05(2.86E-07)\ Thurber & 7.15E+02(8.01E+02) & **5.46E+02(4.03E+02)**\ Rat42 & 1.3186(0.4807) & **0.9483(0.0043)**\ [lllll]{} Model & PSO & MVO & p-value & h\ Misra1a & 1.9869(0.1696) & **0.1381(0.0677)** & 1.17E-06 $\ll$ 0.5 & 1\ Gauss1 & 76.8515(133.73033) & **6.2735(0.7319)** & 1.17E-06 $\ll$ 0.5 & 1\ DanWood & 0.0061(0.0061) & **0.0013(7.04E-05)** & 5.99E-06 $\ll$ 0.5 & 1\ Nelson & 0.0707(0.0567) & **0.0403(5.54E-04)** & 2.12E-04 $\ll$ 0.5 & 1\ Lanczos2 & 8.39E-04(0.0012) & **2.41E-06(4.26E-06)** & 2.56E-06 $\ll$ 0.5 & 1\ Roszman1 & **2.96E-05(5.61E-06)** & 3.78E-05(2.55E-06) & 3.10E-06 $\ll$ 0.5 & -1\ ENSO & 13.4705(6.06E-06) & 13.4705(7.45E-05) & 0.9064 $>$ 0.5 & 0\ MGH09 & 3.91E-05(7.19E-06) & **3.48E-05(1.18E-06)** & 0.004 $<$ 0.5& 1\ Thurber & 9.16E+02(1.05E+03) & **3.57E+02(2.10E+02)** & 3.70E-03 $\ll$ 0.5& 1\ Rat42 & 2.617(0.3246) & **0.4734(0.0349)** & 1.17E-06 $\ll$ 0.5& 1\ \ From Table \[tab:training\], it is observed that MVO performs better in training than PSO for most of the models. The standard deviations of training MSEs of MVO are also lower than that of PSO. It is observed from Table \[tab:1\] that the mean testing MSEs are better than that of PSO for most of the models. To test the significance in the difference of performance of MVO and PSO, a non-parametric statistical test, Wilcoxon’s Signed Ranked Test [@Ref11] has been carried out with significance level ($\alpha$) = $0.05$. The p-values and null hypothesis values ($h$) are given in Table \[tab:1\]. The p-values less than $0.05$ indicates statistically significant difference in the performance whereas p-values greater than or equal to $0.05$ depict no significant difference in the performance of the algorithms. In Table \[tab:1\], $h=1$ indicates MVO statistically outperforms PSO, $h=-1$ indicates PSO statistically outperforms MVO, and $h=0$ indicates no significant difference in the performance. From Table- \[tab:1\], it is observed that the $h$-values come out to be 1 for 8 out of 10 models and it signifies the statistically better performance of MVO over PSO. PSO statistically outperforms MVO for the Roszman1 model. There is no significant difference in the performance of MVO and PSO for ENSO model. The robustness of meta-heuristic algorithms is measured in terms of standard deviations. From Table \[tab:training\] & \[tab:1\], it can be observed that the standard deviation of PSO is lower than PSO that indicates that MVO is more robust than PSO in non-linear regression. Model prediction results using MVO for training and testing data in regression analysis of the Gauss1 model are given in Fig. \[fig:1\] & \[fig:2\] respectively. From these graphs, it is observed that MVO almost fits the training and testing curves for Gauss1 model. The convergence graph of MVO and PSO is given Fig. \[fig:3\]. From this graph, it is observed that MVO has better convergence behavior than PSO. ![Model prediction results using MVO for training data in regression analysis of Gauss1 model.[]{data-label="fig:1"}](gauss1_mvo_train.png){height="6cm" width="13cm"} ![Model prediction results using MVO for testing data in regression analysis of Gauss1 model.[]{data-label="fig:2"}](gauss1_mvo_test.png){height="6cm" width="13cm"} ![Convergence graph of MVO for Danwood model.[]{data-label="fig:3"}](graph_danwood.png){height="6cm" width="13cm"} Conclusion {#sec:4} ========== In this paper, MVO is used for nonlinear regression analysis. MVO is applied to search the parameters of different regression models. For the experiment, 10 well-known benchmark regression models are used. A comparative study has been carried out with PSO. The experimental results demonstrate that the proposed method statistically outperforms PSO in nonlinear regression analysis. In the future, different meta-heuristic algorithms will be studied in nonlinear regression analysis. Franklin, A.G. and Iyer, H.K.: Regression Analysis: Concepts and Application, Duxbury Pr. (1994 ) Bates, D. and Watts, D.: Nonlinear Regression Analysis and Its Applications, John Wiley and Sons, New York.(1988) OZSOY, V.S., ORKCU, H.H.: Estimating the Parameters of Nonlinear Regression Models Through Particle Swarm Optimization, Gazi University Journal of Science, Vol. 29(1),pp. 187–199 (2016) Mohanty, S.D.: Particle Swarm Optimization and regression analysis – I, Astronomical Review, Vol. 7, No. 2, pp 29–35 (2015) Cheng, S., Zhao, C., Wu, J., Shi, Y.: Particle Swarm Optimization in Regression Analysis: A Case Study, In: International Conference in Swarm Intelligence, Part I, LNCS 7928, pp. 55–63 (2013) Erdogmus, P., Ekiz, S.: Nonlinear Regression using Particle Swarm Optimization and Genetic Algorithm, International Journal of Computer Applications (0975 – 8887), Vol. 153, No. 6 (2016) Lu, J., Shen, Qi., Jiang, J., Shen, G., Yu, R.: QSAR analysis of cyclooxygenase inhibitor using particle swarm optimization and multiple linear regression, Journal of Pharmaceutical and Biomedical Analysis, Vol. 35, pp. 679–687 (2004) Barmpalexis, P., Karagianni, A., Karasavvaides, G., Kachrimanis, K.: Comparison of multi-linear regression, particle swarm optimization artificial neural networks and genetic programming in the development of mini-tablets, International Journal of Pharmaceutics, Vol. 551(1-2), pp. 166-176 (2018) Cerny, B., Nelson, P., Zhou, C.: Using Differential Evolution for Symbolic Regression and Numerical Constant Creation, In: Proceedings of the 10th annual conference on Genetic and evolutionary computation (GECCO ’08), pp. 1195–1202 (2008) Park, S., Song, N., Yu, W., Kim2, W.: PSR: PSO-Based Signomial Regression Model, Vol. 19, No. 4, 2019, pp.  307–314(2019) Mishra, S.K.: Performance of Differential Evolution Method in Least Squares Fitting of Some Typical Nonlinear Curves, SSRN-Elsevier, MPRA Paper No. 4656 (2007) Gilli, M., and Schumann, E.: Robust Regression with Optimisation Heuristics, In: A. Brabazon et al. (Eds.): Natural Computing in Computational Finance: Vol. 3, SCI 293, pp. 9-30 (2010) Yang, C., Chuang, C., Jeng, J., Tao, C.: Constructing the linear regression models for the symbolic interval-values data using PSO algorithm, In: IEEE Proceedings of 2011 International Conference on System Science and Engineering, (2011) Mirjalili, S., Mirjalili, S.M., Hatamlou, A.: Multi-Verse Optimizer: a nature inspired algorithm for global optimization, Neural Computing and Applications, Vol. 27, pp. 495–513 (2016) <https://www.itl.nist.gov/div898/strd/nls/nls_main.shtml> Derrac, J., Garcĺa, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms, Swarm and Evolutionary Computation, Vol. 1, pp.3–18 (2011)

ArXiv

--- abstract: 'We study a variant of a problem considered by Dinaburg and Sinaĭ on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.' address: - 'Department of Mathematical Sciences, University of Aarhus, Ny Munkegade Building 530, 8000 Aarhus C, Denmark' - 'School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel' author: - 'Morten S. Risager' - Zeév Rudnick title: On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces --- [^1] Statement of results {#sec:statements} ==================== For a pair of coprime integers $(a,b)$, the linear Diophantine equation $ax-by=1$ is well known to have infinitely many integer solutions $(x,y)$, any two differing by an integer multiple of $(b,a)$. Dinaburg and Sinaĭ [@DinaburgSinaui:1990a] studied the statistics of the “minimal” such solution $v'=(x_0,y_0)$ when the coefficient vector $v=(a,b)$ varies over all primitive integer vectors lying in a large box with commensurate sides. Their notion of “minimality” was in terms of the $L^\infty$-norm $|v'|_\infty:=\max(|x_0|,|y_0|)$, and they studied the ratio $|v'|_\infty/|v|_\infty$, showing that it is uniformly distributed in the unit interval. Other proofs were subsequently given by Fujii [@Fujii:1992a] who reduced the problem to one about modular inverses, and then used exponential sum methods, in particular a non-trivial bound on Kloosterman sums, and by Dolgopyat [@Dolgopyat:1994a], who used continued fractions. In this note, we consider a variant of the question by using minimality with respect to the Euclidean norm $|(x,y)|^2:=x^2+y^2$ and study the ratio $|v'|/|v|$ of the Euclidean norms as the coefficient vector varies over a large ball. In this case too we find uniform distribution, in the interval $[0,1/2]$. However, the methods involved appear quite different, as we invoke an equidistribution theorem of Anton Good [@Good:1983a] which uses harmonic analysis on the modular curve. A lattice point problem ----------------------- We recast the problem in slightly more general and geometric terms. Let $L\subset \C$ be a lattice in the plane, and let ${\operatorname{area}}(L)$ be the area of a fundamental domain for $L$. Any primitive vector $v$ in $L$ can be completed to a basis $\{v,v'\}$ of $L$. The vector $v'$ is unique up to a sign change and addition of a multiple of $v$. In the case of the standard lattice $\Z[\sqrt{-1}]$, taking $v=(a,b)$ and $v'=(x,y)$, the condition that $v$, $v'$ give a basis of $\Z[\sqrt{-1}]$ is equivalent to requiring $ay-bx=\pm 1$. The question is: If we pick $v'$ to minimize the length $|v'|$ as we go through all possible completions, how does the ratio $|v'|/|v|$ between the lengths of $v'$ and $v$ fluctuate? It is easy to see (and we will prove it below) that the ratio is bounded, indeed that for a minimizer $v'$ we have $$\frac{|v'|}{|v|} \leq \frac 12 +O(\frac 1{|v|^4})\;.$$ We will show that the ratio $|v'|/|v|$ is uniformly distributed in $[0,1/2]$ as $v$ ranges over all primitive vectors of $L$ in a large (Euclidean) ball. We refine the problem slightly by requiring that the lattice basis $\{v,v'\}$ is oriented positively, that is ${\operatorname{Im}}(v'/v)>0$. Then $v'$ is unique up to addition of an integer multiple of $v$. For the standard lattice $\Z[\sqrt{-1}]$ and $v=(a,b)$, $v'=(x,y)$ the requirement is then that $ay-bx=+1$. Define the [*signed*]{} ratio by $$\rho(v):=\pm |v'|/|v|$$ where we chose $|v'|$ minimal, and the sign is $+$ if the angle between $v$ and $v'$ is acute, and $-$ otherwise. \[unif dist of rho\] As $v$ ranges over all primitive vectors in the lattice $L$, the signed ratio $\rho(v)$ is uniformly distributed modulo one. Explicitly, let $L_{prim}(T)$ be the set of primitive vectors in $L$ of norm $|v|\leq T$. It is well known that $$\#L_{prim}(T) \sim \frac 1{\zeta(2)} \frac{\pi}{{\operatorname{area}}(L)} T^2, \quad T\to \infty$$ Theorem \[unif dist of rho\] states that for any fixed subinterval $[\alpha,\beta]\in (-1/2,1/2]$, $$\frac 1{\#L_{prim}(T)} \{v\in L_{prim}(T): \alpha<\rho(v)<\beta \} \to \beta-\alpha$$ as $T\to \infty$. Equidistribution of real parts of orbits ---------------------------------------- We will reduce Theorem \[unif dist of rho\] by geometric arguments to a result of Anton Good [@Good:1983a] on uniform distribution of the orbits of a point in the upper half-plane under the action of a Fuchsian group. Let $\G$ be discrete, co-finite, non-cocompact subgroup of $\slr$. The group $\slr$ acts on the upper half-plane $\H=\{z\in \C: {\operatorname{Im}}(z)>0\}$ by linear fractional transformations. We may assume, possibly after conjugation in $\slr$, that $\infty$ is a cusp and that the stabilizer $\G_{\!\infty}$ of $\infty$ in $\G$ is generated by $$\pm {\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) }$$ which as linear fractional transformation gives the unit translation $z\mapsto z+1$. (If $-I\notin \G$ there should be no $\pm$ in front of the matrix). The group $\G=\sl$ is an example of such a group. We note that the imaginary part of $\g(z)$ is fixed on the orbit $\G_{\!\infty}\g z$, and that the real part modulo one is also fixed on this orbit. Good’s theorem is \[equidistribution\] Let $\G$ be as above and let $z\in\H$. Then ${\operatorname{Re}}(\G z)$ is uniformly distributed modulo one as ${\operatorname{Im}}(\g z)\to 0$. More precisely, let $$(\GinfmodG)_{\varepsilon,z}=\{\g\in\GinfmodG : {\operatorname{Im}}{\g z}>\varepsilon\}\;.$$ Then for every continuous function $f\in C(\R\slash \Z)$, as $\varepsilon\to 0$, $$\frac 1{ \#(\GinfmodG)_{\varepsilon,z}} \sum_{\g\in(\GinfmodG)_{\varepsilon,z}}f({\operatorname{Re}}{\g z}) \to\int_{\R\slash\Z}f(t)dt \;.$$ Though the writing in [@Good:1983a] is not easy to penetrate, the results deserve to be more widely known. We sketch a proof of Theorem \[equidistribution\] in appendix \[sec:spectral\], assuming familiarity with standard methods of the spectral theory of automorphic forms. [**Acknowledgements:**]{} We thank Peter Sarnak for his comments on an earlier version and for alerting us to Good’s work. A geometric argument {#sec:Geom} ==================== We start with a basis $\{v,v'\}$ for the lattice $L$ which is oriented positively, that is ${\operatorname{Im}}(v'/v)>0$. For a given $v$, $v'$ is unique up to addition of an integer multiple of $v$. Consider the parallelogram $P(v,v')$ spanned by $v$ and $v'$. Since $\{v,v'\}$ form a basis of the lattice $L$, $P(v,v')$ is a fundamental domain for the lattice and the area of $P(v,v')$ depends only on $L$, not on $v$ and $v'$: ${\operatorname{area}}(P(v,v'))={\operatorname{area}}(L)$. Let $\mu(L)>0$ be the minimal length of a nonzero vector in $L$: $$\mu(L)=\min\{ |v|:0\neq v\in L\}\;.$$ Any minimal vector $v'$ satisfies $$\label{upper bd on v'} |v'|^2\leq (\frac{|v|}2 )^2 + (\frac {{\operatorname{area}}(L)} {|v|})^2 \;.$$ Moreover, if $|v|>2{\operatorname{area}}(L)/\mu(L)$ then the minimal vector $v'$ is unique up to sign. To see , note that the height of the parallelogram $P$ spanned by $v$ and $v'$ is ${\operatorname{area}}(P)/|v| = {\operatorname{area}}(L)/|v|$. If $h$ is the height vector, then the vector $v'$ thus lies on the affine line $h+\R v$ so is of the form $h+tv$. After adding an integer multiple of $v$ we may assume that $|t|\leq 1/2$, a choice that minimizes $|v'|$, and then $$|v'|^2 = t^2|v|^2+ |h|^2\leq \frac 14 |v|^2 + (\frac{{\operatorname{area}}(L)}{|v|})^2 \;.$$ We now show that for $|v|\gg_L 1$, the minimal choice of $v'$ is unique if we assume ${\operatorname{Im}}(v'/v)>0$, and up to sign otherwise: Indeed, writing the minimal $v'$ as above in the form $v'=h+tv$ with $|t|\leq 1/2$, the choice of $t$ is unique unless we can take $t=1/2$, in which case we have the two choices $v'=h\pm v/2$. To see that $t=\pm 1/2$ cannot occur for $|v|$ sufficiently large, we argue that if $v'=h+v/2$ then we must have $2 h=2v'-v\in L$. The length of the nonzero vector $2h$ must then be at least $\mu(L)$. Since $|h|={\operatorname{area}}(L)/|v|$ this gives $2{\operatorname{area}}(L)/|v|\geq \mu(L)$, that is $$|v|\leq \frac{2{\operatorname{area}}(L)}{\mu(L)}$$ Hence $v'$ is uniquely determined if $|v|>2{\operatorname{area}}(L)/\mu(L)$. Let $\alpha=\alpha_{v,v'}$ be the angle between $v$ and $v'$, which takes values between $0$ and $\pi$ since ${\operatorname{Im}}(v'/v)>0$. As is easily seen, for any choice of $v'$, $\sin\alpha_{v,v'}$ shrinks as we increase $|v|$, in fact we have: \[lem:angle\] For any choice of $v'$ we have $$\label{upper bd on alpha} \sin \alpha \leq \frac{{\operatorname{area}}(L)}{\mu(L)}\frac 1{|v|} \;.$$ To see , note that the area of the fundamental parallelogram $P(v,v')$ is given in terms of $\alpha$ and the side lengths by $${\operatorname{area}}(P) =|v| |v'|\sin \alpha$$ and since $v'$ is a non-zero vector of $L$, we necessarily have $|v'|\geq \mu(L)$ and hence, since ${\operatorname{area}}(P)={\operatorname{area}}(L)$ is independent of $v$, $$0<\sin \alpha \leq \frac{{\operatorname{area}}(L)}{\mu(L)|v|}$$ as claimed. Note that if we take for $v'$ with minimal length, then we have a lower bound $\sin\alpha \geq 2{\operatorname{area}}(L)/|v|^2 +O( 1/|v|^6)$ obtained by inserting into the area formula ${\operatorname{area}}(L)=|v||v'|\sin\alpha$. Given a positive basis $\{v,v' \}$, we define a measure of skewness of the fundamental parallelogram as follows: Let $\Pi_v(v')$ be the orthogonal projection of the vector $v'$ to the line through $v$. It is a scalar multiple of $v$: $$\Pi_v(v')= {\operatorname{sk}}(v,v') v$$ where the multiplier ${\operatorname{sk}}(v,v')$, which we call the [*skewness*]{} of the parallelogram, is given in terms of the inner product between $v$ and $v'$ as $$\label{exp ecc} {\operatorname{sk}}(v,v') = \frac{\langle v',v\rangle}{|v|^2} \;.$$ Thus we see that the skewness is the real part of the ratio $v'/v$: $${\operatorname{sk}}(v,v') = {\operatorname{Re}}(v'/v) \;.$$ If we replace $v'$ by adding to it an integer multiple of $v$, then ${\operatorname{sk}}(v,v')$ changes by $${\operatorname{sk}}(v,v'+nv) = {\operatorname{sk}}(v,v') + n \;.$$ In particular, since $v'$ is unique up to addition of an integer multiple of $v$, looking at the fractional part, that is in $\R/\Z$, we get a quantity ${\operatorname{sk}}(v)\in (-1/2,1/2]$ depending only on $v$: $${\operatorname{sk}}(v) : ={\operatorname{sk}}(v,v') \mod 1 \;.$$ This is the least skewness of a fundamental domain for the lattice constructed from the primitive vector $v$. The signed ratio $\rho(v) = \pm |v'|/|v|$ and the least skewness ${\operatorname{sk}}(v)$ are asymptotically equivalent: $$\rho(v) = {\operatorname{sk}}(v)\left(1+O(\frac 1{|v|^2})\right) \;. $$ In terms of the angle $0<\alpha<\pi$ between the vectors $v$ and $v'$, we have $$\label{relation between ecc and alpha} {\operatorname{sk}}(v,v') = \frac{|v'|}{|v|}\cos\alpha \;. $$ Our claim follows from this and the fact $\cos\alpha = \pm 1+O(1/|v|^2)$, which follows from the upper bound of Lemma \[lem:angle\]. Thus the sequences $\{\rho(v)\}$, $\{{\operatorname{sk}}(v)\}$ are asymptotically identical, hence uniform distribution of one implies that of the other. To prove Theorem \[unif dist of rho\] it suffices to show \[unif dist of ecc\] As $v$ ranges over all primitive vectors in the lattice $L$, the least skewness ${\operatorname{sk}}(v)$ become uniformly distributed modulo one. This result, for the standard lattice $\Z[\sqrt{-1}]$, was highlighted by Good in the introduction to [@Good:1983a]. Below we review the reduction of Theorem \[unif dist of ecc\] to Theorem \[equidistribution\]. Proof of Theorem \[unif dist of ecc\] ------------------------------------- Our problems only depend on the lattice $L$ up to scaling. So we may assume that $L$ has a basis $L=\{1,z\}$ with $z=x+iy$ in the upper half-plane. The area of a fundamental domain for $L$ is ${\operatorname{area}}(L)={\operatorname{Im}}(z)$. Any primitive vector has the form $v=cz+d$ with the integers $(c,d)$ co-prime. Now given the positive lattice basis $v=cz+d$ and $v'=az+b$, form the integer matrix $\g=\begin{pmatrix} a&b\\ c&d\end{pmatrix}$ , which has $\det(\g)=+1$ since $\{v,v'\}$ form a positive basis of the lattice. Thus we get a matrix in the modular group $\G=SL_2(\Z)$. Then with $\g$ applied as a Möbius transformation to $z$, the length of $v$ can be computed via $${\operatorname{Im}}(\g z) = \frac{{\operatorname{Im}}(z)}{|cz+d|^2}=\frac{{\operatorname{area}}(L)}{|v|^2}$$ The signed ratio between the lengths of $v$ and $v'$ (when $v'$ is chosen of minimal length) is $$\rho{(v)} = \pm |\gamma z| \;.$$ where the sign is $+$ if ${\operatorname{Re}}(\g z)>0$ and $-$ otherwise. Moreover, we have $${\operatorname{sk}}(v,v') = {\operatorname{Re}}(\g z)$$ Indeed, $${\operatorname{Re}}(\g z) = \frac{ac(x^2+y^2) +(ad+bc)x +bd}{|cz+d|^2}$$ which is ${\operatorname{sk}}(v,v')$ in view of . Consequently, the uniform distribution modulo one of ${\operatorname{sk}}(v)$ as $|v|\to\infty$ is then exactly the uniform distribution modulo one of ${\operatorname{Re}}(\g z)$ as $\g$ varies over $\GinfmodG$ with ${\operatorname{Im}}(\g z )\to 0$, that is Theorem \[equidistribution\]. A sketch of a proof of Good’s theorem {#sec:spectral} ===================================== To prove Theorem \[equidistribution\], we use Weyl’s criterion to reduce it to showing that the corresponding “Weyl sums” satisfy $$\label{character asymptotics} \sum_{\g \in(\GinfmodG)_{\varepsilon,z}}e(m{\operatorname{Re}}{\g z})=\delta_{m=0}\frac{t_\G}{\vol{(\GmodH)}}\frac 1\varepsilon +o(1/\varepsilon)$$ as $\varepsilon\to 0$. Here $t_\G$ equals $2$ if $-I\in \G$ and $1$ otherwise. In turn, will follow, by a more or less standard Tauberian theorem (see e.g. [@PetridisRisager:2004a p. 1035-1038]) from knowing the analytic properties of the series $$V_m(z,s):=\sum_{\g\in\GinfmodG}{\operatorname{Im}}(\g z)^se(m{\operatorname{Re}}(\g z)) \;.$$ studied also in [@Good:1981b; @Neunhoffer:1973a] Here $e(x)=\exp(2\pi i x)$. The series is absolutely convergent for ${\operatorname{Re}}(s)>1$, as is seen by comparison with the standard non-holomorphic Eisenstein series $V_0(z,s)=E(z,s)$ of weight $0$ (See [@Selberg:1989a]). For general $m$ the series is closely related to the Poincaré series $$U_m(z,s) = \sum_{\g\in\GinfmodG}{\operatorname{Im}}(\g z)^s e(m\g z)$$ studied by Selberg [@Selberg:1965a]. For a different application of the series $V_m(z,s)$, see [@Sarnak:2001a]. The analytic properties from which we can conclude Theorem \[equidistribution\] are given by \[continuation\] The series $V_m(z,s)$ admits meromorphic continuation to ${\operatorname{Re}}(s)>1/2.$ If poles exist they are real and simple. If $m\neq 0$ then $V_m(z,s)$ is regular at $s=1$. If $m=0$ the point $s=1$ is a pole with residue $t_\G/\vol{(\GmodH)}$. Moreover, $V_m(z,s)$ has polynomial growth on vertical strips in ${\operatorname{Re}}(s)>1/2$. [**Sketch of proof.**]{} The claim about continuation of $V_0(z,s)=E(z,s)$ is well-known and goes back to Roelcke [@Roelcke:1956a] and Selberg [@Selberg:1963a]. To handle also $m\neq 0$ we may adopt the argument of Colin de Verdière [@Colin-de-Verdiere:1983a Th' eorème 3] and of Goldfeld and Sarnak [@GoldfeldSarnak:1983a] to get the result. This is done as follows: Consider the hyperbolic Laplacian $$\Delta=-y^2\left(\frac{\partial^2 }{\partial x^2}+ \frac{\partial^2 }{\partial y^2}\right) \;.$$ If we restrict $\Delta$ to smooth functions on $\GmodH$ which are compactly supported it defines an essentially self-adjoint operator on $L^2(\GmodH,d\mu)$ where $d\mu(z)=dxdy/y^2$, with inner product $${\left \langle f,g \right\rangle}=\int_{\GmodH}f(z)\overline{g(z)}d\mu(z).$$ We will also denote by ${\Delta}$ the self-adjoint closure. Let $h(y)$ be a smooth function which equals 0 if $y<T$ and 1 if $y>T+1$ where $T$ is sufficiently large. One may check that when ${\operatorname{Re}}(s)>1$ $$V_m(z,s)-h(y)y^se(mx)$$ is square integrable. This is an easy exercise using [@Kubota:1973a Theorem 2.1.2]. The series $V_m(z,s)$ satisfies $$\label{straightforward} (\Delta-s(1-s))V_m(z,s)=(2\pi m)^2V_m(z,s+2) \textrm{ when } {\operatorname{Re}}(s)>1,$$ since $f_s(z)=y^s e^{2\pi i m {\operatorname{Re}}z}$ satisfies this equation and because the Laplacian commutes with isometries, so does $V_m(z,s)$, being a sum of translates of $f_s$. Therefore $$\begin{aligned} \nonumber (\Delta-s(1-s))&(V_m(z,s)-h(y)y^se(mx))\\ =(2\pi m)^2& (V_m(z,s+2)-h(y)y^{s+2}e(mx))\\ \nonumber &-h''(y)y^{s+2}e(mx)-2h'(y)y^{s+1}e(mx)\end{aligned}$$ is also square integrable, since the last two terms are compactly supported. We can therefore use the resolvent $({\Delta}-s(1-s))^{-1}$ to invert this and find $$V_m(z,s)- h(y)y^se(mx)=({\Delta}-s(1-s))^{-1}((2\pi m)^2V_m(z,s+2)-H(z,s))$$ where $$H(z,s)=(2\pi m)^2h(y)y^{s+2}e(mx))+h''(y)y^{s+2}e(mx)+2h'(y)y^{s+1}e(mx)$$ This defines the meromorphic continuation of $V_m(z,s)$ to ${\operatorname{Re}}(s)>1/2$ by the meromorphicity of the resolvent (see e.g [@Faddeev:1967a]). The singular points are simple and contained in the set of $s\in \C$ such that $s(1-s)$ is an eigenvalue of ${\Delta}$. Since ${\Delta}$ is self-adjoint, these lie on the real line (when ${\operatorname{Re}}(s)>1/2$). The potential pole at $s=1$ has residue a constant times $$\int_{\GmodH}(2\pi m)^2V_m(z,3)-H(z,1)d\mu$$ The contribution from $h''(y)y^{s+2}e(mx)+2h'(y)y^{s+1}e(mx)$ is easily seen to be zero if $T$ is large enough using $\int_0^1 e(mx)dx=0$ when $m\neq 0$. To handle the rest we may unfold to get $$\begin{aligned} (2\pi m)^2&\int_{\GmodH}(V_m(z,3)-h(y)y^3e(mx))d\mu(z)\\& = (2\pi m)^2\int_0^\infty\int_0^1(y^3-h(y)y^3)e(mx)y^{-2}dxdy=0 \end{aligned}$$ so $V_m(z,s)$ is analytic at $s=1$. The claim about growth in vertical strips is proved as in [@PetridisRisager:2004a Lemma 3.1]. It is possible to extend the main idea of the proof of Proposition \[continuation\] to prove the meromorphic continuation of $V_m(z,s)$ to $s\in \C$. But since our main aim was to prove Theorem \[equidistribution\] we shall stop here. [10]{} Yves Colin de Verdi[è]{}re. Pseudo-laplaciens. [II]{}. , 33(2):87–113, 1983. Efim I. Dinaburg and Yakov G. Sina[ĭ]{}. The statistics of the solutions of the integer equation [$ax-by=\pm 1$]{}. , 24(3):1–8, 96, 1990. Dimitry Dolgopyat. On the distribution of the minimal solution of a linear [D]{}iophantine equation with random coefficients. , 28(3):22–34, 95, 1994. Ludvig D. Faddeev. The eigenfunction expansion of [L]{}aplace’s operator on the fundamental domain of a discrete group on the [L]{}obačevskiĭ plane. , 17:323–350, 1967. Akio Fujii. On a problem of [D]{}inaburg and [S]{}inaĭ. , 68(7):198–203, 1992. Dorian Goldfeld and Peter Sarnak. Sums of [K]{}loosterman sums. , 71(2):243–250, 1983. A. Good. Beiträge zur [T]{}heorie der [D]{}irichletreihen, die [S]{}pitzenformen zugeordnet sind. , 13(1):18–65, 1981. Anton Good. On various means involving the fourier coefficients of cusp forms. , 183(1):95–129, 1983. Tomio Kubota. . Kodansha Ltd., Tokyo, 1973. H. Neunh[ö]{}ffer. Über die analytische [F]{}ortsetzung von [P]{}oincaréreihen. , pages 33–90, 1973. Yiannis N. Petridis and Morten S. Risager. Modular symbols have a normal distribution. , 14(5):1013–1043, 2004. Walter Roelcke. Analytische [F]{}ortsetzung der [E]{}isensteinreihen zu den parabolischen [S]{}pitzen von [G]{}renzkreisgruppen erster [A]{}rt. , 132:121–129, 1956. Peter Sarnak. Estimates for [R]{}ankin-[S]{}elberg [$L$]{}-functions and quantum unique ergodicity. , 184(2):419–453, 2001. Atle Selberg. Discontinuous groups and harmonic analysis. In [*Proc. Internat. Congr. Mathematicians (Stockholm, 1962)*]{}, pages 177–189. Inst. Mittag-Leffler, Djursholm, 1963. Atle Selberg. On the estimation of [F]{}ourier coefficients of modular forms. In [*Proc. Sympos. Pure Math., Vol. VIII*]{}, pages 1–15. Amer. Math. Soc., Providence, R.I., 1965. Atle Selberg. . Springer-Verlag, Berlin, 1989. With a foreword by K. Chandrasekharan. [^1]: The first author was funded by a Steno Research Grant from The Danish Natural Science Research Council. The second author was supported by the Israel Science Foundation (grant No. 925/06).

ArXiv

--- abstract: | A Gray code for a combinatorial class is a method for listing the objects in the class so that successive objects differ in some prespecified, small way, typically expressed as a bounded Hamming distance. In a previous work, the authors of the present paper showed, among other things, that the $m$-ary Reflected Gray Code Order yields a Gray code for the set of restricted growth functions. Here we further investigate variations of this order relation, and give the first Gray codes and efficient generating algorithms for bounded restricted growth functions. [**Keywords:**]{} [*[Gray code (order), restricted growth function, generating algorithm]{}*]{} author: - Ahmad Sabri - Vincent Vajnovszki title: 'More restricted growth functions: Gray codes and exhaustive generations' --- Introduction ============ In [@SV] the authors shown that both the order relation induced by the generalization of the Binary Reflected Gray Code and one of its suffix partitioned version yield Gray codes on some sets of restricted integer sequences, and in particular for restricted growth functions. These results are presented in a general framework, where the restrictions are defined by means of statistics on integer sequences. In the present paper we investigate two prefix partitioning order relations on the set of [*bounded*]{} restricted growth functions: as in [@SV], the original Reflected Gray Code Order on $m$-ary sequences, and a new order relation which is an appropriate modification of the former one. We show that, according to the parity of the imposed bound, one of these order relations gives a Gray code on the set of bounded restricted growth functions. As a byproduct, we obtain a Gray code for restricted growth functions with a specified odd value for the largest entry; the case of an even value of the largest entry remains an open problem. In the final part we present the corresponding exhaustive generating algorithms. A preliminary version of these results were presented at The Japanese Conference on Combinatorics and its Applications in May 2016 in Kyoto [@SV_Jap]. Notation and definitions ======================== A [*restricted growth function*]{} of length $n$ is an integer sequence ${\boldsymbol{s}}=s_1s_2\ldots s_n$ with $s_1=0$ and $0\leq s_{i+1}\leq \max\{s_j\}_{j=1}^i+1$, for all $i$, $1\leq i\leq n-1$. We denote by $R_n$ the set of length $n$ restricted growth functions, and its cardinality is given by the $n$th Bell number (sequence [A000110]{} in [@sloa]), with the exponential generating function $e^{e^x}-1$. And length $n$ restricted growth functions encode the partitions of an $n$-set. For an integer $b\geq 1$, let $R_n(b)$ denote the set of [*$b$-bounded*]{} sequences in $R_n$, that is, $$R_n(b)=\{s_1s_2\ldots s_n\in R_n\,:\, \max\{s_i\}_{i=1}^n\leq b\},$$ and $$R_n^{*}(b)=\{s_1s_2\ldots s_n\in R_n\,:\, \max\{s_i\}_{i=1}^n= b\}.$$ See Table \[Tb1\] for an example. ---- ----------- -- ----- ----------- --------- ----- ----------------------------------------------------------------------------------------------------- -- 1. 0 0 0 0 0 15. 0 1 0 0 0 [*3*]{} 29. **[0 1 1 1 2]{} & [*1*]{}\ 2. & 0 0 0 0 1 & [*1*]{} & 16. & 0 1 0 0 1 & [*1*]{} & 30. & **[0 1 1 2 2]{} & [*1*]{}\ 3. & 0 0 0 1 0 & [*2*]{} & 17. & **[0 1 0 0 2]{} & [*1*]{} & 31. & **[0 1 1 2 1]{} & [*1*]{}\ 4. & 0 0 0 1 1 & [*1*]{} & 18. & 0 1 0 1 0 & [*2*]{} & 32. & **[0 1 1 2 0]{} & [*1*]{}\ 5. &**[0 0 0 1 2]{} & [*1*]{} & 19. & 0 1 0 1 1 & [*1*]{} & 33. & **[0 1 2 2 0]{} & [*1*]{}\ 6. & 0 0 1 0 0 & [*3*]{} & 20. & **[0 1 0 1 2]{} & [*1*]{} & 34. & **[0 1 2 2 1]{} & [*1*]{}\ 7. & 0 0 1 0 1 & [*1*]{} & 21. & **[0 1 0 2 2]{} & [*1*]{} & 35. & **[0 1 2 2 2]{} & [*1*]{}\ 8. &**[0 0 1 0 2]{} & [*1*]{} & 22. & **[0 1 0 2 1]{} & [*1*]{} & 36. & **[0 1 2 1 2]{} & [*1*]{}\ 9. & 0 0 1 1 0 & [*2*]{} & 23. & **[0 1 0 2 0]{} & [*1*]{} & 37. & **[0 1 2 1 1]{} & [*1*]{}\ 10. & 0 0 1 1 1 & [*1*]{} & 24. & 0 1 1 0 0 & [*2*]{} & 38. & **[0 1 2 1 0]{} & [*1*]{}\ 11. &**[0 0 1 1 2]{} & [*1*]{} & 25. & 0 1 1 0 1 & [*1*]{} & 39. & **[0 1 2 0 2]{} & [*2*]{}\ 12. &**[0 0 1 2 2]{} & [*1*]{} & 26. & **[0 1 1 0 2]{} & [*1*]{} & 40. & **[0 1 2 0 1]{} & [*1*]{}\ 13. &**[0 0 1 2 1]{} & [*1*]{} & 27. & 0 1 1 1 0 & [*2*]{} & 41. & **[0 1 2 0 0]{} & [*1*]{}\ 14. &**[0 0 1 2 0]{} & [*1*]{} & 28. & 0 1 1 1 1 & [*1*]{} & & &\ ************************************************** ---- ----------- -- ----- ----------- --------- ----- ----------------------------------------------------------------------------------------------------- -- : \[Tb1\] The set $R_5(2)$, and in bold-face the set $R^*_5(2)$. Sequences are listed in $\preccdot$ order (see Definition \[de:coRGCorder\]) and in italic is the Hamming distance between consecutive sequences. If a list of same length sequences is such that the Hamming distance between successive sequences (that is, the number of positions in which the sequences differ) is bounded from above by a constant, independent on the sequences length, then the list is said to be a [*Gray code*]{}. When we want to explicitly specify this constant, say $d$, then we refer to such a list as a [*$d$-Gray code*]{}; in addition, if the positions where the successive sequences differ are adjacent, then we say that the list is a [*$d$-adjacent Gray code*]{}. The next two definitions give order relations on the set of $m$-ary integer sequences of length $n$ on which our Gray codes are based. \[de:RGCorder\] Let $m$ and $n$ be positive integers with $m\geq 2$. The [*Reflected Gray Code Order*]{} $\prec$ on $\{0,1,\ldots,m-1\}^n$ is defined as: ${\boldsymbol{s}}=s_1s_2\ldots s_n$ is less than ${\boldsymbol{t}}=t_1t_2\ldots t_n$, denoted by ${\boldsymbol{s}}\prec{\boldsymbol{t}}$, if either $\sum_{i=1}^{k-1} s_i$ is even and $s_k<t_k$, or $\sum_{i=1}^{k-1} s_i$ is odd and $s_k>t_k$ for some $k$ with $s_i=t_i$ ($1\leq i\leq k-1$) and $s_k\neq t_k$. This order relation is the natural extension to $m$-ary sequences of the order induced by the Binary Reflected Gray Code introduced in [@Gray]. See for example [@BBPSV; @SV] where this order relation and its variations are considered in the context of factor avoiding words and of statistic-restricted sequences. \[de:coRGCorder\] Let $m$ and $n$ be positive integers with $m\geq 2$. The [*co-Reflected Gray Code Order*]{}[^1] $\preccdot$ on $\{0,1,\ldots,m-1\}^n$ is defined as: ${\boldsymbol{s}}=s_1s_2\ldots s_n$ is less than ${\boldsymbol{t}}=t_1t_2\ldots t_n$, denoted by ${\boldsymbol{s}}\preccdot\ {\boldsymbol{t}}$, if either $U_k$ is even and $s_k<t_k$, or $U_k$ is odd and $s_k>t_k$ for some $k$ with $s_i=t_i$ ($1\leq i\leq k-1$) and $s_k\neq t_k$ where $ U_k= | \{i\in \{1,2,\ldots,k-1\}:s_i\neq 0, s_i \mbox{ is even}\}|$. See Table \[Tb\_co\] for an example. For a set $S$ of same length integer sequences the [*$\prec$-first*]{} (resp. [*$\prec$-last*]{}) sequence in $S$ is the first (resp. last) sequence when the set is listed in $\prec$ order; and [*$\preccdot$-first*]{} and [*$\preccdot$-last*]{} are defined in a similar way. And for sequence ${\boldsymbol{u}}$, $ {\boldsymbol{u}}\,|\,S$ denotes the subset of $S$ of sequences having prefix ${\boldsymbol{u}}$. Both order relations, Reflected and co-Reflected Gray Code Order produce prefix partitioned lists, that is to say, if a set of sequences is listed in one of these order relations, then the sequences having a common prefix are consecutive in the list. ---- ------- ----- ------- ----- ------- 1. 0 0 0 10. 1 0 0 19. 2 2 0 2. 0 0 1 11. 1 0 1 20. 2 2 1 3. 0 0 2 12. 1 0 2 21. 2 2 2 4. 0 1 0 13. 1 1 0 22. 2 1 2 5. 0 1 1 14. 1 1 1 23. 2 1 1 6. 0 1 2 15. 1 1 2 24. 2 1 0 7. 0 2 2 16. 1 2 2 25. 2 0 2 8. 0 2 1 17. 1 2 1 26. 2 0 1 9. 0 2 0 18. 1 2 0 27. 2 0 0 ---- ------- ----- ------- ----- ------- : \[Tb\_co\] The set $\{0,1,2\}^3$ listed in $\preccdot$ order. The Gray codes ============== In this section we show that the set $R_n(b)$, with $b$ odd, listed in $\prec$ order is a Gray code. However, $\prec$ does not induce a Gray code when $b$ is even: the Hamming distance between two consecutive sequences can be arbitrary large for large enough $n$. To overcome this, we consider $\preccdot$ order instead of $\prec$ order when $b$ is even, and we show that the obtained list is a Gray code. In the proof of Theorem \[k\_odd\] below we need the following propositions which give the forms of the last and first sequence in $R_n(b)$ having a certain fixed prefix, when sequences are listed in $\prec$ order. \[pro:pro\_Rb\_odd1\] Let $b\geq1$ and odd, $k\leq n-2$ and ${\boldsymbol{s}}=s_1\ldots s_{k}$. If ${\boldsymbol{t}}$ is the $\prec$-last sequence in ${\boldsymbol{s}}\,|\,R_n(b)$, then ${\boldsymbol{t}}$ has one of the following forms: 1. ${\boldsymbol{t}}={\boldsymbol{s}}M0\ldots0$ if $\sum_{i=1}^{k} s_i$ is even and $M$ is odd, 2. ${\boldsymbol{t}}={\boldsymbol{s}}M(M+1)0\ldots0$ if $\sum_{i=1}^{k} s_i$ is even and $M$ is even, 3. ${\boldsymbol{t}}={\boldsymbol{s}}0\ldots0$ if $\sum_{i=1}^{k} s_i$ is odd, where $M=\min\{b,\max\{s_i\}_{i=1}^k+1\}$. Let ${\boldsymbol{t}}=s_1\dots s_kt_{k+1}\ldots t_n$ be the $\prec$-last sequence in ${\boldsymbol{s}}\,|\,R_n(b)$. Referring to the definition of $\prec$ order in Definition \[de:RGCorder\], if $\sum_{i=1}^{k} s_i$ is even, then $t_{k+1}=\min\{b,\max\{s_i\}_{i=1}^k+1\}= M$, and based on the parity of $M$, two cases can occur. - If $M$ is odd, then we have that $\sum_{i=1}^k s_i+t_{k+1}=\sum_{i=1}^{k} s_i+M$ is odd, thus $t_{k+2}\ldots t_n=0\ldots 0$, and we retrieve the form prescribed by the first point of the proposition. - If $M$ is even, then $M\neq b$ and $\sum_{i=1}^k s_i+t_{k+1}=\sum_{i=1}^{k} s_i+M$ is even, thus $t_{k+2}=\max\{s_1,\ldots, s_k,t_{k+1}\}+1=M+1$, which is odd. Next, we have $\sum_{i=1}^k s_i+t_{k+1}+t_{k+2}=\sum_{i=1}^k s_i+2M+1$ is odd, and this implies as above that $t_{k+3}\ldots t_n=0\ldots 0$, and we retrieve the second point of the proposition. For the case when $\sum_{i=1}^{k} s_i$ is odd, in a similar way we have $t_{k+1}\dots t_n=0\dots 0$. The next proposition is the ‘first’ counterpart of the previous one. Its proof is similar by exchanging the parity of the summation from ‘odd’ to ‘even’ and vice-versa, and it is left to the reader. \[pro:pro\_Rb\_odd2\] Let $b\geq1$ and odd, $k\leq n-2$ and ${\boldsymbol{s}}=s_1\ldots s_{k}$. If ${\boldsymbol{t}}$ is the $\prec$-first sequence in ${\boldsymbol{s}}\,|\,R_n(b)$, then ${\boldsymbol{t}}$ has one of the following forms: 1. ${\boldsymbol{t}}={\boldsymbol{s}}M0\ldots0$ if $\sum_{i=1}^{k} s_i$ is odd and $M$ is odd, 2. ${\boldsymbol{t}}={\boldsymbol{s}}M(M+1)0\ldots0$ if $\sum_{i=1}^{k} s_i$ is odd and $M$ is even, 3. ${\boldsymbol{t}}={\boldsymbol{s}}0\ldots0$ if $\sum_{i=1}^{k} s_i$ is even, where $M=\min\{b,\max\{s_i\}_{i=1}^k+1\}$. Based on Propositions \[pro:pro\_Rb\_odd1\] and \[pro:pro\_Rb\_odd2\], we have the following theorem. \[k\_odd\] For any $n,b\geq 1$ and $b$ odd, $R_n(b)$ listed in $\prec$ order is a $3$-adjacent Gray code. Let ${\boldsymbol{s}}=s_1s_2\dots s_n$ and ${\boldsymbol{t}}=t_1t_2\dots t_n$ be two consecutive sequences in $\prec$ ordered list for the set $R_n(b)$, with ${\boldsymbol{s}}\prec {\boldsymbol{t}}$, and let $k$ be the leftmost position where ${\boldsymbol{s}}$ and ${\boldsymbol{t}}$ differ. If $k\geq n-2$, then obviously ${\boldsymbol{s}}$ and ${\boldsymbol{t}}$ differ in at most three positions, otherwise let ${\boldsymbol{s}}'=s_1\ldots s_k$ and ${\boldsymbol{t}}'=t_1\ldots t_k$. Thus, ${\boldsymbol{s}}$ is the $\prec$-last sequence in ${\boldsymbol{s}}'\,|\,R_n(b)$ and ${\boldsymbol{t}}$ is the $\prec$-first sequence in ${\boldsymbol{t}}'\,|\,R_n(b)$. Combining Propositions \[pro:pro\_Rb\_odd1\] and \[pro:pro\_Rb\_odd2\] we have that, when $k\leq n-3$, $s_{k+3}s_{k+4}\dots s_n=t_{k+3}t_{k+4}\dots t_n=00\dots 0$. And since $s_i=t_i$ for $i=1\ldots,k-1$, the statement holds. Theorem \[k\] below shows the Graycodeness of $R_n(b)$, $b\geq 1$ and even, listed in $\preccdot$ order, and as for Theorem \[k\_odd\] we need the next two propositions; in its proof we will make use of the Iverson bracket notation: $[P]$ is $1$ if the statement $P$ is true, and 0 otherwise. Thus, for a sequence $s_1s_2\dots s_n$ and a $k\leq n$, $| \{i\in \{1,2,\ldots,k\}:s_i\neq 0 \mbox{ and } s_i \mbox{ is even}\}|=\sum_{i=1}^k[s_i\neq0 {\rm\ and\ } s_i {\rm\ is\ even}]$. \[pro:pro\_Rb\_even1\] Let $b\geq2$ and even, $k\leq n-2$ and ${\boldsymbol{s}}=s_1s_2\ldots s_k$. If ${\boldsymbol{t}}$ is the $\preccdot$-last sequence in ${\boldsymbol{s}}\,|\,R_n(b)$, then ${\boldsymbol{t}}$ has one of the following forms: 1. ${\boldsymbol{t}}={\boldsymbol{s}}M0\ldots0$ if $U_{k+1}$ is even and $M$ is even, 2. ${\boldsymbol{t}}={\boldsymbol{s}}M(M+1)0\ldots0$ if $U_{k+1}$ is even and $M$ is odd, 3. ${\boldsymbol{t}}={\boldsymbol{s}}0\ldots0$ if $U_{k+1}$ is odd, where $M=\min\{b,\max\{s_i\}_{i=1}^k+1\}$ and $U_{k+1}=\sum_{i=1}^k[s_i\neq0 {\rm\ and\ } s_i {\rm\ is\ even}]$. Let ${\boldsymbol{t}}=s_1\dots s_kt_{k+1}\dots t_n$ be the $\preccdot$-last sequence in ${\boldsymbol{s}}\,|\,R_n(b)$. Referring to the definition of $\preccdot$ order in Definition \[de:coRGCorder\], if $U_{k+1}$ is even, then $t_{k+1}=\min\{b,\max\{s_i\}_{i=1}^k+1\}=M>0$, and based on the parity of $M$, two cases can occur. - If $M$ is even, then $U_{k+1}+[t_{k+1}\neq0 {\rm\ and\ } t_{k+1} {\rm\ is\ even}]=U_{k+1}+1$ is odd, thus $t_{k+2}\dots t_n=0\dots 0$, and we retrieve the form prescribed by the first point of the proposition. - If $M$ is odd, then $M\neq b$ and $U_{k+1}+[t_{k+1}\neq0 {\rm\ and\ } t_{k+1} {\rm\ is\ even}]=U_{k+1}$ is even, thus $t_{k+2}=\max\{s_1,\dots, s_k,t_{k+1}\}+1=M+1$, which is even. Next, we have $U_{k+1}+[t_{k+1}\neq0 {\rm\ and\ } t_{k+1} {\rm\ is\ even}]+ [t_{k+2}\neq0 {\rm\ and\ } t_{k+2} {\rm\ is\ even}]=U_{k+1}+1$ is odd, and this implies as above that $t_{k+3}\dots t_n=0\dots 0$, and we retrieve the second point of the proposition. For the case when $U_{k+1}$ is odd, in a similar way we have $t_{k+1}\dots t_n=0\dots 0$. The next proposition is the ‘first’ counterpart of the previous one. \[pro:pro\_Rb\_even2\] Let $b\geq2$ and even, $k\leq n-2$ and ${\boldsymbol{s}}=s_1s_2\ldots s_k$. If ${\boldsymbol{t}}$ is the $\preccdot$-first sequence in ${\boldsymbol{s}}\,|\,R_n(b)$, then ${\boldsymbol{t}}$ has one of the following forms: 1. ${\boldsymbol{t}}={\boldsymbol{s}}M0\ldots0$ if $U_{k+1}$ is odd and $M$ is even, 2. ${\boldsymbol{t}}={\boldsymbol{s}}M(M+1)0\ldots0$ if $U_{k+1}$ is odd and $M$ is odd, 3. ${\boldsymbol{t}}={\boldsymbol{s}}0\ldots0$ if $U_{k+1}$ is even, where $M=\min\{b,\max\{s_i\}_{i=1}^k+1\}$ and $U_{k+1}=\sum_{i=1}^k[s_i\neq0 {\rm\ and\ } s_i {\rm\ is\ even}]$. Based on Propositions \[pro:pro\_Rb\_even1\] and \[pro:pro\_Rb\_even2\] we have the following theorem, its proof is similar with that of Theorem \[k\_odd\]. \[k\_even\] For any $n\geq 1$, $b\geq 2$ and even, $R_n(b)$ listed in $\preccdot$ order is a $3$-adjacent Gray code. It is worth to mention that, neither $\prec$ for even $b$, nor $\preccdot$ for odd $b$ yields a Gray code on $R_n(b)$. Considering $b\geq n$ in Theorem \[k\_odd\] and \[k\_even\], the bound $b$ does not actually provide any restriction, and in this case $R_n(b)=R_n$, and we have the following corollary. For any $n\geq 1$, $R_n$ listed in both $\prec$ and $\preccdot$ order are $3$-adjacent Gray codes. \[k\] For any $b\geq 1$ and odd, $n>b$, $R_n^*(b)$ listed in $\prec$ order is a $5$-Gray code. For two integers $a$ and $b$, $0<a\leq b$, we define $\tau_{a,b}$ as the length $b-a$ increasing sequence $(a+1)(a+2) \ldots (b-1)b$, and $\tau_{a,b}$ is vanishingly empty if $a=b$. Imposing to a sequence ${\boldsymbol{s}}$ in $R_n(b)$ to have its largest element equal to $b$ (so, to belong to $R^*_n(b)$) implies that either $b$ occurs in ${\boldsymbol{s}}$ before its last position, or ${\boldsymbol{s}}$ ends with $b$, and in this case the tail of ${\boldsymbol{s}}$ is $\tau_{a,b}$ for an appropriate $a<b$. More precisely, in the latter case, ${\boldsymbol{s}}$ has the form $s_1s_2\ldots s_j\tau_{a,b}$, for some $j$ and $a$, with $a=\max\{s_i\}_{i=1}^j$ and $j=n-(b-a)$. Now let ${\boldsymbol{s}}=s_1s_2\ldots s_n\prec{\boldsymbol{t}}=t_1t_2\ldots t_n$ be two consecutive sequences in the $\prec$ ordered list for $R^*_n(b)$, and let $k\leq n-3$ be the leftmost position where ${\boldsymbol{s}}$ and ${\boldsymbol{t}}$ differ, thus $s_1s_2\ldots s_{k-1}=t_1t_2\ldots t_{k-1}$. It follows that ${\boldsymbol{s}}$ is the $\prec$-last sequence in $R^*_n(b)$ having the prefix $s_1s_2\ldots s_k$, and using Proposition \[pro:pro\_Rb\_odd1\] and the notations therein, by imposing that $\max\{s_i\}_{i=1}^n$ is equal to $b$, we have: - if $\sum_{i=1}^ks_i$ is odd, then ${\boldsymbol{s}}$ has the form $s_1s_2\ldots s_k0\ldots0\,\tau_{a,b}$, where $a=\max\{s_i\}_{i=1}^k$, - if $\sum_{i=1}^ks_i$ is even, then ${\boldsymbol{s}}$ has one of the following forms: - $s_1s_2\ldots s_k M0\ldots0\,\tau_{M,b}$, or - $s_1s_2\ldots s_kM(M+1)0\ldots0\,\tau_{M+1,b}$. When the above $\tau$’s suffixes are empty, we retrieve precisely the three cases in Proposition \[pro:pro\_Rb\_odd1\]. Similarly, ${\boldsymbol{t}}$ is the $\prec$-first sequence in $R^*_n(b)$ having the prefix $t_1t_2\ldots t_{k-1}t_k=s_1s_2\ldots s_{k-1}t_k$. Since by the definition of $\prec$ order we have that $t_k=s_k+1$ or $t_k=s_k-1$, it follows that $\sum_{i=1}^kt_i$ and $\sum_{i=1}^ks_i$ have different parity (that is, $\sum_{i=1}^kt_i$ is odd if and only if $\sum_{i=1}^ks_i$ is even), and by Proposition \[pro:pro\_Rb\_odd2\] and replacing for notational convenience $M$ by $M'$, we have: - if $\sum_{i=1}^ks_i$ is odd, then ${\boldsymbol{t}}$ has the form $t_1t_2\ldots t_k0\ldots0\,\tau_{a',b}$, where $a'=\max\{t_i\}_{i=1}^k$, - if $\sum_{i=1}^ks_i$ is even, then ${\boldsymbol{t}}$ has one of the following forms: - $t_1t_2\ldots t_k M'0\ldots0\,\tau_{M',b}$, or - $t_1t_2\ldots t_kM'(M'+1)0\ldots0\,\tau_{M'+1,b}$. With these notations, since $t_k\in \{s_k+1,s_k-1\}$, it follows that - if $\sum_{i=1}^ks_i$ is odd, then $a'\in\{a-1,a,a+1\}$, and so the length of $\tau_{a,b}$ and that of $\tau_{a',b}$ differ by at most one; and - if $\sum_{i=1}^ks_i$ is even, then $M'\in\{M-1,M,M+1\}$, and the length of the non-zero tail of ${\boldsymbol{s}}$ and that of ${\boldsymbol{t}}$ (defined by means of $\tau$ sequences) differ by at most two. Finally, the whole sequences ${\boldsymbol{s}}$ and ${\boldsymbol{t}}$ differ in at most five (not necessarily adjacent) positions, and the statement holds. Generating algorithms ===================== An exhaustive generating algorithm is one generating all sequences in a combinatorial class, with some predefined properties ([*e.g.*]{}, having the same length). Such an algorithm is said to run in [*constant amortized time*]{} if it generates each object in $O(1)$ time, in amortized sense. In [@Ruskey] the author called such an algorithm [*CAT algorithm*]{} and shows that a recursive generating algorithm satisfying the following properties is precisely a CAT algorithm: - Each recursive call either generates an object or produces at least two recursive calls; - The amount of computation in each recursive call is proportional to the degree of the call (that is, to the number of subsequent recursive calls produced by current call). Procedure [Gen1]{} in Fig. \[fig:alg\_Rnb\_odd\] generates all sequences belonging to $R_n(b)$ in Reflected Gray Code Order. Especially when $b$ is odd, the generation induces a 3-adjacent Gray code. The bound $b$ and the generated sequence $s=s_1s_2\ldots s_n$ are global. The $k$ parameter is the position where the value is to be assigned (see line 8 and 13); the $dir$ parameter represents the direction of sequencing for $s_k$, whether it is up (when $dir$ is even, see line 7) or down (when $dir$ is odd, see line 12); and $m$ is such that $m+1$ is the the maximum value that can be assigned to $s_k$, that is, $\min\{b-1,\max\{s_i\}_{i=1}^{k-1}\}$ (see line 5). The algorithm initially sets $s_1=0$, and the recursive calls are triggered by the initial call [Gen1]{}$(2,0,0)$. For the current position $k$, the algorithm assigns a value to $s_k$ (line 8 or 13) followed by recursive calls in line 10 or 15. This scheme guarantees that each recursive call will produce subsequent recursive calls until $k=n+1$ (line 4), that is, when a sequence of length $n$ is generated and printed out by [Type()]{} procedure. This process eventually generates all sequences in $R_n(b)$. In addition, by construction, algorithm [Gen1]{} satisfies the previous CAT desiderata, and so it is en efficient exhaustive generating algorithm. +-----------------------------------------------------------------------+ | \ | | [01]{} [**procedure**]{} [Gen1]{}($k$, $dir$, $m$: integer)\ | | [02]{} [**global**]{} $s$, $n$, $b$: integer;\ | | [03]{} [**local**]{} $i$, $u$: integer;\ | | [04]{} [**if**]{} $k=n+1$ [**then**]{} [Type()]{};\ | | [05]{} [**else**]{} **if** $m=b$ [**then**]{} $m:=b-1$; | | [**endif**]{}\ | | [06]{} $dir$ mod $2=0$\ | | [07]{} **for** =$i:=0$ [**to**]{} $m+1$ [**do**]{}\ | | [08]{} $s_k:=i$;\ | | [09]{} $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; | | [**endif**]{}\ | | [10]{} ;\ | | [11]{}\ | | [12]{} $i:=m+1$ [**downto**]{} $0$ [**do**]{}\ | | [13]{} $s_k:=i$;\ | | [14]{} $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; | | [**endif**]{}\ | | [15]{} $(k+1,i+1,u)$;\ | | [16]{}\ | | [17]{}\ | | [18]{} [**endif**]{}\ | | [19]{} [**end procedure.**]{} | +:=====================================================================:+ +-----------------------------------------------------------------------+ Similarly, the call [Gen2]{}$(2,0,0)$ of the algorithm in Fig. \[fig:alg\_Rnb\_even\] generates sequences in $R_n(b)$ in co-Reflected Gray Code Order, and in particular when $b$ is even, a 3-adjacent Gray code for these sequences. And again it satisfies the CAT desiderata, and so it is en efficient exhaustive generating algorithm. Finally, algorithm [Gen3]{} in Fig. \[fig:alg\_Pnb\_odd\] generates the set $R^*_n(b)$ in Reflected Gray Code Order and produces a 5-Gray code if $b$ is odd. It mimes algorithm [Gen1]{} and the only differences consist in an additional parameter $a$ and lines 5, 6, 13 and 19, and its main call is [Gen3]{}$(2,0,0,0)$. Parameter $a$ keeps track of the maximum value in the prefix $s_1s_2\dots s_{k-1}$ of the currently generated sequence, and it is updated in lines 13 and 19. Furthermore, when the current position $k$ belongs to a $\tau$-tail (see the proof of Theorem \[k\]), that is, condition $k=n+1+a-b$ in line 5 is satisfied, then the imposed value is written in this position, and similarly for the next two positions. Theorem \[k\] ensures that there are no differences between the current sequence and the previous generated one beyond position $k+2$, and thus a new sequence in $R^*_n(b)$ is generated. And as previously, [Gen3]{} is a CAT generating algorithm. +-----------------------------------------------------------------------+ | \ | | [**procedure**]{} [Gen2]{}($k$, $dir$, $m$: integer)\ | | [**global**]{} $s$, $n$, $b$: integer;\ | | [**local**]{} $i$, $u$: integer;\ | | [**if**]{} $k=n+1$ [**then**]{} [Type()]{};\ | | [**else**]{} **if** $m=b$ [**then**]{} $m:=b-1$; [**endif**]{}\ | | $dir$ mod 2$=0$\ | | **for** = $i:=0$ [**to**]{} $m+1$ [**do**]{}\ | | $s_k:=i$;\ | | $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; [**endif**]{}\ | | $s_k=0$ **then** [Gen2]{}$(k+1, 0,u)$;\ | | $(k+1, i+1,u)$;\ | | \ | | \ | | $i:=m+1$ [**downto**]{} $0$ [**do**]{}\ | | $s_k:=i$;\ | | $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; [**endif**]{}\ | | $s_k=0$ **then** [Gen2]{}$(k+1, 1,u)$;\ | | $(k+1, i,u)$;\ | | \ | | \ | | \ | | [**endif**]{}\ | | [**end procedure.**]{} | +:=====================================================================:+ +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | \ | | [01]{} [**procedure**]{} [Gen3]{}($k$, $dir$, $m$, $a$: integer)\ | | [02]{} [**global**]{} $s$, $n$, $b$: integer;\ | | [03]{} [**local**]{} $i$, $u$, $\ell$: integer;\ | | [04]{} [**if**]{} $k=n+1$ [**then**]{} [Type()]{};\ | | [05]{} [**else**]{} **if** $k=n+1 + a-b$\ | | [06]{} **for** $i:=0$ [**to**]{} $2$ [**do**]{} [**if**]{} | | $k+i\leq n$ [**then**]{} $s_{k+i}:=a+1+i$; [**endif**]{} | | [**endfor**]{}\ | | [07]{} ;\ | | [08]{} **if** $m=b$ [**then**]{} $m:=b-1$; [**endif**]{}\ | | [09]{} $dir$ mod 2$=0$\ | | [10]{} **for** = $i:=0$ [**to**]{} $m+1$ [**do**]{}\ | | [11]{} $s_k:=i$;\ | | [12]{} $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; | | [**endif**]{}\ | | [13]{} $a<s_k$ [**then**]{} $\ell:=s_k$; [**else**]{} $\ell:=a$; | | [**endif**]{}\ | | [14]{} ;\ | | [15]{}\ | | [16]{} $i:=m+1$ [**downto**]{} $0$ [**do**]{}\ | | [17]{} $s_k:=i$;\ | | [18]{} $m<s_k$ [**then**]{} $u:=s_k$; [**else**]{} $u:=m$; | | [**endif**]{}\ | | [19]{} $a<s_k$ [**then**]{} $\ell:=s_k$; [**else**]{} $\ell:=a$; | | [**endif**]{}\ | | [20]{} $(k+1,i+1,u,\ell)$;\ | | [21]{}\ | | [22]{}\ | | [23]{}\ | | [24]{} [**endif**]{}\ | | [25]{} [**end procedure.**]{} | +:=====================================================================:+ +-----------------------------------------------------------------------+ [**Final remarks.**]{} We suspect that the upper bounds 3 in Theorems \[k\_odd\] and \[k\_even\], and 5 in Theorem \[k\] are not tight, and a natural question arises: are there more restrictive Gray codes for $R_n(b)$ and for $R^*_n(b)$ with $b$ odd? Finally, is there a natural order relation inducing a Gray code on $R^*_n(b)$ when $b$ is even? A. Bernini, S. Bilotta, R. Pinzani, A. Sabri, V. Vajnovszki, Reflected Gray codes for $q$-ary words avoiding a given factor, Acta Informatica, 52(7), 573-592 (2015). F. Gray, Pulse code communication, U.S. Patent 2632058 (1953). F. Ruskey, Combinatorial generation, Book in preparation. A. Sabri, V. Vajnovszki, Reflected Gray code based orders on some restricted growth sequences, The Computer Journal, 58(5), 1099-1111 (2015). A. Sabri, V. Vajnovszki, Bounded growth functions: Gray codes and exhaustive generation, The Japanese Conference on Combinatorics and its Applications, May 21-25, 2016, Kyoto, Japan. N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences, available electronically at [http://oeis.org]{}. [^1]: In [@SV] a similar terminology is used for a slightly different notion

ArXiv

--- address: 'Max Planck Institute for the Physics of Complex Systems, D-01187, Dresden, Germany' author: - 'M. V. Fistul and J. B. Page[@byline]' title: | Penetration of dynamic localized states in DC-driven\ Josephson junction ladders by discrete jumps --- Introduction ============ The subject of large-amplitude anharmonic dynamics in lattices has received widespread attention over the past decade. In particular, intense theoretical focus has centered on so-called intrinsic localized modes (ILMs), also known as discrete breathers, with the result that many of their properties are now well understood[@ILMs]. These excitations result from the interplay between nonlinearity and discreteness, and they can be highly localized in perfect lattices, with or without external driving. They can occur in a variety of different lattices: recent experiments have reported vibrational ILMs in a quasi-1D charge-density wave system[@Bishop], spin-wave ILMs in a quasi-1D antiferromagnetic system[@Sievers], and discrete breathers in Josephson junction (JJ) ladders[@Ustinov1; @Orlando]. The latter systems are noteworthy, in that arrays of coupled JJs have served for many years as reliable laboratory systems for studying diverse nonlinear phenomena[@StrogLikh]. The nonlinear dynamics are particularly rich. A single “small” JJ subject to an applied constant DC bias current can be mapped onto the problem of a damped pendulum driven by a constant torque, with the dynamical degree of freedom being the Josephson phase difference[@BandC]. There are thus two qualitatively different states, namely a static (superconducting) state and a dynamic (whirling or resistive) state, with the latter producing a readily measured voltage $V \propto \dot \varphi$ across the whirling junction. When several junctions are assembled to form a regular array, such as the ladder shown in Fig. \[Fig1\], they become inductively coupled. In the coupled system, junctions in the superconducting state can also exhibit steady-state librations, when JJs in the whirling state are present. In view of the mapping onto the pendulum problem, JJ ladders share features with lattices of nonlinearly coupled electric dipole rotors, driven by an external monochromatic AC electric field[@BonPage], but with the important simplification that they can be driven with purely DC bias currents. Figure \[Fig1\] sketches an anisotropic JJ ladder consisting of small JJs of two types, “horizontal” and “vertical,” which are, respectively, perpendicular and parallel to the applied bias current (arrows). The anisotropy arises from the different areas of the horizontal and vertical junctions and is characterized by the parameter $\eta=A_h/A_v=I_c^h/I_c^v$, where $I_c^h$ and $I_c^v$ are the the critical currents for each type of junction. References [@Ustinov1; @Orlando; @Ustinov2] report experimental observations of various discrete breathers in ladders driven by a [*homogeneously*]{} applied DC bias current, represented by the dashed arrows in Fig. \[Fig1\]. For these states, the localized voltage patterns have a simple structure involving only two nonzero steady-state voltages (rotational frequencies). The breathers were found to be stable in the limit of small coupling ($\eta \alt 1$) and for bias currents $I_{\rm ext} \alt I_c^v$. For the case of large 2-D JJ arrays subject to a homogeneously applied DC bias, more complicated inhomogeneous states, with [*meandering*]{} voltage patterns, have also been reported[@Misha1]. Here, we study the dynamics of a JJ ladder with an external DC bias current applied at only one edge (solid arrows, Fig. \[Fig1\]). For increasing bias ($I_{\rm ext} \agt I_c^v$) and over a wide range of anisotropies, we find by direct numerical simulations that the dynamic state expands into the ladder one cell at a time, by a sequence of abrupt jumps. This behavior is in marked contrast to the well-known cases of long JJs and JJ parallel arrays ($\eta~=~\infty$), where the entire system abruptly switches to the resistive state at a particular value of the DC bias. It is also different than the breather case, since all of the junctions within the boundary of this localized dynamic state whirl, and at each expansion the number of different frequencies (voltages) grows. The sequence of $I$-$V$ characteristics and threshold currents can be modeled analytically, yielding very good agreement with the numerical results. Numerical Simulations ===================== We consider a ladder with a large but finite number of cells $N$. The ladder’s state is specified by the time-dependent Josephson phases $\{\varphi_i\}$, $\{\psi_i\}$, and $\{\tilde \psi_i\}$ for the vertical, upper horizontal and lower horizontal junctions, respectively, where $i$ denotes the cell. We have found in our simulations that the symmetry condition ${\tilde \psi_i}=-\psi_i$ holds for the phenomena to be discussed here. The ladder dynamics are then determined by the coupled nonlinear equations of motion obtained in Refs.  and : $$\begin{aligned} \label{GenEq} \hat L(\varphi_i)&=&\gamma_i+\frac{1}{\beta_L}[\varphi_{i-1}-2\varphi_i+ \varphi_{i+1}+2(\psi_i-\psi_{i-1})], \nonumber \\ &&i=2,\ldots,N-1, \\ \hat L(\psi_i)&=&\frac{1}{\eta \beta_L}(\varphi_i-\varphi_{i+1}-2\psi_i), \qquad \quad i=1,\ldots,N, \nonumber\end{aligned}$$ where the operator $\hat L(\varphi) \equiv \ddot \varphi + \alpha \dot\varphi +\sin(\varphi)$. The equations for the vertical junctions at $i=1$ and $N$ are $$\begin{aligned} \label{BC1} \hat L(\varphi_1)&=&\gamma_1+\frac{1}{\beta_L}(\varphi_2-\varphi_1+2\psi_1), \\ \hat L(\varphi_N)&=&\gamma_N+ \frac{1}{\beta_L}(\varphi_{N-1}-\varphi_N-2\psi_{N-1}). \nonumber\end{aligned}$$ Equations (\[GenEq\]) and (\[BC1\]) describe each junction within the resistively and capacitively shunted junction (RCSJ) model[@BandC], and the unit of time is the inverse of the plasma frequency $\omega_J \equiv \sqrt{2eI_c/C\hbar}$. Since each junction’s critical current and capacitance scale with the area, $\omega_J$ is independent of the anisotropy parameter $\eta$, as is the effective damping constant $\alpha \equiv 1/(\omega_J RC)$. The normalized bias current $\gamma_i$ is defined as $I_{i,{\rm ext}}/I_c^v$. The inductive coupling between junctions is determined by the parameter $\beta_L \equiv 2\pi L I_c^v/\Phi_0$, where $L$ is the self-inductance of a single cell and $\Phi_0 = hc/2e$ is the elementary flux quantum. Coupling beyond that described by $\beta_L$ is not included. We performed direct numerical integration of the equations of motion for ladders with $N=20$ cells, using a fifth-order Gear predictor-corrector algorithm[@AandT], for a range of anisotropies: $\eta =$ 0.5, 1.0, 2.0, 3.0, and 5.0. The arrays were underdamped, with $\alpha=0.1$, and we used a moderate value of the coupling parameter $\beta_L=0.5$. The external DC bias was applied at one edge, i.e. $\gamma_1=\gamma$ and all other $\gamma_i=0$. To simulate the $I$-$V$ curves, we started with all phases at zero and gradually increased the external bias $\gamma$ from zero to 50, in increments of 0.005. When junctions were present in the whirling state, the MD time scale was set by the time-average period of the fastest rotating phase. For a given value of gamma, we waited for at least 100 of these reference periods before averaging, in order to avoid transients, following which we computed the time-averages $\langle \dot \varphi_i \rangle$ and $\langle \dot \psi_i\rangle$ over at least 100 additional reference periods. These averages are proportional to the average voltages across the junctions. The current was then incremented, with the initial phase configuration being that from the preceding MD time step. In all runs, the time step was 1/200 of the reference period. Our simulated $I$-$V_i$ curves for an anisotropy of $\eta =2$ are shown in Fig. \[Fig2\]. The most striking finding is the occurrence of extremely sharp voltage jumps. At each of the corresponding threshold currents $\gamma^{thr}_n$, a new cell is added to the ladder’s dynamic state. With the applied current below the first threshold $\gamma^{thr}_1$, all junctions are in a static (zero voltage) state. When $\gamma$ exceeds $\gamma^{thr}_1$, the first vertical junction and its adjacent top and bottom horizontal junctions abruptly switch into the dynamic state, with all other junctions remaining in the zero voltage state—the rotating phases are confined to the first cell. As the bias is increased further, all three average voltages for this 1-cell dynamic state increase linearly until the next threshold current $\gamma^{thr}_2$ is reached, at which point the dynamic state suddenly expands into the second cell, accompanied by sharp changes of the voltages in the first cell. This process continues, yielding successive transitions from $n$-cell dynamic localized states to $(n+1)$-cell dynamic states. The distribution of threshold currents and voltage ratios depends on the ladder’s anisotropy. Over the range $0<\gamma <50$, the ladder reached a 3-cell state for $\eta=0.5$ and 1, a 4-cell state for $\eta=2$, and a 5-cell state for $\eta=3$ and 5. In the following, these states will be termed $n$-cell edge states. An $n$-cell edge state is in striking contrast to an n-cell breather. The breather occurs away from the ladder’s edge and is homogeneously driven by same DC bias current ($I_{\rm ext} \alt I_c$) applied at every cell, whereas the edge states are driven by a DC bias ($I_{\rm ext} \agt I_c$) applied at just one edge. The edge states have a richer internal structure—[*all*]{} of the junctions within an edge state are in a nonzero voltage state (see Fig. \[Fig3\]), whereas in a breather state, all of the horizontal junctions are in the zero voltage state except for those on the breather’s boundary[@Ustinov1; @Orlando; @Ustinov2]. Moreover, all of a breather’s vertical junction phases rotate at the same average frequency, whereas the $n$-cell edge state exhibits a peculiar distribution of average frequencies. This frequency (voltage) distribution depends on both $n$ and the ladder’s anisotropy. For example, our simulations for the $\eta=2$, 3-cell edge state of Fig. \[Fig3\](d) yield the ratios given in second column of Table \[Table1\]. Comparison with the third column shows that they are in very good agreement with analytic predictions derived below. ------------------------- ------- --------------------------------------- Ratio MD Predicted $\omega_1^v/\omega_2^v$ 2.667 $(3\eta^2+8\eta+4)/2\eta(1+\eta)=8/3$ $\omega_1^v/\omega_3^v$ 8.006 $(3\eta^2+8\eta+4)/\eta^2=8$ $\omega_1^h/\omega_2^h$ 2.499 $(\eta^2+6\eta+4)/\eta(2+\eta)=5/2$ $\omega_1^h/\omega_3^h$ 5.017 $(\eta^2+6\eta+4)/\eta^2=5$ $\omega_3^v/\omega_3^h$ 2.005 2 ------------------------- ------- --------------------------------------- : Average frequency (voltage) ratios for 3-cell edge states. The MD ratios are for the $\eta=2$, 3-cell edge state of Fig. \[Fig3\](d), and the predicted ratios were calculated from Eqs. (\[Vert\]) and (\[Horiz\]). \[Table1\] The superconducting state forming ahead of the $n$-cell edge state is also unusual. Fig. \[Fig3\] gives snapshot images of the Josephson phase distribution for several values of the applied DC current bias, for the anisotropy $\eta=2$. In panel (a), the current is just below the first threshold, and one sees a single [*Josephson vortex*]{} in the superconducting part of the ladder. The remaining panels (b)–(e) show the phases just after a new cell is added to the dynamic state. At the threshold currents $\gamma^{thr}_n$, the superconducting state becomes unstable, and the vortex jumps into the next cell as the edge state expands. Our simulations show that in general the superconducting state is sensitive to the anisotropy. Thus for rather small values of $\eta \lesssim 1$, there are no vortices trapped in the superconducting part of the ladder over the range of currents studied. For these cases, the Josephson phases of the vertical junctions in the superconducting portion of the ladder simply decrease with distance from the boundary of the resistive portion, corresponding to the [*Meissner state*]{} of the superconductor. With increasing anisotropy, single Josephson vortices appear in the superconducting portion, as in Fig. \[Fig3\]. For large anisotropies ($\eta \sim 5$) more complex [*vortex trains*]{} are observed, and we also find that the penetration of the edge state changes the nature of the superconducting vortex state, rather than simply pushing it ahead as for $\eta=2$. A detailed discussion of the superconducting state will be given elsewhere (Ref. [@FP]). Theoretical Analysis ==================== The unusual voltage distributions in the $n$-cell edge states can be explained analytically by making use of Kirchhoff’s laws, applied to the time-average currents (normalized to $I^v_c$) and corresponding dimensionless voltages in each cell. The key assumption is the coexistence of the resistive and superconducting states in different portions of the ladder. For a cell $i$ within an $n$-cell edge state, current conservation gives $I^v_i+I^h_i=I^h_{i-1}$, while the voltage condition is $I^v_i-\frac{2}{\eta}I^h_i- I^v_{i+1}=0$. Combining these yields an equation for just the horizontal currents: $$\label{DifEq} I^h_{i+1}+I^h_{i-1}-2\left(1+\frac{1}{\eta}\right)I^h_i=0.$$ This equation and the two from which it was derived apply to all cells $i$ in $1 \le i \le n$, provided we define $I^h_0 \equiv \gamma,\; I^v_{n+1} \equiv 0$, and $I^h_{n+1} \equiv I^h_n$, in order to take proper account of the $n$-cell edge state’s boundaries. Equation (\[DifEq\]) is readily solved by substituting $I^h_i = \lambda^i$, which yields two roots, namely $\lambda \equiv 1+\frac{1}{\eta} + \sqrt{(1+\frac{1}{\eta})^2-1}$ and $1/\lambda$. Hence the general solution of Eq. (\[DifEq\]) is $I^h_i = c_1 \lambda^i+ c_2 \lambda^{-i}$, where the constants $c_1$ and $c_2$ are obtained from the above definitions of $I^h_0$ and $I^h_{n+1}$. With the horizontal currents thus determined, the vertical currents may be computed from $I^v_i=I^h_{i-1} - I^h_i$. The currents are then converted into the average voltages via $V^v_i = I^v_i/\alpha$ and $V^h_i=I^h_i/(\alpha \eta)$. The resulting voltage distribution within an $n$-cell edge state is ($1\le i\le n$): $$\label{Vert} V^v_i=\frac{\gamma(1-\lambda)(\lambda^{i-1} -\lambda^{2n+1-i})} {\alpha(\lambda^{2n+1}+1)},$$ $$\label{Horiz} V^h_i=\frac{\gamma (\lambda^{i} +\lambda^{2n+1-i})} {\alpha \eta (\lambda^{2n+1}+1)}.$$ Equations (\[Vert\]) and (\[Horiz\]) give the predicted voltage ratios in Table \[Table1\], which for $\eta=2$ are seen to be in very good agreement with our MD simulations. Indeed, we find that for all of the values of $\eta$ studied, the predicted $I$-$V$ curves are in very good agreement with the MD curves, such as those of Fig. \[Fig2\]. Only the values of the current thresholds for the jumps are left undetermined by these equations. We can also predict the distribution $\{\gamma^{thr}_n\}$ of threshold currents for each $\eta$ by assuming that the superconducting state associated with the $(n-1)$-cell edge state becomes unstable and converts to the $n$-cell edge state when the current $I^h_n$ reaches a depinning current $I_{dp}$, which we take to be independent of $n$. This yields an expression for the threshold currents $$\label{Ithr} \gamma^{thr}_n = I_{dp}\frac{\cosh{[(n-\frac{1}{2}) \ln \lambda]}} {\cosh{(\frac{1}{2} \ln \lambda)}}.$$ The ratios $\gamma^{thr}_n/I_{dp}$ predicted by Eq. (\[Ithr\]) for $n=$ 2, 3, 4 and 5 are plotted versus $\eta$ in Fig. \[Fig4\]. To compare with the MD results, we fit $I_{dp}$ to the first observed MD threshold current for each $\eta$, namely $\gamma^{thr}_1 =$ 1.295, 1.510, 2.040, 2.525, and 4.010, for $\eta =$ 0.5, 1.0, 2.0, 3.0, and 5.0, respectively. The resulting MD ratios are shown by the circles in Fig. \[Fig4\] and agree well with the predictions. Hysteresis ========== Despite their rich structure of frequency ratios, the $n$-cell edge states are found to be highly stable in our simulations, for the case of increasing current. However, we also find notable hysteresis effects when the simulations are started with an $n$-cell edge state for large $n$ and the applied DC current is gradually [*decreased*]{} to zero. Figure \[Fig5\] is representative of the hysteretic behavior encountered. The threshold currents and stability properties for the sequence of down-conversions $\{n \rightarrow n-1\}$ are very different than for the increasing-current case. In particular, we observe resonant steps, switching processes, and nonlinear regions in the $I$-$V$ curves. We believe that all of these features arise from the resonant interaction between the n-cell edge states and other excitations, both localized and delocalized, as will be discussed elsewhere[@FP]. Summary ======= In summary, our numerical simulations have revealed unusual localized dynamic states in anisotropic JJ ladders subject to a DC bias current at one edge. Increasing the bias causes these states to expand by adding single cells in a sequence of sudden jumps, giving rise to a diverse set of voltage distributions and sharp changes in the $I$-$V$ curves. This behavior occurs for a wide range of parameters and should be observable through the $I$-$V$ characteristics or by direct visualization using low temperature scanning laser microscopy techniques[@Ustinov1; @Ustinov2; @Misha1]. We thank A. V. Ustinov and S. Flach for useful discussions. J. B. Page gratefully acknowledges the Max Planck Institute for the Physics of Complex Systems, Dresden, for their support and hospitality. Permanent address: Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504. See for instance, S. Flach and C. R. Willis, Phys. Rep. [**295**]{}, 181 (1998); A. J. Sievers and J. B. Page, in [*Dynamical Properties of Solids VII Phonon Physics*]{}, edited by G. K. Horton and A. A. Maradudin (Elsevier, Amsterdam, 1995); and references therein. B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop, W.-Z. Wang, and M. I. Salkola, Phys. Rev. Lett. [**82**]{}, 3288 (1999). U. T. Schwarz, L. Q. English, and A. J. Sievers, Phys. Rev. Lett. [**83**]{}, 223 (1999). P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, Phys. Rev. Lett. [**84**]{}, 745 (2000). E. Trias, J. J. Mazo, and T. P. Orlando, Phys. Rev. Lett. [**84**]{}, 741 (2000). S. H. Strogatz, [*Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering*]{} (Addison-Wesley, Reading, MA, 1994); K .K. Likharev, Rev. Mod. Phys. [**51**]{}, 101 (1979). A. Barone and G. Paternò, [*Physics and Applications of the Josephson Effect*]{} (Wiley, New York, 1982). D. Bonart and J. B. Page, Phys. Rev. E [**60**]{}, R1134 (1999). P. Binder, D. Abraimov, and A. V. Ustinov, Phys. Rev. E [**62**]{}, 2858 (2000). D. Abraimov, P. Caputo, G. Filatrella, M. V. Fistul, G. Yu. Logvenov, and A. V. Ustinov, Phys. Rev. Lett. [**83**]{}, 5354 (1999). P. Caputo, M. V. Fistul, A. V. Ustinov, B. A. Malomed, and S. Flach, Phys. Rev. B [**59**]{}, 14050 (1999). G. Grimaldi, G. Filatrella, S. Pace, and U. Gambardella, Phys. Lett. A [**223**]{}, 463 (1996). See, for instance, M. P. Allen and D. J. Tildesly, [ *Computer Simulations of Liquids*]{} (Clarendon, Oxford, 1987). M. Fistul and J. B. Page, unpublished.

ArXiv

--- abstract: 'In the standard scenario of isolated low-mass star formation, strongly magnetized molecular clouds are envisioned to condense gradually into cores, driven by ambipolar diffusion. Once the cores become magnetically supercritical, they collapse to form stars. Most previous studies based on this scenario are limited to axisymmetric calculations leading to single supercritical core formation. The assumption of axisymmetry has precluded a detailed investigation of cloud fragmentation, generally thought to be a necessary step in the formation of binary and multiple stars. In this contribution, we describe the non-axisymmetric evolution of initially magnetically subcritical clouds using a newly-developed MHD code. It is shown that non-axisymmetric perturbations of modest fractional amplitude ($\sim 5\%$) can grow nonlinearly in such clouds during the supercritical phase of cloud evolution, leading to the production of either a highly elongated bar or a set of multiple dense cores.' author: - Fumitaka Nakamura - 'Zhi-Yun Li' title: On the Formation of Binary Stars and Small Stellar Groups in Magnetically Subcritical Clouds --- \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Introduction ============ Over the last few decades, a basic framework has been developed for the formation of low-mass stars in relative isolation (Shu, Adams, & Lizano 1987). In this by now “standard” picture, a molecular cloud, which is initially supported by strong magnetic field against its self-gravity, gradually contracts as the magnetic support weakens by ambipolar diffusion. Magnetically supercritical cores are formed, which collapse to produce stars. Quantitative studies based on this scenario have been carried out by many authors. In most of such studies, axisymmetry has been adopted. However, observations have shown that binary and multiple stars are common product of star formation. We need to understand how such (non-axisymmetric) stellar systems are formed in magnetically supported clouds. To elucidate the formation mechanism of binary stars and stellar groups, we have begun a systematic numerical study of the non-axisymmetric evolution initially magnetically subcritical clouds, by removing the restriction of axisymmetry. In this contribution, we present some of our recent results on this investigation. Model and Numerical Method ========================== As a first step, we adopted the thin-disk approximation often used in axisymmetric calculations (e.g., Basu & Mouschovias 1994; Li 2001). The disk is assumed in hydrostatic equilibrium in the vertical direction. The vertically-integrated MHD equations are solved numerically for the cloud evolution in the disk plane, with a 2D MHD code (see Li & Nakamura 2002 for code description). The magnetic structure is solved in 3D space. The initial conditions for star formation are not well determined either observationally and theoretically. Following Basu & Mouschovias (1994), we prescribe an axisymmetric reference state. See Nakamura & Li and Li & Nakamura (2002) for the details of the reference cloud model. The reference cloud is allowed to evolve into an equilibrium configuration, with the magnetic field frozen-in. Once the equilibrium state is obtained, we reset the time to $t=0$ and add a non-axisymmetric perturbation to the surface density distribution. Then, the cloud evolution is followed with the ambipolar diffusion turned on. Numerical Results ================= From axisymmetric calculations, Li (2001) classified the evolution of magnetically subcritical clouds into two cases, depending mainly on the initial cloud mass and the initial density distribution. When the initial cloud is not so massive and/or has a centrally-condensed density distribution, it collapses to form a single supercritical core ([core-forming cloud]{}). On the other hand, when the initial cloud has many thermal Jeans masses and/or a relatively flat density distribution near the center, it collapses to form a ring after the central region becomes magnetically supercritical ([ring-forming cloud]{}). In the following, we show that the core-forming cloud doesn’t fragment during the dynamic collapse phase, but becomes unstable to the bar mode ([*bar growth*]{}), whereas the ring-forming cloud can break up into several blobs ([*multiple fragmentation*]{}). Bar growth: Implication for Binary Formation -------------------------------------------- In Fig. 1 we show an example of the bar growth models. In this model, we adopted the reference density distribution of Basu & Mouschovias (1994), which is more centrally-condensed than the model to be shown in the next subsection, and the rotation profile of Nakamura & Hawana (1997). It has a characteristic radius of $r_0=7.5\pi c_s^2/(2\pi G\Sigma_{0,\rm ref})$ (where $c_s$ is the effective isothermal sound speed and $\Sigma_{0, \rm ref}$ the central cloud surface density in the reference state), initial flux-to-mass ratio of $\Gamma _0 = 1.5 B_{\infty}/(2\pi G^{1/2}\Sigma_{0,\rm ref})$ (where $B_\infty$ is the strength of the initially uniform background field), and a dimensionless rotation rate of $\omega=0.1$. We added to the equilibrium state an $m=2$ perturbation of surface density, with a fractional amplitude of merely 5%. During the initial quasi-static contraction phase, a central core condenses gradually out of the magnetically subcritical cloud, with no apparent tendency for the mode to grow. Rather, the iso-density contours appear to oscillate, changing the direction of elongation along $x$-axis in the disk plane to $y$-axis. After a supercritical core develops, the contraction becomes dynamic and the bar mode grows significantly. During the intermediate stages \[panels (c) and (d)\], the aspect ratio of the bar remains more or less frozen at $R\sim 2$. As the collapse continues, the growth rate of the bar increases dramatically by the very end of the starless collapse. The density distribution along the minor axis of the bar is well reproduced by a power-law profile of $r^{-2}$, which is different from that of an isothermal equilibrium filament ($\propto r^{-4}$). When the volume density exceeds a critical value of $10^{12}$ cm$^{-3}$, we changed the equation of state from isothermal to adiabatic, to mimic the transition to the optically thick regime. The bar is surrounded by an accretion shock, which is analogous to the first core of spherical calculations \[panel (f)\]. The aspect ratio of this “first” bar continues to increase during the early optically thick regime. The highly elongated first bar is expected to break up into two or more pieces. We suspect that bar fragmentation is an important, perhaps the dominant, route for binary and small multiple-star formation. We have also followed the evolution of this model cloud perturbed by other (higher) $m$ modes ($m\ge3$), and found no significant mode growth. The reason why the cloud is unstable only to the bar mode appears to be the following. In the absence of nonaxisymmetric perturbations, the supercritical collapse approaches a self-similar solution derived approximately by Nakamura & Hanawa (1997). In the self-similar solution, the effective radius of the central plateau is at most 3-4 times the effective Jeans length, making the cloud unstable to dynamic contraction but not to multiple fragmentation. Indeed, Nakamura & Hanawa (1997) showed that the self-similar solution is unstable only to the $m=2$ mode, consistent with our result. The tendency for the supercritical collapse to approach the self-similar solution is responsible for the bar formation during the dynamic collapse. Detailed numerical results on bar formation will appear elsewhere (Nakamura & Li 2002, in preparation). Multiple Fragmentation and Formation of Small Stellar Groups ------------------------------------------------------------ In Fig. 2 we show an example of the multiple fragmentation models. In this model, we adopted the reference density profile of Li (2001) with $n=8$, which is less centrally-condensed than the model shown in the previous subsection. The model has a characteristic radius of $r_0=7\pi c_s^2/(2\pi G\Sigma_{0,\rm ref})$, initial flux-to-mass ratio of $\Gamma _0 = 1.5B_{\infty}/(2\pi G^{1/2}\Sigma_{0,\rm ref})$, and rotation rate of $\omega=0.1$. Random density perturbations are added to the axisymmetric equilibrium state. The maximum fractional amplitude of the perturbations is set to 10%. During the quasi-static contraction phase, the infall motions are subsonic, and there is no sign of fragmentation. Once the flux-to-mass ratio in the central high-density region drops below the critical value, the contraction is accelerated near the center. As the collapse continues, the central supercritical region begins to fragment into five blobs. By the time shown in panel (f), the blobs are well separated from the background material and are significantly elongated. Subsequent dynamic collapse of each blob is similar to that of the bar growth case. Individually, we expect each core to produce a highly elongated bar, which could further break up into pieces, producing perhaps binary or multiple stars. Together, the formation of a small stellar group is the most likely outcome. Detailed numerical results on multiple fragmentation are given in Li & Nakamura (2002). Summary ======= Our main conclusion is that despite (indeed because of) the presence of the strong magnetic field, the initially magnetically subcritical clouds are unstable to non-axisymmetric perturbations during the supercritical phase of cloud evolution. The cloud evolution is classified into two cases, depending mainly on the initial cloud mass and density distribution. When the initial cloud is not so massive and has a centrally condensed density distribution, it doesn’t break into pieces but becomes unstable to a bar mode ([*bar growth*]{}). This bar is expected to fragment into two or more pieces to form binary or small multiple stars, when the bar becomes opaque to dust emission and is surrounded by an accretion shock. On the other hand, when the initial cloud has many Jeans masses and a relatively flat density distribution near the center, it can fragment into several or many cores after a supercritical region develops near the center ([*multiple fragmentation*]{}). This fragmentation may be responsible for small cluster formation in relatively isolated regions. Boss (2000) showed the fragmentation of 3D magnetic clouds numerically, treating the magnetic forces and ambipolar diffusion in an approximate way (see also the contribution by Boss). He concluded that magnetic fields (magnetic tension force) can enhance cloud fragmentation by reducing the tendency for the development of a central singularity, which would make fragmentation more difficult. We also find that magnetic fields can have beneficial effects on fragmentation. Strong magnetic fields can support clouds with many Jeans masses and flatten mass distribution, both of which are conducive to fragmentation once the magnetic support weakens through ambipolar diffusion. Numerical computations in this work were carried out at the Yukawa Institute Computer Facilities, Kyoto University. F.N. gratefully acknowledges the support of the JSPS Postdoctoral Fellowships for Research Abroad.

ArXiv

--- abstract: 'We study the set $M(X)$ of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set $X$. For an infinite measure $\mu \in M(X)$, the set $\mathfrak{M}_\mu = \{x \in X : \mbox{for any compact open set } U \ni x \mbox{ we have } \mu(U) = \infty \}$ is called defective. We call $\mu$ *non-defective* if $\mu(\mathfrak{M}_\mu) = 0$. The class $M^0(X) \subset M(X)$ consists of probability measures and infinite non-defective measures. We classify measures $\mu$ from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and compact open values set $S(\mu)$ are defined. A criterion when two good measures from $M^0(X)$ are homeomorphic is given. For any group-like $D \subset [0,1)$ we find a good probability measure $\mu$ on $X$ such that $S(\mu) = D$. For any group-like $D \subset [0,\infty)$ and any locally compact, zero-dimensional, metric space $A$ we find a good non-defective measure $\mu$ on $X$ such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. We consider compactifications $cX$ of $X$ and give a criterion when a good measure $\mu \in M^0(X)$ can be extended to a good measure on $cX$.' author: - | O. Karpel\ Institute for Low Temperature Physics,\ 47 Lenin Avenue, 61103 Kharkov, Ukraine\ (e-mail: [email protected]) title: Good measures on locally compact Cantor sets --- Introduction ============ The problem of classification of Borel finite or infinite measures on topological spaces has a long history. Two measures $\mu$ and $\nu$ defined on Borel subsets of a topological space $X$ are called *homeomorphic* if there exists a self-homeomorphism $h$ of $X$ such that $\mu = \nu\circ h$, i.e. $\mu(E) = \nu(h(E))$ for every Borel subset $E$ of $X$. The topological properties of the space $X$ are important for the classification of measures up to a homeomorphism. For instance, Oxtoby and Ulam [@Oxt-Ul] gave a criterion for a Borel probability measure on the finite-dimensional cube to be homeomorphic to the Lebesgue measure. Similar results were obtained for various manifolds (see [@Alp-Pr; @Oxt-Pr]). A Cantor set (or Cantor space) is a non-empty zero-dimensional compact perfect metric space. For Cantor sets the situation is much more difficult than for connected spaces. During the last decade, in the papers [@Akin3; @Austin; @S.B.O.K.; @D-M-Y; @Yingst] the Borel probability measures on Cantor sets were studied. In [@K], infinite Borel measures on Cantor sets were considered. For many applications in dynamical systems the state space is only locally compact. In this paper, we study Borel both finite and infinite measures on non-compact locally compact Cantor sets. It is possible to construct uncountably many full (the measure of every non-empty open set is positive) non-atomic measures on the Cantor set $X$ which are pairwise non-homeomorphic (see [@Akin1]). This fact is due to the existence of a countable base of clopen subsets of a Cantor set. The *clopen values set* $S(\mu)$ is the set of finite values of a measure $\mu$ on all clopen subsets of $X$. This set provides an invariant for homeomorphic measures, although it is not a complete invariant. For the class of the so called *good* probability measures, $S(\mu)$ *is* a complete invariant. By definition, a full non-atomic probability or non-defective measure $\mu$ is good if whenever $U$, $V$ are clopen sets with $\mu(U) < \mu(V)$, there exists a clopen subset $W$ of $V$ such that $\mu(W) = \mu(U)$ (see [@Akin2; @K]). Good probability measures are exactly invariant measures of uniquely ergodic minimal homeomorphisms of Cantor sets (see [@Akin2], [@GW]). For an infinite Borel measure $\mu$ on a Cantor set $X$, denote by $\mathfrak{M}_\mu$ the set of all points in $X$ whose clopen neighbourhoods have only infinite measures. The full non-atomic infinite measures $\mu$ such that $\mu(\mathfrak{M}_\mu) = 0$ are called *non-defective*. These measures arise as ergodic invariant measures for homeomorphisms of a Cantor set and the theory of good probability measures can be extended to the case of non-defective measures (see [@K]). In Section 2, we define a good probability measure and a good non-defective measure on a non-compact locally compact Cantor set $X$ and extend the results concerning good measures on Cantor sets to non-compact locally compact Cantor sets. For a Borel measure $\mu$ on $X$, the set $S(\mu)$ is defined as a set of all finite values of $\mu$ on the compact open sets. The defective set $\mathfrak{M}_\mu$ is the set of all points $x$ in $X$ such that every compact open neighbourhood of $x$ has infinite measure. We prove the criterion when two good measures on non-compact locally compact Cantor sets are homeomorphic. For every group-like subset $D \subset [0,1)$ we find a good probability measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$. For every group-like subset $D \subset [0,\infty)$ and any locally compact, zero-dimensional, metric space $A$ (including $A = \emptyset$) we find a good non-defective measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. In Section 3, compactifications of non-compact locally compact Cantor sets are studied. We investigate whether compactification can be used to classify measures on non-compact locally compact Cantor sets. We consider only the compactifications which are Cantor sets and extend measure $\mu$ by giving the remainder of compactification a zero measure. It turns out that in some cases good measure can be extended to a good measure on a Cantor set, while in other cases the extension always produces a measure which is not good. The extensions of a non-good measure are always non-good. After compactification of a non-compact locally compact Cantor set, new compact open sets are obtained. We study how the compact open values set changes. Based on this study, we give a criterion when a good measure on a non-compact locally compact Cantor set stays good after the compactification. Section 4 illustrates the results of Sections 2 and 3 with the examples. For instance, the Haar measure on the set of $p$-adic numbers and the invariant measure for $(C,F)$-construction are good. We give examples of good ergodic invariant measures on the generating open dense subset of a path space of stationary Bratteli diagrams such that any compactification gives a non-good measure. Measures on locally compact Cantor sets ======================================= Let $X$ be a non-compact locally compact metrizable space with no isolated points and with a (countable) basis of compact and open sets. Hence $X$ is totally disconnected. The set $X$ is called a *non-compact locally compact Cantor set*. Every two non-compact locally compact Cantor sets are homeomorphic (see [@D2]). Take a countable family of compact open subsets $O_n \subset X$ such that $X = \bigcup_{n=1}^\infty O_n$. Denote $X_1 = O_1$, $X_2 = O_2 \setminus O_1$, $X_3 = O_3 \setminus (O_1 \cup O_2)$,... The subsets $X_n$ are compact, open, pairwise disjoint and $X = \bigcup_{n=1}^\infty X_n$. Since $X$ is non-compact, we may assume without loss in generality that all $X_n$ are nonempty. Since $X$ has no isolated points, every $X_n$ has the same property. Thus, we represent $X$ as a disjoint union of a countable family of compact Cantor sets $X_n$. Recall that a Borel measure on a locally compact Cantor space is called *full* if every non-empty open set has a positive measure. It is easy to see that for a non-atomic measure $\mu$ the support of $\mu$ in the induced topology is a locally compact Cantor set. We can consider measures on their supports to obtain full measures. Denote by $M(X)$ the set of full non-atomic Borel measures on $X$. Then $M(X) = M_f(X) \sqcup M_\infty(X)$, where $M_f(X) = \{\mu \in M(X) : \mu(X) < \infty\}$ and $M_\infty(X) = \{\mu \in M(X) : \mu(X) = \infty\}$. For a measure $\mu \in M_\infty(X)$, denote $\mathfrak{M}_\mu = \{x \in X : \mbox{for any compact and open set } U \ni x \mbox{ we have } \mu(U) = \infty \}$. It will be shown that $\mathfrak{M}_\mu$ is a Borel set. Denote by $M_\infty^{0}(X) = \{\mu \in M_\infty(X) : \mu(\mathfrak{M}_\mu) = 0\}$. Let $M^0(X) = M_f(X) \sqcup M_\infty^{0}(X)$. Throughout the paper we will consider only measures from $M^0(X)$. We normalize the measures from $M_f(X)$ so that $\mu(X) = 1$ for any $\mu \in M_f(X)$. Recall that $\mu \in M^0(X)$ is *locally finite* if every point of X has a neighbourhood of finite measure. The properties of measures from the class $M^0(X)$ are collected in the following proposition. \[basic\_prop\] Let $\mu \in M^0(X)$. Then \(1) The measure $\mu$ is locally finite if and only if $\mathfrak{M}_\mu = \emptyset$, \(2) The set $X \setminus \mathfrak{M}_\mu$ is open. The set $\mathfrak{M}_\mu$ is $F_\sigma$. \(3) For any compact open set $U$ with $\mu(U) = \infty$ and any $a > 0$ there exists a compact open subset $V \subset U$ such that $a \leq \mu(V) < \infty$. \(4) The set $\mathfrak{M}_\mu$ is nowhere dense. \(5) $X = \bigsqcup_{i = 1}^\infty V_i \bigsqcup \mathfrak{M}_\mu$, where each $V_i$ is a compact open set of finite measure and $\mathfrak{M}_\mu$ is a nowhere dense $F_\sigma$ and has zero measure. The measure $\mu$ is $\sigma$-finite. \(6) $\mu$ is uniquely determined by its values on the algebra of compact open sets. **Proof.** (1) The condition $\mathfrak{M}_\mu = \emptyset$ means that every point $x \in X$ has a compact open neighbourhood of finite measure. Hence $\mu$ is locally finite and vise versa. \(2) We have $X \setminus \mathfrak{M}_\mu = \{x \in X : \mbox{ there exists a compact open set } U_x \ni x \mbox{ such that } \mu(U_x) < \infty \}$. Then for every point $x \in X \setminus \mathfrak{M}_\mu$ we have $U_x \subset X \setminus \mathfrak{M}_\mu$. Hence $X \setminus \mathfrak{M}_\mu$ is open. Therefore, for every $n \in \mathbb{N}$ the set $X_n \setminus \mathfrak{M}_\mu$ is open and $X_n \cap \mathfrak{M}_\mu$ is closed. Then $\mathfrak{M}_\mu = \bigsqcup_{n \in \mathbb{N}} \; (X_n \cap \mathfrak{M}_\mu)$ is $F_\sigma$ set. \(3) Let $U$ be a non-empty compact open subset of $X$ such that $\mu(U) = \infty$. Since $\mu \in M^{0}(X)$, we have $\mu(U) = \mu(U \setminus \mathfrak{M}_{\mu})$. Since $U$ is open, the set $U \setminus \mathfrak{M}_{\mu} = U \cap (X \setminus \mathfrak{M}_{\mu})$ is open. There are only countably many compact open subsets in $X$, hence the open set $U \setminus \mathfrak{M}_{\mu}$ can be represented as a disjoint union of compact open subsets $\{U_i\}_{i \in \mathbb{N}}$ of finite measure. We have $\mu(U) = \sum_{i=0}^{\infty} \mu(U_i) = \infty$, hence for every $a \in \mathbb{R}$ there is a compact open subset $V = \bigsqcup_{i=0}^{N}U_i$ such that $a \leq \mu(V) < \infty$. \(4) Let $U$ be a compact open subset of $X$. It suffices to show that there exists a non-empty compact open subset $V \subset U$ such that $V \cap \mathfrak{M}_{\mu} = \emptyset$. If $\mu(U) < \infty$ then $U \cap \mathfrak{M}_{\mu} = \emptyset$. Otherwise, by (3), there exists a compact open subset $V \subset U$ such that $0 < \mu(V) < \infty$. Obviously, $V \cap \mathfrak{M}_{\mu} = \emptyset$. \(5) follows from the proof of (3). \(6) follows from (5). $\blacksquare$ For a measure $\mu \in M^0(X)$ define the *compact open values set* as the set of all finite values of the measure $\mu$ on the compact open sets: $$S(\mu) = \{\mu(U):\,U\mbox{ is compact open in } X \mbox{ and } \mu(U) < \infty\}.$$ For each measure $\mu \in M^0(X)$, the set $S(\mu)$ is a countable dense subset of the interval $[0, \mu(X))$. Indeed, the set $S(\mu)$ is dense in $[0, \mu(V)]$ for every compact open set $V$ of finite measure (see [@Akin1]). By Proposition \[basic\_prop\], $S(\mu)$ is dense in $[0, \mu(X))$. Let $X_{1}$, $X_{2}$ be two non-compact locally compact Cantor sets. It is said that measures $\mu_{1} \in M(X_{1})$ and $\mu_{2} \in M(X_{2})$ are *homeomorphic* if there exists a homeomorphism $h \colon X_{1} \rightarrow X_{2}$ such that $\mu_{1}(E) = \mu_{2}(h(E))$ for every Borel subset $E \subset X_1$. Clearly, $S(\mu_{1}) = S(\mu_{2})$ for any homeomorphic measures $\mu_1$ and $\mu_2$. We call two Borel infinite measures $\mu_1 \in M^0_\infty(X_{1})$ and $\mu_2 \in M^0_\infty(X_{2})$ *weakly homeomorphic* if there exists a homeomorphism $h \colon X_{1} \rightarrow X_{2}$ and a constant $C>0$ such that $\mu_{1}(E) = C \mu_{2}(h(E))$ for every Borel subset $E \subset X_1$. Then $S(\mu_{1}) = C S(\mu_{2})$. Let $D$ be a dense countable subset of the interval $[0,a)$ where $a \in (0, \infty]$. Then $D$ is called *group-like* if there exists an additive subgroup $G$ of $\mathbb{R}$ such that $D = G \cap [0, a)$. It is easy to see that $D$ is group-like if and only if for any $\alpha, \beta \in D$ such that $\alpha \leq \beta$ we have $\beta - \alpha \in D$ (see [@Akin2; @K]). Let $X$ be a locally compact Cantor space (either compact or non-compact) and $\mu\in M^0(X)$. A compact open subset $V$ of $X$ is called *good* for $\mu$ (or just good when the measure is understood) if for every compact open subset $U$ of $X$ with $\mu(U) < \mu(V)$, there exists a compact open set $W$ such that $W \subset V$ and $\mu(W) = \mu(U)$. A measure $\mu$ is called *good* if every compact open subset of $X$ is good for $\mu$. If $\mu \in M^0(X)$ is a good measure and $\nu \in M^0(X)$ is (weakly) homeomorphic to $\mu$ then, obviously, $\nu$ is good. It is easy to see that in the case of compact Cantor set the definition of a good measure coincides with the one given in [@Akin2]. For a compact open subset $U \subset X$ let $\mu|_U$ be the restriction of the measure $\mu$ to the Cantor space $U$. Then the set $U$ is good if and only if $S(\mu|_U) = S(\mu|_X) \cap [0, \mu(U)]$. Denote by $H_{\mu}(X)$ the group of all homeomorphisms of a space $X$ preserving the measure $\mu$. The action of $H_{\mu}(X)$ on $X$ is called *transitive* if for every $x_{1}, x_{2} \in X$ there exists $h \in H_{\mu}(X)$ such that $h(x_{1}) = x_{2}$. The action is called *topologically transitive* if there exists a dense orbit, i.e. there is $x \in X$ such that the set $O(x) = \{h(x) : h \in H_\mu(X)\}$ is dense in $X$. We extend naturally the notion of partition basis introduced in [@Akin3]. A *partition basis* $\mathcal{B}$ for a non-compact locally compact Cantor set $X$ is a collection of compact open subsets of $X$ such that every non-empty compact open subset of $X$ can be partitioned by elements of $\mathcal{B}$. The properties of good measures on non-compact locally compact Cantor sets are gathered in the following proposition. The proofs for the measures on compact Cantor spaces can be found in [@Akin2; @Akin3; @K]. \[many\] Let $X$ be a locally compact Cantor space (either compact or non-compact). Let $\mu \in M^0(X)$. Then \(a) If $\mu$ is good and $C > 0$ then $C \mu$ is good and $S(C \mu) = C S(\mu)$. \(b) If $\mu$ is good and $U$ is a non-empty compact open subset of $X$ then the measure $\mu|_U$ is good and $S(\mu|_U) = S(\mu) \cap [0, \mu(U)]$. \(c) $\mu$ is good if and only if every compact open subset of finite measure is good. \(d) $\mu$ is good if and only if for every non-empty compact open subset $U$ of finite measure, the measure $\mu|_U$ is good. \(e) If $\mu$ is good then $S(\mu)$ is group-like. \(f) If a compact open set $U$ admits a partition by good compact open subsets then $U$ is good. \(g) The measure $\mu$ is good if and only if there exists a partition basis $\mathcal B$ consisting of compact open sets which are good for $\mu$. \(h) If $\mu$ is good, then the group $H_\mu(X)$ acts transitively on $X \setminus \mathfrak{M_\mu}$. In particular, the group $H_\mu(X)$ acts topologically transitively on $X$. \(i) If $\mu$ is a good measure on $X$ and $\nu$ is the counting measure on $\{1,2,...,n\}$ then $\mu \times \nu$ is a good measure on $X \times \{1,2,...,n\}$. **Proof**. (a), (b) are clear. \(c) Suppose that every compact open subset of finite measure is good. Let $V$ be any compact open set with $\mu(V) = \infty$ and $U$ be a compact open set with $\mu(U) < \infty$. By Proposition \[basic\_prop\], there exists a compact open subset $W \subset V$ such that $\mu(U) \leq \mu(W) < \infty$. By assumption, $W$ is good. Hence there exists a compact open set $W_1\subset W$ with $\mu(W_1) = \mu(U)$ and $V$ is good. \(d) Suppose that for every non-empty compact open subset $U$ of finite measure, the measure $\mu|_U$ is good. We prove that every compact open subset of finite measure is good, then use (c). Let $U$, $V$ be compact open sets with $0 < \mu(U) < \mu(V) < \infty$. Set $W = U \cup V$. Then $W$ is a compact open set of finite measure. Since $\mu|_W$ is good, there exists $W_1 \subset V$ such that $\mu(W_1) = \mu(U)$. \(e) If $\mu$ is good then for any $\alpha, \beta \in S(\mu)$ such that $\beta - \alpha \geq 0$, we have $\beta - \alpha \in S(\mu)$. Hence $S(\mu)$ is group-like. (f) See [@Akin3] for the case of finite measure and [@K] for infinite measure. \(g) If there exists a partition basis $\mathcal B$ consisting of compact open sets which are good for $\mu$, then, by (f), every compact open set is good. \(h) For any $x, y \in X \setminus \mathfrak{M_\mu}$ there exists a compact open set $U$ of finite measure such that $x, y \in U$. By (d), the measure $\mu|_U$ is a good finite measure on a Cantor space $U$. By Theorem 2.13 in [@Akin2], there exists a homeomorphism $h \colon U \rightarrow U$ which preserves $\mu$ and $h(x) = y$. Define $h_1 \in H_\mu(X)$ to be $h$ on $U$ and the identity on $X \setminus U$. For every $x \in X\setminus\mathfrak{M}_\mu$ we have $O(x) = X \setminus\mathfrak{M}_\mu$. By Proposition \[basic\_prop\], the set $X \setminus\mathfrak{M}_\mu$ is dense in $X$. Hence $H_\mu(X)$ acts topologically transitively on $X$. \(i) The rectangular compact open sets $U \times \{z\}$, where $U$ is compact open in $X$ and $z \in \{1,2,...,n\}$, form a partition basis for $X \times \{1,2,...,n\}$. Since $\mu \times \nu (U \times \{z\}) = \mu(U)$, these sets are good. The measure $\mu$ is good by (g). $\blacksquare$ For $G$ an additive subgroup of $\mathbb{R}$ we call a positive real number $\delta$ a *divisor* of $G$ if $\delta G = G$. The set of all divisors of $G$ is called $Div(G)$. By a full measure on a discrete countable topological space $Y$ we mean a measure $\nu$ such that $0 < \nu(\{y\}) < \infty$ for every $y \in Y$. We will use the following theorem for $Y = \mathbb{Z}$, but the proof stays correct for any discrete countable topological space $Y$. \[good\_product\] Let $\mu$ be a good measure on a non-compact locally compact Cantor space $X$. Let $\nu$ be a full measure on $\mathbb{Z}$, where $\mathbb{Z}$ is endowed with discrete topology. Let $G$ be an additive subgroup of $\mathbb{R}$ generated by $S(\mu)$. Then $\mu \times \nu$ is good on $X \times \mathbb{Z}$ if and only if there exists $C > 0$ such that $\nu(\{i\}) \in C \cdot Div(G)$ for every $i \in \mathbb{Z}$. **Proof.** Lets prove the “if” part. Suppose $\mu$ is good on $X$ and $\nu(\{i\}) \in C \cdot Div(G)$ for some $C > 0$ and every $i \in \mathbb{Z}$. By Proposition \[many\] (g), it suffices to prove that a compact open set of the form $U \times \{i\}$ is good for any compact open $U \subset X$ and any $i \in \mathbb{Z}$. Thus, it suffices to show that $S(\mu \times \nu|_{U \times \{i\}}) = S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$. The inclusion $S(\mu \times \nu|_{U \times \{i\}}) \subset S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$ is always true, hence we need to prove the inverse inclusion. We have $S(\mu \times \nu|_{U \times\{i\}}) = \nu(\{i\}) S(\mu|_U) = C \delta S(\mu|_U)$ for some $\delta \in Div(G)$. Since $\mu$ is good on $X$, we obtain $S(\mu|_U) = G \cap [0, \mu(U)]$. Hence $S(\mu \times \nu|_{U \times\{i\}}) = C G \cap [0, C \delta \mu(U)] = C G \cap [0, \mu \times \nu (U \times \{i\})]$. Note that $C \delta \mu(U) \in CG$ because $\delta \in Div(G)$. Therefore, it suffices to prove that $S(\mu \times \nu|_{X \times \mathbb{Z}}) \subset C G$. The set $S(\mu \times \nu|_{X \times \mathbb{Z}})$ consists of all finite sums $\sum_{i,j} \mu(U_i) \nu(\{j\})$, where each $U_i$ is a compact open set in $X$ and $j \in \mathbb{Z}$. We have $\sum_{i,j} \mu(U_i) \nu(\{j\}) = \sum_{i,j} \mu(U_i) C \delta_j \subset CG$, here $\delta_i \in Div(G)$. Hence $S(\mu \times \nu|_{U \times\{i\}}) \supset S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu (U \times \{i\})]$ and $U \times \{i\}$ is good. Now we prove the “only if part”. Suppose that $\mu \times \nu$ is good on $X \times \mathbb{Z}$. Then for any $i \in \mathbb{Z}$ we have $S(\mu \times \nu|_{X \times\{i\}}) = S(\mu \times \nu|_{X \times \mathbb{Z}}) \cap [0, \mu \times \nu(X \times\{i\})]$. Note that $S(\mu \times \nu|_{X \times\{i\}} = \nu(\{i\}) S(\mu|_X)$. Denote by $\widetilde{G}$ the additive subgroup of $\mathbb{R}$ generated by $S(\mu \times \nu|_{X \times \mathbb{Z}})$. Let $\alpha = \nu(\{i\})$. Then $\alpha G = \widetilde{G}$. Let $j \in \mathbb{Z}$ and $\beta = \nu(\{j\})$. By the same arguments, we have $\beta G = \widetilde{G}$. Then $\frac{\alpha}{\beta} \in Div(G)$. Indeed, $\frac{\alpha}{\beta} G = \frac{1}{\beta} \widetilde{G} = G$. Hence $\alpha = \beta \delta$, where $\delta \in Div(G)$. Set $C = \nu(\{j\})$. Then for every $i \in \mathbb{Z}$ we have $\nu(\{i\}) = C \delta_i$ where $\delta_i = \frac{\nu(\{i\})}{\nu(\{j\})} \in Div (G)$. $\blacksquare$ Let $X$, $Y$ be non-compact locally compact Cantor sets. If $\mu \in M^0(X)$, $\nu \in M^0(Y)$ are good measures, then the product $\mu \times \nu$ is a good measure on $X \times Y$ and $$S(\mu \times \nu) = \left\{\sum_{i=0}^N \alpha_i \cdot \beta_i : \alpha_i \in S(\mu), \beta_i \in S(\nu), N \in \mathbb{N}\right\} \cap [0, \mu(X)\times \nu(Y)).$$ **Proof.** Let $X = \bigsqcup_{m = 1}^{\infty} X_n$ and $Y = \bigsqcup_{n = 1}^{\infty} Y_n$, where each $X_n$, $Y_n$ is a Cantor set. Then $X \times Y = \bigsqcup_{m, n = 1}^{\infty} X_m \times Y_n$ and $\mu \times \nu |_{X_m \times Y_n} = \mu|_{X_n} \times \nu|_{Y_n}$. Since $\mu|_{X_n}$ and $\nu|_{Y_n}$ are good finite or non-defective measures on a Cantor set, the measure $\mu \times \nu |_{X_m \times Y_n}$ is good by Theorem 2.8 ([@Akin3]), Theorem 2.10 ([@K]). By Proposition \[many\], $\mu \times \nu$ is good on $X \times Y$. $\blacksquare$ \[krit\_homeo\_good\] Let $X$, $Y$ be non-compact locally compact Cantor spaces. Let $\mu \in M^0(X)$ and $\nu \in M^0(Y)$ be good measures. Let $S(\mu) = S(\nu)$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. Assume that there is a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$ where the sets $\mathfrak{M}$ and $\mathfrak{N}$ are endowed with the induced topologies. Then there exists a homeomorphism $\widetilde{h} \colon X \rightarrow Y$ which extends $h$ such that $\mu = \nu \circ \widetilde{h}$. Conversely, if $\mu \in M^0(X)$ and $\nu \in M^0(Y)$ are good homeomorphic measures then $S(\mu) = S(\nu)$ and there is a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$. **Proof.** The second part of the Theorem is clear. We prove the first part. Let $X = \bigsqcup_{i=1}^\infty X_i$ and $Y = \bigsqcup_{j=1}^\infty Y_j$ where $X_i$, $Y_j$ are compact Cantor spaces. First, consider the case when $\mathfrak{M} = \mathfrak{N} = \emptyset$, i.e. the measures $\mu$, $\nu$ are either finite of infinite locally finite measures. Since $S(\mu) = S(\nu)$, we have $\mu(X_1) \in S(\nu)$. There exists $n \in \mathbb{N}$ such that $\nu(\bigsqcup_{j=1}^{n-1} Y_j) \leq \mu(X_1) < \nu(\bigsqcup_{j=1}^{n} Y_j)$. Since $S(\nu)$ is group-like, we see that $\mu(X_1) - \nu(\bigsqcup_{j=1}^{n-1} Y_j) \in S(\nu)$. Since $\nu$ is good, there exists a compact open subset $W \subset Y_n$ such that $\nu(W) = \mu(X_1) - \nu(\bigsqcup_{j=1}^{n-1} Y_j)$. Hence $Z = \bigsqcup_{j=1}^{n-1} Y_j \sqcup W$ is a compact Cantor set and $\mu(X_1) = \nu(Z)$. By Theorem 2.9 ([@Akin2]), there exists a homeomorphism $h_1 \colon X_1\rightarrow Z$ such that $\mu |_{X_1} = \nu |_{Z} \circ h_1$. Set $\widetilde{h}|_{X_1} = h_1$. Consider $(Y_n \setminus W) \bigsqcup_{j=n+1}^\infty Y_j$ instead of $Y$ and $\bigsqcup_{i=2}^\infty X_i$ instead of $X$. Reverse the roles of $X$ and $Y$. Proceed in the same way using $Y_n \setminus W$ instead of $X_1$. Thus, we obtain countably many homeomorphisms $\{h_i\}_{i=1}^\infty$. Given $x \in X$, set $\widetilde{h}(x) = h_i(x)$ for the corresponding $h_i$. Then $\widetilde{h} \colon X \rightarrow Y$ is a homeomorphism which maps $\mu$ into $\nu$. Now, let $\mathfrak{M} \neq \emptyset$. If $\mu(X_1) < \infty$, we proceed as in the previous case. If $\mu(X_1) = \infty$ then $X_1 \cap \mathfrak{M} \neq \emptyset$. Then $h(X_1 \cap \mathfrak{M})$ is a compact open subset of $\mathfrak{N}$ in the induced topology. Hence there exists a compact open set $W \subset Y$ such that $W \cap \mathfrak{N} = h(X_1 \cap \mathfrak{M})$. Then, by Theorem 2.11 ([@K]), the sets $X_1$ and $W$ are homeomorphic via measure preserving homeomorphism $h_1$ and $h_1|_{X_1 \cap \mathfrak{M}} = h$. Since $W$ is compact, there exists $N$ such that $W \subset \bigsqcup_{n=1}^N Y_n$. Reverse the roles of $X$ and $Y$ and consider $\bigsqcup_{n=1}^N Y_n \setminus W$ instead of $X_1$. $\blacksquare$ The corollary for weakly homeomorphic measures follows: \[krit\_weak\_homeo\_good\] Let $\mu \in M_{\infty}^0(X)$ and $\nu \in M_{\infty}^0(Y)$ be good infinite measures on non-compact locally compact Cantor sets $X$ and $Y$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. Then $\mu$ is weakly homeomorphic to $\nu$ if and only if the following conditions hold: \(1) There exists $c > 0$ such that $S(\mu) = c S(\nu)$, \(2) There exists a homeomorphism $h \colon \mathfrak{M} \rightarrow \mathfrak{N}$ where the sets $\mathfrak{M}$ and $\mathfrak{N}$ are endowed with the induced topologies. Let $\mu \in M^0_{\infty}(X)$ be a good measure on a non-compact locally compact Cantor set $X$ and $V$ be any compact open subset of $X$ with $\mu(V) < \infty$. Then $\mu$ on $X$ is homeomorphic to $\mu$ on $X \setminus V$. Let $S(\mu) = G \cap [0, \infty)$. Then $\mu$ is homeomorphic to $c \mu$ if and only if $c \in Div(G)$. Let $\mu$ be a good finite or non-defective measure on a non-compact locally compact Cantor set $X$. Let $U$, $V$ be two compact open subsets of $X$ such that $\mu(U) = \nu(V) < \infty$. Then there is $h \in H_\mu(X)$ such that $h(U) = V$. **Proof.** Set $Y = U \cup V$. Then $Y$ is a Cantor set with $\mu(Y) < \infty$. By Proposition 2.11 in [@Akin2], there exists a self-homeomorphism $h$ of $Y$ such that $h(U) = V$ and $h$ preserves $\mu$. Set $h$ to be identity on $X \setminus Y$. $\blacksquare$ Let $\mu$ and $\nu$ be good non-defective measures on non-compact locally compact Cantor sets $X$ and $Y$. Let $\mathfrak{M}$ be the defective set for $\mu$ and $\mathfrak{N}$ be the defective set for $\nu$. If there exist compact open sets $U \subset X$ and $V \subset Y$ such that $\mu(U) = \nu(V) < \infty$ and $\mu|U$ is homeomorphic to $\nu|V$, then $\mu$ is homeomorphic to $\nu$ if and only if $\mathfrak{M}$ and $\mathfrak{N}$ (with the induced topologies) are homeomorphic. **Proof.** Let $\gamma = \mu(U) = \nu (V)$. Since $\mu|U$ is homeomorphic to $\nu|V$, we have $S(\mu|U) = S(\nu|V)$. Since $\mu$ and $\nu$ are good, we have $S(\mu) \cap [0, \gamma] = S(\nu) \cap [0, \gamma]$ by Proposition \[many\]. Since $S(\mu)$ and $S(\nu)$ are group-like, we obtain $S(\mu) = S(\nu)$. $\blacksquare$ \[goodSmu\] Let $\mu \in M^{0}(X)$ be a good measure on a non-compact locally compact Cantor set $X$. Then the compact open values set $S(\mu)$ is group-like and the defective set $\mathfrak{M}_\mu$ is a locally compact, zero-dimensional, metric space (including $\emptyset$). Conversely, for every countable dense group-like subset $D$ of $[0, 1)$, there is a good probability measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$. For every countable dense group-like subset $D$ of $[0, \infty)$ and any locally compact, zero-dimensional, metric space $A$ (including $A = \emptyset$) there is a good non-defective measure $\mu$ on a non-compact locally compact Cantor set such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. **Proof**. The first part of the theorem follows from Propositions \[basic\_prop\], \[many\]. We prove the second part. First, consider the case of finite measure. Let $D \subset [0,1)$ be a countable dense group-like subset. Then there exist a strictly increasing sequence $\{\gamma_n\}_{n=1}^{\infty} \subset D$ such that $\lim_{n \rightarrow \infty} \gamma_n = 1$. For $n = 1,2,...$ set $\delta_n = \gamma_{n} - \gamma_{n-1}$. Denote by $S_n = D \cap [0, \delta_n]$. Then $D_n = \frac{1}{\delta_n} (D \cap [0,\delta_n])$ is a group-like subset of $[0,1]$ with $1 \in D_n$. In [@Akin2], it was proved that there exists a good probability measure $\mu_n$ on a Cantor set $X_n$ such that $S(\mu_n|_{X_n}) = D_n$. The measure $\nu_n = \delta_n \mu_n$ is a good finite measure on $X_n$ with $S(\nu_n|_{X_n}) = D \cap [0,\delta_n]$. Set $X = \bigsqcup_{n=1}^{\infty} X_n$ and let $\mu|_{X_n} = \nu_n$. Then $\mu$ is a good probability measure on a non-compact locally compact Cantor space $X$ and $S(\mu|_{X}) = D$. Now consider the case of infinite measure. Let $\gamma \in D$. Since $D \subset [0, \infty)$ is group-like, we see that $\frac{1}{\gamma}D \cap [0,1]$ is a group-like subset of $[0,1]$. In [@Akin2] it was proved that there exists a good probability measure $\mu_1$ on a Cantor space $Y$ with $S(\mu_1) = \frac{1}{\gamma}D \cap [0,1]$. Set $\mu = \gamma \mu_1$. Then $\mu$ is a good finite measure on $Y$ and $S(\mu) = D \cap [0,\gamma]$. Endow the set $\mathbb{Z}$ with discrete topology. Let $\nu$ be a counting measure on $\mathbb{Z}$. Set $X = Y \times \mathbb{Z}$ and $\widetilde{\mu} = \mu \times \nu$. Then, by Theorem \[good\_product\], $\widetilde{\mu}$ is good with $S(\widetilde{\mu}) = D$ and $\mathfrak{M}_{\widetilde{\mu}} = \emptyset$. Suppose $A$ is a non-empty compact zero-dimensional, metric space. Then, by Theorem 2.15 ([@K]), there exists a good non-defective measure $\mu$ on a Cantor space $Y$ such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. By the above, there exists a good locally finite measure $\nu$ on a non-compact locally compact set $X$ with $S(\nu) = D$ and $\mathfrak{M}_\nu = \emptyset$. Set $Z = Y \sqcup X$ and $\widetilde{\mu}|_{Y} = \mu$, $\widetilde{\mu}|_{X} = \nu$. Then $\widetilde{\mu}$ is good on a non-compact locally compact Cantor set $Z$ with $S(\widetilde{\mu}) = D$ and $\mathfrak{M}_{\widetilde{\mu}}$ is homeomorphic to $A$. Suppose that $A$ is a non-empty, non-compact, locally compact, zero-dimensional metric space. Then $A = \bigsqcup_{n=1}^{\infty} A_n$ where each $A_n$ is a non-empty, compact, zero-dimensional metric space. By Theorem 2.15 ([@K]), for every $n = 1,2,...$ there exists a good non-defective measure $\mu_n$ on a Cantor set $Y_n$ such that $S(\mu_n) = D$ and $\mathfrak{M}_{\mu_n}$ is homeomorphic to $A_n$. Set $X = \bigsqcup_{n=1}^{\infty} Y_n$ and $\mu|_{Y_n} = \mu_n$. Then $\mu$ is good on a non-compact locally compact Cantor set $X$ with $S(\mu) = D$ and $\mathfrak{M}_{\widetilde{\mu}}$ is homeomorphic to $A$. $\blacksquare$ \[invarhomeo\] Let $D$ be a countable dense group-like subset of $[0, \infty)$. Then there exists an aperiodic homeomorphism of a non-compact locally compact Cantor set with good non-defective invariant measure $\widetilde{\mu}$ such that $S(\widetilde{\mu}) = D$. **Proof**. We use the construction similar to the one in the proof of Theorem \[goodSmu\]. Let $\mu$ be a good measure on a Cantor set $Y$ with $S(\mu) = D \cap [0,\gamma]$ for some $\gamma \in D$. Let $\nu$ be a counting measure on $\mathbb{Z}$. Set $\widetilde{\mu} = \mu \times \nu$ on $X = Y \times \mathbb{Z}$. Then $\widetilde{\mu}$ is a good non-defective measure on a non-compact locally compact Cantor set $X$ with $S(\widetilde{\mu}) = D$. Since the measure $\mu$ is a good finite measure on $Y$, there exists a minimal homeomorphism $T \colon Y \rightarrow Y$ such that $\mu$ is invariant for $T$ (see [@Akin2]). Let $T_1(x,n) = (Tx, n+1)$. Then $T_1$ is aperiodic homeomorphism of $X$. The measure $\widetilde{\mu}$ is invariant for $T_1$. $\blacksquare$ The measure $\widetilde{\mu}$ built in Corollary \[invarhomeo\] is invariant for any skew-product with the base $(Y,T)$ and cocycle acting on $\mathbb{Z}$. Let $X$ be a non-compact locally compact Cantor set. Then there exist continuum distinct classes of homeomorphic good measures in $M_f(X)$. There also exist continuum distinct classes of weakly homeomorphic good measures in $M_\infty^0(X)$. **Proof.** There exist uncountably many distinct group-like subsets $\{D_\alpha\}_{\alpha \in \Lambda}$ of $[0,1]$. By Theorem \[goodSmu\], for each $D_\alpha$ there exists a good probability measure $\mu_\alpha$ on $X$ such that $S(\mu_\alpha) = D_\alpha$. By Theorem \[krit\_homeo\_good\], the measures $\{\mu_\alpha\}_{\alpha \in \Lambda}$ are pairwise non-homeomorphic. Let $Y$ be a compact Cantor set. Let $\mu$ be a non-defective measure on $Y$. Denote by $[\mu]$ the class of weak equivalence of $\mu$ in the set of all non-defective measures on $Y$. There exist continuum distinct classes $[\mu_\alpha]$ of weakly homeomorphic good non-defective measures on a Cantor set $Y$ (see Theorem 2.18 in [@K]). Moreover, if there exists $C>0$ such that $G(S(\mu_\alpha)) = C G(S(\mu_\beta))$ then $\mu_\beta \in [\mu_\alpha]$. Let $\nu$ be a counting measure on $\mathbb{Z}$. Then, by Theorem \[good\_product\], $\mu_\alpha \times \nu$ is a good measure on a non-compact locally compact Cantor set $Y \times \mathbb{Z}$ and $G(S(\mu_\alpha \times \nu)) = G(S(\mu_\alpha))$. Hence, by Corollary \[krit\_weak\_homeo\_good\], the measures $\mu_\alpha \times \nu$ and $\mu_\beta \times \nu$ are weakly homeomorphic if and only if $\mu_\beta \in [\mu_\alpha]$. $\blacksquare$ If $\mu$ is Haar measure for some topological group structure on a non-compact locally compact Cantor space $X$ then $\mu$ is a good measure on $X$. **Proof.** The ball $B$ centered at the identity in the invariant ultrametric is a compact open subgroup of $X$. Since $\mu$ is translation-invariant, by Proposition \[many\], it suffices to show that $\mu|_B$ is good for every such ball $B$. Since the restriction of $\mu$ on $B$ is a Haar measure on a compact Cantor space, $\mu|_B$ is good by Proposition 2.4 in [@Akin3]. $\blacksquare$ From measures on non-compact spaces to measures on compact spaces and back again ================================================================================ Let $X$ be a non-compact locally compact Cantor space. A *compactification* of $X$ is a pair $(Y,c)$ where $Y$ is a compact space and $c \colon X \rightarrow Y$ is a homeomorphic embedding of $X$ into $Y$ (i.e. $c \colon X \rightarrow c(X)$ is a homeomorphism) such that $\overline{c(X)} = Y$, where $\overline{c(X)}$ is the closure of $c(X)$. In the paper, by compactification we will mean not only a pair $(Y,c)$ but also the compact space $Y$ in which $X$ can be embedded as a dense subset. We will denote the compactifications of a space $X$ by symbols $cX$, $\omega X$, etc., where $c$, $\omega$ are the corresponding homeomorphic embeddings. Let $\mu \in M^0(X)$. We will consider only such compactifications $cX$ that $cX$ is a Cantor set. Since $c$ is a homeomorphism, the measure $\mu$ on $X$ passes to a homeomorphic measure on $c(X)$. Since we are interested in the classification of measures up to homeomorphisms, we can identify the set $c(X)$ with $X$. Hence $X$ can be considered as an open dense subset of $cX$. The set $cX \setminus X$ is called the *remainder* of compactification. As far as $X$ is locally compact, the remainder $cX \setminus X$ is closed in $cX$ for every compactification $cX$ (see [@E]). Since $\overline{X} = cX$, the set $cX \setminus X$ is a closed nowhere dense subset of $cX$. Compactifications $c_1X$ and $c_2X$ of a space $X$ are *equivalent* if there exists a homeomorphism $f \colon c_1X \rightarrow c_2X$ such that $fc_1(x) = c_2(x)$ for every $x \in X$. We shall identify equivalent compactifications. For any space $X$ one can consider the family $\mathcal{C}(X)$ of all compactifications of $X$. The order relation on $\mathcal{C}(X)$ is defined as follows: $c_2X \leq c_1X$ if there exists a continuous map $f \colon c_1X \rightarrow c_2X$ such that $fc_1 = c_2$. Then we have $f(c_1(X)) = c_2(X)$ and $f(c_1X \setminus c_1(X)) = c_2X \setminus c_2(X)$. \[Alex\] Every non-compact locally compact space $X$ has a compactification $\omega X$ with one-point remainder. This compactification is the smallest element in the set of all compactifications $\mathcal{C}(X)$ with respect to the order $\leq$. The topology on $\omega X$ is defined as follows. Denote by $\{\infty\}$ the point $\omega X \setminus X$. Open sets in $\omega X$ are the sets of the form $\{\infty\} \cup (X \setminus F)$, where $F$ is a compact subspace of $X$, together with all sets that are open in $X$. For any Borel measure $\nu$ on the set $cX \setminus X$ with the induced topology, $\widetilde{\mu} = \mu + \nu$ is a Borel measure on $cX$ such that $\widetilde{\mu}|_X = \mu$. Since the aim of compactification is the study of a measure $\mu$ on a locally compact set $X$, we will consider only such extensions $\widetilde{\mu}$ on $cX$ that $\mu(cX \setminus X) = 0$. \[Smu\_diff\_comp\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$. Let $c_1X$, $c_2X$ be the compactifications of $X$ such that $c_1X \leq c_2X$. Denote by $\mu_1$ the extension of $\mu$ on $c_1 X$ and by $\mu_2$ the extension of $\mu$ on $c_2 X$. Then $S(\mu) \subseteq S(\mu_1) \subseteq S(\mu_{2})$. **Proof.** Since $c_1X \leq c_2X$, there exists a continuous map $f \colon c_2X \rightarrow c_1 X$ such that $f(c_2 X \setminus X) = c_1 X \setminus X$ and $fc_2(x) = c_1(x)$ for any $x \in X$. Since $f$ is continuous, it suffices to prove that $f$ preserves measure, that is $\mu_1(V) = \mu_2(f^{-1}(V))$ for any compact open $V \subset X$. Recall that we can identify $c_i(X)$ with $X$. Hence we can consider $f$ as an identity on $X \subset c_iX$ and $f$ preserves measure. That is, for every compact open subset $U$ of $X$ we have $\mu(U) = \mu_1(U) = \mu_2(U)$. Hence $S(\mu) \subseteq S(\mu_1)$. Since $\mu(c_iX \setminus X) = 0$, the measure of any clopen subset of $c_iX$ is the sum of measures of compact open subsets of $X$. Hence the measures of all clopen sets are preserved. Thus, $S(\mu_1) \subseteq S(\mu_{2})$. $\blacksquare$ We can consider the homeomorphic embedding of a set $X$ into a non-compact locally compact Cantor set $Y$ such that $\mu(Y \setminus X) = 0$. Then, by the same arguments as above, the inclusion $S(\mu|_X) \subseteq S(\mu|_Y)$ holds. \[krit\_good\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$ be a good measure. Let $cX$ be any compactification of $X$. Then $\mu$ is good on $cX$ if and only if $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$. **Proof**. First, we prove the “if” part. Let $V$ be a clopen set in $cX$. Consider two cases. First, let $V \cap (cX \setminus X) = \emptyset$. Then $V$ is a compact open subset of $X$. Since $\mu$ is good on $X$ and $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$, we see that $V$ stays good in $cX$. Now, suppose that $V \cap (cX \setminus X) \neq \emptyset$. Then $V \cap X$ is an open set and $\mu(V) = \mu(V \cap X) = \mu(\bigsqcup_{n=1}^{\infty} V_n)$ where each $V_n$ is a compact open set in $X$. Let $U$ be any compact open subset of $X$ with $\mu(U) < \mu(V)$. Then there exists $N \in \mathbb{N}$ such that $\mu(U) < \mu(\bigsqcup_{n = 1}^{N} V_n)$. The set $Z = \bigsqcup_{n = 1}^{N} V_n$ is a compact open subset of $X$. Since $S(\mu|_{cX}) \cap [0, \mu(X)) = S(\mu|_X)$, we have $\mu(U) \in S(\mu|_X)$. Since $\mu$ is good on $X$, there exists a compact open subset $W \subset Z$ such that $\mu(W) = \mu(U)$. Now we prove the “only if” part. Assume the converse. Suppose that $\mu$ is good and the equality does not hold. Then there exists $\gamma \in (0,\mu(X))$ such that $\gamma \in S(\mu|_{cX}) \setminus S(\mu|_X)$. Since $S(\mu|_X)$ is dense in $(0,\mu(X))$, there exists a compact open subset $U \subset X$ such that $\mu(U) > \gamma$. Hence $\gamma \in S(\mu|_{cX}) \cap [0, \mu(U)]$ and $\gamma \not \in S(\mu|_U)$. Thus $U$ is not good and we get a contradiction. $\blacksquare$ By Proposition \[basic\_prop\], the set $X \setminus \mathfrak{M}_\mu$ is a non-compact locally compact Cantor set and $\overline{X \setminus \mathfrak{M}_\mu} = X$. Thus, the set $X \setminus \mathfrak{M}_\mu$ can be homeomorphically embedded into $X$ and then into some compactification $cX$. After embedding $X \setminus \mathfrak{M}_\mu$ into $X$, we add only compact open sets of infinite measure. Hence if $\mu$ was good on $X \setminus \mathfrak{M}_\mu$, it remains good on $X$ and $S(\mu|_{X \setminus \mathfrak{M}_\mu}) = S(\mu|_X)$. We can consider $X$ as a step towards compactification of $X\setminus \mathfrak{M}_\mu$ and include $\mathfrak{M}_\mu$ into $cX \setminus X$. The measure $\mu \in M^0(X)$ is locally finite on $X \setminus \mathfrak{M}_\mu$, so we can consider only locally finite measures among infinite ones. If $\mu$ is not good on a locally compact Cantor set $X$ then clearly $\mu$ is not good on any compactification $cX$. Let $\mu$ be a good infinite locally finite measure on a non-compact locally compact Cantor set $X$. Then $\mu$ is good on $\omega X$. **Proof.** By definition of topology on $\omega X$, the “new” open sets have compact complement. Since $\mu$ is locally finite on $X$, the measure of compact subsets of $X$ is finite. Hence the measure of each new clopen set is infinite. By Theorem \[krit\_good\], $\mu$ is good on $\omega X$. $\blacksquare$ \[gamma\] Let $\mu$ be a good measure on a non-compact locally compact Cantor set $X$. Then for any $\gamma \in [0,\mu(X))$ there exists a compactification $cX$ such that $\gamma \in S(\mu|_{cX})$. **Proof.** The set $S(\mu|_{X})$ is dense in $[0,\mu(X))$. Hence for every $\gamma \in [0,\mu(X)$ there exist $\{\gamma_n\}_{n=1}^{\infty} \subset S(\mu|_{X})$ such that $\lim_{n\rightarrow \infty} \gamma_n = \gamma$. Since $\mu$ is good, there exist disjoint compact open subsets $\{U_n\}_{n=1}^{\infty}$ such that $\mu(U_n) = \gamma_n$. Then $U = \bigsqcup_{n=1}^{\infty} U_n$ is a non-compact locally compact Cantor set. Consider the compactification $cX = \omega U \sqcup c(X \setminus U)$, where $c(X \setminus U)$ is any compactification of $X \setminus U$. Then $\omega U$ is a clopen set in $cX$ and $\mu(\omega U) = \gamma \in S(\mu|_{cX})$. $\blacksquare$ From Theorems \[krit\_good\], \[gamma\] the corollary follows: For any measure $\mu$ on a non-compact locally compact Cantor space $X$ there exists a compactification $cX$ such that $\mu$ is not good on $cX$. If a measure $\mu \in M^0(X)$ is a good probability measure then, by Theorem \[krit\_good\], the measure $\mu$ is good on $cX$ if and only if $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$. \[1\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M_f(X)$. If there exists a compactification $cX$ such that $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$ then $1 \in G(S(\mu|_X))$. **Proof**. Let $\gamma \in S(\mu|_{cX}) \cap (0,1)$. Since the complement of a clopen set is a clopen set, we have $1 - \gamma \in S(\mu|_{cX})$. Since $S(\mu|_{cX}) = S(\mu|_X) \cup \{1\}$, we have $\gamma, 1 - \gamma \in S(\mu|_{X})$. Hence $1 \in G(S(\mu|_X))$. $\blacksquare$ Thus, if $1 \not \in G(S(\mu|_X))$ then for any compactification $cX$ the set $S(\mu|_X)$ cannot be preserved after the extension. The examples are given in the last section. The corollary follows from Proposition \[1\] and Theorem \[krit\_good\]. Let $\mu$ be a probability measure on a non-compact locally compact Cantor set $X$ and $1 \not \in G(S(\mu|_X))$. Then for any compactification $cX$ of $X$, $\mu$ is not good on $cX$. \[goodAlex\] Let $\mu$ be a good probability measure on a non-compact locally compact Cantor set $X$. Then $\mu$ is good on Alexandroff compactification $\omega X$ if and only if $1 \in G(S(\mu|_X))$. **Proof**. By Proposition \[1\] and Theorem \[krit\_good\], if $\mu$ is good on $\omega X$ then $1 \in G(S(\mu|_X))$. Suppose $\mu$ is good on $X$ and $1 \in G(S(\mu|_X))$. Since $\mu$ is good, any compact open subset of $\mu$ is good, hence for every compact open $U \subset X$ we have $\mu(U) = G(S(\mu|_X)) \cap [0, \mu(U)] = S(\mu)\cap [0, \mu(U)]$. Every clopen subset of $\omega X$ has a compact open subset of $X$ as a complement. Hence for every clopen $V \subset \omega X$ we see that $\mu(V) = 1 - \mu(X \setminus V) \in G(S(\mu|_X)) \cap (0,1) = S(\mu|_X)$. So, $S(\mu|_{\omega X}) = S(\mu|_X) \cup \{1\}$. Hence $\mu$ is good on $\omega X$ by Theorem \[krit\_good\]. $\blacksquare$ For a Cantor set $Y$ denote by $M^0(Y)$ the set of all either finite or non-defective measures on $Y$ (see [@K]). Since an open dense subset of a Cantor set is a locally compact Cantor set, the corollary follows: \[good\_subs\] Let $Y$ be a (compact) Cantor set and measure $\mu \in M^0(Y)$. Let $X \subset Y$ be an open dense subset of $Y$ of full measure. If $\mu$ is good on $Y$ then $\mu$ is good on $X$. **Proof**. The set $X$ is a locally compact Cantor set and $Y$ is a compactification of $X$. Any compact open subset $U$ of $X$ is a clopen subset of $Y$ and all clopen subsets of $U$ are compact open sets. Thus, a $\mu|_Y$-good compact open set in $X$ is, a fortiory, $\mu|_X$-good. $\blacksquare$ Thus, the extensions of a non-good measure are always non-good. The corollary follows from Lemma \[Smu\_diff\_comp\], Theorem \[krit\_good\] and Corollary \[good\_subs\]. \[2compactifications\] Let $X$ be a non-compact locally compact Cantor set and $\mu \in M^0(X)$. Let $c_1X$, $c_2X$ be compactifications of $X$ such that $c_1X \geq c_2X$. Let $\mu$ be good on $c_1X$. Then $\mu$ is good on $c_2X$. Moreover, if $\mu \in M_f(X)$ then $\mu|_{c_1X}$ is homeomorphic to $\mu|_{c_2X}$. Recall that Alexandroff compactification $\omega X$ is the smallest element in the set of all compactifications of $X$. Hence, if $\mu$ is not good on $\omega X$ then $\mu$ is not good on any compactification $cX$ of $X$. The following theorem can be proved using the results of Akin [@Akin2] for measures on compact sets. Let $X$, $Y$ be non-compact locally compact Cantor spaces, and $\mu \in M^0_f(X)$, $\nu \in M^0_f(Y)$ be good measures such that their extensions to $\omega X$, $\omega Y$ are good. Then $\mu|_X$ and $\nu|_Y$ are homeomorphic if and only if $S(\mu|_X) = S(\nu|_Y)$. **Proof.** The “only if” part is trivial, we prove the “if” part. Since $\mu|_{\omega X}$ and $\nu|_{\omega Y}$ are good by Theorem \[krit\_good\], we have $S(\mu|_{\omega X}) = S(\nu|_{\omega Y})$. Denote by $x_0 = \omega X \setminus X$ and $y_0 = \omega Y \setminus Y$. By Theorem 2.9 [@Akin2], there exists a homeomorphism $f \colon \omega X \rightarrow \omega Y$ such that $f_*\mu = \nu$ and $f(x_0) = y_0$. Hence $f(X) = Y$ and the theorem is proved. $\blacksquare$ In Example 1, we present a class of good measures on non-compact locally compact Cantor sets such that these measures are not good on the Alexandroff compactifications. Thus, these measures are not good on any compactification of the corresponding non-compact locally compact Cantor sets. Examples ======== Let $B$ be a non-simple stationary Bratteli diagram with the matrix $A$ transpose to the incidence matrix. Let $\mu$ be an ergodic $\mathcal{R}$-invariant measure on $B$ (see [@S.B.O.K.; @S.B.; @K]). Let $\alpha$ be the class of vertices that defines $\mu$. Then $\mu$ is good as a measure on a non-compact locally compact set $X_\alpha$. The measure $\mu$ on $X_\alpha$ can be either finite or infinite, but it is always locally finite. The set $X_B$ is a compactification of $X_\alpha$. Since $\mu$ is ergodic, we have $\mu(X_B \setminus X_\alpha) = 0$. In [@S.B.O.K.; @K] the criteria of goodness for probability or non-defective measure $\mu$ on $X_B$ were proved in terms of the Perron-Frobenius eigenvalue and eigenvector of $A$ corresponding to $\mu$ (see Theorem 3.5 [@S.B.O.K.] for probability measures and Corollary 3.4 [@K] for infinite measures). It is easy to see that these criteria are particular cases of Theorem \[krit\_good\]. We consider now a class of stationary Bratteli diagrams and give a criterion when a measure $\mu$ from this class is good on the Alexandroff compactification $\omega X_\alpha$. Fix an integer $N \geq 3$ and let $$F_N = \begin{pmatrix} 2 & 0 & 0\\ 1 & N & 1\\ 1 & 1 & N \\ \end{pmatrix}$$ be the incidence matrix of the Bratteli diagram $B_N$. For $A_N = F_N^T$ we easily find the Perron-Frobenius eigenvalue $\lambda = N+1$ and the corresponding probability eigenvector $$x = \left(\frac{1}{N},\ \frac{N-1}{2N},\ \frac{N-1}{2N}\right)^T.$$ Let $\mu_N$ be the measure on $B_N$ determined by $\lambda$ and the eigenvector $x$. The measure $\mu_N$ is good on $\omega X_\alpha$ if and only if for there exists $R \in \mathbb{N}$ such that $\frac{2(N+1)^R}{N-1}$ is an integer. This is possible if and only if $N = 2^k+1$, $k \in \mathbb{N}$. For instance, the measure $\mu_N$ is good on $\omega X_\alpha$ for $N = 3, 5$ but is not good for $N = 4$. Note that the criterion for goodness on $\omega X_\alpha$ here is the same as for goodness on $X_B$. This example is a particular case of more general result (the notation from [@S.B.O.K.] is used below): Let $B$ be a stationary Bratteli diagram defined by a distinguished eigenvalue $\lambda$ of the matrix $A = F^T$. Denote by $x = (x_1,...,x_n)^T$ the corresponding reduced vector. Let the vertices $2, \ldots, n$ belong to the distinguished class $\alpha$ corresponding to $\mu$. Then $\mu$ is good on $X_B$ if and only if $\mu$ is good on $\omega X_\alpha$. **Proof.** By Theorem \[Alex\] and Corollary \[2compactifications\], if $\mu$ is good on $X_B$ then $\mu$ is good on $\omega X_\alpha$. We prove the converse. By Theorem 3.5 in [@S.B.O.K.] and Theorem \[goodAlex\], it suffices to prove that $1 \in G(S(\mu|_{X_\alpha}))$ only if there exists $R \in \mathbb{N}$ such that $\lambda^R x_1 \in H(x_2,...,x_n)$. Note that $G(S(\mu|_{X_\alpha})) = \left(\bigcup_{N=0}^\infty \frac{1}{\lambda^N} H(x_2,...,x_n) \right)$, where $H(x_2,...,x_n)$ is an additive group generated by $x_2,...,x_n$. Suppose that $1 \in G(S(\mu|_{X_\alpha}))$. Since $\sum_{i=1}^n x_i = 1$, we see that $x_1 \in G(S(\mu|_{X_\alpha}))$, hence there exists $R \in \mathbb{N}$ such that $\lambda^R x_1 \in H(x_2,...,x_n)$. $\blacksquare$ Return to a general case of ergodic invariant measures on stationary Bratteli diagrams. If $\mu$ is a probability measure on $X_\alpha$ and $S(\mu|_{X_\alpha}) \cup \{1\} = S(\mu|_{X_B})$ then, by Lemma \[Smu\_diff\_comp\], we have $S(\mu|_{\omega X_\alpha}) = S(\mu|_{X_B})$. By Theorem \[krit\_good\], the measure $\mu$ is good on $\omega X_\alpha$. Hence $\mu|_{\omega X_\alpha}$ is homeomorphic to $\mu|_{X_B}$ (see [@Akin2]). If $\mu$ is infinite, then the measures $\mu|_{\omega X_\alpha}$ and $\mu|_{X_B}$ are not homeomorphic since $\mathfrak{M}_{\mu|_{\omega X_\alpha}}$ is one point and $\mathfrak{M}_{\mu|_{X_B}}$ is a Cantor set (see [@K]). Let $X$ be a Cantor space and $\mu$ be a good probability measure on $X$ with $S(\mu) = \{\frac{m}{2^n} : m \in \mathbb{N} \cap [0,2^n], n \in \mathbb{N}\}$ (for example a Bernoulli measure $\beta (\frac{1}{2},\frac{1}{2})$). Clearly, $\mu_n = \frac{1}{2^n}\mu$ is a good measure for $n \in \mathbb{N}$ with $S(\mu_n) = \frac{1}{2^n} S(\mu) \subset S(\mu)$. Let $\{X_n, \mu_n\}_{n=1}^{\infty}$ be a sequence of Cantor spaces with measures $\mu_n$. Let $A = \bigsqcup_{n=1}^{\infty} X_n$ be the disjoint union of $X_n$. Denote by $\nu$ a measure on $A$ such that $\nu|_{X_n} = \mu_n$. Then $\nu$ is a good measure on a locally compact Cantor space $A$ with $S(\nu) = S(\mu) \cap [0,1)$. Consider the one-point compactification $\omega A$ and the extension $\nu_1$ of $\nu$ to $\omega A$. We add to $S(\nu)$ the measures of sets which contain $\{\infty\}$ and have a compact open complement. Hence we add the set $\Gamma = \{1 - \gamma : \gamma \in S(\nu)\}$. Since $\Gamma \subset S(\nu) \cup \{1\}$, we have $S(\nu_1) = S(\nu) \cup \{1\}$. By Theorem \[krit\_good\], the measure $\nu_1$ is good on $\omega A$. Consider the two-point compactification of $A$. Let $A = A_1 \sqcup A_2$ where $A_1 = \bigsqcup_{k = 1}^{\infty} X_{2k - 1}$ and $A_2 = \bigsqcup_{j = 1}^{\infty} X_{2j}$. Then $cA = \omega A_1 \sqcup \omega A_2$ is a two-point compactification of $A$. Let $\nu_2$ be the extension of $\nu$ to $cA$. Then $\nu_2(A_1) = \frac{2}{3} \not \in S(\nu)$. Hence, by Theorem \[krit\_good\], the measure $\nu_2$ is not good on $cA$. In the same example, we can make a two-point compactification which preserves $S(\nu|_A)$. Since $\mu_n$ is good for $n \in \mathbb{N}$, there is a compact open partition $X_n^{(1)} \sqcup X_n^{(2)} = X_n$ such that $\mu_n (X_n^{(i)}) = \frac{1}{2^{n+1}}$ for $i = 1,2$. Let $B_i = \bigsqcup_{n=1}^{\infty} X_n^{(i)}$ for $i = 1,2$. Consider $\widetilde{c}A = \omega B_1 \sqcup \omega B_2$. Then it can be proved the same way as above that $S(\nu|_{\widetilde{c}A}) = S(\nu|_A) \cup \{1\}$. Let $\mu = \beta(\frac{1}{3}, \frac{2}{3})$ be a Bernoulli (product) measure on Cantor space $Y = \{0,1\}^\mathbb{N}$ generated by the initial distribution $p(0) = \frac{1}{3}$, $p(1) = \frac{2}{3}$. Then $\mu$ is not good but $S(\mu) = \{\frac{a}{3^n} : a \in \mathbb{N} \cap [0, 3^n], n \in \mathbb{N}\}$ is group-like (see [@Akin1]). Let $X$ be any open dense subset of $Y$ such that $\mu(Y \setminus X) = 0$. Thus, $Y$ is a compactification of a non-compact locally compact Cantor space $X$ and $\mu$ extends from $X$ to $Y$. Then $\mu$ is not good on $X$. The compact open subsets of $X$ are exactly the clopen subsets of $Y$ that lie in $X$. The compact open subset of $X$ is a union of the finite number of compact open cylinders. Consider any compact open cylinder $U = \{a_0,...,a_n,*\}$ which consists of all points in $z \in Y$ such that $z_i = a_i$ for $0 \leq i \leq n$. Then $U$ is a disjoint union of two subcylinders $V_1 = \{a_0,...,a_n,0,*\}$ and $V_2 = \{a_0,...,a_n,1,*\}$ with $\mu(V_2) = 2 \mu(V_1)$. Let the numerator of the fraction $\mu(V_1)$ be $2^k$. Then the numerator of the fraction $\mu(V_2)$ is $2^{k+1}$. Moreover, for any compact open $W \subset V_2$ the numerator of the fraction $\mu(W)$ will be divisible by $2^{k+1}$. Since $S(\mu)$ contains only finite sums of measures of cylinder compact open sets and the denominators of elements of $S(\mu)$ are the powers of $3$, there is no compact open subset $W \subset V_2$ such that $\mu(W) = \mu(V_1)$. Hence $\mu$ is not good on $X$. Moreover, let $x = \{00...\}$ be a point in $Y$ which consists only of zeroes. Then $S(\mu |_{Y \setminus\{x\}})\varsubsetneq S(\mu |_Y)$ while $S(\mu |_{Y \setminus\{y\}}) = S(\mu |_Y)$ for any $y \neq x$. Consider the case $y \neq x$. Let, for instance, $y = \{111....\}$, all other cases are proved in the same way. Let $U_n = \{z \in Y : z_0 = ... = z_{n-1} = 1, z_{n} = 0\}$. Then $Y \setminus\{y\} = \bigsqcup_{n=1}^\infty U_n \sqcup \{0*\}$. Denote by $S_N = \mu(\bigsqcup_{n=1}^N U_n)$ and $S_0 = 0$. Then $\lim_{N \rightarrow \infty} S_N = \frac{2}{3}$. Let $G = \{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. Then $G$ is an additive subgroup of reals and $S(\mu|_Y) = G \cap [0,1]$. We prove that $S(\mu |_{Y \setminus\{y\}}) = G \cap [0,1)$, i.e. for every $n \in \mathbb{N}$ and $a = 0,..., 3^n - 1$ there exists a compact open set $W$ in $Y \setminus \{y\}$ such that $\mu(W) = \frac{a}{3^n}$. Indeed, we have $S(\mu|_{\{0*\}}) = G \cap [0, \frac{1}{3}]$ and $[0,1) = \cup_{n=0}^\infty [S_N, S_N + \frac{1}{3}]$. Hence $G \cap [0,1) = \cup_{n=0}^\infty (G \cap [S_N, S_N + \frac{1}{3}])$. For every $\gamma \in G$ there exists $N\in \mathbb{N}$ such that $\gamma \in [S_N, S_N + \frac{1}{3}]$. There exists a compact open subset $W_0$ of $\{0*\}$ such that $\mu(W_0) = \gamma - S_N$. Set $W = U_N \sqcup W_0$. Then $W$ is a compact open subset of $Y \setminus \{y\}$ and $\mu(W) = \gamma$. Now consider the set $Y \setminus\{x\}$. Every cylinder that lies in $Y \setminus\{x\}$ has even numerator, hence $S(\mu |_{Y \setminus\{x\}})\varsubsetneq S(\mu |_Y)$. It can be proved in the same way as above that $S(\mu |_{Y \setminus\{x\}}) = \{\frac{2k}{3^n} : k \in \mathbb{N}\} \cap [0,1)$. Denote by $|A|$ the cardinality of a set $A$. Given two subsets $E, F \subset \mathbb{Z}$, by $E+F$ we mean $\{e+f | e \in E, f \in F\}$ (for more details see [@D1; @D2]). Let $\{F_n\}_{n=1}^\infty, \{C_n\}_{n=1}^\infty \subset \mathbb{Z}$ such that for each $n$ \(1) $|F_n|<\infty$, $|C_n|<\infty$, \(2) $|C_n| > 1$, \(3) $F_n + C_n + \{-1,0,1\} \subset F_{n+1}$, \(4) $(F_n + c) \cap (F_n + c') = \emptyset$ for all $c \neq c' \in C_{n+1}$. Set $X_n = F_n \times \prod_{k > n} C_k$ and endow $X_n$ with a product topology. By (1),(2), each $X_n$ is a Cantor space. For each $n$, define a map $i_{n,n+1} \colon X_{n} \rightarrow X_{n + 1}$ such that $$i_{n,n+1}(f_n, c_{n+1}, c_{n+2},...) = (f_n + c_{n+1}, c_{n+2},...).$$ By (1), (2) each $i_{n,n+1}$ is a well defined injective continuous map. Since $X_n$ is compact, we see that $i_{n,n+1}$ is a homeomorphism between $X_n$ and $i_{n,n+1}(X_n)$. So the embedding $i_{n,n+1}$ preserves topology. The set $i_{n, n+1}(X_n)$ is a clopen subset of $X_{n+1}$. Let $i_{m,n} \colon X_m \rightarrow X_n$ such that $i_{m,n} = i_{n, n-1} \circ i_{n-1, n-2} \circ ... \circ i_{m+1,m}$ for $m < n$ and $i_{n,n} = id$. Denote by $X$ the topological inductive limit of the sequence $(X_n, i_{n,n+1})$. Then $X = \bigcup_{n=1}^\infty X_n$. Since $i_{m,n} = i_{n, n-1} \circ i_{n-1, n-2} \circ ... \circ i_{m+1,m}$ for $m < n$, we can write $X_1 \subset X_2 \subset ...$ The set $X$ is a non-compact locally compact Cantor set. The Borel $\sigma$-algebra on $X$ is generated by cylinder sets $[A]_n = \{x \in X : x = (f_n, c_{n+1},c_{n+2},...) \in X_n \mbox{ and } f_n \in A\}$. There exists a canonical measure on $X$. Let $\kappa_n$ stand for the equidistribution on $C_n$ and let $\nu_n = \frac{|F_n|}{|C_1|...|C_n|}$ on $F_n$. The product measure on $X_n$ is defined as $\mu_n = \nu_n \times \kappa_{n+1} \times \kappa_{n+2}\times ...$ and a $\sigma$-finite measure $\mu$ on $X$ is defined by restrictions $\mu|_{X_n} = \mu_n$. The measure $\mu$ is a unique up to scaling ergodic locally finite invariant measure for a minimal self-homeomorphism of $X$ (for more details see [@D1; @D2]). For every two compact open subsets $U,V \subset X$ there exists $n \in \mathbb{N}$ such that $U, V \subset X_n$. The measure $\mu$ is obviously good, since the restriction of $\mu$ onto $X_n$ is just infinite product of equidistributed measures on $F_n$ and $C_m$, $m > n$. We have $S(\mu) = \{\frac{a}{|C_1|...|C_n|} : a, n \in \mathbb{N}\} \cap [0, \mu(X))$. Let $p$ be a prime number and $\mathbb{Q}_p$ be the set of $p$-adic numbers. Endowed with the $p$-adic norm, the set $\mathbb{Q}_p$ is a non-compact locally compact Cantor space. Then the Haar measure $\mu$ on $\mathbb{Q}_p$ is good and $S(\mu) = \{n p^{\gamma} | n \in \mathbb{N}, \gamma \in \mathbb{Z}\}$. **Acknowledgement** I am grateful to my advisor Sergey Bezuglyi for giving me the idea of this work, for many helpful discussions and for reading the preliminary versions of this paper. [99]{} *E.Akin*, Measures on Cantor space, “Topology Proc.”, **24** (1999), 1 - 34. *E.Akin*, Good measures on Cantor space, “Trans. Amer. Math. Soc.”, **357** (2005), 2681 - 2722. *E. Akin, R. Dougherty, R.D. Mauldin, A. Yingst*, Which Bernoulli measures are good measures? “Colloq. Math.”, **110** (2008), 243 – 291. *S.Alpern, V.S.Prasad,* Typical Dynamics of Volume Preserving Homeomorphisms, Cambridge Tracts in Mathematics, 139, Cambridge Univercity Press, Cambridge 2000. *T.D.Austin*, A pair of non-homeomorphic product measures on the Cantor set, “Math. Proc. Cam. Phil. Soc.”, **142** (2007), 103 - 110. *S. Bezuglyi and O. Karpel*, *Homeomorphic Measures on Stationary Bratteli Diagrams*, J. Funct. Anal. 261(2011), 3519-3548. *S. Bezuglyi, J.Kwiatkowski, K.Medynets, and B.Solomyak,* Invariant measures on stationary Bratteli diagrams, “Ergodic Theory Dynam. Syst.”, **30** (2010), 973 - 1007. *A. Danilenko,* $(C,F)$-actions in ergodic theory, Geometry and Dynamics of Groups and Spaces, Progr. Math., **265** (2008), 325-351. *A. Danilenko,* Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math., **12** (2001), 113 - 123. *R.Dougherty, R.Daniel Mauldin, and A.Yingst,* On homeomorphic Bernoulli measures on the Cantor space, “Trans. Amer. Math. Soc.” **359** (2007), 6155 - 6166 *R. Engelking,* General Topology, Berlin: Heldermann, 1989. *E. Glasner, B. Weiss,* Weak orbital equivalence of minimal Cantor systems, “Internat. J. Math.” **6** (1995), 559 - 579. *O. Karpel,* Infinite measures on Cantor spaces, J Difference Equ. Appl., DOI:10.1080/10236198.2011.620955. *J.C.Oxtoby, V.S.Prasad,* Homeomorphic measures in the Hilbert Cube, “Pacific J. Math.”, **77** (1978), 483 - 497. *J.C.Oxtoby, S.M.Ulam*, Measure preserving homeomorphisms and metrical transitivity, “Ann. Math.”, **42** (1941), 874 - 920. *Andrew Q. Yingst*, A characterization of homeomorphic Bernoulli trial measures, “Trans. Amer. Math. Soc.”, **360** (2008), 1103 - 1131.

ArXiv

--- title: 'Modeling of Transport through Submicron Semiconductor Structures: A Direct Solution of the Coupled Poisson-Boltzmann Equations' --- We report on a computational approach based on the self-consistent solution of the steady-state Boltzmann transport equation coupled with the Poisson equation for the study of inhomogeneous transport in deep submicron semiconductor structures. The nonlinear, coupled Poisson-Boltzmann system is solved numerically using finite difference and relaxation methods. We demonstrate our method by calculating the high-temperature transport characteristics of an inhomogeneously doped submicron GaAs structure where the large and inhomogeneous built-in fields produce an interesting fine structure in the high-energy tail of the electron velocity distribution, which in general is very far from a drifted-Maxwellian picture. [2]{} The carrier dynamics in submicron structures is far from thermal equilibrium due to strong and rapidly varying external and built-in electric fields. Hot electron and ballistic effects dominate the transport characteristics and the electron velocity distribution function in such systems is far from a drifted-Maxwellian description. In order to fully take into account the nonequilibrium nature of the transport, a full solution of the semiclassical Boltzmann transport equation (BTE) is required. Although the Monte Carlo method has been very popular for the solution of the BTE in semiconductor device simulation [@jacoboni], several works [@barangerPRB87]-[@majoranaCOMPEL04] have recently solved the BTE by direct methods, thus allowing noise-free spatial and temporal resolution of the electron distribution function, which in the Monte Carlo method may be difficult to obtain due to the statistical nature of the approach. In this paper, we present a straight-forward approach to calculate the electron distribution function, $f(x,v)$, for submicron inhomogeneous semiconductor structures by solving the steady-state BTE self-consistently with the Poisson equation. We solve the strictly two-dimensional (2D) BTE (one dimension corresponding to position and one to velocity) and treat scattering within the relaxation time approximation (RTA) where each individual scattering mechanism is represented by a characteristic scattering rate that can be derived from quantum mechanical scattering theory. We demonstrate our approach for submicron, inhomogeneously doped structures and discuss the general nonequilibrium transport characteristics. Basic equations =============== The Boltzmann equation describes the dynamics of the semiclassical distribution function, $f({\bf r}, {\bf v}, t)$, under the influence of electric and magnetic fields, as well as different scattering processes. In the absence of a magnetic field, the 2D phase-space, steady-state BTE in the RTA is written according to: $$-\frac{eE(x)}{m^{\ast}}\frac{\partial f(x,v)}{\partial v}+ v\frac{\partial f(x,v)}{\partial x}=-\frac{f(x,v)-f_{LE}(x,v)}{\tau(\varepsilon)}~, \label{bte}$$ where $m^{\ast}$ is the electron effective mass in the parabolic band approximation, and $f_{LE}(x,v)$ is a local equilibrium distribution function appropriate to a local density, applied field and equilibrium lattice temperature, $T_{0}$, to which the distribution function $f(x,v)$ relaxes at a relaxation rate $\tau(\varepsilon)^{-1}$. As the local equilibrium function, we choose in the following a Maxwell-Boltzmann (MB) distribution at $T_{0}$, normalized to the local density $n(x)$ $$f_{LE}(x,v)=n(x)\left [ \frac{m^{\ast}}{2\pi kT_{0}} \right ]^{1/2} e ^{-\frac{m^{\ast} v^{2}}{2k_{B}T_{0}}}~. \label{mb}$$ The inhomogeneous electric field, $E(x)$, in the BTE, originating from the spatially dependent electron and doping densities, $n(x)$ and $N_{D}(x)$, is obtained from the Poisson equation $$\frac{d ^{2} \phi}{d x^{2}} = -\frac{dE}{dx}= -e \frac{N_{D}(x) - n(x)}{\epsilon \epsilon_{0}} = -\rho(x), \label{poisson}$$ where $\epsilon$ is the static dielectric constant. Since the electron density is related to the distribution function by $$n(x)=\int^{\infty}_{-\infty} f(x,v)dv~, \label{density}$$ the Poisson and Boltzmann equations constitute a coupled, nonlinear set of equations, and thus, Eqs. (\[bte\]-\[density\]) have to be solved self-consistently. Numerical procedure =================== The numerical procedure consists, in short, of initializing the system parameters, discretizing Eqs. (\[bte\]-\[density\]) on a 2D grid in phase-space, performing the self-consistent Poisson-Boltzmann loop and, upon convergence, calculate and output the electron distribution function, electric field and the desired moments of the BTE. In the calculations, after initialization, we rescale the system parameters and the equations according to $$x^{\prime}=x/L_{D},~v^{\prime}=v\tau/L_{D}, \label{scaling}$$ where $L_{D}=\sqrt{\epsilon \epsilon_{0}k_{B}T_{0}/e^{2}N}$ is the Debye length, $N=\max[N_{D}(x)]$ and $\tau$ is a characteristic scattering time. The choice of grid size and resolution depends to a large extent on the system parameters and the electrostatics present in the device. In order to reproduce details due to strong and rapidly varying electric fields, we choose the spatial grid step size to be smaller than the Debye length, $L_{D}$, defined above. In velocity space, on the other hand, the discrete grid step size needs to be small enough to resolve fine structure in the distribution function, as well as give accurate results for the moments of the BTE. In addition, the grid needs to be large enough, in velocity, in order to capture the full information in the high-energy tail of the distribution function, and in position, in order to damp out the effects of the contact boundaries. The Poisson and Boltzmann equations are solved by finite difference and iterative relaxation methods [@numrec]. For the Poisson equation (\[poisson\]), we use forward and backward Euler differences according to $$L^{+}_{x}L^{-}_{x}\phi_{j}=\frac{\phi_{j+1}-2\phi_{j}+ \phi_{j-1}}{(\delta x)^{2}}= -\rho_{j}~, \label{poissondifference}$$ where $L^{+}_{x}\phi(x)=(\phi_{j+1}-\phi_{j})/\delta x$ and $L^{-}_{x}\phi(x)=(\phi_{j}-\phi_{j-1})/\delta x$ denote forward and backward Euler steps, respectively. The resulting matrix equation is solved iteratively using successive overrelaxation (SOR) [@numrec]. For the solution of the BTE, we adopt an upwind finite difference scheme [@fatemiJCP93] which amounts to the following discretization of the partial derivatives in Eq. (\[bte\]): $$\begin{aligned} \frac{\partial f}{\partial v} & = & L_{v}^{+[-]}f(x,v)~~E(x)>0~[E(x)\leq 0] \\ \frac{\partial f}{\partial x} & = & L_{x}^{+[-]}f(x,v)~~v<0~[v\geq 0]~. \label{btedifference}\end{aligned}$$ As for the Poisson equation, we use SOR to solve the matrix equation resulting from the discretization of Eq. (\[bte\]). For the boundary conditions of the Poisson-Boltzmann equations we adopt the following: For the potential, the values at the system boundaries, denoted (l)eft and (r)ight are fixed to $\phi(x_{l})=U_{0}$ and $\phi(x_{r})=0$, respectively, corresponding to an externally applied voltage $U_{0}$. The electron density is allowed to fluctuate freely around the boundaries, subject to the condition of global charge neutrality, which is enforced between each successive iteration in the self-consistent Poisson-Boltzmann loop. We choose the size of the highly-doped contacts to be large enough such that the electron density and the electric field deep inside the contacts is constant. For the electron distribution function four boundary conditions can be defined in the 2D phase-space. At the velocity cut-off in phase-space, we choose $f(x,v_{max})=f(x,-v_{max})= f_{LE}(x,v)$ which is reasonable since we assume $v_{max}\geq 30k_{B}T_{0}$ in the calculations. At such high velocities, the electron population is negligible and of the same order as the local equilibrium distribution $f_{LE}(x,v)$. At the contact boundaries, we assume that the electric field is low and constant (as verified in the calculations), and thus, the homogeneous solution to the BTE in the linear response regime of transport applies. Hence, $$f(x_{i},v)=f_{LE}(x_{i},v)[1-vE(x_{i})\tau(\varepsilon)/k_{B}T_{0}],$$ where $i=l,r$. The iterative Poisson-Boltzmann loop consists of an updating procedure for the electric field, electron distribution function and electron density using Eqs. (\[poisson\], \[bte\], \[density\]), until convergence. The convergence criterion is determined and checked in terms of the evolution of the $L_{2}$ norm of the potential and density variations between subsequent iterations. Typically, the results are converged when the $L_{2}$ norms for the potential and density are on the order of $10^{-3}$ of the original values. Between subsequent iterations, we employ linear mixing in the electron density, according to $$n^{\prime}(x)=(1-\alpha) n^{old}(x)+\alpha n^{new}(x)~,$$ where $n^{old}(x)$ is the input density to the Poisson solver, $n^{new}(x)$ is the new density obtained from the solution of the BTE using the new electric field obtained from the Poisson solver, and $n^{\prime}(x)$ is the final density that is used as an input to the next iteration in the Poisson-Boltzmann loop. The convergence and stability of the self-consistent loop are strongly dependent on the system parameters and the nonequilibrium nature of the electronic system. If the system is strongly out of equilibrium, displaying large variations and strengths of the electric field, the mixing parameters $\alpha$ may have to be chosen as small as a few percent, thus affecting the overall runtime. Furthermore, for highly doped structures, the convergence is slower, partly due to the required small grid size in position due to the small Debye length, but also due to the slow convergence in the SOR procedure in the BTE, where the stability of the numerical scheme is given in terms of a Courant-Friedrich-Levy type condition [@numrec]. Still, the computational demands for the calculations reported in this paper are modest. Numerical results ================= In the following, we demonstrate our numerical approach with calculations of the transport characteristics of a model GaAs $n^{+}-n^{-}-n^{+}-n^{-}-n^{+}$ structure with the doping densities $n^{+}$=10$^{23}$ m$^{-3}$ and $n^{-}$=10$^{19}$ m$^{-3}$. In order to highlight the effects of inhomogeneities and scattering while keeping the nature of the scattering structureless, we use a constant scattering time $\tau=2.5\cdot 10^{-13}$ s, which corresponds to realistic mobilities of GaAs at room temperature for which the calculations have been performed. The central $n^{-}-n^{+}-n^{-}$ region has the dimensions 200/200/200 nm, whereas the contacts are 1 $\mu$m long. {width="70.00000%"} In Fig. \[fig1\] we show the electric field and potential energy around the central region of the system described above, subject to an applied bias voltage $V_{b}=-0.5$ V. Due to the charge imbalance, electrons diffuse towards the lightly doped regions, where potential barriers are formed and, correspondingly, a large and inhomogeneous electric field on the order of 10 kV/cm is formed, even in the absence of an external applied voltage. As a finite voltage is applied to the device, the majority of the potential drop occurs over the submicron central region, giving rise to a strongly inhomogeneous field distribution, in contrast to the $n^{+}$ contact regions, where the field in comparison is very low and constant. The electron velocity distribution in the central region of the structure is shown in Fig. \[fig2\](a), for five specific spatial points as depicted in Fig. \[fig1\]. Figure \[fig2\](b) shows a contour plot of the full spatial dependence of $f(x,v)$ in that region. It is clear that the inhomogeneous electric field gives rise to a strong spatial dependence of the velocity distribution function along the direction of transport, and that the distribution function in the central region is very far from thermal equilibrium. In the outermost highly doped $n^{+}$ regions, where the field is low and constant the distribution is simply a shifted Maxwellian. In the lightly doped $n^{-}$ regions on the other hand, the velocity distribution is highly asymmetric and develops a narrow peak that rapidly shifts toward higher velocity along the direction of transport. This peak contains quasi-ballistic electrons which are accelerated by the strong electric field in the central region, and thus, have a considerably larger average velocity compared to the electrons in the contacts. Close to the potential barrier the distribution function is suppressed at low velocities due to the skimming of the distribution of incoming electrons, as well as the restriction of drain induced electron flow with $v<0$ due to the potential barrier. However, the low-velocity contribution to the distribution function gradually increases away from the barrier, as thermionically injected electrons gradually are thermalized and the lower effective barrier height allows electrons from the $n^{+}$ regions to penetrate the lightly-doped region. Thus, the total distribution function consists of a quasi-ballistic, high-velocity and a diffusive, low-velocity contributions, which gives the total distribution function a highly non-Maxwellian broad and asymmetric shape. Furthermore, the presence of the two barriers creates an additional quasi-ballistic structure in the high-energy tail of the distribution function in the second $n^{-}$ region, as electrons that traverse the intermediate $n^{+}$ region ballistically get an additional acceleration toward higher velocities by the electric field in the second $n^{-}$ region, thus creating two high-velocity electron beams. These features emphasize the highly nonequilibrium nature of the electron transport in these type of systems and demonstrate that our method is capable of taking them fully into account. {width="70.00000%"} Conclusions =========== We have presented a numerical method for the solution of the steady-state, coupled Poisson-Boltzmann equations for the study of inhomogeneous, submicron semiconductor structures and demonstrated our approach on a submicron GaAs structure with strong built-in electric fields. We have shown that our method is capable of taking into account the strong nonequilibrium transport properties that arise in such systems due to the presence of very large and inhomogeneous electric fields, and that interesting structure is present in the high-energy tail of the distribution function, caused by quasi-ballistic electrons.  \ [**Acknowledgments**]{}\  \ This work was supported by the Indiana 21st Century Research and Technology Fund. [99]{} C. Jacoboni, and P. Lugli, [*The Monte Carlo Method for Semiconductor Device Simulation*]{} (Springer-Verlag, Wien, 1989). H. U. Baranger, and J. W. Wilkins, “Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends,” [*Physical Review B*]{} [ **36**]{}, 1487 (1987). E. Fatemi, and F. Odeh, “Upwind finite difference solution of Boltzmann Equation applied to electron transport in semiconductor devices”, [*Journal of Computational Physics*]{} [**108**]{}, 209 (1993). A. Majorana, and R. M. Pitadella, “A finite difference scheme solving the Boltzmann-Poisson system for semiconductor devices”, [*Journal of Computational Physics*]{} [ **174**]{}, 649 (2001). J.-H. Rhew [*et al.*]{}, “A numerical study of ballistic transport in nanoscale MOSFET”, [*Solid State Electronics*]{} [ **46**]{}, 1899 (2002). J. A. Carrillo [*et al.*]{}, “A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods”, [*Journal of Computational Physics*]{} [ **184**]{}, 498 (2003). A. Majorana [*et al.*]{}, “Charge transport in 1D silicon devices via Monte Carlo simulation and Boltzmann-Poisson solver”, [*COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering*]{} [ **23**]{}, 410 (2004). W. H. Press [*et al.*]{}, [*Numerical Recipes in C*]{} (Cambridge University Press, Cambridge, 1992).

ArXiv

Ł[[L]{}]{} [$\tilde{\phantom{a}}$]{} [**Bessel Beams**]{}\ Kirk T. McDonald\ [*Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544*]{}\ (June 17, 2000) Problem ======= Deduce the form of a cylindrically symmetric plane electromagnetic wave that propagates in vacuum. A scalar, azimuthally symmetric wave of frequency $\omega$ that propagates in the positive $z$ direction could be written as $$\psi({\bf r},t) = f(\rho) e^{i(k_z z - \omega t)}, \label{eq1}$$ where $\rho = \sqrt{x^2 + y^2}$. Then, the problem is to deduce the form of the radial function $f(\rho)$ and any relevant condition on the wave number $k_z$, and to relate that scalar wave function to a complete solution of Maxwell’s equations. The waveform (\[eq1\]) has both wave velocity and group velocity equal to $\omega / k_z$. Comment on the apparent superluminal character of the wave in case that $k_z < k = \omega / c$, where $c$ is the speed of light. Solution ======== As the desired solution for the radial wave function proves to be a Bessel function, the cylindrical plane waves have come to be called Bessel beams, following their introduction by Durnin [@Durnin1; @Durnin2]. The question of superluminal behavior of Bessel beams has recently been raised by Mugnai [@Mugnai]. Bessel beams are a realization of super-gain antennas [@Schelkunoff; @Bouwkamp; @Yaru] in the optical domain. A simple experiment to generate Bessel beams is described in [@McQueen]. Sections 2.1 and 2.2 present two methods of solution for Bessel beams that satisfy the Helmholtz wave equation. The issue of group and signal velocity for these waves is discussed in sec. 2.3. Forms of Bessel beams that satisfy Maxwell’s equations are given in sec. 2.4. Solution via the Wave Equation ------------------------------ On substituting the form (\[eq1\]) into the wave equation, $$\nabla^2 \psi = { 1 \over c^2} {\partial^2 \psi \over \partial t^2}, \label{eq2}$$ we obtain $${d^2 f \over d\rho^2} + {1 \over \rho} {d f \over d \rho} + (k^2 - k_z^2) f = 0. \label{eq3}$$ This is the differential equation for Bessel functions of order 0, so that $$f(\rho) = J_0(k_r \rho), \label{eq4}$$ where $$k_\rho^2 + k_z^2 = k^2. \label{eq5}$$ The form of eq. (\[eq5\]) suggests that we introduce a (real) parameter $\alpha$ such that $$k_\rho = k \sin \alpha, \qquad \mbox{and} \qquad k_z = k \cos\alpha. \label{eq6}$$ Then, the desired cylindrical plane wave has the form $$\psi({\bf r},t) = J_0(k \sin\alpha \, \rho) e^{i(k \cos\alpha \, z - \omega t)}, \label{eq7}$$ which is commonly called a Bessel beam. The physical significance of parameter $\alpha$, and that of the group velocity $$v_g = {d \omega \over d k_z} = {\omega \over k_z} = v_p = {c \over \cos\alpha} \label{eq8}$$ will be discussed in sec. 2.3. While eq. (\[eq7\]) is a solution of the Helmholtz wave equation (\[eq2\]), assigning $\psi({\bf r},t)$ to be a single component of an electric field, say $E_x$, does not provide a full solution to Maxwell’s equations. For example, if ${\bf E} = \psi \hat{\bf x}$, then $\nabla \cdot {\bf E} = \partial \psi / \partial x \neq 0$. Bessel beams that satisfy Maxwell’s equations are given in sec. 2.4. Solution via Scalar Diffraction Theory -------------------------------------- The Bessel beam (\[eq7\]) has large amplitude only for $\abs{\rho} \lsim 1/ k \sin\alpha$, and maintains the same radial profile over arbitrarily large propagation distance $z$. This behavior appears to contradict the usual lore that a beam of minimum transverse extent $a$ diffracts to fill a cone of angle $1/a$. Therefore, the Bessel beam (\[eq7\]) has been called “diffraction free” [@Durnin2]. Here, we show that the Bessel beam does obey the formal laws of diffraction, and can be deduced from scalar diffraction theory. According to that theory [@Jackson], a cylindrically symmetric wave $f(\rho)$ of frequency $\omega$ at the plane $z = 0$ propagates to point [**r**]{} with amplitude $$\psi({\bf r},t) = {k \over 2 \pi i} \int \int \rho' d\rho' d\phi f(\rho') {e^{i(k R - \omega t)} \over R}, \label{eq9}$$ where $R$ is the distance between the source and observation point. Defining the observation point to be $(\rho,0,z)$, we have $$R^2 =z^2 + \rho^2 + \rho^{'2} - 2 \rho \rho' \cos\phi, \label{eq10}$$ so that for large $z$, $$R \approx z + {\rho^2 + \rho^{'2} - 2 \rho \rho' \cos\phi \over 2 z}. \label{eq11}$$ In the present case, we desire the amplitude to have form (\[eq1\]). As usual, we approximate $R$ by $z$ in the denominator of eq. (\[eq9\]), while using approximation (\[eq11\]) in the exponential factor. This leads to the integral equation $$\begin{aligned} f(\rho) e^{i k_z z} & = & {k \over 2 \pi i} {e^{ik z} e^{i k \rho^2 / 2 z} \over z} \int_0^\infty \rho' d\rho' f(\rho') e^{i k \rho^{'2} / 2z} \int_0^{2 \pi} d\phi e^{-i k \rho \rho' \cos\phi / z} \nonumber \\ & = & {k \over i} {e^{ik z} e^{i k \rho^2 / 2 z} \over z} \int_0^\infty \rho' d\rho' f(\rho') J_0(k \rho \rho' / z) e^{i k \rho^{'2} / 2z}, \label{eq12}\end{aligned}$$ using a well-known integral representation of the Bessel function $J_0$. It is now plausible that the desired eigenfunction $f(\rho)$ is a Bessel function, say $J_0(k_\rho \rho)$, and on consulting a table of integrals of Bessel functions we find an appropriate relation [@Gradshteyn], $$\int_0^\infty \rho' d\rho' J_0(k_{\rho} \rho') J_0(k \rho \rho' / z) e^{i k \rho^{'2} / 2z} = {i z \over k} e^{-i k \rho^2 / 2 z} e^{- i k_\rho^2 z / 2 k} J_0(k_\rho \rho). \label{eq13}$$ Comparing this with eq. (\[eq12\]), we see that $f(\rho) = J_0(k_\rho \rho)$ is indeed an eigenfunction provided that $$k_z = k - {k_\rho^2 \over 2 k}. \label{eq14}$$ Thus, if we write $k_\rho = k \sin\alpha$, then for small $\alpha$, $$k_z \approx k (1 - \alpha^2 / 2) \approx k \cos\alpha, \label{eq15}$$ and the desired cylindrical wave again has form (\[eq7\]). Strictly speaking, the scalar diffraction theory reproduces the “exact” result (\[eq7\]) only for small $\alpha$. But the scalar diffraction theory is only an approximation, and we predict with confidence that an “exact” diffraction theory would lead to the form (\[eq7\]) for all values of parameter $\alpha$. That is, “diffraction-free” beams are predicted within diffraction theory. It remains that the theory of diffraction predicts that an infinite aperture is needed to produce a beam whose transverse profile is invariant with longitudinal distance. That a Bessel beam is no exception to this rule is reviewed in sec. 2.3. The results of this section were inspired by [@Jiang]. One of the first solutions for Gaussian laser beams was based on scalar diffraction theory cast as an eigenfunction problem [@Boyd]. Superluminal Behavior --------------------- In general, the group velocity (\[eq8\]) of a Bessel beam exceeds the speed of light. However, this apparently superluminal behavior cannot be used to transmit signals faster than lightspeed. An important step towards understanding this comes from the interpretation of parameter $\alpha$ as the angle with respect to the $z$ axis of the wave vectors of an infinite set of ordinary plane waves whose superposition yields the Bessel beam [@Eberly]. To see this, we invoke the integral representation of the Bessel function to write eq. (\[eq7\]) as $$\begin{aligned} \psi({\bf r},t) & = &J_0(k \sin\alpha \, \rho) e^{i(k \cos\alpha \, z - \omega t)} \nonumber \\ & = & {1 \over 2 \pi} \int_0^{2 \pi} d \phi e^{i(k \sin\alpha \, x \cos\phi + k \sin\alpha \, y \sin\phi + k \cos\alpha \, z - \omega t)} \label{eq16} \\ & = & {1 \over 2 \pi} \int_0^{2 \pi} d \phi e^{i({\bf q} \cdot {\bf r} - \omega t)}, \nonumber\end{aligned}$$ where the wave vector [**q**]{}, given by $${\bf q} = k (\sin\alpha \cos\phi, \sin\alpha \sin\phi, \cos\alpha), \label{eq17}$$ makes angle $\alpha$ to the $z$ axis as claimed. We now see that a Bessel beam is rather simple to produce in principle [@Durnin2]. Just superpose all possible plane waves with equal amplitude and a common phase that make angle $\alpha$ to the $z$ axis, According to this prescription, we expect the $z$ axis to be uniformly illuminated by the Bessel beam. If that beam is created at the plane $z = 0$, then any annulus of equal radial extent in that plane must project equal power into the beam. For large $\rho$ this is readily confirmed by noting that $J_0^2(k \sin\alpha\, \rho) \approx \cos^2(k \sin\alpha\, \rho + \delta)/ (k \sin\alpha\, \rho)$, so the integral of the power over an annulus of one radial period, $\Delta \rho = \pi / (k \sin\alpha)$, is independent of radius. Thus, from an energy perspective a Bessel beam is not confined to a finite region about the $z$ axis. If the beam is to propagate a distance $z$ from the plane $z = 0$, it must have radial extent of at least $ \rho = z \tan\alpha$ at $z = 0$. An arbitrarily large initial aperture, and arbitrarily large power, is required to generate a Bessel beam that retains its “diffraction-free” character over an arbitrarily large distance. Each of the plane waves that makes up the Bessel beam propagates with velocity $c$ along a ray that makes angle $\alpha$ to the $z$ axis. The intersection of the $z$ axis and a plane of constant phase of any of these wave moves forward with superluminal speed $c / \cos\alpha$, which is equal to the phase and group velocities (\[eq8\]). This superluminal behavior does not represent any violation of special relativity, but is an example of the “scissors paradox" that the point of contact of a pair of scissors could move faster than the speed of light while the tips of the blades are moving together at sublightspeed. A ray of sunlight that makes angle $\alpha$ to the surface of the Earth similarly leads to a superluminal velocity $c / \cos\alpha$ of the point of contact of a wave front with the Earth. However, we immediately see that a Bessel beam could not be used to send a signal from, say, the origin, $(0,0,0)$, to a point $(0,0,z)$ at a speed faster than light. A Bessel beam at $(0,0,z)$ is made of rays of plane waves that intersect the plane $z = 0$ at radius $\rho = z \tan\alpha$. Hence, to deliver a message from $(0,0,0)$ to $(0,0,z)$ via a Bessel beam, the information must first propagate from the origin out to at least radius $\rho = z \tan\alpha$ at $z = 0$ to set up the beam. Then, the rays must propagate distance $z/\cos\alpha$ to reach point $z$ with the message. The total distance traveled by the information is thus $z(1 + \sin\alpha)/\cos\alpha$, and the signal velocity $v_s$ is given by $$v_s \approx c {\cos\alpha \over 1 + \sin\alpha}, \label{eq18}$$ which is always less than $c$. The group velocity and signal velocity for a Bessel beam are very different. Rather than being a superluminal carrier of information at its group velocity $c / \cos\alpha$, a modulated Bessel beam could be used to deliver messages only at speeds well below that of light. Solution via the Vector Potential --------------------------------- To deduce all components of the electric and magnetic fields of a Bessel beam that satisfies Maxwell’s equation starting from a single scalar wave function, we follow the suggestion of Davis [@Davis] and seek solutions for a vector potential [**A**]{} that has only a single component. We work in the Lorentz gauge (and Gaussian units), so that the scalar potential $\Phi$ is related by $$\nabla \cdot {\bf A} + {1 \over c} {\partial \Phi \over \partial t} = 0. \label{e1}$$ The vector potential can therefore have a nonzero divergence, which permits solutions having only a single component. Of course, the electric and magnetic fields can be deduced from the potentials via $${\bf E} = - \nabla \Phi - {1 \over c} {\partial {\bf A} \over \partial t}, \label{e2}$$ and $${\bf B} = \nabla \times {\bf A}. \label{e3}$$ For this, the scalar potential must first be deduced from the vector potential using the Lorentz condition (\[e1\]). We consider waves of frequency $\omega$ and time dependence of the form $e^{-i \omega t}$, so that $\partial \Phi / \partial t = - i k \Phi$. Then, the Lorentz condition yields $$\Phi = - {i \over k} \nabla \cdot {\bf A}, \label{e4}$$ and the electric field is given by $${\bf E} = ik \left[ {\bf A} + {1 \over k^2} {\bf \nabla} ({\bf \nabla} \cdot {\bf A}) \right]. \label{e5}$$ Then, $\nabla \cdot {\bf E} = 0$ since $\nabla^2 (\nabla \cdot {\bf A}) + k^2 (\nabla \cdot {\bf A}) = 0$ for a vector potential [**A**]{} of frequency $\omega$ that satifies the wave equation (\[eq2\]), We already have a scalar solution (\[eq7\]) to the wave equation, which we now interpret as the only nonzero component, $A_j$, of the vector potential for a Bessel beam that propagates in the $+z$ direction, $$A_j({\bf r},t) = \psi({\bf r},t) \propto J_0(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e6}$$ We consider five choices for the meaning of index $j$, namely $x$, $y$, $z$, $\rho$, and $\phi$, which lead to five types of Bessel beams. Of these, only the case of $j = z$ corresponds to physical, azimuthally symmetric fields, and so perhaps should be called the Bessel beam. ### $j = x$ In this case, $$\nabla \cdot {\bf A} = {\partial \psi \over \partial x} = - {k \sin\alpha \, x \over \rho} J_1(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e7}$$ In calculating $\nabla(\nabla \cdot {\bf A})$ we use the identity $J_1' = (J_0 - J_2)/2$. Also, we divide [**E**]{} and [**B**]{} by the factor $ik$ to present the results in a simpler form. We find, $$\begin{aligned} E_x & = & \left\{ J_0(\varrho) - {\sin^2\alpha\ \over \rho^2} \left[ {y^2 J_1(\varrho) \over \varrho} - {x^2 \over 2} \left(J_0(\varrho) - J_2(\varrho) \right) \right] \right\} e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_y & = & {\sin^2\alpha\, x y \over \rho^2} \left[ { J_1(\varrho) \over \varrho} - {1 \over 2} \left(J_0(\varrho) - J_2(\varrho) \right) \right] e^{i(k \cos\alpha\, z - \omega t)}, \label{e8} \\ E_z & = & - i \sin 2\alpha {x \over 2 \rho} J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ where $$\varrho \equiv k \sin\alpha\, \rho, \label{e9}$$ and $$\begin{aligned} B_x & = & 0, \nonumber \\ B_y & = & \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e10} \\ B_z & = & - i \sin\alpha {x \over \rho} J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}. \nonumber\end{aligned}$$ A Bessel beam that obeys Maxwell’s equations and has purely $x$ polarization of its electric field on the $z$ axis includes nonzero $y$ and $z$ polarization at points off that axis, and does not exhibit the azimuthal symmetry of the underlying vector potential. ### $j = y$ This case is very similar to that of $j = x$. ### $j = z$ In this case the electric and magnet fields retain azimuthal symmetry, so that it is convenient to display the $\rho$, $\phi$ and $z$ components of the fields. First, $$\nabla \cdot {\bf A} = {\partial \psi \over \partial z} = i k \cos\alpha \, J_0(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e11}$$ Then, we divide the electric and magnetic fields by $k \sin\alpha$ to find the relatively simple forms: $$\begin{aligned} E_\rho & = & \cos\alpha\, J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_\phi & = & 0, \label{e12} \\ E_z & = & i \sin\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & 0, \nonumber \\ B_\phi & = & J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e13} \\ B_z & = & 0. \nonumber\end{aligned}$$ This Bessel beam is a transverse magnetic (TM) wave. The radial electric field $E_\rho$ vanishes on the $z$ axis (as it must if that axis is charge free), while the longitudinal electric field $E_z$ is maximal there. Cylindrically symmetric waves with radial electric polarization are often called axicon beams [@McLeod]. ### $j = \rho$ In this case, $$\nabla \cdot {\bf A} = {1 \over \rho} {\partial \rho \psi \over \partial \rho} = \left[ {J_0(k \sin\alpha\, \rho) \over \rho} - k \sin\alpha\, J_1(k \sin\alpha\, \rho) \right] e^{i(k \cos\alpha\, z - \omega t)}. \label{e14}$$ After dividing by $ik$, the electric and magnetic fields are $$\begin{aligned} E_\rho & = & \left\{ J_0(\varrho) - \sin^2\alpha \left[ {J_0(\varrho) \over \varrho^2} + {J_1(\varrho) \over \varrho} + {1 \over 2} (J_0(\varrho - J_2(\varrho)) \right] \right\} e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_\phi & = & 0, \label{e15} \\ E_z & = & i \cos\alpha \sin\alpha \left[ {J_0(\varrho) \over \varrho} - J_1(\varrho) \right] e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & 0, \nonumber \\ B_\phi & = & \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e16} \\ B_z & = & 0. \nonumber\end{aligned}$$ The radial electric field diverges as $1 / \rho^2$ for small $\rho$, so this case is unphysical. ### $j = \phi$ Here, $$\nabla \cdot {\bf A} = {1 \over \rho} {\partial \psi \over \partial \phi} = 0. \label{e17}$$ After dividing by $ik$, the electric and magnetic fields are $$\begin{aligned} E_\rho & = & 0, \nonumber \\ E_\phi & = & J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e18} \\ E_z & = & 0, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & - \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ B_\phi & = & 0, \label{e19} \\ B_z & = & - i \sin\alpha\ \left[ {J_0(\varrho) \over \varrho} - J_1(\varrho) \right] e^{i(k \cos\alpha\, z - \omega t)}. \nonumber\end{aligned}$$ These fields are unphysical due to the finite value of $E_\phi$ at $\rho = 0$, and the divergence of $B_z$ as $\rho \to 0$. [99]{} J. Durnin, [*Exact solutions for nondiffracting beams. I. The scalar theory*]{}, J. Opt. Soc. Am. [**A**]{} 4, 651-654 (1987). J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, [*Diffraction-free beams*]{}, Phys. Rev. Lett. [**778**]{},1499-1501 (1987). D. Mugnai, A. Ranfagni and R. Ruggeri, [*Observation of Superluminal Behavior in Wave Propagation*]{}, Phys. Rev. Lett. [**84**]{}, 4830-4833 (2000). S.A. Schelkunoff, [*A methematical theory of linear arrays*]{}, Bell. Sys. Tech. J. [**22**]{}, 80-107 (1943). C.J. Bouwkamp and N.G. deBruijn, [*The problem of optimum antenna current distribution*]{}, Philips Res. Rep. [**1**]{}, 135-158 (1946). N. Yaru, [*A Note on Super-Gain Antenna Arrays*]{}, Proc. I.R.E. [**39**]{}, 1081-1085 (1951). C.A. McQueen, J. Arlt and K. Dholkia, [*An experiment to study a “nondiffracting" light beam*]{}, Am. J. Phys. [**67**]{}, 912-915 (1999). J.D. Jackson, [*Classical Electrodynamics*]{}, 3d ed.(Wiley, New York, 1999). I.S. Gradshteyn and I.M. Ryzhik, [*Table of Integrals, Series, and Products*]{}, 5th ed.(Academic Press, San Diego, 1994), integral 6.633.2. A. Zhiping, Q. Lu and Z. Liu, [*Propagation of apertured Bessel beams*]{}, Appl. Opt. [**34**]{}, 7183-7185 (1995). G.D. Boyd and J.P. Gordon, [*Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers*]{}, Bell Sys. Tech. J. [**40**]{}, 489-509 (1961). P.W. Milonni and J.H. Eberly, [*Lasers*]{} (Wiley Interscience, New York, 1988), sec. 14.14. L.W. Davis, [*Theory of electromagnetic beams*]{}, Phys. Rev. A [**19**]{}, 1177-1179 (1979). J.H. McLeod, [*The Axicon: A New Type of Optical Element*]{}, J. Opt. Soc. Am. [**44**]{}, 592-597 (1954).

ArXiv

--- author: - 'Jan-e Alam' title: In search of quark gluon plasma in nuclear collisions --- [**Abstract**]{}\ At high temperatures and densities the nuclear matter undergoes a phase transition to a new state of matter called quark gluon plasma (QGP). This new state of matter which existed in the universe after a few microsecond of the big bang can be created in the laboratory by colliding two nuclei at relativistic energies. In this presentation we will discuss how the the properties of QGP can be extracted by analyzing the spectra of photons, dileptons and heavy flavours produced in nuclear collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies. Introduction ============ The theory of strong interaction - Quantum Chromodynamics (QCD) has a unique feature - it possess the property of asymptotic freedom which implies that at very high temperatures and/or densities nuclear matter will convert to a deconfined state of quarks and gluons [@collins]. Recent lattice QCD based calculations [@lqcd] indicate that the value of the temperature for the nuclear matter to QGP transition $\sim 175$ MeV. It is expected that such high temperature can be achieved in the laboratory by colliding nuclei at RHIC and LHC energies. A high multiplicity system of deconfined quarks and gluons with power law type of momentum distributions can created just after the nuclear collisions at high energies. Interactions among these constituents may alter the momentum distribution of quarks and gluons from a power law to an exponential one - resulting in a thermalized state of quarks and gluons with initial temperature, $T_i$. This thermalized system with high internal pressure expands very fast as a consequence it cools and reverts to hadronic matter at a temperature, $T_c\sim 175$ MeV. The hadrons formed after the hadronization of quarks may maintain thermal equilibrium among themselves until the expanding system becomes too dilute to support collectivity at a temperature, $T_F (\sim 120$ MeV) called freeze-out temperature from where the hadrons fly freely from the interaction zone to the detector. The electromagnetic (EM) probes [@mclerran] (see  [@rapp; @alam1; @alam2] for review) [*i.e.*]{} real photons and dileptons can be used to follow the evolution of the system from the pristine partonic stage to the final hadronic stage through an intermediary phase transition or cross over. In the state of QGP some of the symmetries of the physical vacuum may either be restored or broken - albeit transiently. The electromagnetic probes, especially the lepton pairs can be used very effectively to investigate whether these symmetries in the system are restored/broken at any stage of the evolving matter. Results from theoretical calculations will be shown in the presentation to demonstrate this aspect of the electromagnetically interacting probes. We will demonstrate that lepton pairs can be used very effectively to probe the collective motion (radial and elliptic) of the system. The other promising probe of the QGP that will be discussed here - is the depletion of the transverse momentum spectra of energetic quarks (and gluons) in QGP. The magnitude of the depletion can be used to estimate the transport coefficients of QGP which is turn can be used to understand the fluidity of the matter. The transport coefficients of QGP and hot hadrons calculated by using perturbative QCD and effective field theory respectively have been applied to evaluate the nuclear suppression ($R_{\mathrm AA}$) of heavy flavours. Theoretical results on $R_{\mathrm AA}$ will be compared with the experimental data available from RHIC and LHC energies. The azimuthal asymmetry of the system estimated through the single leptons originating from the decays of open heavy flavours produced from the fragmentation of heavy quarks will also be discussed. The electromagnetic probes ========================== The dilepton production per unit four-volume from a thermal medium produced in heavy ion collisions is well known to be given by: &=&- L(M\^2)f\_[BE]{}(p\_0) g\^\ &&W\_(p\_0,[p]{}) \[eq1\] where the factor $L(M^2)=(1+{2m_l^2}/{M^2})~ (1-4m_l^2/M^2)^{1/2}~$ is of the order of unity for electrons, $M(=\sqrt{p^2})$ being the invariant mass of the pair and the hadronic tensor $W_{\mu\nu}$ is defined by W\_(p\_0,[p]{})=d\^4xe\^[ipx]{}\[eq2\] where $J^{em}_\mu(x)$ is the electromagnetic current and $\langle.\rangle$ indicates ensemble average. For a deconfined thermal medium such as the QGP, Eq. (\[eq1\]) leads to the standard rate for lepton pair productions from $q\bar q$ annihilation at lowest order. The production of low mass dileptons from the decays of light vector mesons in the hadronic matter can be obtained as (see  [@sabya] for details): &=&- f\_[BE]{}(p\_0) g\^\ &&\_[R=,,]{}K\_R \^R\_(p\_0,[p]{}) \[eq2\] where $f_{BE}$ is the thermal distributions for bosons, $\rho^R_{{\mu\nu}}(q_0,{\vec q})$ is the spectral function of the vector meson $R (=\rho,\omega,\phi)$ in the medium, $K_R=F_R^2 m_R^2$, $m_R$ is the mass of $R$ and $F_R$ is related to the decay of $R$ to lepton pairs. The interaction of the vector mesons with the hadrons in the thermal bath will shift the location of both the pole and the branch cuts of the spectral function - resulting in mass modification or broadening - which can be detected through the dilepton measurements and may be connected with the restoration of chiral symmetry in the thermal bath. In the present work the interaction of $\rho$ with thermal $\pi$,$\omega$, $a_1$, $h_1$ [@sabya; @sabya2] and nucleons  [@ellis] have been considered to evaluate the in-medium spectral function of $\rho$. The finite temperature width of the $\omega$ spectral function has been taken from  [@Weise]. To evaluate the dilepton yield from a dynamically evolving system produced in heavy ion collisions (HIC) one needs to integrate the fixed temperature production rate given by Eq. \[eq2\] over the space time evolution of the system - from the initial QGP phase to the final hadronic freeze-out state through a phase transition in the intermediate stage. We assume that the matter is formed in QGP phase with zero net baryon density at temperature $T_i$ in HIC. Ideal relativistic hydrodynamics with boost invariance [@bjorken] has been applied to study the evolution of the system. The EoS required to close the hydrodynamic equations is constructed by taking results from lattice QCD for high $T$  [@lqcd] and hadron resonance gas comprising of all the hadronic resonances up to mass of $2.5$ GeV  [@victor; @bmja] for lower $T$. The system is assumed to get out of chemical equilibrium at $T=T_{ch}=170$ MeV [@tsuda]. The kinetic freeze-out temperature $T_{F}=120$ MeV fixed from the $p_T$ spectra of the produced hadrons. Invariant mass spectra of lepton pairs -------------------------------------- The $M$ distribution of lepton pairs originating from quark matter (QM) and hadronic matter (HM) with and without medium effects on the spectral functions of $\rho$ and $\omega$ are displayed in Fig. \[fig1\]. We observe that for $M\,>\,M_{\phi}$ the QM contributions dominate. For $M_{\rho}\lesssim M\lesssim M_{\phi}$ the HM shines brighter than QM. For $M\,<M_{\rho}$, the HM (solid line) over shines the QM due to the enhanced contributions primarily from the medium induced broadening of $\rho$ spectral function. However, the contributions from QM and HM become comparable in this $M$ region if the medium effects on $\rho$ spectral function is ignored (dotted line). Therefore, the results depicted in Fig. \[fig1\] indicate that a suitable choice of $M$ window will enable us to unravel the contributions from a particular phase (QM or HM). An appropriate choice of $M$ window will also allow us to extract the medium induced effects. To further quantify these points we evaluate the following [@payalv2]: $$\begin{aligned} &&F= \frac{\int^\prime \left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2} {\int\left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2}\nn \label{eq3}\end{aligned}$$ where the $M$ integration in both the numerator and denominator are performed for selective windows from $M_1$ to $M_2$ with mean $M$ defined as $\langle M\rangle = (M_1+M_2)/2$. While in the denominator the integration is done over the entire lifetime, the prime in $\int^\prime$ in the numerator indicates that the $\tau$ integration in the numerator is done from $\tau_1=\tau_i$ to $\tau_2=\tau_i+\Delta\tau$ with incremental $\Delta\tau$ until $\tau_2$ attains the life time of the system. In the inset of Fig. \[fig1\] $F$ is plotted against $\tau_{\mathrm av} (=(\tau_1+\tau_2)/2)$. The results substantiate that pairs with high $\langle M\rangle\sim 2.5$ GeV originate from early time ($\tau_{\mathrm av}\lesssim 3$ fm/c, QGP phase) and pairs with $\langle M\rangle\sim 0.77$ GeV mostly emanate from late hadronic phase ($\tau_{\mathrm av}\geq 4$ fm/c). The change in the properties of $\rho$ due to its interaction with thermal hadrons in the bath is also visible through $F$ evaluated for $\langle M\rangle\sim 0.3$ GeV with and without medium effects. ![Invariant mass distribution of dileptons from hadronic matter (HM) for modified and unmodified $\rho$ meson. []{data-label="fig1"}](fig1.eps) ![\[a\] and \[b\] indicate elliptic flow of lepton pairs as a function of $p_T$ for various $M$ windows. \[c\] displays the effect of the broadening of $\rho$ spectral function on the elliptic flow for $\langle M\rangle = 300$ MeV. \[d\] shows the variation of $R$ (see text) with $p_T$ for $\langle M\rangle=0.3$ GeV, 0.77 GeV and 2.5 GeV. []{data-label="fig2"}](fig2.eps) ![Variation of dilepton elliptic flow as function of $\langle M\rangle$ for QM, HM (with and without medium effects) and for the entire evolution. The inset shows the variation of momentum space anisotropy with proper time.[]{data-label="fig3"}](fig3.eps) ![The dilepton yield plotted against $M_T-M_{av}$ for different $M$ windows for LHC initial condition.[]{data-label="fig4"}](fig4.eps) Elliptic flow of lepton pairs ----------------------------- The elliptic flow of dilepton, $v_2(p_T,M)$ can be defined as: $$\begin{aligned} &&v_2= \frac{\sum\int cos(2\phi) \left(\frac{dN}{d^2p_TdM^2dy}\arrowvert_{y=0}\right) d\phi} {\sum\int\left(\frac{dN}{d^2p_TdM^2dy}\arrowvert_{y=0}\right)d\phi } \label{eqv2}\end{aligned}$$ where the $\sum$ stands for summation over QM and HM phases. Fig. \[fig2\] (\[a\] and \[b\]) show the differential elliptic flow, $v_2(p_T)$ of dileptons arising from various $\langle M\rangle$ domains. We observe that for $\langle M\rangle=2.5$ GeV $v_2$ is small for the entire $p_T$ range because these pairs arise from the early epoch (see inset of Fig. \[fig1\]) when the flow is not developed entirely. However, the $v_2$ is large for $\langle M\rangle=0.77$ GeV as these pairs originate predominantly from the late hadronic phase when the flow is fully developed. It is also interesting to note that the medium induced enhancement of $\rho$ spectral function provides a visible modification in $v_2$ for dileptons below $\rho$ peak (Fig. \[fig2\] \[c\]). The medium-induced effects lead to an enhancement of $v_2$ of lepton pairs which is culminating from the ‘extra’ interaction (absent when a vacuum $\rho$ is considered) of the $\rho$ with other thermal hadrons in the bath. In Fig. \[fig2\] \[d\] we depict the variation of $R$ with $p_T$ for $\langle M\rangle=0.3$ GeV (solid circle) 0.77 GeV (solid line) and 2.5 GeV (open circle), the quantity $R$ is defined as $R=v_2^{\mathrm QM}/(v_2^{\mathrm QM}+v_2^{\mathrm HM})$ where $v_2^{\mathrm i}$ is the elliptic flow of the phase $i(=QM+HM$. The results clearly illustrate that $v_2$ of lepton pairs in the large $\langle M\rangle$ domain originate from QM. Fig. \[fig3\] shows $p_T$ integrated elliptic flow, $v_2(\langle M\rangle)$ evaluated for different $\langle M\rangle$ windows defined above. The $v_2$ (which is proportional to momentum space anisotropy, $\epsilon_p$) of QM is small because the pressure gradient is not fully developed in the QGP phase as evident from the inset plot of $\epsilon_p$ with $\tau$. The hadronic phase $v_2$ has a peak around $\rho$ pole indicating large flow at late times. For $\langle M\rangle\>>\, m_\phi$ the $v_2$ obtained from the combined phases approach the value corresponding to the $v_2$ for QGP. Therefore, measurement of $v_2$ for large $\langle M\rangle$ will bring information of the properties of the QGP. It is important to note that the $p_T$ integrated $v_2(\langle M\rangle)$ of lepton pairs with $\langle M\rangle\,\sim m_\pi, m_K$ is close to the hadronic $v_2^\pi$ and $v_2^K$ if the thermal effects on $\rho$ properties are included. Exclusion of medium effects give lower $v_2$ for lepton pairs compared to hadrons. We also observe that the variation of $v_2(\langle M\rangle)$ with $\langle M\rangle$ has a structure similar to $dN/dM$ vs $M$. As indicated by Eq. \[eq1\] we can write $v_2(\langle M\rangle)\sim \sum v_2^{\mathrm i}\times f_{\mathrm i}$. The structure of $dN/dM$ is reflected in $v_2(\langle M\rangle)$ through $f_i$. Radial flow of dileptons ------------------------ The transverse mass distributions of the lepton pairs at LHC is displayed in Fig. \[fig4\]. The variation of inverse slope (deduced from the from the transverse mass distributions of lepton pairs, Fig. \[fig4\]) with $\langle M\rangle$ for LHC is depicted in Fig. \[fig5\]. The radial flow in the system is responsible for the rise and fall of $T_{\mathrm eff}$ with $\langle M\rangle$ (solid line) in the mass region ($0.5<$ M(GeV)$<1.3$), for $v_T=0$ (dashed line) a completely different behaviour is obtained. ![$T_{eff}$ for different values of the $M$-bins for LHC conditions. The dashed line is obtained by setting $v_T=0$.[]{data-label="fig5"}](fig5.eps) ![$R_{side}$ and $R_{out}$ as a function of $\langle M \rangle$. The dashed, dotted and the solid line (with asterisk) indicate the HBT radii for the QGP, hadronic and total dilepton contributions from all the phases respectively. The solid circles are obtained by switching off the contributions from $\rho$ and $\omega$. []{data-label="fig6"}](fig6.eps) Radial flow from HBT interferometry of lepton pairs --------------------------------------------------- It was shown in Ref. [@payal] that the variation of HBT radii ($R_{side}$ and $R_{out}$) extracted from the correlation of dilepton pairs with $\langle M\rangle$ can used to extract collective properties of the evolving QGP. While the radius ($R_{\mathrm side}$) corresponding to $q_{side}$ is closely related to the transverse size of the system and considerably affected by the collectivity, the radius ($R_{\mathrm out}$) corresponding to $q_{out}$ measures both the transverse size and duration of particle emission. The extracted $R_{\mathrm side}$ and $R_{\mathrm out}$ for different $\langle M\rangle$ are shown in Fig. \[fig6\]. The $R_{\mathrm side}$ shows non-monotonic dependence on $M$, starting from a value close to QGP value (indicated by the dashed line) it drops with increase in $M$ finally again approaching the QGP value for $\langle M\rangle \,>\,m_\phi$. It can be shown that $R_{side}\sim 1/(1+E_{\mathrm collective}/E_{\mathrm thermal})$. In the absence of radial flow, $R_{\mathrm side}$ is independent of $q_{\mathrm side}$. With the radial expansion of the system a rarefaction wave moves toward the center of the cylindrical geometry as a consequence the radial size of the emission zone decreases with time. Therefore, the size of the emission zone is larger at early times and smaller at late time. The high $\langle M\rangle$ regions are dominated by the early partonic phase where the collective flow has not been developed fully [*i.e.*]{} the ratio of collective to thermal energy is small hence show larger $R_{\mathrm side}$ for the source. In contrast, the lepton pairs with $M\sim m_\rho$ are emitted from the late hadronic phase where the size of the emission zone is smaller due to larger collective flow giving rise to a smaller $R_{\mathrm side}$. The ratio of collective to thermal energy for such cases is quite large, which is reflected as a dip in the variation of $R_{\mathrm side}$ with $\langle M\rangle$ around the $\rho$-mass region (Fig. \[fig6\] upper panel). Thus the variation of $R_{\mathrm side}$ with $M$ can be used as an efficient tool to measure the collectivity in various phases of matter. The dip in $R_{\mathrm side}$ at $\langle M\rangle\sim m_\rho$ is due to the contribution dominantly from the hadronic phase. The dip, in fact vanishes if the contributions from $\rho$ and $\omega$ is switched off (circle in Fig. \[fig6\]). We observe that by keeping the $\rho$ and $\omega$ contributions and setting radial velocity, $v_r=0$, the dip in $R_{\mathrm side}$ vanishes, confirming the fact that the dip is caused by the radial flow of the hadronic matter. Therefore, the value of $R_{\mathrm side}$ at $\langle M\rangle\sim m_\rho$ may be used to estimate the average $v_r$ in the hadronic phase. The $R_{\mathrm out}$ probes both the transverse dimension and the duration of emission as a consequence unlike $R_{\mathrm side}$ it does not remain constant even in the absence of radial flow and its variation with $M$ is complicated. The large $M$ regions are populated by lepton pairs from early partonic phase where the effect of flow is small and the duration of emission is also small - resulting in smaller values of $R_{\mathrm out}$. For lepton pair from $M\sim m_\rho$ the flow is large which could have resulted in a dip as in $R_{\mathrm side}$ in this $M$ region. However, $R_{\mathrm out}$ probes the duration of emission too which is large for hadronic phase. The larger duration compensates the reduction of $R_{\mathrm out}$ due to flow in the hadronic phase resulting in a bump in $R_{\mathrm out}$ for $M\sim m_\rho$ (Fig. \[fig6\] lower panel). Both $R_{\mathrm side}$ and $R_{\mathrm out}$ approach QGP values for $\langle M\rangle\sim 2.5$ GeV implying dominant contributions from partonic phase. ![$R_{AA}$ as a function of $p_T$ for $D$ and $B$ mesons at LHC. Experimental data taken from  [@ALICE].[]{data-label="fig7"}](fig7.eps) Suppression of heavy flavours in QGP ==================================== The depletion of hadrons with high transverse momentum ($p_T$) produced in Nucleus + Nucleus collisions with respect to those produced in proton + proton (pp) collisions has been considered as a signature of QGP formation. The two main processes which cause the depletion are (i) the elastic collisions and (ii) the radiative loss or the inelastic collisions of the high energy partons with the quarks, anti-quarks and gluons in the thermal bath. In the present work we focus on the energy loss of heavy quarks in QGP in deducing the properties of the medium. Because (i) the abundance of charm and bottom quarks in the partonic plasma for the expected range of temperature to be attained in the experiments is small, consequently the bulk properties of the plasma is not decided by them and (ii) they produce early and therefore, can witness the entire evolution history. Hence heavy quarks may act as an efficient probe for the diagnosis of QGP. The depletion of heavy quarks in QGP has gained importance recently in view of the measured nuclear suppression in the $p_T$ spectra of non-photonic single electrons [@stare; @phenixe]. We assume here that the light quarks and gluons thermalize before heavy quarks. The charm and bottom quarks execute Brownian motion [@we1] (see references therein) in the heat bath of QGP. Therefore, the interaction of the heavy quarks with QGP may be treated as the interactions between equilibrium and non-equilibrium degrees of freedom. The Fokker-Planck (FP) equation provide an appropriate framework for the evolution of the heavy quark in the expanding QGP heat bath which can be written as [@we1]: &=&C\_[1]{}(p\_[x]{},p\_[y]{},t) +C\_[2]{}(p\_[x]{},p\_[y]{},t)\ &+& C\_[3]{}(p\_[x]{},p\_[y]{},t) +C\_[4]{}(p\_[x]{},p\_[y]{},t)\ &+& C\_[5]{}(p\_[x]{},p\_[y]{},t)f +C\_[6]{}(p\_[x]{},p\_[y]{},t). \[fpeqcartesian\]  . where, C\_[1]{}&=& D\ C\_[2]{}&=& D\ C\_[3]{}&=&  p\_[x]{} +2  \ C\_[4]{}&=&  p\_[y]{} +2  \ C\_[5]{}&=& 2  +  +  \ C\_[6]{}&=& 0  . where the momentum, $\textbf{p}=(\textbf{p}_T,p_z)=(p_x,p_y,p_z)$, $\gamma$ is the drag coefficient and $D$ is the diffusion coefficient. We numerically solve Eq. \[fpeqcartesian\] [@antia] with the boundary conditions: $f(p_x,p_y,t)\ra 0$ for $p_x$,$p_y\ra \infty$ and the initial (at time $t=\tau_i$) momentum distribution of charm and bottom quarks are taken MNR code [@MNR]. The system under study has two components. The equilibrium component, the QGP comprising of the light quarks and the gluons. The non-equilibrium component, the heavy quarks produced due to the collision of partons of the colliding nuclei has momentum distribution determined by the perturbative QCD (pQCD), which evolves due to their interaction with the expanding QGP background. The evolution of the heavy quark momentum distribution is governed by the FP equation. The interaction of the heavy quarks with the QGP is contained in the drag and diffusion coefficients. The drag and diffusion coefficients are provided as inputs, which are, in general, dependent on both temperature and momentum. The evolution of the temperature of the background QGP with time is governed by relativistic hydrodynamics. The solution of the FP equation at the (phase) transition point for the charm and bottom quarks gives the (quenched) momentum distribution of hadrons ($B$ and $D$ mesons) through fragmentation. The fragmentation of the initial momentum distribution of the heavy quarks results in the unquenched momentum distribution of the $B$ and $D$ mesons. The ratio of the quenched to the unquenched $p_T$ distribution is the nuclear suppression factor which is experimentally measured. The quenching is due to the dragging of the heavy quark by QGP. Hence the properties of the QGP can be extracted from the suppression factor. Nuclear suppression factor -------------------------- The variation of the nuclear suppression factor, $R_{AA}$  [@we1] with $p_T$ of the electron originating from the decays of $D$ and $B$ mesons have been displayed in Fig. \[fig7\] for RHIC initial condition ($T_i=300$ MeV). A less suppression of $B$ is observed compared to $D$. The theoretical results show a slight upward trend for $p_T$ above 10 GeV both for mesons containing charm and bottom quarks. Similar trend has recently been experimentally observed for light mesons at LHC energy [@CMS]. This may originate from the fact that the drag (and hence the quenching) for charm and bottom quarks are less at higher momentum. The same formalism is extended to evaluate the nuclear suppression factor, $R_{AA}$ both for charm and bottom at LHC energy. Result has been compared with the recent ALICE data(Ref. [@ALICE]) in Fig. \[fig8\]. The data is reproduced well by assuming formation of QGP at an initial temperature $\sim 550$ MeV after Pb+Pb collisions at $\sqrt{s_{\mathrm NN}}=2.76$ TeV. Elliptic flow of heavy flavours ------------------------------- In Fig. \[fig9\] we compare the experimental data obtained by the PHENIX [@phenixemb] collaborations for Au + Au minimum bias collisions at $\sqrt{s_{\mathrm NN}}=200$ GeV with theoretical results obtain in the present work. We observe that the data can be reproduced by including both radiative and collisional loss with $c_s=1/\sqrt{4}$. In this case $v_2^{HF}$ first increases and reaches a maximum of about 7% then saturates for $p_T>2$ GeV. However, with ideal equation of state ($c_s^2=1/3$) we fail to reproduce data. This is because with larger value of $c_s$ the system expands faster as a result has shorter life time for fixed $T_i$ and $T_c$. Consequently the heavy quarks get lesser time to interact with the expanding thermal system and fails to generate enough flow. From the energy dissipation we have evaluated the shear viscosity ($\eta$) to entropy ($s$) density ratio using the relation [@mmw]: $\eta/s\sim 1.25T^3/\hat{q}$, where $\hat{q}= \langle p^2_T\rangle/L$ and $dE/dx\sim \alpha_s\langle p^2_T\rangle$  [@RB], $L$ is the length traversed by the heavy quark. The average value of $\eta/s\sim 0.1-0.2$, close to the AdS/CFT bound [@KSS]. ![$R_{AA}$ as a function of $p_T$ for $D$ and $B$ mesons at LHC. Experimental data taken from  [@ALICE].[]{data-label="fig8"}](fig8.eps) ![Elliptic flow of single electrons originating from the heavy mesons decays. []{data-label="fig9"}](fig9.eps) Summary ======= In this work we have discussed the productions of lepton pairs from nuclear collisions at relativistic energies and shown that lepton pairs can trace the evolution of collectivity of the system. The elliptic flow and the nuclear suppression factor of the electrons originating from the heavy flavour decays have been studied by including both the radiative and the collisional processes of energy loss in evaluating the effective drag and diffusion coefficients of the heavy quarks. The results have been compared with the available experimental data and properties of QGP expected to be formed at RHIC collisions have been extracted. [**Acknowledgment:**]{} The author is grateful to Trambak Bhattacharyya, Santosh K Das, Sabyasachi Ghosh, Surasree Mazumder, Bedangadas Mohanty, Payal Mohanty and Sourav Sarkar for collaboration and to Tetsufumi Hirano for providing hadronic chemical potentials. [99]{} J. C. Collins and M.J. Perry, Phys. Rev. Lett. [**34**]{}, 1353 (1975). S. Borsanyi [*et al.*]{} JHEP [**1011**]{}, 077 (2010). L. D. McLerran and T. Toimela, Phys. Rev. D [**31**]{} (1985) 545 R. Rapp and J. Wambach, Adv. Nucl. Phys. [**25**]{} (2000) 1. J. Alam, S. Raha and B. Sinha, Phys. Rep. [**273**]{} (1996) 243. J. Alam, S. Sarkar, P. Roy, T. Hatsuda and B. Sinha, Ann. Phys. [**286**]{} (2001) 159. S.Ghosh, S. Sarkar and J. Alam, Eur. Phys. J. C [**71**]{}, 176 (2011). S. Ghosh, S. Mallik and S. Sarkar, Eur. Phys. J. C [**70**]{} (2010) 251. V. L. Eletsky, M. Belkacem, P. J. Ellis and J. I. Kapusta, Phys. Rev.  C [**64**]{}, 035202 (2001). R. A. Schneider and W. Weise, Phys. Lett.  B [**515**]{}, 89 (2001). J. D. Bjorken, Phys. Rev. D [**27** ]{}, 140(1983). V. Roy, A. K. Choudhuri, Phys. Lett. B, [**703**]{}, 313 (2011). B. Mohanty and J. Alam, Phys. Rev. C [**68**]{}, 064903 (2003). T. Hirano and K. Tsuda, Phys. Rev. C [**66**]{}, 054905 (2002). P. Mohanty [*et al.*]{}, arXiv:1111.2159 \[nucl-th\]. P. Mohanty, J. Alam and B. Mohanty, Phys.Rev. C [**84**]{}, 024903 (2011). B. I. Abeleb [*et al.*]{} (STAR Collaboration), Phys. Rev. Lett. [**98**]{}, 192301 (2007). A. Adare [*et al.*]{} (PHENIX Collaboration), Phys. Rev. Lett. [**98**]{}, 172301 (2007). S. Mazumder, T. Bhattacharyya, J. Alam and S. K. Das, Phys.Rev. C [**84**]{}, 044901 (2011). H. M. Antia, Numerical Methods for Scientists and Engineers, Tata McGraw-Hill, 1991. M. L. Mangano, P. Nason and G. Ridolfi, Nucl. Phys. B [**373**]{}, 295 (1992). Y. J. Lee (for CMS collaboration), Quark Matter, 2011, May 23-28, Annecy, France. A. Rossi (for ALICE collaboration), J. Phys. G, [**38**]{}, 124139 (2011). S. S. Adler [*et al.*]{} (PHENIX Collaboration), Phys. Rev. Lett. [**98**]{}, 172301 (2007) A. Majumder, B. Müller and X. N. Wang, Phys. Rev. Lett. [**99**]{}, 192301 (2007). R. Baier, arXiv hep-ph/0209038. P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. [**94**]{}, 111601 (2005).

ArXiv

--- address: - 'Konkoly Observatory, Budapest, HUNGARY' - 'Kapteyn Institute, Groningen, The NETHERLANDS' - 'Physics Department, University of Florida, Gainesville, FL, USA' -   author: - Zoltán Kolláth - 'Jean-Philippe Beaulieu' - 'J. Robert Buchler & Phil Yecko' title: '[ASTROPHYSICAL JOURNAL LETTERS, in press, accepted May 18, 1998]{}   Nonlinear Beat Cepheid Models' --- \#1 2 2 The Fourier analysis of the observational data of the [*beat Cepheid*]{} light curves and radial velocities shows constant power in two basic frequencies and in their linear combinations which indicates that the stars pulsate in two modes (or more if resonances are involved). Since the beginning of theoretical Cepheid studies in the early 1960s numerical hydrodynamical attempts at modelling the phenomenon of beat pulsation have failed, and beat Cepheids have been a bane of stellar pulsation theory. In Cepheids energy is carried through the pulsating envelope to the surface by radiation transport as well as by turbulent convection (TC). Even though convection can transport almost all the energy in the hydrogen partial ionization region, this convection is inefficient in the sense that it only mildly affects the structure of the envelope. It was thus generally thought that convection, while important for providing a red edge to the instability strip, would play a minor role the appearance of the nonlinear pulsation. Purely radiative models did indeed give good overall agreement with the observed light and radial velocities. However, recently it has become increasingly clear that there are a number of severe problems with radiative models (Buchler 1998), in addition to their inability to account for beat behavior. We have recently implemented in our hydrodynamics codes a one dimensional model diffusion equation for turbulent energy (Yecko, Kolláth & Buchler 1998) similar to those advocated by Stellingwerf (1982), Kuhfuss (1986), Gehmeyr & Winkler (1992) and Bono & Stellingwerf (1994). In contrast to these authors, however, we have developed additional tools that allow us to find beat behavior without having to rely on very time-consuming and sometimes inconclusive hydrodynamic integrations to determine if a model undergoes stable, or steady beat pulsations. These are (a) a linear stability analysis which yields the frequencies and growth rates of [*all modes*]{}, (b) a relaxation method (based on the general algorithm of Stellingwerf with the modifications of Kovács & Buchler, 1987) to obtain nonlinear periodic pulsations (limit cycles) when they exist, (c) a stability analysis of the limit cycles that gives their (Floquet) stability exponents. The 1D turbulent diffusion equation, and the concomitant eddy viscosity, the turbulent pressure and the convective and turbulent fluxes contain (seven) order unity parameters that need to be calibrated through a comparison to observations. In a first paper (Yecko, Kolláth & Buchler 1998) in which we performed a broad survey of the linear properties of TC Cepheid models we found that of these the mixing length, the strengths of the convective flux and of the eddy viscosity play a dominant role and that broad combinations of these three parameters exist that give agreement with the observed widths of both the fundamental and first overtone instability strips. In this Letter we show that the inclusion of TC produces pulsating beat Cepheid models that satisfy the observational constraints, in particular those of period ratios, of modal pulsation amplitudes and of their ratios. Furthermore the models are very robust with respect to the numerical and physical parameters. Our discovery of beat Cepheid models has been partially serendipitous. When we started to investigate the nonlinear pulsations of a typical Small Magellanic Cloud Cepheid model ($M\ngth =\ngth 4.0M_\odot$, $L\ngth =\ngth 1100L_\odot$, = 5900K, $X\ngth =\ngth 0.73$ and $Z\ngth =\ngth 0.004$) with the the turbulent convective hydrocode, we encountered beat pulsations that appeared steady. We use the OPAL opacities of Iglesias & Rogers (1996) combined with those of Alexander & Ferguson (1994). The values of the TC parameters – for a definition cf. Yecko et al. (1998) – are $\alpha_c\ngth =\ngth 3$, $\alpha_\Lambda \ngth =\ngth0.41$, $\alpha_p\ngth =\ngth 0.667$, $\alpha_t\ngth =\ngth 1$, $\alpha_D\ngth =\ngth 4$, $\alpha_s\ngth =\ngth 0.75$, $\alpha_\nu \ngth =\ngth 1.2$. The steadiness of these beat pulsations was confirmed when several nonlinear hydrodynamics calculations, each initiated with a different admixture of fundamental and first overtone eigenvectors, converged towards the same final steady beat pulsational state. This convergence could be corroborated when we extracted the slowly varying amplitudes with the help of a time-dependent Fourier decomposition, and plotted the resulting phase portraits ($A_0(t)$ vs. $A_1(t)$) that are shown in Fig. 1 where all initializations are seen to converge toward a fixed point located at $A_0 \ngth =\ngth 0.0104$ and $A_1\ngth =\ngth 0.0200$ (These radial displacement amplitudes assume the eigenvectors to be normalized to unity at the stellar surface, $\delta r/r_*\ngth =\ngth 1$). While the observed transient behavior of the models provides a conclusive proof of the presence of steady beat pulsations, it is important to explain and describe the behavior on a more fundamental level. The phase portrait of Fig. 1 is very similar to those found for nonresonant mode interaction on the basis of amplitude equations. (Buchler & Kovács 1986, 1987, hereafter BK86 and BK87). We show here that indeed the nonlinear behavior of the hydrodynamical model pulsations can be understood very simply that way. The amplitudes of the two nonresonantly interacting modes obey remarkably simple equations $$\begin{aligned} \quad\quad {dA_0 \over dt} = & A_0 \th (\kappa_0 - q_{00} A_0^2 - q_{01} A_1^2) \nonumber \quad\quad\quad\quad\quad (1a)\\ \quad\quad {dA_1\over dt} = & A_1 \th (\kappa_1 - q_{10} A_0^2 - q_{11} A_1^2) \nonumber \quad\quad\quad\quad\quad (1b) \end{aligned}$$ These amplitude equations are ‘normal forms’ and are therefore generic for any dynamical system in which two modes interact nonresonantly. The assumptions underlying these amplitude equations are satisfied for Cepheids: (a) The lowest modes (fundamental and first overtone here) are weakly nonadiabatic, i.e. the ratios of linear growth rates $\kappa$ to periods are small, a condition that is readily confirmed by our linear stability analysis; (b) The pulsations are weakly nonlinear which allows a truncation of the amplitude equations in the lowest permissible (third) order; weak nonlinearity can be established by comparing the linear and nonlinear periods which differ less than a tenth of a percent. Furthermore both the nonlinear self-saturation coefficients $q_{00}$ and $q_{11}$ as well as the cross-coupling coefficients $q_{01}$ and $q_{10}$ have always been found to be positive in Cepheid models so that amplitude saturation can occur in third order, and it is sufficient to keep terms up to cubic in the amplitudes. (c) In the range of interest there is no important low order resonance between the fundamental and the first overtone modes, and possibly a higher mode. In the following discussion we look at the regime where both modes are linearly unstable, $\kappa_0>0$ and $\kappa_1> 0$. Eqs. (1) then have two [*single mode*]{} fixed points. The amplitude of the single mode fundamental (0) fixed point is $A_0=\sqrt{\kappa_0/q_{00}}$ and its coefficient for stability to first overtone perturbations is $\bar\kappa_{1(0)} = \kappa_1 - q_{10} A_0^2 \th$. In our notation a positive coefficient implies growth and thus instability. The corresponding first overtone (1) limit cycle amplitude is $A_1=\sqrt{\kappa_1/q_{11}}$ and its coefficient of stability to fundamental perturbations is $\bar\kappa_{0(1)} = \kappa_0 - q_{01} A_1^2 \th$. The $\bar\kappa$’s, when multiplied by the periods $P_k$ of their limit cycles, are equal to the corresponding Floquet exponents (Buchler, Moskalik & Kovács 1991). Eqs. (1) can also have [*a double mode fixed point*]{} whose amplitudes satisfy $A_{0\th DM}^2 = \bar\kappa_{0(1)} q_{11}/ {\cal D} < A_0^2$, $A_{1\th DM}^2 = \bar\kappa_{1(0)} q_{00}/ {\cal D} < A_1^2$, where ${\cal D} = q_{00} q_{11}$ – $q_{01} q_{10}$. This fixed point exists provided $A_{0\th DM}^2>0$ and $A_{1\th DM}^2>0$. Then, if ${\cal D} <$ 0, the double mode limit cycle is unstable. Stable pulsations occur either in the fundamental [*or*]{} first overtone, and the pulsational mode is determined by the evolutionary history of the model (hysteresis). On the other hand, if ${\cal D}>0$, the double mode fixed point is stable, and steady double mode pulsations occur (BK86). One can show that these conditions are equivalent to requiring $\bar\kappa_{0(1)}>0$ and $\bar\kappa_{1(0)}>0$, conditions which also imply that both [*single mode*]{} limit cycles (fundamental and first overtone) are individually unstable. This validates Stellingwerf’s (1975) suggestion that the simultaneous instability of the fundamental and the first overtone leads to steady beat pulsations (in the absence of a resonance). It provides an economical tool to search for double mode behavior, because we can now relatively easily compute single mode limit cycles and their stability. As a further confirmation that the nonresonant scenario applies to the pulsating Cepheid model, we have determined the coefficients of Eqs. (1) as in BK87 by fitting time-dependent solutions of these equations to the temporal variation of the amplitudes in their approach to the limit cycle as shown in Fig. 1. The fitted trajectories in the phase portrait are practically undistinguishable from the hydro results, confirming the applicability and accuracy of the amplitude equation formalism and the absence of any relevant resonances. We mention in passing that the expression ‘double mode Cepheids’ is often used cavalierly for beat Cepheids. Since no additional, resonant overtone is involved in the beat pulsations, the latter are thus truly double mode pulsations. With the relaxation code we are able to compute both the fundamental and the first overtone limit cycles with their respective amplitudes and Floquet stability exponents $\lambda_{1(0)} = P_0 \bar\kappa_{1(0)}$ and $\lambda_{0(1)} = P_1 \bar\kappa_{0(1)}$. The above discussion then shows that from these four quantities we can extract the four nonlinear $q_{jk}$ coefficients when we have already computed the linear periods and growth rates. The values we obtain this way for this beat Cepheid model agree quite well with those that we obtain from the fit described in the previous paragraph. Note that these two determinations rely on independent numerical hydrodynamical input, the first on two periodic limit cycles (that are linearly unstable), the second on transient evolution toward the stable double mode pulsation. In order to investigate the robustness of the observed beat behavior we now explore the pulsational behavior of a [*sequence*]{} of Cepheid models in which the effective temperature of the equilibrium modes of the sequence varies from =6200K to 5800K. Such a sequence is approximately along an evolutionary path. The eddy viscosity parameter $\alpha_\nu$ is treated as an additional variable parameter to explore the effect of TC on the behavior. In Fig. 2 the stability coefficients of the sequence are plotted versus , with open/filled circles for those of the fundamental/overtone single mode cycles. The curves are labelled with the corresponding strengths $\alpha_\nu$ of the eddy viscosity. As discussed above we expect double mode behavior where both Floquet exponents are positive. (The stability exponents due to perturbations with other modes are always smaller in this sequence and are therefore irrelevant here). For the low value of $\alpha_\nu \ngth =\ngth 0.5$ (dotted lines) the two stability coefficients are never positive simultaneously, thus excluding double-mode behavior. On the other hand, for $\alpha_\nu = 1.2$ a double mode region appears between $\sim$ 5875 – 5915K and for $\alpha_\nu \ngth =\ngth 2.0$ this broadens to $\sim$ 5965 – 6050K. How does turbulent convection bring about double mode behavior? Fig. 2 shows that, in the region of interest, an increase in the turbulent eddy viscosity causes a rapid decrease in the stability of the fundamental limit cycle ($\bar\kappa_{1(0)}$, filled circles), but an increase in that of the first overtone limit cycle ($\bar\kappa_{0(1)}$, open circles). This description, though, does not tell us whether it is the effect of TC on the linear $\kappa$’s or on the nonlinear $q$’s, or on both, that is responsible for the beat pulsations. Table 1 shows the variation with $\alpha_\nu$ of the relevant model quantities, viz. the linear growth rates, the nonlinear coupling coefficients, the discriminant ${\cal D}= q_{00} q_{11} - q_{01}q_{10}$, the amplitude of the fundamental cycle and its stability coefficient with respect to overtone perturbations, and the same for the first overtone. The [*necessary*]{} condition for stable double mode pulsations, viz. ${\cal D} >$ 0, is never found to be satisfied in radiative models. In these models the cross-coupling always dominates over the self-saturation coefficients. Table 1 shows that an increase in the strength of the eddy viscosity causes $q_{00}$ and $q_{11}$ to increase faster than $q_{01}$ and $q_{10}$. It is therefore the resultant change in the sign of ${\cal D}$ that makes double mode behavior possible for sufficiently large $\alpha_\nu$. The condition for beat behavior is thus seen to be rather subtle in that it involves the effects of convection beyond the linear regime for which it seems difficult to give a ‘simple’ physical explanation. Fig. 3 gives the overall modal selection picture in the $\alpha_\nu$ –  plane. The linear edges of the instability region ($\kappa_0\ngth =\ngth 0$ and $\kappa_1\ngth =\ngth 0$) are shown as dashed lines. By computing the fundamental and first overtone limit cycles for a number of $\alpha_\nu$ and  values, by interpolation, we can obtain $\bar\kappa_{0(1)}$ or $\bar\kappa_{1(0)}$ as a function of $\alpha_\nu$ and , and in particular the loci where they vanish. The solid curves give the nonlinear pulsation edges and are marked ORE and FBE. It is straightforward to show that if the two linear growth rates vanish at the same point, the four curves will intersect in a single point on this diagram, that we label [*critical point*]{}. The curve marked OBE is the linear blue edge of the first overtone mode and it coincides with the overtone nonlinear blue edge up to and on the left of the critical point. The linear fundamental blue edge becomes also the fundamental blue edge above the critical point. Above the line ORE we have $\bar\kappa_{0(1)} \ngth > \ngth 0$ and the first overtone limit cycle is unstable. Below the line FBE the quantity $\bar\kappa_{1(0)}\ngth > \ngth 0$ and the fundamental limit cycle is unstable. Thus in the region marked ‘dm’ both single mode limit cycles are unstable, and this is the region of double-mode pulsation. In the small triangular region at the bottom, on the other hand, both limit cycles are stable, and [*either*]{} fundamental or first overtone limit cycles can occur. [ ]{} In summary, stable first overtone pulsations occur in the dotted region, delineated by the lines OBE and ORE. The fundamental limit cycle is stable in the region marked by open squares, delineated by FBE and FRE (not shown on the far right). This figure makes it particularly evident how TC favors double mode pulsations and why all efforts with radiative codes have failed in modelling beat Cepheids. We have seen that when TC effects are sufficiently large then the Cepheids should run into the double mode regime in both their crossings of the instability strip. Furthermore, as a Cepheid crosses the double mode regime redward, say, the first overtone amplitude should gradually go to zero while the fundamental amplitude increases from zero to the value it attains as a fundamental mode Cepheid (BK86). The question arises whether this nonresonant scenario that is derived from the amplitude equations is in agreement with the observations. The four SMC beat Cepheids from the EROS survey (analyzed by Beaulieu and reproduced in Buchler 1998) all have the same amplitude ratios, $A_0/A_1 \sim 0.45$, a priori in disagreement with the nonresonant scenario shown in Fig. 1. of BK86 that suggests that Cepheids with all amplitudes ratios should occur. In Fig. 4 we display the behavior of the component modal amplitudes of the beat Cepheid models for the $\alpha_\nu\ngth =\ngth 1.2$ sequence of Fig. 2. The amplitudes of the stable single mode limit cycles are shown as solid lines with solid dots for the fundamental and open dots for the first overtone, and as dashed lines where they are unstable. The fundamental and first overtone component amplitudes of the stable double mode pulsators are shown as solid squares and open diamonds, respectively. It is seen that although the modal amplitudes do indeed vary continuously throughout as the double mode regime is traversed, the behavior is very rapid near the cooler side. The reason for this unexpected behavior is that the q’s are [*not*]{} constant in this sequence, and what is more, they vary in such a way that ${\cal D}$ happens go through zero around 5850K. It is the presence of this nearby pole that causes a change in the curvature of $A_0$. According to Fig. 4 it is therefore much more likely to find beat Cepheids in the slowly varying regime where the ratio $A_0/A_1\approxlt 0.5$. The computed behavior of the modal amplitudes is thus in agreement with the observed SMC Cepheids, and the nonresonant scenario is consistent with the observations. We have demonstrated that turbulent convection leads naturally to beat behavior in Cepheids, which does not occur with purely radiative models. The reason is that the nonlinear effects of TC dissipation can create a region in which both the fundamental and the first overtone cycles are unstable, and the model undergoes stable double mode pulsations. At a more basic level the amplitude equation formalism shows that turbulent convection modifies the nonlinear coupling between the fundamental and first overtone modes in such as way as to allow beat behavior. The development of a relaxation code (TC) to find periodic pulsations, and a Floquet stability analysis of these limit cycles has made the search quite efficient, and a broader survey of beat Cepheids, with wide ranges of metallicities is in progress. This will also search for beat Cepheid models that pulsate in the first and second overtones. [ This research has been supported by the Hungarian OTKA (T-026031), AKP (96/2-412 2,1) and by the NSF (AST95–28338) at UF. Two of us (JPB) and (ZK) thank the French Académie des Sciences for financial support. ]{} [rrrrrrrrrrrr]{}  $\alpha_\nu$ &$\kappa_0$   &$\kappa_1$    &$q_{00}$ &$q_{01}$ &$q_{10}$ &$q_{11}$  &${\cal D}$ &$A_0$   &$\bar\kappa_{0(1)}$   &$A_1$   &$\bar\kappa_{1(0)}$  \ 0.5 &2.382e-3 &1.124e-2 &1.199 &3.532 & 4.898 &13.311 &–1.343 & 4.458e-2 &1.510e-3 &2.906e-2 &–6.012e-4\ 1.0 & 2.169e-3 & 8.636e-3 & 1.815 & 3.865 & 6.645 & 14.924 & 1.407 & 3.457e-2 & 6.959e-4 & 2.406e-2 & –6.752e-5\ 1.2 &2.082e-3 &7.582e-3 &2.179 &4.200 & 7.554 &16.001 &3.140 & 3.091e-2 &3.650e-4&2.177e-2 &9.169e-5\ 1.5 &1.947e-3 &5.983e-3 &2.893 &4.917 & 9.212 & 18.172 & 7.269 & 2.595e-2 & –2.188e-4 & 1.814e-2 & 3.285e-4\ 2.0 &1.720e-3 &3.282e-3 &4.685 &7.194 & 13.039 &24.229 &19.704 & 1.916e-2 &–1.506e-3 &1.164e-2 & 7.458e-4\ Alexander, D. R., Ferguson, J. W. 1994, ApJ 437, 879 Bono, G., Stellingwerf, R.F. 1994, ApJ Suppl 93, 233–269 Buchler, J.R. 1998, in [*A Half Century of Stellar Pulsation Interpretations: A Tribute to Arthur N. Cox*]{}, ed. P.A. Bradley and J.A. Guzik, ASP Conf. Ser. 135, 220 Buchler, J.R., Kolláth, Z., Beaulieu, J.P. , Goupil. M.J. 1996, ApJ Letters, 462, L83 Buchler, J.R., , Kovács, G. 1986, ApJ 308, 661, \[BK86\]; 1987, ibid. 318, 232 \[BK87\] Buchler, J.R., Moskalik, P., Kovács, G. 1991, ApJ 380, 185. Gehmeyr, M. , Winkler, K.-H. A. 1992, AA 253, 92–100; ibid. 253, 101–112 Iglesias, C. A.& Rogers, F. J. 1996, ApJ 464, 943 Kovács, G. , Buchler, J.R. 1987, ApJ 324, 1026. Kuhfuss, R. 1986, AA 160, 116 Stellingwerf, R.F. 1975, ApJ 199, 705 Stellingwerf, R.F. 1982, ApJ 262, 330 Yecko, P., Kolláth Z., Buchler, J. R. 1998, A&A, in press

ArXiv

--- abstract: 'We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, ${\mathcal{F}}(G,d)$, satisfies the metric approximation property. We show also that, given a finitely generated group $G$, with its word metric $d$, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, ${\mathcal{F}}(G,d)$ has a Schauder basis. Examples and applications are discussed. In particular, for any net $N$ in a real hyperbolic $n$-space $\mathbb{H}^n$, ${\mathcal{F}}(N)$ has a Schauder basis.' address: - | Institute of Mathematics\ Czech Academy of Sciences\ Žitná 25\ 115 67 Praha 1\ Czech Republic - 'Instituto de Ciência e Tecnologia da Universidade federal de São Paulo, Av. Cesare Giulio Lattes, 1201, ZIP 12247-014 São José dos Campos/SP, Brasil' author: - Michal Doucha - 'Pedro L. Kaufmann' bibliography: - 'references.bib' title: 'Approximation properties in Lipschitz-free spaces over groups' --- [^1] Introduction ============ Lipschitz-free spaces form by now a fundamental class of Banach spaces, whose study has been revitalized since the appearance of the seminal paper of Godefroy and Kalton ([@godefroy2003lipschitz]). There are two main important properties that both characterize these spaces. Namely, they are free objects in the category of Banach spaces over the metric spaces. Second, they are canonical isometric preduals to the Banach spaces of pointed Lipschitz real-valued functions. Another appealing feature is that their study connects Banach space theory to several other areas of mathematics, including optimal transport and geometry, and, as we demonstrate here, also harmonic analysis. We recall some basic facts about Lipschitz-free spaces in Section \[section:preliminaries\]. Approximation properties in Lipschitz-free spaces have been one of the main research directions since the publication of [@godefroy2003lipschitz]. It has become clear since then that there are metric spaces such that the corresponding Lipschitz-free space does not have the approximation property, since by [@godefroy2003lipschitz Theorem 5.3], a Banach space $X$ has the bounded approximation property if and only if the Lipschitz-free space ${\mathcal{F}}(X)$ does. The attention was therefore shifted to certain amenable classes of metric spaces, in particular compact metric spaces and, to some extent, also to uniformly discrete metric spaces. The compact metric case is particularly important since it has been shown by Godefroy in [@God2015] that the bounded approximation property of Lipschitz-free space over a compact metric space $M$ is equivalent to the existence of linear almost extension operators of Lipschitz functions over subsets of $M$, a subject currently receiving high attention in both geometry and computer science (see e.g. [@BruBru] and [@LN05]). The question whether Lipschitz-free space over any uniformly discrete metric space has the bounded approximation property is perhaps the most serious and still open, we refer to [@godefroy2014free Question 1] for a motivation and to [@Kalton] for the proof that such a space has the approximation property. Regarding compact metric spaces, the first compact metric space such that the corresponding Lipschitz-free space fails the approximation property has been found in [@godefroy2014free], and later, even a compatible metric on the Cantor space has been found so that the Lipschitz-free space lacks the approximation property (see [@HaLaPe]). Positive results have been however obtained in [@Dalet-compact] and [@dalet2015free] when one restricts to countable compact, resp. proper metric spaces. The goal of this paper is to consider certain fundamental classes of compact metric spaces, resp. uniformly discrete metric spaces, which are amenable to methods of harmonic analysis, resp. geometry. Namely, compact groups with left-invariant (or right-invariant) metrics, resp. finitely generated groups with word metrics. It turns out that harmonic analytic methods, resp. certain combinatorial and geometric methods, go hand in hand with our goal of showing approximation properties in Lipschitz-free spaces over compact metric groups, resp. finitely generated groups. In the compact group case we obtain a satisfactory definitive solution. \[thm:intro1\] Let $G$ be a compact metrizable group with an arbitrary compatible left-invariant (or right-invariant) metric $d$. Then ${\mathcal{F}}(G,d)$ has the metric approximation property. Just to show a meager application, we recall that there has been interest in for which compatible metrics of the Cantor space the corresponding Lipschitz-free space has some approximation property. Godefroy and Ozawa show in [@godefroy2014free] that for certain ‘small Cantor spaces’, the free space has the metric approximation property. On the other hand, H' ajek, Lancien, and Perneck' a show in [@HaLaPe] that there are ‘fat Cantor spaces’ for which the free space does not have the approximation property. We recall that there is a very large and thoroughly studied class of compact (metrizable) groups, the *profinite (metrizable) groups*, which are inverse limits of finite groups. So in the infinite metrizable case, they are topologically totally disconnected uncountable metrizable spaces without isolated points - therefore homeomorphic to the Cantor space (see the monograph [@profinite] for more information on profinite groups). It turns out that for any compatible and left-invariant metric on any such group structure on the Cantor space we get a free space with the metric approximation property. In case of finitely generated groups, the metric approximation property follows from known results (see Section \[section:fingengrps\]), so we aim for much stronger property, having the Schauder basis, at the cost of having less general result that applies just to a certain subclass of finitely generated groups. We state the result and postpone the definition of the new notions to the corresponding section. \[thm:intro2\] Let $G$ be a shortlex combable group with its word metric $d$. Then ${\mathcal{F}}(G,d)$ has the Schauder basis [*(see Theorem \[thm:shortlex\])*]{}. We mention that the theorem applies in particular to hyperbolic groups and Artin groups of large type. One of the applications (see Corollary \[cor:hyperbolicnet\]) is that the Lipschitz-free space over any net in the real hyperbolic $n$-space ${\mathbb{H}}^n$ has the Schauder basis.\ The paper is organized as follows. In Section \[section:preliminaries\] we present a characterization of the $\lambda$-bounded approximation property tailored for Lipschitz-free spaces (Proposition \[tool\]). In Section \[section:cpt\] we prove Theorem \[thm:intro1\]; first we tackle Lie groups using harmonic analysis tools in Subsection, then in Subsection \[subsection:generalCpt\] we prove the general case by approximating compact groups by compact Lie ones. In this last subsection we also generalize of Theorem \[thm:intro1\] to compact homogeneous spaces equipped with quotient metrics (Theorem \[thm:homogeneousspace\]). Section \[section:fingengrps\] is dedicated to finitely generated groups; we prove Theorem \[thm:intro2\] and provide some examples and applications. We conclude with some remarks and questions in Section \[section:problems\], and presenting in Appendix \[appendSphere\] a generalization of Theorem \[thm:homogeneousspace\] for the specific case of the Euclidean sphere. Preliminaries {#section:preliminaries} ============= Lipschitz-free spaces --------------------- Let $M$ be a metric space and $0$ be some distinguished point in $M$. Let $\mathrm{Lip}_0(M)$ denote the Banach space of real-valued Lipschitz functions defined on $M$ which vanish at $0$, equipped with the norm $\|\cdot\|_{\mathrm{Lip}}$ which assigns to each function its minimal Lipschitz constant. The Lipschitz-free space over $M$, denoted by ${\mathcal{F}}(M)$, is the canonical isometric predual of $\mathrm{Lip}_0(M)$ given by the closed linear span of $\{\delta(x):x\in M\}$ in $\mathrm{Lip}_0(M)^*$, where each $\delta(x)$ is the evaluation functional defined by $\delta(x)(f):=f(x)$. This gives $\mathrm{Lip}_0(M)$ a $w^*$ topology which coincides, on bounded sets of $\mathrm{Lip}_0(M)$, with the topology of pointwise convergence. ${\mathcal{F}}(M)$ satisfies a powerful universal property: given a Banach space $X$ and a Lipschitz function $F:M\to X$ vanishing at $0$, there exists a unique bounded linear operator $T:{\mathcal{F}}(M)\to X$ such that $T\circ \delta = F$. Its operator norm coincides with the Lipschitz constant of $F$. We refer to Weaver’s book [@weaver1999lipschitz] for a thorough introduction to the subject. There, Lipschitz-free spaces are denominated Arens-Eells spaces. Verifying if a Lipschitz-free space has the BAP ----------------------------------------------- In this subsection we present a characterization of $\lambda$-bounded approximation property suited for Lipschitz-free spaces (Proposition \[tool\] below). Let us briefly recall and comment the definition of this property: Let $X$ be a Banach space, and $\lambda\geq 1$. We say that $X$ has the $\lambda$-approximation property ($\lambda$-BAP for short) if one of the following equivalent assertions holds: 1. For each compact subset $K$ of $X$ and each $\epsilon>0$, there is a $\lambda$-bounded finite rank operator $T$ on $X$ such that $\|Tx-x\|<\epsilon$, for each $x\in K$. 2. For each finite subset $F$ of $X$ and each $\epsilon>0$, there is a $\lambda$-bounded finite rank operator $T$ on $X$ such that $\|Tx-x\|<\epsilon$, for each $x\in F$. 3. There is a $\lambda$-bounded net of finite rank operators $(T_\alpha)$ on $X$ such that $\langle \varphi, T_\alpha x\rangle \rightarrow \langle \varphi,x\rangle$ for each $\varphi\in X^*$ and each $x\in X$ (that is, $T_\alpha$ converges to the identity operator in the *weak operator topology*). If a Banach space $X$ has the 1-BAP, we say that $X$ has the metric approximation property (MAP for short). \[defBAP\] Formulations (1) and (2) are classic; their equivalence with (3) is shown for instance in [@kim2008characterizations]. Recall that a Banach space $X$ has the $\lambda$-BAP if and only if, for each $\delta>0$, $X$ has the $(\lambda+\delta)-BAP$. To see this, fix a compact set $K\subset X$ and $\varepsilon>0$, and take $\delta>0$ small enough so that $M\delta(\lambda+\delta)/\lambda < \epsilon/2$, where $M=\sup_{x\in K}\|x\|$. Let $T$ be a finite rank, $(\lambda+\delta)$-bounded operator on $X$ such that $\|Tx - x\|<\epsilon/2$, for all $x\in K$. Then it is immediately verified that the $\lambda$-bounded operator $S=\lambda T/\|T\|$ satisfies, for each $x\in K$, $\|Sx - x\| <\varepsilon.$\ In what follows $\mathrm{Lip}(M)$ denotes the space of real-valued functions defined on the metric space $M$. We will still use $\|f\|_{\mathrm{Lip}}$ to denote the Lipschitz constant of $f\in \mathrm{Lip}(M)$, keeping in mind that $\|\cdot\|_{\mathrm{Lip}}$ defines only a seminorm in $\mathrm{Lip}(M)$. Let $K$ be a compact metric space and $\lambda\geq 1$. The following assertions are equivalent: 1. ${\mathcal{F}}(K)$ has the $\lambda$-BAP; 2. for each $\varepsilon>0$ there is a net $T_\alpha$ of bounded operators on $C(K)$ such that 1. $T_\alpha$ are of finite rank, 2. each $T_\alpha$ maps Lipschitz functions to Lipschitz functions, 3. $\|T_\alpha f\|_{\mathrm{Lip}}\leq (\lambda+\varepsilon)\|f\|_{\mathrm{Lip}}$, for each $\alpha$ and each $f\in \mathrm{Lip}(K)$, and 4. $T_\alpha f$ converges pointwise to $f$, for each $f\in \mathrm{Lip}(K)$. \[tool\] For the proof we will need the following: Let $K$ be a compact metric space, fix $0\in M$, and let $T$ be a bounded operator on $C(K)$ such that $T(B_{\mathrm{Lip}_0(K)})\subset A.B_{ \mathrm{Lip}_0(K)}$ for some $A>0$. Then $T$ restricted to $\mathrm{Lip}_0(K)$, when seen as an operator on $\mathrm{Lip}_0(K)$, is $w^*$-continuous. \[Trest\] Consider $S=T|_{\mathrm{Lip}_0(K)}$ as an operator on $\mathrm{Lip}_0(K)$. Let $f_\nu$ be a bounded net $w^*$-converging to $f$ in $B_{\mathrm{Lip}_0(K)} $. Since on bounded sets the $w^*$ topology coincides with the topology of pointwise convergence and $f_\nu$ are equicontinuous, it follows that $f_\nu$ converges to $f$ uniformly. By the norm continuity of $T$, $Sf_\nu$ converges to $Sf$ uniformly and, since $\{Sf_\nu\}$ is $A$-bounded in the Lipschitz norm, $Sf_\nu$ $w^*$-converges to $Sf$. That is, $S$ is $w^*$-continuous when restricted to $B_{\mathrm{Lip}_0(K)}$. It follows by Banach-Dieudonné’s theorem that $S$ is $w^*$-continuous in $\mathrm{Lip}_0(K)$. ((1)$\Rightarrow$(2)) Let $T_\alpha$ be a net of $\lambda$-bounded finite rank operators on ${\mathcal{F}}(K)$ converging to the identity in the weak operator topology. Their adjoints $T_\alpha^*:\mathrm{Lip}_0(K)\rightarrow \mathrm{Lip}_0(K)$ can be extended to operators on $C(K)$ that clearly satisfy properties (a)–(d). ((2)$\Rightarrow$(1)) Choose some $0\in K$, and $T_\alpha$ satisfying (a)–(d). It is easily checked that the operators $S_\alpha$ defined on $C(K)$ by $$S_\alpha (f)(x) = T_\alpha (f)(x) - T_\alpha (f)(0)$$ are bounded, satisfy the same assumptions (a)–(d) and $S_\alpha$ are moreover ($\lambda+\varepsilon$)-bounded operators on $\mathrm{Lip}_0(K)$. Each $S_\alpha$ thus satisfies the assumptions of Lemma \[Trest\], and it follows that there are finite-rank, ($\lambda+\varepsilon$)-bounded operators $R_\alpha$ on ${\mathcal{F}}(K)$ with $R_\alpha^*=S_\alpha$, that is: $$\langle S_\alpha f,\gamma\rangle = \langle f, R_\alpha \gamma\rangle,\,\forall f\in \mathrm{Lip}_0(K),\,\forall \gamma \in {\mathcal{F}}(K).$$ Now since $S_\alpha f$ is a bounded net in $\mathrm{Lip}_0(K)$ converging pointwise to $f$, it must converge $w^*$, thus $R_\alpha$ converges to $Id_{{\mathcal{F}}(M)}$ in the weak operator topology. The conclusion follows from the remark after Definition \[defBAP\]. We point out that the above characterization can be generalized to an arbitrary metric space $M$ if we forget the part of $T_\alpha$ being bounded operators on some $C(K)$, and assume that $T_\alpha$ are ($\lambda+\varepsilon)$-bounded operators on $\mathrm{Lip}(M)$ which are moreover pointwise continuous. We omit the proof, which follows the same lines. Approximation properties of Lipschitz-free spaces over compact groups {#section:cpt} ===================================================================== The classical Birkhoff-Kakutani theorem says that a topological group is metrizable if and only if it is metrizable by a left-invariant (equivalently right-invariant) metric if and only if it is Hausdorff and first countable. Compatible left-invariant metrics on a fixed metrizable group are easily seen to be unique up to uniform equivalence. However, a group can admit compatible and left-invariant metrics which are not bi-Lipschitz equivalent. It is even possible that the corresponding Lipschitz-free spaces are not isomorphic. For example, if $d$ is the arc length metric on the torus $\mathbb{T}$, the Lipschitz-free space ${\mathcal{F}}(\mathbb{T},d)$ is isomorphic to $L^1$, but if we substitute $d$ by $d^\alpha$ with $0<\alpha<1$ (in the process often referred to as *snowflaking*), ${\mathcal{F}}(\mathbb{T},d^\alpha)$ is isomorphic to $\ell_1$ (see [@Kalton Theorem 6.5]). In this section we focus on compact metrizable groups (or equivalently, compact first countable groups) and the issue of many non-Lipschitz equivalent left-invariant metrics does not bother us. We use methods of harmonic analysis that are robust enough to make our proofs work for arbitrary compactible left or right-invariant metrics on any compact metrizable group. The main goal of the section is to prove Theorem \[thm:intro1\], however we will have something to say also about free spaces over certain homogeneous spaces of compact groups (see Theorem \[thm:homogeneousspace\]). The section is divided into two subsections. One is dealing with compact Lie groups where the machinery of harmonic analysis is used. The other is dealing with general compact metrizable groups using the fact each such a group can be approximated by compact Lie groups. This approximation then lifts to the corresponding Lipschitz-free space; indeed, we shall show that Lipschitz-free space over a compact metric group can be approximated by Lipschitz-free spaces over compact Lie groups. Lipschitz-free spaces over compact Lie groups {#subsection:cptLie} --------------------------------------------- \[cpt\] From this point on, unless stated otherwise, we always assume that each locally compact group $G$ is equipped with a left-invariant Haar measure, denoted by $\mu$. When $G$ is compact, $\mu$ is assumed to be bi-invariant and of course normalized. Let us start some basic definitions and facts from representation theory of compact groups, which can be found in any standard textbook on harmonic analysis or representation theory of compact groups. Let $G$ be a compact group. A *unitary representation* $U$ is a strongly continuous group homomorphism $x\in G\mapsto U_x \in B(H_U)$ (i.e. on $B(H_U)$ we consider the strong operator topology), where $H_U$ is a complex Hilbert space, such that each operator $U_x$ is unitary on $H_U$. Such unitary representation is said to be *irreducible* if the only invariant subspaces of $H_U$, i.e. subspaces preserved by all $U_x$, are $\{0\}$ and $H_U$. In our specific case where $G$ is compact, all continuous irreducible unitary representations $x\in G\mapsto U_x \in B(H_U)$ satisfy $\dim H_U <\infty$ (see [@hewitt2012abstract Theorem 22.13]). Given an irreducible unitary representation $U:G\rightarrow B(H_U)$ and non-zero vectors $\xi,\eta\in H_U$, the function of the form $$x\in G\mapsto \langle U_x \xi,\eta\rangle \in \mathbb{C}$$ is called a *matrix coefficient* (associated to $U$). If $\xi=\eta$, we call such matrix coefficient a *positive-definite function*. If $H_U$ is equipped with a fixed orthonormal basis $\{\zeta^U_1,...,\zeta^U_{d_U}\}\subseteq H_U$, then the matrix coeeficients $$\varphi^U_{jk}(x) = \langle U_x \zeta^U_j,\zeta^U_k\rangle$$ are called *coordinate functionals* (associated to $U$ and $\{\zeta^U_1,...,\zeta^U_{d_U}\}$). They satisfy the following property, as a consequence of [@hewitt2013abstract Theorem 27.20]: $$\begin{aligned} \forall f\in C(G)\,\forall x\in G,\, f\ast \varphi^U_{jk}(x) = \sum_{r=1}^{d_U} \int f(y)\,\overline{\varphi^U_{rj}(y)}\,d\lambda(y)\, \varphi^U_{rk}(x). \label{coordfunct}\end{aligned}$$ A *trigonometric polynomial* on $G$ is a linear combination of matrix coefficients. It is straightforward to see that all trigonometric polynomials are in fact linear combinations of coordinate functionals, independently of the choice of bases $\{\zeta^U_1,...,\zeta^U_{d_U}\}$, which are assumed to be fixed. Thus, for any given trigonometric polynomial $P$, there is a finite set $F$ of continuous irreducible unitary representations such that $P$ is in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$. It follows from (\[coordfunct\]) that the operator $f\in C(G) \mapsto f\ast P\in C(G)$ has its range contained in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$, thus in particular it is of finite rank. The following proposition is our crucial tool from harmonic analysis. \[crucialharmonic\] Suppose that $G$ is a compact Lie group. Then there exists a sequence $F_n$ of positive real functions on $G$ satisfying 1. each $F_n$ is a positive definite *central* (commutes under convolution with any function in $L_1(G)$) trigonometric polynomial, 2. $F_n(g^{-1}) = F_n(g)$, $g\in G$, for each $n$, 3. for each $n$, $\int F_n \,d\lambda =1$, and 4. $f\ast F_n(x) \rightarrow f(x)$ $\lambda$-almost everywhere for every $f\in L_p(G),\,1\leq p <\infty$. By [@Knapp Corollary IV.4.22], every compact Lie group is isomorphic to a matrix group, so we may suppose that $G\subseteq \mathrm{GL}(n,\mathbb{C})$ for some $n\in {\mathbb{N}}$. By the following standard ‘unitarization trick’, we may assume that $G$ is a (necessarily closed) subgroup of the unitary group $U(n)$: let $\langle \cdot,\cdot\rangle'$ be an arbitrary inner product on $\mathbb{C}^n$. Define a new inner product $\langle \cdot,\cdot\rangle$ by setting $$\langle \xi,\eta\rangle:=\int_{g\in G} \langle g\xi,g\eta\rangle'd\mu(g),$$ where $\mu$ is an invariant probability Haar measure on $G$. It is standard and straightforward to check that $\langle \cdot,\cdot\rangle$ is still an inner product, which is moreover invariant by the action of $G$. That implies that $G$ is a subgroup of a finite-dimensional unitary group. Now by [@hewitt2013abstract Theorem 44.29], $G$ satisfies all conditions needed to apply directly [@hewitt2013abstract Theorem 44.25], which gives us positive real functions $F_n$ on $G$ satisfying the conditions (1)–(4). Another ingredient we will need is the following version of Young’s convolution inequality, suitable for Lipschitz functions defined on a locally compact group: Let $G$ be a locally compact group equipped with a compatible left-invariant metric $d$. Suppose that $f\in L^1(G)$ and $g\in \mathrm{Lip}(G)$. Then $f\ast g\in \mathrm{Lip}(G)$ and $$\|f\ast g \|_{\mathrm{Lip}} \leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}}.$$ \[lipconv\] Given arbitrary $x,y\in G$, $$\begin{aligned} |f*g(x) - f*g(y)| & = \left| \int f(z)[g(z^{-1}x) - g(z^{-1}y)]\,d\mu(z) \right| \\ & \leq \int |f(z)| |g(z^{-1}x) - g(z^{-1}y)|\,\mu(z)\\ & \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(z^{-1}x,z^{-1}y)\, \mu(z)\\ & \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(x,y) \,\mu(z)\\ &\leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}} d(x,y).\end{aligned}$$ We are now ready to prove the main result from this subsection. We emphasize that in the following we are equipping a Lie group with an *arbitrary* left-invariant metric inducing its topology, not Riemannian metric as it is common in Lie theory. Suppose that $G$ is a compact Lie group equipped with a compatible left-invariant metric. Then ${\mathcal{F}}(G)$ has the MAP. \[proptorus\] Define $T_n: C(G) \rightarrow C(G)$ by $T_n(f) = f\ast F_n$, where $F_n$ are the functions established in Theorem (\[crucialharmonic\]). It suffices to see that these operators satisfy conditions (a)–(d) from Proposition (\[tool\]) with $\lambda=1$, from which follows that ${\mathcal{F}}(G)$ satisfies the MAP. Indeed, each $T_n$ has finite rank, since $F_n$ is a trigonometric polynomial. Thus condition (a) is satisfied. Young’s inequality gives us that $\|T_n f \|_{\mathrm{Lip}} \leq \|F_n\|_{L^1}\|f\|_{\mathrm{Lip}}\leq \|f\|_{\mathrm{Lip}}$, for each $f\in Lip(G)$. Thus condition (b) and (c) are satisfied with constant $\lambda = 1$. (d) follows from condition (4) and the fact that, for each fixed $f\in Lip(G)$, $\{T_n f\}_n$ is equicontinuous, which yields uniform convergence since $G$ is compact. Lipschitz-free spaces over general compact metric groups and homogeneous spaces {#subsection:generalCpt} ------------------------------------------------------------------------------- We start with some remarks on quotient metrics on homogeneous spaces. Let $G$ be a topological group equipped with a compatible left-invariant metric $d$ and let $H$ be a closed subgroup. We want to define a quotient metric on the homogeneous space $G/H$ of left-cosets of $H$. There are two cases. Either $H$ is normal, so $G/H$ is a group. Then the formula $$\begin{aligned} \label{quotmetric} D(gH,fH):=\inf_{h_1,h_2\in H} d(gh_1,fh_2)\end{aligned}$$ defines a compatible left-invariant metric on the quotient group $G/H$, which we refer to as *quotient metric*. We leave the easy verification to the reader. Or $H$ is not normal, so $G/H$ is just a $G$-homogeneous space, i.e. a homogeneous space equipped with a continuous transitive action of $G$. Then the same formula $D(gH,fH):=\inf_{h_1,h_2\in H} d(gh_1,fh_2)$ does not in general define a metric; it does define a compatible $G$-invariant metric provided that $d$ is additionally right $H$-invariant, i.e. $d(g,f)=d(gh,fh)$ for $g,f\in G$ and $h\in H$. We show this. We only verify the triangle inequality, the compatibility and $G$-invariance are easier and left to the reader. Pick $g_1,g_2,g_3\in G$ and let us check that $D(g_1H,g_3H)\leq D(g_1H,g_2H)+D(g_2H,g_3H)$. Choose an arbitrary $\varepsilon>0$ and and some $h_1,h_2, h'_2,h_3\in H$ such that $D(g_1H,g_2H)\geq d(g_1h_1,g_2h_2)-\varepsilon$ and $D(g_2H,g_3H)\geq d(g_2h'_2,g_3h_3)-\varepsilon$. Then $$D(g_1H,g_3H)\leq d(g_1h_1,g_3h_3(h'_2)^{-1}h_2)\leq d(g_1h_1,g_2h_2)+d(g_2h_2,g_3h_3(h'_2)^{-1}h_2)=$$ $$d(g_1h_1,g_2h_2)+d(g_2h'_2,g_3h_3)\leq D(g_1H,g_2H)+D(g_2H,g_3H)-2\varepsilon.$$ Since $\varepsilon$ was arbitrary, we are done. We note that when $G$ is a metrizable group and $H$ is a compact subgroup, then a compatible left-invariant and right $H$-invariant metric on $G$ always exists. Indeed, let $d$ be an arbitrary compatible left-invariant metric on $G$. We define, for $g,f\in G$, $D(g,f):=\max_{h\in H} d(gh,fh)$. Alternatively, using a normalized invariant Haar measure $\mu$ on $H$, we can define $D$ by averaging as follows: $D(g,f):=\int_H d(gh,fh)d\mu(h)$. We leave to the reader to check that both formulas define a compatible left-invariant and right $H$-invariant metrics. Our main tool in this subsection will be the following proposition. We note that simultaneously while writing this paper, the content of the proposition is being developed into a more general form in [@AACD]. \[prop:projection\] Let $G$ be a topological group equipped with a compatible metric $d$ and a compact subgroup $H$. Suppose, additionally, that at least one of the following conditions holds: (i) $d$ is left-invariant and $H$ is normal, (ii) $d$ is right-invariant and $H$ is normal, or (iii) $d$ is left-invariant and right $H$-invariant. Then there exists a norm one projection $P:{\mathcal{F}}(G,d)\rightarrow {\mathcal{F}}(G,d)$ ranging onto a linearly isometric copy of ${\mathcal{F}}(G/H,D)$, where $D$ is the quotient metric as defined in . By the discussion preceding the statement of the proposition, it is verified that $D$ is a well-defined metric. Let $\mu$ be the normalized invariant Haar measure on $H$. If (ii) or (iii) holds, we define a map $P':G\rightarrow {\mathcal{F}}(G)$ by setting for any $g\in G$ $$\begin{aligned} P'(g):=\int_H \delta(g\cdot h)-\delta(h)d\mu(h). \label{P'}\end{aligned}$$In case only (i) holds, we define for any $g\in G$ $$P'(g):=\int_H \delta(h\cdot g)-\delta(h)d\mu(h).$$ We will treat only the case where (ii) or (iii) holds and $P'$ is defined as in , the remainder case is completely analogous. We claim that $P'$ is a $1$-Lipschitz map preserving the distinguished point. The latter is clear, we show that it is $1$-Lipschitz. Pick $g,f\in G$, we have $$\|P'(g)-P'(f)\|=\|\int_H \delta(g\cdot h)-\delta(f\cdot h)d\mu(h)\|\leq \int_H \|\delta(g\cdot h)-\delta(f\cdot h)\|d\mu(h)=d(g,f),$$ where in the last equality we used that $\mu$ is probability and $d$ is invariant. It follows that $P'$ extends to a norm one linear operator $P:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$. We claim it is the desired projection. First we show that it is a projection. For every $h\in H$ let $P'_h:G\rightarrow {\mathcal{F}}(G)$ be the map defined for every $g\in G$ by $P'_h(g):=\delta(g\cdot h)-\delta(h)$. The following are straightforward to verify: - $P'_h$ is an isometry with $P'_h(1)=0$, thus it extends to a norm one linear map $P_h:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$. - For every $g\in G$ we have $P'(g)=\int_H P'_h(g)d\mu(h)$ and so also for every $x\in{\mathcal{F}}(G)$ we have $P(x)=\int_H P_h(x)d\mu(h)$. It follows that in order to show that $P^2=P$ it suffices to check that for every $g\in G$ and $h\in H$ we have $P_h\circ P(\delta(g))=P(\delta(g))$. Indeed, the previous equality implies $$P^2(\delta(g))=\int_H P_h\circ P(\delta(g))d\mu(h)=\int_H P(\delta(g))d\mu(h)=P(\delta(g)).$$ Since the set $\{x\in{\mathcal{F}}(G)\colon P^2(x)=P(x)\}$ is a closed linear subspace, we get that $P^2(x)=P(x)$ for all $x\in{\mathcal{F}}(G)$ since ${\mathcal{F}}(G)$ is the closed linear span of $\{\delta(g)\colon g\in G\}$. Let us thus fix $g\in G$ and $h\in H$. We have $$\begin{aligned} P_h\circ P(\delta(g))&=P_h\circ\int_H \delta(g\cdot f)-\delta(f)d\mu(f)\\ &=\int_H \Big(\delta(g\cdot f\cdot h)-\delta(h)\Big)-\Big(\delta(f\cdot h)-\delta(h)\Big)d\mu(f)\\ &=\int_H \delta(g\cdot f\cdot h)-\delta(f\cdot h)d\mu(f\cdot h)=P(\delta(g)),\end{aligned}$$ which finishes the claim. Let $X$ denote the range of $P$ and let us show that it is linearly isometric with ${\mathcal{F}}(G/H,D)$. We define a map $T':G/H\rightarrow {\mathcal{F}}(G)$ by setting for any left coset $gH$ $$T'(gH):=P'(g).$$ We check that it is correctly defined and $1$-Lipschitz. For the former, we need to check that for any $g\in G$ and $h\in H$ we have $P'(g)=P'(gh)$, i.e. $\int_H \delta(g\cdot f)d\mu(f)=\int_H \delta(gh\cdot f)d\mu(f)$. But the equality follows from the invariance of $\mu$. To check that $T'$ is $1$-Lipschitz, pick two cosets $g_1H$ and $g_2H$ and suppose that $f\in H$ is such that $D(g_1H,g_2H)=d(g_1,g_2f)$. Then we have $$\begin{aligned} \|T'(g_1H)-T'(g_2H)\|&=\|\int_H \delta(g_1\cdot h)-\delta(g_2 f\cdot h)d\mu(h)\| \\ &=\int_H D(g_1H,g_2H)d\mu(f)=D(g_1H,g_2H),\end{aligned}$$ showing that $T'$ is actually isometric. It follows that $T'$ extends to a norm one linear surjection $T:{\mathcal{F}}(G/H)\rightarrow X$. In order to show that $T$ is isometric, it suffices to prove that for any finite linear combination $x=\sum_i \alpha_i \delta(g_i H)$ we have $\|x\|_{{\mathcal{F}}(G/H)}=\|T(x)\|_{{\mathcal{F}}(G)}$. One inequality already follows from the fact that $\|T\|=1$, so we only need to prove $\|x\|_{{\mathcal{F}}(G/H)}\leq\|T(x)\|_{{\mathcal{F}}(G)}$. Let $f\in\rm{Lip}_0(G/H)$ be a $1$-Lipschitz function satisfying $\|x\|_{{\mathcal{F}}(G/H)}=|\sum_i \alpha_i f(g_iH)|$. Let $\tilde f$ denote its lift to $G$. That is, for any $g\in G$ and $h\in H$, $\tilde f(gh)=f(gH)$. It is clear that $\tilde f$ is $1$-Lipschitz. In the following, we shall not notationally distinguish between $\rm{Lip}_0(G/H)$-functions and their unique extension to linear functionals. Since we have $\|T(x)\|\geq \tilde f(T(x))$, it suffices to check that $\tilde f(T(x))=f(x)$. For that, in turn, it suffices to check that for every $gH\in G/H$ we have $f(gH)=\tilde f(T'(g))$. We have $$\tilde f(T'(g))=\langle \int_H \delta(g\cdot h)-\delta(h)d\mu(h),\tilde f\rangle=\int_H \tilde f(g\cdot h)-\tilde f(h)d\mu(h)=f(gH)-f(H)=f(gH),$$ which finishes the proof. \[thm:MAPcompactgrp\] Let $G$ be a compact group with a compatible left-invariant metric $d$. Then ${\mathcal{F}}(G,d)$ has the MAP. Before embarking on the proof, we state the following folklore fact, which we prove for the convenience of the reader. \[fact2\] For every compatible left-invariant metric $d$ on a compact metrizable group $G$ there exists a compatible bi-invariant metric $D$ such that $d(g,h)\leq D(g,h)$ for all $g,h\in G$. Let $d$ be an arbitrary compatible left-invariant metric on $G$. We define a compatible bi-invariant metric $D$ by setting, for any $g,f\in G$, $$D(g,f):=\max_{h\in G} d(gh,fh).$$ Clearly, $d\leq D$ and $D$ is bi-invariant. Let us check that $D$ satisfies the triangle inequality and it is compatible with the topology of $G$. For the triangle inequality, pick $g,h,f\in G$ and let us show that $D(g,f)\leq D(g,h)+D(h,f)$. Let $x\in G$ be such that $D(g,f)=d(gx,fx).$ Then we have $$D(g,f)=d(gx,fx)\leq d(gx,hx)+d(hx,fx)\leq D(g,h)+D(h,f),$$ showing the traingle inequality. In order to show that it is compatible with the topology, using left-invariance, it suffices to show that for every sequence $(g_n)_n\subseteq G$ we have $d(g_n,1)\to 0$ if and only if $D(g_n,1)\to 0$. One direction follows immediately from the fact that $d\leq D$. Thus we only need to show that if $d(g_n,1)\to 0$, then $D(g_n,1)\to 0$. Suppose it is not the case and assume without loss of generality that $\lim_n D(g_n,1)=r>0$. For each $n$, let $h_n\in G$ be such that $D(g_n,1)=d(g_nh_n,h_n)$. Again, without loss of generality, we may assume that $(h_n)_n$ converges to some $h\in G$. Then, using that $(g_n)_n$ converges to $1$ since $d$ is compatible, we have $$r=\lim_n D(g_n,1)=\lim_n d(g_nh_n,h_n)=d(h,h)=0,$$ a contradiction finishing the proof. Let $G$ be a compact metrizable group with compatible left-invariant metric $d$. We shall also fix some compatible bi-invariant metric $D$ on $G$ which majorizes $d$, i.e. $d\leq D$, which exists by Fact \[fact2\]. It is a standard consequence of Peter-Weyl’s theorem ([@Knapp Theorem 4.20]) that $G$ can be topologically embedded into an infinite direct product $\prod_{i\in\mathbb{N}} U_i$, where each $U_i$ is a finite-dimensional unitary group. Indeed, by [@Knapp Theorem 4.20] finite-dimensional unitary representations of $G$ separate points. Since $G$ is separable, one can find a sequence $\{\pi_n:G\rightarrow B(H_n)\}_{n\in{\mathbb{N}}}$ of finite dimensional unitary representations separating the points, and their product $\prod_{n\in{\mathbb{N}}} \pi_n$ is then, using also compactness of $G$, a topological embedding of $G$ into a countable product of unitary groups. In particular, $G$ is an inverse limit of a sequence of compact Lie groups $(G_n)_n$. Indeed, let $\Psi:=\prod_{n\in{\mathbb{N}}} \pi_n: G\rightarrow \prod_{i\in\mathbb{N}} U_i$ be the embedding and let, for each $n\in\mathbb{N}$, $P_n:\prod_{i\in\mathbb{N}} U_i\rightarrow \prod_{i\leq n} U_i$ be the projection on the first $n$-coordinates. Then for each $n\in\mathbb{N}$, $G_n:=P_n\circ \Psi[G]$ is a compact Lie group (a closed subgroup of $\prod_{i\leq n} U_i$), and $(G_n)_n$ form an inverse system whose limit is $G$. It follows that there exists a decreasing sequence of compact normal subgroups $(H_n)_n$ of $G$ such that $\bigcap_n H_n=\{1\}$, and for every $n\in\mathbb{N}$, $G/H_n=G_n$. For each $n$, we denote by $d_n$ the quotient metric on $G_n$, which is then compatible and left-invariant on $G_n$. In the following, ${\mathcal{F}}(G)$ is meant to be ${\mathcal{F}}(G,d)$ and ${\mathcal{F}}(G_n)$ is meant to be ${\mathcal{F}}(G_n,d_n)$, for all $n$. Notice that in the following claim we rather work with the bi-invariant metric $D$ majorizing $d$, established in Fact \[fact2\].\ [**Claim 1.**]{} $\rm{diam}_D(H_n)\to 0$, where $\rm{diam}_D(H_n):=\sup_{g,h\in H_n} D(g,h)=\sup_{g\in H_n} D(g,1)$. Suppose on the contrary that there exist $\varepsilon>0$ and a sequence $(h_n)_n\subseteq G$ such that $h_n\in H_n$ and $D(h_n,1)\geq \varepsilon$. Since $G$ is compact, the sequence without loss of generality converges to some $h\in G$. By the continuity of the metric, we have $D(h,1)\geq \varepsilon$, thus in particular $h\neq 1$. On the other hand, since the sequence $(H_n)_n$ is decreasing and each of the subgroups is closed, $h\in \bigcap_n H_n$. This contradicts that $\bigcap_n H_n=\{1\}$. For each $n$, let $\mu_n$ be the normalized invariant Haar measure on the compact subgroup $H_n$, and let $P'_n$, resp. $P_n$ be the map, resp. projection from Proposition \[prop:projection\] applied to the groups $G$ and $H_n$ with the metric $d$ and the quotient metric $d_n$ on $G/H_n$. [**Claim 2.**]{} For every $x\in {\mathcal{F}}(G)$, we have $x=\lim_{n\to\infty} P_n(x)$. Suppose first that $x\in {\mathcal{F}}(G)$ is a finite linear combination of Dirac elements. That is, there are $m\in {\mathbb{N}}$, $g_1,\dots,g_m \in G$ and $\alpha_1,\dots,\alpha_m\in{\mathbb{R}}$ with $x=\sum_{i=1}^m \alpha_i \delta(g_i)$. For a fixed $n$, let us compute $\|x-P_n(x)\|$. We have $$\begin{aligned} \|x-P_n(x)\|& = \|x-\int_{H_n} h\cdot x - (\sum_{i=1}^m \alpha_i)\delta(h)\,d\mu_n(h)\|\\ &\leq \int_{H_n}\|x- h\cdot x \|\,d\mu_n(h) + (\sum_{i=1}^m |\alpha_i|)\|\int_{H_n}\delta(h)\,d\mu_n(h)\|.\end{aligned}$$ Suppose that $\rm{diam}_D(H_n)= \varepsilon_n$. Then for every $h\in H_n$ we have $$\begin{aligned} \|x-h\cdot x\|& =\|\sum_{i=1}^m \alpha_i \delta(g_i)-\sum_{i=1}^m \alpha_i \delta(h\cdot g_i)\|\leq \sum_{i=1}^m |\alpha_i| d(h\cdot g_i,g_i)\\ & \leq \sum_{i=1}^m |\alpha_i| D(h\cdot g_i,g_i)\leq \sum_{i=1}^m |\alpha_i| D(h,1)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n,\end{aligned}$$ and it follows that $\int_{H_n}\|x-h\cdot x\|d\mu_n(h)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n$. On the other hand, for each $f\in B_{\rm{Lip}_0(G)}$, $$|\langle \int_{H_n}\delta(h)\,d\mu_n(h),f \rangle| = |\int_{H_n}f(h)\,d\mu_n(h)|\leq \int_{H_n}\rm{diam}_d(H_n)\,d\mu_n(h)\leq \int_{H_n} \rm{diam}_D(H_n)\leq \varepsilon_n,$$ so $\|\int_{H_n}\delta(h)\,d\mu_n(h)\|\leq \varepsilon_n$. It follows that $$\|x-P_n(x)\|\leq 2\sum_{i=1}^m |\alpha_i|\varepsilon_n,$$ so by **Claim 1**, $\lim_{n\rightarrow\infty} P_n(x)=x$.\ Now let $x\in {\mathcal{F}}(G)$ be an arbitrary element and note that, for each finitely supported $y$, $$\begin{aligned} \|x-P_n(x)\|&\leq \|x-y\|+\|y-P_n(y)\| + \|P_n(y) - P_n(x)\|\\ & = \|x-y\| +\|y-P_n(y)\| + \|P_n\|\|y - x\|=2\|x-y\| +\|y-P_n(y)\|.\end{aligned}$$ Hence, $$\limsup_{n\rightarrow\infty}\|x-P_n(x)\|\leq 2\|x-y\| +\lim_{n\rightarrow\infty}\|y-P_n(y)\| = 2\|x-y\|.$$ Since $y$ can be chosen arbitrarily close to $x$, the result follows. We are ready to show that ${\mathcal{F}}(G)$ has MAP. Pick finitely many $x_1,\ldots,x_m\in {\mathcal{F}}(G)$ and some $\varepsilon$. By the previous claim we can find $n$ so that for all $i\leq m$ we have $\|x_i-P_n(x_i)\|<\varepsilon/2$. By Theorem \[proptorus\], $F(G_n)$ has the MAP. Thus there exists a norm one finite rank operator $T':F(G_n)\rightarrow F(G_n)$ such that for all $i\leq m$ we have $\|P_n(x_i)-T'\circ P_n(x_i)\|<\varepsilon/2$. Set $T:=T'\circ P_n$. It is a norm one finite rank operator from $F(G)$ into $F(G_n)\subseteq F(G)$ such that for all $i\leq n$ we have $$\|T(x_i)-x_i\|\leq \|x_i-P_n(x_i)\|+\|P_n(x_i)-T(x_i)\|<\varepsilon/2+\varepsilon/2=\varepsilon.$$ This finishes the proof. Finally, we show the metric approximation property also for Lipschitz-free spaces over homogeneous spaces of compact metrizable groups. We recall that a *homogeneous space for a group* $G$ is a topological space on which $G$ acts transitively. It can be identified with the left coset space $G/H$, where $H$ is a subgroup, with the quotient topology. To ensure some regularity, we must restrict to compatible metrics which are $G$-invariant, i.e. those so that the action of $G$ is by isometries. \[thm:homogeneousspace\] Let $M$ be a $G$-homogeneous space, where $G$ is a compact group. Suppose that $D$ is a $G$-invariant metric on $M$ that is a quotient of some bi-invariant metric $d$ on $G$. Then ${\mathcal{F}}(M,D)$ has the MAP. We have that $M$ is isomorphic to $G/H$, where is a stabilizer of some point $0\in M$. $H$ is then a closed subgroup. By Theorem \[thm:MAPcompactgrp\], ${\mathcal{F}}(G,d)$ has the MAP. By Proposition \[prop:projection\], there exists a norm one projection $P:{\mathcal{F}}(G)\rightarrow X$, where $X$ is linearly isometric to ${\mathcal{F}}(G/H,D)$. Since MAP is inherited by $1$-complemented subspaces, we are done. Lipschitz-free spaces over finitely generated groups {#section:fingengrps} ==================================================== The goal of this section is to prove Theorem \[thm:intro2\], show examples of situations to which the theorem applies, and provide applications. We mention that the class of Lipschitz-free spaces for which it is known they have the Schauder basis is still rather limited. It is proved in [@HaPe] that ${\mathcal{F}}({\mathbb{R}}^n)$ and ${\mathcal{F}}(\ell_1)$ have a Schauder basis, in [@CuDo] that free spaces over any separable ultrametric space have a monotone Schauder basis, and in [@hajek2017some] that ${\mathcal{F}}(N)$ has a Schauder basis if $N$ is a net in a separable $C(K)$-space. Obviously, it also follows that free spaces isomorphic to these Banach spaces have basis as well, so e.g. by [@kaufmann2015products], ${\mathcal{F}}(B_{\ell_1})$ and ${\mathcal{F}}(B_{{\mathbb{R}}^n})$ have bases. We start by recalling the fundamental idea of geometric group theory - how to view finitely generated groups as metric spaces. Our standard reference for geometric group theory is [@DruKap]. In contrast to the case of compact metrizable groups, we will not consider arbitrary compatible left-invariant metrics on such groups, just certain canonical and maximal, in a sense, ones, called word metrics. Let $G$ be a finitely generated group. Let $S\subseteq G$ be a finite symmetric generating subset (recall that ‘symmetric’ means that for each $s\in S$, also $s^{-1}\in S$). Recall that we can then define a (left-invariant) metric $d_S$, called *word metric*, on $G$ by defining, for $g\neq h\in G$, $$d_S(g,h):=\min\{n\in{\mathbb{N}}\colon \exists s_1,\ldots,s_n\in S\; (g=hs_1\ldots s_n)\}.$$ It is well known and easy to check that by chosing another finite symmetric generating set $T\subseteq G$, the identity map between $(G,d_S)$ and $(G,d_T)$ is bilipschitz. In particular, the isomorphism class of the Banach space ${\mathcal{F}}(G)$ is well-defined. Since every finitely generated group with its word metric is a countable proper metric space, it immediately follows from [@dalet2015free] that ${\mathcal{F}}(G)$ has the MAP. Therefore we will aim for stronger properties and indeed we shall present a class of finitely generated group such that free spaces over any of them has the Schauder basis. Fix now some finitely generated group $G$ and a finite symmetric generating set $S$. Next choose arbitrarily a linear order on the set $S$. Consider now $S$ as an alphabet, i.e. its elements are considered to be letters, and denote by $W$ the set of all *reduced words* over the alphabet $S$. That is, each element $w$ of $W$ is a string of symbols $s_1 s_2\ldots s_n$ from $S$ such that for no $i<n$, the letters $s_i$ and $s_{i+1}$ are inverses of each other when viewed as group elements. For each $w\in W$, - by $|w|$ we denote the length of the word, i.e. the number of its letters; - by $w_G$ we denote the corresponding group element of $G$, i.e. we evaluate the letters of $w$ as elements of $G$; - for every $i<|w|$, by $w(i)$, we denote the $i$-th letter of $w$, and by $w(\leq i)$, we denote the word obtained from $w$ by taking the first $i$ letters. The fixed linear order on the set $S$ defines a lexicographical order on $W$ which we shall denote by $\preceq'$. We define another linear order $\preceq$ on $W$, called the *shortlex* order, by setting, for $w,v\in W$, $$w\preceq v \text{ if either }|w|<|v|, \text{ or }|w|=|v|\text{ and }w\preceq' v.$$ For an element $g\in G$, by $W_g$ we denote the set $\{w\in W\colon w_G=g\}$ and by $w_g$ the minimal element of the set $W_g$, which is easily verified to exist, in $\preceq$. If there is no danger of confusion, for an element $g\in G$ we shall denote by $|g|$ the number $|w_g|$ which is equal to $d_S(g,1_G)$. Finally, we use the linear order $\preceq$ on $W$ to define a linear order $\leq$ on $G$. For $g,h\in G$ we set $$g\leq h\text{ if }w_g\preceq w_h.$$ We call $G$ *shortlex combable*, with respect to a fixed symmetric generating set $S$ and a linear order on $S$, if there exists a constant $K\geq 1$ such that for every $g,h\in G$ with $d_S(g,h)=1$ and for every $i\leq \min\{d_S(g,1_G),d_S(h,1_G)\}$ we have $$d_S((w_g(\leq i))_G, (w_h(\leq i))_G)\leq K.$$ The constant $K$ will be called the *combability constant* of $G$. First we show how such groups are useful for our purposes. Then we provide examples and show some applications. The following is the main result of this section. \[thm:shortlex\] Let $G$ be a finitely generated shortlex combable group (with respect to $S$ equipped with some linear order). Then ${\mathcal{F}}(G,d_S)$ has a Schauder basis. Since the linear order $\leq$ on $G$ is clearly isomorphic to the standard order on ${\mathbb{N}}$, we can use it to enumerate $G$ as $(g_n)_{n\in{\mathbb{N}}}$. For each $n\in{\mathbb{N}}$, set $G_n:=\{g_i\colon i\leq n\}$. For each $n$ we now define maps $P_n: G\rightarrow G_n$ as follows. First, set $m=\max\{|h|\colon h\in G_n\}$, then for $g\in G$, set $$P_n(g):=\begin{cases} g & \text{if }g\in G_n\\ (w_g(\leq m))_G & \text{if }g\notin G_n, (w_g(\leq m))_G\in G_n\\ (w_g(\leq m-1))_G & \text{otherwise}. \end{cases}$$ We leave to the reader the straightforward verification that $P_n$ is well defined.\ [**Claim.**]{} The maps $(P_n)_{n\in{\mathbb{N}}}$ are uniformly bounded Lipschitz commutting retractions.\ First we check that for every $n,m\in{\mathbb{N}}$, $P_n\circ P_m=P_m\circ P_n$, i.e. the maps commute. For every $g\in G$ and $0\leq i\leq |g|$, denote by $g_i$ the element $(w_g(\leq i))_G$. That is, $g_0=1_G$, $g_{|g|}=g$, and $g_0,g_1,\ldots,g_{|g|}$ is a geodesic path in $G$ from the identity element $1_G$ to $g$. It is easy to see that for each $n\in{\mathbb{N}}$, $P_n(g)=g_i$, where $i$ is the largest integer so that $g_i\in G_n$. With this observation it is now clear that the maps $(P_n)_n$ commute. Let $K$ be the combability constant of $G$. We show that each $P_n$ is a $K+1$-Lipschitz retraction on its image. It is obviously a retraction, so it suffices to show that for every $n\in{\mathbb{N}}$ and $g,h\in G$ with $d_S(g,h)=1$ we have $d_S(P_n(g),P_n(h))\leq K+1$. Let such $n$ and $g,h\in G$ be fixed. We distinguish two cases.\ [*Case 1.*]{} At least one of $g,h$ lies in $G_n$: Say that $g\in G_n$. If also $h\in G_n$, then there is nothing to prove since $P_n(g)=g$ and $P_n(h)=h$. So suppose that $h\notin G_n$. Necessarily we have $h_{|h|-1}\leq g$ since otherwise the path $1_G, g_1,\ldots, g, h$ would be a geodesic path from $1_G$ to $h$ of length $|h|$ smaller in the lexicographical ordering than the path $1_G, h_1,\ldots,h_{|h|-1},h$. It follows that $h_{|h|-1}\in G_n$, so $P_n(h)=h_{|h|-1}$. Since $P_n(g)=g$ we get $$d_S(P_n(g),P_n(h))\leq 2.$$\ [*Case 2.*]{} We have $g,h\notin G_n$. Note that then $P_n(g)=g_i$ and $P_n(h)=h_j$, for some $i,j< \max\{|g|,|h|\}$, where $|i-j|\leq 1$. Since by the definition of shortlex combability, $d_S(g_i,h_i)\leq K$ and $d_S(g_j,h_j)\leq K$, we get that $$d_S(P_n(g),P_n(h))=d_S(g_i,h_j)\leq K+1.$$ This finishes the proof of the claim.\ Finally, for each $n\in{\mathbb{N}}$ we denote by $L_n: {\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G_n)$ the lift of $P_n$, the unique linear operator extending $P_n$. The properties of $(P_n)_n$ imply that - for each $n\in{\mathbb{N}}$, the map $L_n$ is a linear projection onto a finite-dimensional subspace of norm bounded by $K+1$; - the projections $(L_n)_n$ commute; - the dimension of the range $L_n[{\mathcal{F}}(G)]$ is the dimension of ${\mathcal{F}}(G_n)$, which is equal to $n$. Since now obviously for every $x\in {\mathcal{F}}(G)$ (notice that it suffices to verify it for the dense set $\mathrm{span}\{\delta_g\colon g\in G\}$) we have $$\lim_{n\to\infty} L_n(x)=x,$$ then by [@AlbiacKalton Proposition 1.1.7] ${\mathcal{F}}(G)$ possesses a Schauder basis. Examples {#subsec:examples} -------- [*Finitely generated abelian groups*]{}. Recall that every infinite finitely generated abelian group $A$ is of the form ${\mathbb{Z}}^n\oplus F$, where $n\geq 1$ and $F=\{0,f_1,\ldots,f_m\}$ is a finite abelian group. Let $e_1,\ldots,e_n$ be the canonical generators of ${\mathbb{Z}}^n$. Then $e_1\leq -e_1\leq e_2\leq -e_2\leq\ldots\leq e_n\leq -e_n\leq f_1\leq \ldots\leq f_m$ is a linearly ordered finite symmetric generating set. We leave to the reader to verify that with this ordered generating set $A$ is shortlex combable.\ [*Free groups*]{}. Let $n\geq 1$ and let $F_n$ be a free group on $n$ generators (which is ${\mathbb{Z}}$ for $n=1$). Let $a_1,\ldots,a_n$ be the free generators. It is again an exercise that with the order $a_1\leq a^{-1}_1\leq\ldots\leq a_n\leq a_n^{-1}$, the group $F_n$ is shortlex combable. We note however, that ${\mathcal{F}}(F_n)$ is linearly isometric to $\ell_1$, thus admits monotone Schauder basis. Indeed, this can be verified directly by noticing that the set $\{\delta(g)-\delta(h)\in{\mathcal{F}}(F_n)\colon d(g,h)=1, d(g,1)>d(h,1)\}$ is equivalent to the standard basis of $\ell_1$.\ [*Hyperbolic groups*]{}. Recall that a geodesic metric space $(M,d)$ is (Rips)-hyperbolic if there exists a constant $K\geq 0$, *hyperbolicity constant*, such that for any triple of points $x,y,z\in M$ and geodesic segments $S_1,S_2,S_3$ connecting each pair of the triple we have $d_{H}(S_i,S_j\cup S_k)\leq K$, where $d_H$ is the Hausdorff distance and $i,j,k$ are pairwise different from $\{1,2,3\}$. In other words, for each $i\leq 3$ and each point $x\in S_i$, $d(x,S_j\cup S_k)\leq K$, where $S_j$ and $S_k$ are the other geodesics besides $S_i$. The notion of hyperbolicity makes sense even for metric spaces which are not literally geodesic, but when a reasonable notion of geodesic segment can be defined. This is the case e.g. for finitely generated groups with word metrics, where a geodesic segment between elements $x,y\in G$ is a sequence $g_0=x,\ldots,g_n=y$, where $n=d(x,y)$, and $d(g_i,g_{i+1})=1$, for $i<n$. Let $G$ be a finitely generated hyperbolic group (with hyperbolicity constant $K$), generated by elements $a_1,\ldots,a_n$. We claim that $G$ with the ordered generating set $a_1\leq a^{-1}_1\leq\ldots\leq a_n\leq a_n^{-1}$ is shortlex combable. Indeed, pick $g,h\in G$ with $d(g,h)=1$, and $i<\max\{|g|,|h|\}$. We show that $d(g_i,h_i)\leq 2K+2$. We have two cases. 1. Either $g_i=g$ or $h_i=h$ (or both). Then it is clear that $d(g_i,h_i)\leq 2$. 2. We have $g_i\neq g$ and $h_i\neq h$. There are geodesic segments $g_0=1_G,\ldots,g_i,\ldots,g$, $h_0=1_G,\ldots,h_i,\ldots,h$, and $g,h$ (of length $1$). We consider thr triple of points $1_G,x,y$ and the geodesic segments between them as above. By definition of hyperbolicity with constant $K$, there exists point $z\in\{1_G=h_0,\ldots, h,g\}$ such that $d(g_i,z)\leq K$. Assume first that $z=h_j$, for some $j\leq |h|$. - $j\geq i$: Since $d(g_i,h_j)\leq K$, by triangle inequality we get that $j\leq i+K$, therefore $d(g_i,h_i)\leq 2K$. - $j<i$: Again by triangle inequality we get that $j\geq i-K$, so $d(g_i,h_i)\leq 2K$. Finally, if $z=g$, then $d(g_i,h)\leq K+1$, so again by triangle inequality we get $|h|\leq i+K+1$, so $d(g_i,h_i)\leq 2(K+1)$.\ [*Large-type Artin groups.*]{} Holt and Rees in [@HoRe] prove that large-type Artin groups are shortlex automatic which immediately from the definition implies being shortlex combable. Artin groups in general belong to one of the most studied classes of groups in geometric group theory. We refer the reader to [@HoRe] for the notion of shortlex automaticity and for the definition of large-type Artin groups. We do not know whether there are groups that are shortlex combable but not shortlex automatic. We refer to [@HoRe] for details. Applications ------------ Our main goal is to prove that for any net $N$ in a real hyperbolic $n$-space ${\mathbb{H}}^n$ (whose definition we recall later), we have that ${\mathcal{F}}(N)$ has the Schauder basis. This will be an immediate consequence of Theorem \[thm:shortlex\] and several standard more general results that we present below. We recall that an action of a group $G$ on a metric space $X$ by isometries is *free* if for every $g\in G$ and $x\in X$, $g\cdot x\neq x$, and *cocompact* if there exists a compact set $K\subseteq X$ such that $\bigcup_{g\in G} g\cdot K=X$. These actions, or rather more generally properly discontinuous cocompact actions (see [@DruKap Chapter 5]), are one of the most studied in geometric group theory. \[prop:actinggroup\] Let $G$ be a finitely generated group acting freely and cocompactly on a proper geodesic metric space $X$ by isometries. Then for every net $N\subseteq X$, we have ${\mathcal{F}}(N)\simeq {\mathcal{F}}(G)$. Let $G$, $X$ and an action of $G$ on $X$ as in the statement of the proposition be fixed. First we invoke [@hajek2017some Proposition 5] which says that for any two nets $N_1,N_2\subseteq X$ we have ${\mathcal{F}}(N_1)\simeq{\mathcal{F}}(N_2)$. It follows that it suffices to find one net $N\subseteq X$ such that ${\mathcal{F}}(N)\simeq{\mathcal{F}}(G)$. Let $0\in X$ be a distinguished point and let $N$ be the $G$-orbit of $0$. Since $X$ is proper and the action is cocompact, it follows that $N$ is a net in $X$. To show that ${\mathcal{F}}(N)\simeq{\mathcal{F}}(G)$ it suffices to prove that $N$ with the restriction of the metric on $X$ is bi-Lipschitz to $G$ (with its word metric). By the Milnor-Schwarz lemma (see [@DruKap Theorem 8.37]), the map $\phi:G\rightarrow N$ defined by $\phi(g):=g\cdot 0$ is a quasi-isometry (we refer reader not familiar with quasi-isometries again to [@DruKap]). Since the action is free, it is also a bijection. It is easy to check that a bijective quasi-isometry between two uniformly discrete metric spaces is in fact a bi-Lipschitz equivalence. This finishes the proof. We now recall the definition of the real hyperbolic $n$-space. There are many definitions and we refer the reader to [@DruKap Chapter 4] for a more thorough discussion. We define ${\mathbb{H}}^n$, for $n\geq 2$, as follows. First we define the following quadratic form; for $x,y\in{\mathbb{R}}^{n+1}$: $$\langle x,y\rangle:=\sum_{i=1}^n x_iy_i-x_{n+1}y_{n+1},$$ and we set $${\mathbb{H}}^n:=\{x\in{\mathbb{R}}^{n+1}\colon \langle x,x\rangle=-1,x_{n+1}>0\}.$$ A metric $d$ on ${\mathbb{H}}^n$ can be defined using the formula, for $x,y\in{\mathbb{H}}^n$, $$\cosh d(x,y)=-\langle x,y\rangle.$$ We now state the main result of this subsection. \[cor:hyperbolicnet\] Let $n\geq 2$ and let $N\subseteq {\mathbb{H}}^n$ be a net. Then ${\mathcal{F}}(N)$ has a Schauder basis. It suffices to find a group $G$ acting freely and cocompactly on ${\mathbb{H}}^n$. Indeed, again by the Milnor-Schwarz lemma ([@DruKap Theorem 8.37]), such $G$ is then finitely generated and quasi-isometric to ${\mathbb{H}}^n$. It follows that $G$ is hyperbolic ([@DruKap Observation 11.125]). Therefore, as we demonstrated in Subsection \[subsec:examples\], $G$ is shortlex combable. Applying Theorem \[thm:shortlex\], we get that ${\mathcal{F}}(G)$ has a Schauder basis. Finally, an application of Proposition \[prop:actinggroup\] finishes the argument. In order to find a group acting freely and cocompactly on ${\mathbb{H}}^n$, one can use several standard results from Riemannian geometry. Let $M$ be an arbitrary $n$-dimensional compact Riemannian manifold without boundary of constant sectional curvature $-1$ equipped with some Riemannian metric. By [@BrHa Theorem 3.32], its universal cover (refer to any standard textbook on algebraic topology, e.g. [@Hatch]) is isometric to ${\mathbb{H}}^n$. Another standard argument from algebraic topology (see e.g. [@Hatch Proposition 1.39]) shows that the fundamental group $\pi_1(M)$ acts on the universal cover ${\mathbb{H}}^n$ by deck transformations, which is a free cocompact action by isometries. This finishes the proof. We remark that in contrast to the Euclidean space ${\mathbb{R}}^n$, in which there exist two non Lipschitz equivalent nets ([@BuKl]), it has been proved in [@Bog] that all nets in ${\mathbb{H}}^n$ are bi-Lipschitz equivalent. Let $\mathcal{G}$ denote the set of all finitely generated hyperbolic groups. A fair question is how many isomorphism types of Banach spaces the set $\{{\mathcal{F}}(G)\colon G\in\mathcal{G}\}$ contains. As we mentioned, since the free group $F_n$ is hyperbolic, it contains ${\mathcal{F}}(F_n)\simeq \ell_1$. It is unknown to us whether or not there exists $G\in\mathcal{G}$ such that ${\mathcal{F}}(G)\not\simeq \ell_1$. Thus it is relevant at this point to reiterate [@candido2019isomorphisms Question 1], which asks precisely about an example of $G\in\mathcal{G}$ with ${\mathcal{F}}(G)\not\simeq\ell_1$, and the discussion following the question. Either answer would bring interesting consequences. If there are such $G$, then we have, potentially many, new examples of Lipschitz-free spaces with Schauder basis. If on the other hand for every $G\in\mathcal{G}$, ${\mathcal{F}}(G)\simeq \ell_1$, then we have an example of a group with Kazhdan’s property (T) that has a metrically proper action by isometries on a renorming of $\ell_1$. We refer to [@candido2019isomorphisms Question 1] where this is properly discussed and the importance of such a result is explained. Problems and Notes {#section:problems} ================== Let us finish by posing some natural questions that follow up this work. The first is whether or not we can generalize Theorem \[thm:intro1\] to locally compact groups: Let $G$ be a locally compact group equipped with a compatible and left-invariant metric. Does ${\mathcal{F}}(G)$ have the MAP? In the noncompact case, we have positive answer for finite dimensional Banach spaces with their norm induced metrics. Although this would be a consequence of the mentioned result from [@godefroy2003lipschitz] which states that a Banach space has $\lambda$-BAP if and only if ${\mathcal{F}}(X)$ does, actually the order of the proofs is reversed. First, Godefroy and Kalton prove that, for finite dimensional Banach spaces $X$, ${\mathcal{F}}(X)$ has the MAP, and then use this to prove the result for general Banach spaces. The proof of the finite dimensional part also involves harmonic averaging. Since Lipschitz-free spaces over finite dimensional Banach spaces even possess a Schauder basis ([@HaPe]), we suggest to turn the attention to locally compact metric groups that are very closely related to such Banach spaces; that is, Carnot groups. We note that a Carnot group is both analytically and algebraically a mild generalization of a finite dimensional Banach space and many results from geometric analysis on Euclidean spaces have been generalized to the setting of Carnot groups (see [@LeD]). The duals of Lipschitz-free spaces over Carnot groups have been already investigated in [@candido2019isomorphisms] and their Lipschitz-free spaces in [@albiac2020lipschitz]. Let $G$ be a Carnot group with a Carnot-Carath' eodory metric. Does ${\mathcal{F}}(G)$ has the Schauder basis? In the case of connected Lie groups, one has a canonical compatible left-invariant metric: the left-invariant Riemannian metric. Since such a metric is locally bi-Lipschitz equivalent to the Euclidean metric, and isomorphic properties of Lipschitz-free spaces often depend only on the local behaviour of the metric, the following question is very natural. We note that one could even replace Lie group there with a general connected Riemannian manifold (without boundary). \[quest:liegrp\] Let $G$ be a connected (real) Lie group equipped with a left-invariant Riemannian metric. Do we have ${\mathcal{F}}(G)\simeq {\mathcal{F}}({\mathbb{R}}^n)$, where $n$ is the dimension of the (real) Lie algebra $\mathfrak{g}$ of $G$? Still in the compact setting, we note that we only required in Theorem \[thm:homogeneousspace\] that the metric in $G$-homogeneous space is a quotient of a metric in $G$ so that we could apply directly Proposition \[prop:projection\]. So one could ask if we can drop this condition. Let $G$ be a (locally) compact group, $M$ be a compact $G$-homogeneous space, and $d$ be a compatible $G$-invariant metric on $M$. Does ${\mathcal{F}}(G)$ have the MAP? In Appendix A, the reader will find a positive answer in the specific case of the sphere ${\mathbb{S}}^{n-1}=O(n)/O(n-1)$. It is natural also to ask about generalizations of Theorem \[thm:intro2\]. Let $G$ be a finitely generated group equipped with a word metric. Does ${\mathcal{F}}(G)$ admit a Schauder basis, or at least a finite dimensional decomposition? In Corollary \[cor:hyperbolicnet\], we have shown that for any net $N\subseteq {\mathbb{H}}^n$, ${\mathcal{F}}(N)$ has a Schauder basis. Given the prominence of the space ${\mathbb{H}}^n$ in geometry and beyond, it is important to understand the Lipschitz-free space of ${\mathbb{H}}^n$ itself. We also note that the hyperbolic spaces ${\mathbb{H}}^n$ together with the Euclidean spaces ${\mathbb{R}}^n$ and Euclidean spheres ${\mathbb{S}}^n$ are important as the model spaces of spaces of constant curvature (see e.g. [@BrHa] for a thorough treatment). Since for ${\mathbb{R}}^n$ and ${\mathbb{S}}^n$ with their canonical Euclidean metrics we have by [@albiac2020lipschitz Theorem 4.21], ${\mathcal{F}}({\mathbb{R}}^n)\simeq{\mathcal{F}}({\mathbb{S}}^n)$, the answer to the following would be desirable. Does ${\mathcal{F}}({\mathbb{H}}^n)$ have the Schauder basis? Do we have ${\mathcal{F}}({\mathbb{H}}^n)\simeq {\mathcal{F}}({\mathbb{R}}^n)$? We note that the previous question is related to the stronger version of Question \[quest:liegrp\] that considers a general Riemannian manifold since ${\mathbb{H}}^n$ is a Riemannian manifold with Riemannian metric (see [@BrHa Proposition 6.17]). On the MAP for ${\mathcal{F}}({\mathbb{S}}^n)$ {#appendSphere} ============================================== Let $d\geq 2$ and let ${\mathbb{S}}^{d-1}=O(d)/O(d-1)$ be the $(d-1)$-dimensional sphere equipped with a rotation-invariant metric $D$ which is compatible with the usual topology. Here, we are not assuming that $D$ is a quotient metric, as the ones described in Subsection \[subsection:generalCpt\]. Let us show that ${\mathcal{F}}({\mathbb{S}}^{d-1},D)$ has the MAP. The proof will also follow from Proposition \[tool\] and summability results. We recall the definitions and results from harmonic analysis that we will use, and establish again a Lipschitz version of Young’s convolution inequality for our setting. Equip ${\mathbb{S}}^{d-1}$ with its surface area measure $\sigma$, and denote $\omega_d=\sigma({\mathbb{S}}^{d-1})$. Convolution on ${\mathbb{S}}^{d-1}$ can be defined as follows. First let $\Lambda = (d-2)/2$, and consider the weighted $L^1$ space $L^1(w_\Lambda,[-1,1])$, where $w_\Lambda (x) = (1-x^2)^{\Lambda - 1/2}$ for each $x\in ]-1,1[$. Recall that the norm in $L^1(w_\Lambda,[-1,1])$ is defined by $$\|f\|_{\Lambda,1} = c_\Lambda \int_{-1}^1 |f(x)|w_\Lambda (x)dx,$$ where $c_\Lambda$ is the normalization constant such that $c_\Lambda \int_{-1}^1w_\Lambda (x)dx =1$. For each $f\in L^1({\mathbb{S}}^{d-1})$ and $g\in L^1(w_\Lambda,[-1,1])$, the convolution $f\ast g: {\mathbb{S}}^{d-1}\rightarrow {\mathbb{R}}$ is defined by $$(f\ast g)(x)= \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} f(y)g(x\cdot y)d\sigma(y).$$ We now prove the validity of Young’s inequality. Let $f\in \mathrm{Lip}({\mathbb{S}}^{d-1})$ and $g \in L^1(w_\Lambda,[-1,1])$, and let $x,y\in {\mathbb{S}}^{d-1}$. There is a rotation $R$ with $y=Rx$, thus $$\begin{aligned} |f*g(x) - f*g(y)| & = \left| \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \frac{1}{\omega_n}\int_{{\mathbb{S}}^{d-1}} f(z)g(Rx\cdot z)d\sigma(z) \right|\\ & = \frac{1}{\omega_d}\left| \int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \int_{{\mathbb{S}}^{d-1}} f(Rz)g(Rx\cdot Rz)d\sigma(z) \right|\\ & = \frac{1}{\omega_d}\left| \int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \int_{{\mathbb{S}}^{d-1}} f(Rz)g(x\cdot z)d\sigma(z) \right|\\ & \leq \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |f(z)-f(Rz)||g(x\cdot z)|d\sigma(z) \\ & \leq \|f\|_{\mathrm{Lip}}\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} D(z,Rz)|g(x\cdot z)|d\sigma(z)\\ & = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |g(x\cdot z)|d\sigma(z)\\ & = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_n}\int_{-1}^1 |g(t)|w_\Lambda (t) dt \leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}D(x,y).\end{aligned}$$ It follows that $f\ast g\in\mathrm{Lip}({\mathbb{S}}^{d-1})$, and $\|f\ast g\|_{\mathrm{Lip}}\leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}.$ The finite dimensional space of real homogeneous harmonic polynomials of degree $n$ on ${\mathbb{R}}^d$ restricted to ${\mathbb{S}}^{d-1}$ is denoted by $\mathcal{H}^d_n$. These spaces are mutually orthogonal with respect to the inner product $$\langle f,g\rangle_{{\mathbb{S}}^{d-1}} = \frac{1}{\omega_n}\int_{{\mathbb{S}}^{d-1}} f(x)g(x)d\sigma(x)$$ and they densely span $L^2({\mathbb{S}}^{d-1})$ (see e.g. [@dai2013approximation], Theorem 2.2.2). Denoting by proj$_n$ the corresponding projection operators, we can associate the partial sum operators $$S_n f=\sum_{k=0}^n \mbox{proj}_nf.$$ These are finite-rank and satisfy $S_nf = f\ast K_n$, where $$K_n(t) = \sum_{k=0}^n \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t)$$ and $C_k^\Lambda$ are the Gegenbauer polynomials ([@dai2013approximation], Proposition 2.2.1). Fix $\delta\geq d-1$ and consider the averages $$K^\delta_n(t) = \frac{1}{A_n^\delta}\sum_{k=0}^n A_{n-k}^\delta \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t),$$ where $A_k^\delta = \binom{k+\delta}{k}=\frac{(\delta+k)(\delta+k-1)\dots(\delta+1)}{k!}$. These give rise to a sequence of finite-rank operators on $L^2({\mathbb{S}}^{d-1})$ defined by $S_n^\delta: f\mapsto f\ast K_n^\delta$. Write $\Lambda_n^\delta:=\|S_n^\delta\|_1= \sup \{\|S_n^\delta h\|_1: h\in B_{L^1({\mathbb{S}}^{d-1})}\}$. By [@dai2013approximation Theorem 2.4.3], for each $n\in {\mathbb{N}}$, $S_n^\delta$ is a nonnegative operator, which implies that $K_n^\delta(t)\geq 0$, $t\in [-1,1]$. It follows that $$\Lambda_n^\delta=\sup_{\|f\|_1 \leq 1} \|f\ast K_n^\delta\|_1 \geq \|1\ast K_n^\delta\|_1 = \|K_n^\delta\|_{\Lambda,1}.$$ It is clear that also $S_n^\delta(C({\mathbb{S}}^{d-1}))\subset C({\mathbb{S}}^{d-1})$ and that its restriction to $C({\mathbb{S}}^{d-1})$ is continuous in the uniform norm. Moreover, by [@dai2013approximation Corollary 2.4.5], when $f$ is continuous (and in particular when it is Lipschitz) on ${\mathbb{S}}^{d-1}$, we have that $S_n^\delta f$ converges to $f$ uniformly. On the other hand, by Young’s inequality, for each $f\in \mathrm{Lip}({\mathbb{S}}^{d-1})$ we have that $$\|S_n^\delta f\|_{\mathrm{Lip}} = \|f\ast K_n^\delta \|_{\mathrm{Lip}} \leq \|f\|_{\mathrm{Lip}} \| K_n^\delta \|_{\Lambda,1}\leq \Lambda_n^\delta \|f\|_{\mathrm{Lip}}.$$ To conclude, let $\varepsilon>0$. By [@dai2013approximation Theorem2.4.4], there is some $n_\varepsilon$ such that $\Lambda_n^\delta<1+\varepsilon$, for each $n\geq n_\varepsilon$. Thus $S_n^\delta$, $n\geq n_\varepsilon$ satisfy conditions (a)–(d) in Proposition \[tool\] and we are done. [**Acknowledgement.**]{} We would like to thank to Gilles Godefroy who suggested to us to study these problems. [^1]: P. L. Kaufmann was supported by grants 2017/18623-5 and 2016/25574-8, São Paulo Research Foundation (FAPESP). M. Doucha was supported by the GAČR project 19-05271Y and RVO: 67985840.

ArXiv

--- abstract: 'Recently, Odrzywolek and Rafelski [@exoplanetclass] have found three distinct categories of exoplanets, when they are classified based on density. We first carry out a similar classification of exoplanets according to their density using the Gaussian Mixture Model, followed by information theoretic criterion (AIC and BIC) to determine the optimum number of components. Such a one-dimensional classification favors two components using AIC and three using BIC, but the statistical significance from both the tests is not significant enough to decisively pick the best model between two and three components. We then extend this GMM-based classification to two dimensions by using both the density and the Earth similarity index [@Kashyap], which is a measure of how similar each planet is compared to the Earth. For this two-dimensional classification, both AIC and BIC provide decisive evidence in favor of three components.' address: - '$^1$Dept. of Physics, University of Florida, Gainesville, Florida, 32611, USA' - '$^2$Dept. of Physics, Indian Institute of Technology, Hyderabad, Kandi, Telangana 502285, India' author: - Soham Kulkarni$^1$ - Shantanu Desai$^2$ bibliography: - 'exoplanet.bib' title: Classifying Exoplanets with Gaussian Mixture Model --- Introduction {#sec:intro} ============ Over the past two decades, there has been a revolution in the field of exoplanet astronomy following the confirmation of more than 3000 planets orbiting stars other than the Sun (see Ref. [@Rice] for a recent review and Ref. [@Wei18] for a summary of exoplanet detection techniques). Lot of work has been done to characterize the properties of the exoplanets discovered using the myriad techniques [@lunine]. Recently, Odrzywolek and Rafelski [@exoplanetclass] (hereafter OR16) have carried out the classification of exoplanets according to their density, following a suggestion long time back [@Weisskopf]. OR16 fitted the exoplanet density data to lognormal distributions to determine the optimum number of components. They found three lognormal components with peak densities at 0.71 $~\rm{gm/cm^3}$, 6.9 $~\rm{gm/cm^3}$, and 29.1 $~\rm{gm/cm^3}$ [@exoplanetclass]. These three components correspond to ice/gas giants, iron/rock super-Earths, and brown dwarfs respectively. The optimum number of components was determined by maximizing the log-likelihood and then checking the goodness of fit for different number of components by calculating the $p$-value from three distinct non-parametric tests. We would like to do a variant of the above analysis by carrying out a similar classification according to density using Gaussian mixture models, followed by information theoretic criterion to determine the optimum number of classes. We have previously used this procedure, to perform a unified classification of all GRB datasets using three different model comparison techniques [@Kulkarni]. We then extend the analysis of OR16 by considering two-dimensional classification using both the density and Earth similarity index. This paper is organized as follows. In Section \[sec:data\], we describe the dataset and the physical quantities used for the classification. The mathematical basis for the classification is discussed in Section \[sec:analysis\]. Our results are shown in Section \[sec:results\] and we conclude in Section \[sec:conclusions\]. Data {#sec:data} ==== Exoplanet Catalog ----------------- In the one-dimensional classification, we shall study the trends obtained from the confirmed detections, by classifying the exoplanet database according to their densities, for which we need the mass and radius of the planets. We obtain the mass and radius information from the catalogs uploaded on the NASA Exoplanet archive[^1] and the Extrasolar planet encyclopedia[^2] as of **February 18, 2017**. From these datasets, we consider only those planets with measured values of mass and density, and which exist in both the datasets with the same observed values to avoid any irregularities and to maintain consistency in the dataset. The NASA Exoplanet archive is a NASA funded public data service, which is hosted by the Infrared Processing and Analysis Center. This catalog lists only those objects, for which their detection and planetary status is sacrosanct. As of Feb 18, 2017, it contained a total of 3440 planets out of which 531 have measured mass and radius values detected. Most of the planets listed in this catalog have been detected using transit photometry. The Extrasolar planet encyclopedia is maintained by the Meudon Observatory in Paris and as of Feb 18, 2017 contained total of 3567 planets (most of which were also detected using transit photometry), of which 615 have measured values for all the parameters. The data provided by the two catalogs is similar except for some differences in their selection criteria. The Extrasolar planet encyclopedia allows planets weighing from 60 Jupiter Mass onwards, whereas the NASA Exoplanet archive uses 30 Jupiter mass as the lower limit, which is also the reason for the smaller number of confirmed exoplanets in the latter. However, one caveat is that the catalog is continuously updated and sometimes false detections are removed as the data gets subjected to more scrutiny. Therefore, in order to obtain a gold sample, we have selected 450 observations, which are common to both the datasets for our study in this paper. Both the datasets used for this analysis as well as the code which looks for common planets between the two catalogs have been uploaded on [github]{} and can be found at <https://github.com/IITH/Exoplanet-Classification>\ In addition to the one-dimensional classification using only density, we also carry out a two-dimensional classification, wherein we use both the density and the Earth Similarity Index (or ESI) [@Kashyap] for the classification. For this, we need some additional parameters for the calculation of ESI. The additional parameters that we need apart from the radius and density are the surface temperature and time period of revolution, as other parameters can be derived from the mass and radius. The escape velocity and surface gravity are calculated by positing that the shape of the planet is a perfect sphere, wherein the total mass is distributed uniformly throughout the volume. We only consider planets for which we have the observed values for all four of these parameters. Calculations for the data: -------------------------- Assuming the planet is a perfect sphere with a uniform mass distribution, the expression for density is: $$\bar{\rho} = \frac{M}{\frac{4}{3}\pi R^{3}}$$ The escape velocity is given by: $$v_{esc} = \sqrt{\frac{2GM}{R}}.$$ The surface gravity is obtained from: $$g_{surf} = \frac{GM}{R^2}.$$ where $G$ is the Gravitational Constant, $M$ is the mass of the planet and $R$ is the radius.\ ESI is a figure of merit used to ascertain how habitable is the planet for life to develop compared to the Earth. More details on the theory behind ESI can be found in the work by Kashyap [@Kashyap], which in turn follows the prescription from Schulze-Makuch et al. [@Schulze] (See also  [@Moya] for alternate indices proposed similar in spirit to ESI). The ESI is based on six different parameters, viz. density, radius, temperature, surface gravity, escape velocity, and the time period of revolution around their Sun. All these parameters are normalized to Earth units, as it is convenient for the index calculation. The ESI is calculated based on the Bray-Curtis Similarity index [@Bray] and is given by: $$ESI_{x} = \left(1- \left|\frac{x-x_0}{x+x_0}\right|\right)^w \label{eq:ESI}$$ where $x$ is the parameter for which the index has to be calculated, $x_0$ is the reference values which in our case is one, as we have expressed all parameters in Earth units and $w$ is the weight exponent. The total ESI is given by: $$ESI = \left(ESI_{g}\times ESI_{temp}\times ESI_{vesc}\times ESI_{p}\times ESI_{r}\times ESI_{d}\right)^{1/6}$$ The values of ESI range from 0 (completely different from Earth) to 1 (resembling a clone of Earth). Analysis Methods: {#sec:analysis} ================= We outline the method used for both the one-dimensional classification using density and the two-dimensional classification using density and ESI. For finding the best-fit parameters, we use the Gaussian-mixture Model (GMM) [@astroml], which is part of the [Scikit-learn]{} package, used for a variety of machine learning applications in python. The GMM fits the data to a mixture of multiple ($k$) lognormal Gaussian distributions, which are characterized by their mean, covariance and their respective weights in the fit data. The GMM method uses the Expectation Maximization (EM) algorithm [@EM] to maximize the likelihood function over the given parameter space. The GMM method can also be generalized to include error bars and this generalized GMM algorithm is referred to in the astrophysics literature as Extreme Deconvolution [@ED]. However, since we are using a planet catalog measured in two separate datasets having negligible error bars, we stick to the ordinary GMM method. Given the probability distribution function $f(x,\theta)$, where $x$ are the observed datapoints, $\theta$ are the parameters used to define the function, $N$ being the total number of exoplanets in our study, and $w_k$ denotes the weights associated with each of the $k$ log normal distributions, the likelihood can be defined as: $$\mathcal{L} = \sum\limits_{i=1}^{N} \sum_{j=1}^{k} w_{j}f_{j}(x_{i},\theta), \label{eq:likelihood}$$ and the probability distribution function for a univariate Gaussian as: $$f(x,\theta) = \frac{1}{\sqrt{2\pi} \sigma} \exp\left(- \frac{(x-\log \rho_{planet})^{2}}{2\sigma^{2}}\right).$$ A generalized bivariate Gaussian distribution can be defined as: $$f(x,\theta) = \frac{1}{2\pi \sigma_1 \sigma_2 \sqrt{1-\rho'^2}} \exp\left[- \frac{1}{2(1-\rho'^2)}\left( \frac{(x-\mu_{x})^{2}}{\sigma_1^2} + \frac{2\rho (x-\mu_{x})(y-\mu_y)}{\sigma_1 \sigma_2} + \frac{(y-\mu_y)^{2}}{\sigma_2^2}\right)\right] \label{eq:2dfit}$$ where $\rho' = \frac{V_{12}}{\sigma_1 \sigma_2}$ is the correlation, $V$ is the covariance of the two variables and $\mu_x$ is the mean log(density) and $\mu_y$ is the mean ESI. An additional condition being used in the EM algorithm is the normalization condition: $$\sum\limits^{k}_{i=1} w_{i} = 1$$ In this study, we use the GMM method for the $k=2$ and $k=3$ lognormal fits to the data followed by information theory based model comparison methods to assess the best fit amongst these two models. Model Comparison ---------------- Once we have obtained the best-fit parameters for each model, we need to select the optimum model from all the possibilities being considered. Naively, the simplest way to do model comparison would be by carrying out likelihood comparison between the competing models and choosing the model with the highest likelihood as the best model. However, the maximization of likelihood could lead to an overfitting of the model to the data with additional degrees of freedom and hence we need a more robust and accurate criterion, which will penalize the use of extra free parameters. This can be done by using the Information criterion tests, such as Akaike Information Criteria (AIC) and the Bayesian Information Criteria (BIC), which are commonly used in Astrophysics literature [@Shi; @Shafer; @Desai16a; @Ganguly; @Liddle] (and references therein). These information criteria-based methods provide a way to penalize the excess free parameters and determine the best model accordingly. ### AIC: {#sec:aic} Akaike Information Criteria or AIC [@Burnham] penalizes lightly the excess free parameters and is defined as: $$AIC = 2p + 2 \ln L \label{eq:aic}$$ where $p$ is the number of free parameters in the model and $L$ is the likelihood. The AIC defined in Eq. \[eq:aic\] is valid when the ratio $N/p$ is very large i.e. $>40$. For a ratio less than this, a first order correction is included and the modified expression is given by: $$AIC = 2p + 2 \ln L + \frac{2p(p+1)}{N-p-1}$$ Throughout our data, the ratio is greater than the value prescribed and hence we do not account for this correction in our study. The preferred model is the one with a lower value of AIC and the efficacy of this hypothesis is determined using the quantity: $$\Delta AIC_i = AIC_{i} - AIC_{min},$$ where $\Delta AIC_i$ value corresponds to the preference of the model $i$ over the model with the lower AIC value and hence is the null hypothesis. The confidence in the model can be determined by the magnitude of the $\Delta AIC$ value. Although one cannot formally calculate $p$-values from $\Delta AIC$, one usually uses qualitative strength of evidence rules to judge the efficacy of a given model [@Shi; @Liddle; @Liddle07]. As pointed out by Liddle [@Liddle], the value for the best model will be, $\Delta AIC_i = 0$. Now, if $0 < \Delta AIC_i < 2$, then we can say that we have a weak or no statistical evidence to reject the $i^{th}$ model over the null hypothesis. $2 < \Delta AIC_i < 6$ implies that the model has only weak support and has evidence against this model. For models with $\Delta AIC_i > 6$ there exists strong evidence against the model and $\Delta AIC_i > 10$ implies a very strong or decisive evidence against the $i^{th}$ model. These rules can be applied directly for the $BIC$ criterion (next subsection) as well. ### BIC: Bayesian Information Criterion or BIC was used by Schwarz [@Schwarz] and is used to penalize the free parameters much more harshly than the AIC criterion and is defined as: $$BIC = p \ln N + 2 \ln L$$ Again, the preferred model is the one with the lower values of BIC and is taken as the null hypothesis for further determining the significance of different models. $$\Delta BIC_i = BIC_{i} - BIC_{min}$$ Similar to the significance test for the AIC criterion, the $\Delta BIC_i$ value acts as the significance measure for the BIC test and follows the same values as for AIC. The only difference being that according to BIC criterion, the penalty for a model with extra number of free parameters is harsher compared to AIC. Results: {#sec:results} ======== 1D classification ----------------- We first describe our results for the one-dimensional classification using only the density. We apply the techniques and methods described in the previous sections to the exoplanet catalog, generated by filtering the data from the NASA Exoplanet archive and the Extrasolar Planet encyclopedia as mentioned earlier. For the density function, we find the best-fit model parameters for $k$ lognormal distributions according to the density from Eq. \[eq:likelihood\]. Each distribution is characterized by its mean, standard deviation and the weight of the distribution indicating the number of planets that have been classified under that particular distribution. We apply the GMM routine to the density functions after varying the number of Gaussians from 1 to 14. ![The GMM based fit for the density of the exoplanets using the best-fit parameters from Eq. \[eq:likelihood\] for $k=2$. Details of the fit can be found in Tab. \[tab:aicbic\].[]{data-label="fig:2ghist"}](2ghist.png) ![The GMM based fit for the density of the exoplanets using the best-fit parameters from Eq. \[eq:likelihood\] for $k=3$. Details of the fit can be found in Table \[tab:aicbic\].[]{data-label="fig:3ghist"}](3ghist.png) ![The AIC and BIC values as a function of the number of Gaussian components used to fit the density of exoplanets. AIC shows a preference for two components, whereas BIC shows a preference for three.[]{data-label="fig:aicbic"}](aicbic_including_points.png) $k$ $ \mu$ $\sigma $ $w_{i}$ $AIC$ $BIC$ $\Delta(AIC) $ $ \Delta(BIC)$ ----- -------- ----------- --------- ------- ------- ---------------- ---------------- 0.88 0.20 322 9.69 1.08 128 0.71 0.17 225 2.03 0.36 175 88.0 0.82 50 \[tab:aicbic\] The scatter plot in Fig. \[fig:fig1\] shows all the selected planets for the study as a function of their mass and radius. The distribution looks clustered in certain areas with lots of outliers. The density distribution of 450 exoplanets with their histograms can be found in Fig. \[fig:2ghist\] and Fig. \[fig:3ghist\] for the 2-Gaussian and 3-Gaussian fits respectively and we can see intuitively that no difference can be discerned by eye from the two figures. Both the models fit well the distribution of the density function, and hence we have to rely on quantitative model comparison tests that have been carried out on the data, viz. the AIC and the BIC test. As seen in Fig. \[fig:aicbic\], the BIC test indicates that the 2-component model is the optimum model as it has the minimum BIC value followed by the 3-component model, which has a larger value than the two component model. This trend is different from the AIC test, as the AIC has a minimum for three Gaussians, indicating that this is the best model, followed by the two-component model. These results if compared to the previous attempts at one-dimensional classification done by OR16 are very similar in both, the two Gaussian and the three Gaussian models proposed in this paper. From the 2-component model, the mean density values are at $0.88 ~\rm{gm/cm^3}$ and $9.69 ~\rm{gm/cm^3}$, with each class containing 322 and 128 exoplanets respectively. The inferred mean values of the density for the 3-component model are at $0.71 ~\rm{gm/cm^3}$, $2.03 ~\rm{gm/cm^3}$ and $88.1 ~\rm{gm/cm^3}$ with 225, 175, and 50 in each of the classes respectively. In the previous study by OR16, the mean density values are at $0.7 ~\rm{gm/cm^3}$ and $6.3 ~\rm{gm/cm^3} $ with 320 and 106 respectively in each class for 2 components and at $0.71 ~\rm{gm/cm^3}$, $6.9 ~\rm{gm/cm^3}$ and $29.1 ~\rm{gm/cm^3}$ with 340, 80, and 7 exoplanets respectively in each class for the 3 components. From Tab. \[tab:aicbic\], we see that for the AIC test, the best model preferred is the 3-Gaussian model but there is sufficient confidence shown in both, the 2-Gaussian model and surprisingly the 4-Gaussian model (see Fig. \[fig:aicbic\]) while rejecting all other models by a huge margin. As described in Sect. \[sec:aic\], the intervals of $\Delta_i$ are well within the range of not having sufficient evidence to reject the 2-component and 4-component models over our null hypothesis of a 3-component model. The BIC test prefers the 2-component model and has weak confidence in the 3-Gaussian model while rejecting all other models by a significant margin and hence rejecting the 4-component model as well from further consideration. Therefore, the results from the two information criterion tests do not agree. However, $\Delta$AIC and $\Delta$BIC are both less than 10 between the two and three component model, so no one model among these is decisively favored between the two. 2D Classification: ------------------ We now proceed to a 2-dimensional GMM based classification using both the logarithm of the density and ESI. We use the combined data from ESI and density using the datasets specified earlier in the manuscript and perform a two-dimensional GMM analysis. We consider only the planets that have measured values for all the quantities required for the calculation of ESI. A total of 450 exoplanets were analyzed for a range of lognormal components.  ![The scatter plot of the distribution using the two components, log (density) and total ESI. The three ellipses represent the $1\sigma$ confidence level region for the 3-component model, which are centered at the means of the distribution acquired from best-fit of Eq. \[eq:likelihood\] and Eq. \[eq:2dfit\].[]{data-label="fig:denesi"}](density2.png) ![AIC and BIC values for the two dimensional GMM analysis (as a function of log(density) and ESI) over the combined data. Both AIC and BIC attain a minimum value for three components. []{data-label="fig:denesiaic"}](esi_den2.png) As we can see from Fig. \[fig:denesiaic\], we get a result that is similar to the one we saw in the above case of 1-D classification, where the 3-component component was preferred only with AIC, albeit with marginal significance, using only the density as a parameter. From this two-dimensional analysis using the total ESI along with the density, we have both AIC and BIC preferring the 3-component distribution over all the other ones and by a substantial margin. The best-fit values of the parameters along with their covariance, as well as the $\Delta AIC$ and $\Delta BIC$ values for the two and three component distributions can be found in Tab. \[tab:aicbic2d\] $k$ $\mu$ $\Sigma$ $w_{i}$ AIC BIC $\Delta(AIC) $ $ \Delta(BIC)$ ----- ----------------- --------------------------------------------------------------------------------------------------------------------- --------- ----- ----- ---------------- ---------------- (-0.063, 0.046) $\left(\begin{array}{cc} 0.24 &0.005\\0.005 &0.0012 \\ \end{array}\right)$ 332 (1.07,0.052) $\left(\begin{array}{cc} 0.902 &-0.01\\ -0.01 &0.0013 \\ \end{array}\right)$ 118 (-0.22,0.042) $\left(\begin{array}{cc} 0.17 &0.0036\\ 0.0036 & 0.012 \\ \end{array}\right)$ 270 (0.57,0.06) $\left(\begin{array}{cc} 0.33 &0.0056\\ 0.0056& 0.0013 \\ \end{array}\right)$ 143 (2.27,0.04) $\left(\begin{array}{cc} 0.68 &-0.011\\ -0.011 & 0.0013 \\ \end{array}\right)$ 37 The AIC and BIC tests both point to definitive evidence for one model (three components) and give concordant results. From the statistical confidence measures $\Delta AIC $ and $\Delta BIC$, we can assert our confidence in the hypothesis of the three-Gaussian model over all other model fits. We find that for the next preferred models in the analysis, the $\Delta AIC = 14 $ and $\Delta BIC = 22$, which is significant enough to reject the respective models in favor of our null hypothesis with strong confidence. Conclusions {#sec:conclusions} =========== In this manuscript, we have undertaken a classification of the exoplanet catalog using clustering based on the logarithm of the planet density (similar to a recent analysis in OR16 [@exoplanetclass]), followed by a 2-dimensional analysis using both the log of density and Earth Similarity Index (ESI) [@Kashyap] for each of the exoplanets. We use Gaussian Mixture Model to classify the data for both the one-dimensional and two-dimensional classifications based on log(density) and {log(density), ESI} respectively. For both of these classifications, we determine the best-fit parameters for each model using the EM algorithm. We then use information theoretic criterion, such as AIC and BIC to determine the optimum number of free parameters. Our results are as follows: 1. For the one-dimensional approach, our analysis does not provide a conclusive evidence between a two-component and a three-component model, since neither of the information criterion tests cross the threshold ($>10$) needed for decisive evidence. As stated in Tab. \[tab:aicbic\], the $\Delta AIC$ test weakly favors the three component Gaussian model, whereas the $\Delta BIC$ test weakly favors the two component Gaussian model. The 2 Gaussian model has the mean values of the density at $0.88 ~\rm{gm/cm^3}$ and $9.69 ~\rm{gm/cm^3}$, whereas the corresponding values for the 3 Gaussian model are located at $0.71 ~\rm{gm/cm^3}$, $2.03 ~\rm{gm/cm^3}$ and $88.1 ~\rm{gm/cm^3}$. 2. The two-dimensional classification on the other hand provide robust and consistent results from both the tests. As is summarized in Tab. \[tab:aicbic2d\], both the tests give decisive evidence for the three component Gaussian model with $\Delta$AIC and $\Delta$BIC $> 10$ in both the cases. The catalogs used for this analysis (which were downloaded on Feb 18, 2017) along with the code used can be found online at <https://github.com/IITH/Exoplanet-Classification>. [^1]: <https://exoplanetarchive.ipac.caltech.edu/> [^2]: <http://exoplanet.eu>

ArXiv

--- abstract: | We develop a nonlinear semi-parametric Gaussian process model to estimate periods of Miras with sparsely sampled light curves. The model uses a sinusoidal basis for the periodic variation and a Gaussian process for the stochastic changes. We use maximum likelihood to estimate the period and the parameters of the Gaussian process, while integrating out the effects of other nuisance parameters in the model with respect to a suitable prior distribution obtained from earlier studies. Since the likelihood is highly multimodal for period, we implement a hybrid method that applies the quasi-Newton algorithm for Gaussian process parameters and search the period/frequency parameter space over a dense grid. A large-scale, high-fidelity simulation is conducted to mimic the sampling quality of Mira light curves obtained by the M33 Synoptic Stellar Survey. The simulated data set is publicly available and can serve as a testbed for future evaluation of different period estimation methods. The semi-parametric model outperforms an existing algorithm on this simulated test data set as measured by period recovery rate and quality of the resulting Period-Luminosity relations. author: - 'Shiyuan He, Wenlong Yuan, Jianhua Z. Huang, James Long & Lucas M. Macri' bibliography: - 'm33gp.bib' title: | Period estimation for sparsely sampled quasi-periodic\ light curves applied to Miras --- Introduction ============ The determination of reliable periods for variable stars has been an area of interest in astronomy for at least four centuries, since the discovery of the variability of Mira ($o$ Ceti) by Fabricius in 1596 and the first attempts to determine its period by Holwarda & Bouillaud in the mid-1600s. The availability of electronic computers for astronomical research half a century ago enabled the development of many algorithms to estimate periods quickly and reliably, such as @Lafler1965 [@Lomb1976; @Scargle1982]. The aforementioned algorithms work best in the case of periodic variations with constant amplitude and Mira variables present several challenges in this regard. While their periods of pulsation are stable except for a few intriguing cases [@Templeton2005], Mira light curves can exhibit widely varying amplitudes from cycle to cycle [see, for example, the historical light curve of Mira compiled by @Templeton2009]. In the case of C-rich Miras, the stochastic changes in mean magnitude across cycles [e.g., @Marsakova1999] only complicate the problem further. The wide variety of light curves for long-period variables, already recognized by @Campbell1925 and @Ludendorff1928, may complicate the identification of Miras among other stars. Lastly, from a purely practical standpoint, it is simpler to obtain light curves spanning several cycles for RR Lyraes or Cepheids (with periods ranging from $\sim 0.5$ to $\sim 100$ d) than for Miras (with periods ranging from $\sim 100$ to $\sim 1500$ d). Despite these challenges, the identification and determination of robust periods for Miras — especially in the regime of sparsely sampled, low signal-to-noise light curves — would be very beneficial for the determination of distances to galaxies of any type. Thanks to the unprecedented temporal coverage of the Large Magellanic Cloud (LMC) by microlensing surveys, the availability of large samples of extremely well-observed Miras has led to a thorough characterization of their period-luminosity relations at various wavelengths [@Wood1999; @Ita2004; @Soszynski2007]. The dispersion of the $K$-band period-luminosity relation [@Glass2003 $\sigma=0.13$ mag], is quite comparable to that of Cepheids at the same wavelength [@Macri2015 $\sigma=0.09$ mag] and makes them competitive distance indicators. The third phase of the OGLE survey [@Udalski2008] imaged most of the LMC with little interruption over 7.5 years and resulted in the discovery of 1663 Miras [@Soszynski2009] with a median of 466 photometric measurements per object. The temporal sampling of these light curves and their photometric precision are exceptional relative to typical astronomical surveys and make period estimation relatively easy. In comparison, a similar span of observations of M33 by the DIRECT [@Macri2001] and M33SSS projects [@Pellerin2011] in the $I$-band consists of a median number of 44 somewhat noisy measurements, heavily concentrated in a few observing seasons. Representative Mira light curves from the OGLE & DIRECT/M33SSS surveys are shown in Fig. \[fig:example.mira.lc\]. There are several reasons for the striking difference in quality between these two data sets. The LMC Miras are among the brightest objects in the OGLE fields, whereas their M33 counterparts are among the faintest in the aforementioned surveys of this galaxy. While the effective exposure times of all these surveys are quite comparable, after taking into account differences in collecting area of their respective telescopes, M33 lies approximately 6.2 mag farther in terms of its $I$-band apparent distance modulus. Furthermore, the main goal of the OGLE project (detection of microlensing events) requires a very dense temporal sampling of the survey fields; this is achieved by using a dedicated telescope and is helped by the fact that the LMC is observable nearly all year long from the site. In contrast, the observations of M33 were carried out using shared facilities (available only a few nights per month) with the primary purpose of studying Cepheids and eclipsing binaries (which do not require exceptionally dense temporal sampling), and the galaxy is only observable all night long for $\sim 1/3$ of the year. Standard period estimation algorithms, which work well for high signal-to-noise, well sampled light curves such as those obtained by OGLE, will fail on more typical data sets represented by the M33 observations. The purpose of this work is to develop and test a methodology for estimating periods for sparsely sampled, noisy, quasi-periodic light curves such as those of Miras observed in M33 by the aforementioned projects. \[fig:example.mira.lc\] {width="49.00000%"} {width="49.00000%"} The rest of the paper is organized as follows. In §\[sec.background\] we review several existing period estimation methods. In §\[sec.model\] we introduce a new semi-parametric (SP) model for Mira variables which uses a Gaussian process to account for deviations from strict periodicity. We use maximum likelihood to estimate the period and the parameters of the Gaussian process, while other nuisance parameters in the model are integrated out with respect to some prior distributions using earlier studies. Since the likelihood is highly multimodal for the period/frequency parameter, we implement a hybrid method that applies the quasi-Newton algorithm for Gaussian process parameters and a grid search for the period/frequency parameter. In order to assess the effectiveness of the SP model, in §\[sec.construct.test\] we carefully construct a simulated data set by fitting smooth functions to the light curves of well-observed OGLE LMC Miras and resampling them at the cadence, noise level, and completeness limits of the aforementioned M33 observations. Using the simulated data, in §\[sec.evaluation\] we compare the performance of existing period estimation methods to our SP model. We find that our proposed model shows an improvement over the generalized Lomb-Scargle (GLS) model under various metrics. In §\[sec.discussion\], we conclude and discuss some future applications. Simulated light curves for reproducing the results in the paper and performance benchmarking are made publicly available as supplementary material. Period estimation techniques {#sec.background} ============================ Let $y_i$ be the magnitude of a variable star observed at time $t_i$ (in units of days) with uncertainty $\sigma_i$. The data set for this object, obtained as part of a time-series survey with $n$ epochs is $\{(t_i,y_i,\sigma_i)\}_{i=1}^n$. One common approach to estimate the primary frequency of such an object is to assume some parametric model for brightness variation and then use maximum likelihood to estimate parameters. @Zechmeister2009 define the GLS model as $$\label{eq:gls} y_i = m + a\sin(2\pi f t_i + \phi) + \sigma_i\epsilon_i,$$ where $\epsilon_i \sim \mathcal{N}(0,1)$, $m$ is the mean magnitude, $a$ is the amplitude, $\phi \in [-\pi,\pi]$ is the phase, and $f$ is the frequency [see @Reimann1994 for early work in this model]. Using the sine angle addition formula and letting $\beta_1 = a\cos(\phi)$ and $\beta_2 = a\sin(\phi)$ one obtains $$\label{eq:gls2} y_i = m + \beta_1\sin(2\pi f t_i) + \beta_2\cos(2\pi f t_i) + \sigma_i\epsilon_i.$$ The likelihood function of this model is highly multimodal in $f$. However at a fixed $f$ the model is linear in the parameters $(m,\beta_1,\beta_2)$. These two facts motivate the computation strategy of performing a grid search across frequency and minimizing a weighted least squares $$\begin{split} & (\widehat{m}(f),\widehat{\beta}_1(f),\widehat{\beta}_2(f) ) \\ & \qquad = \operatorname*{arg\,min}_{m,\beta_1,\beta_2} \sum_{i=1}^n \frac{1}{\sigma_i^2} \left\{y_i - m \right.\\ & \qquad \qquad \left. -\beta_1\sin(2\pi f t_i) - \beta_2\cos(2\pi f t_i)\right\}^2, \end{split}$$ at every frequency $f$ on the grid. Under the normality assumption, the weighted least squares minimization is equivalent to maximizing the likelihood. Since the model is linear, computation of $\widehat{m}(f),\widehat{\beta}_1(f),\widehat{\beta}_2(f)$ is straightforward. The residual sums of squares at $f$ is $$\begin{split} {\rm RSS}(f) & = \sum_{i=1}^n \frac{1}{\sigma_i^2} \{y_i - \widehat{m}(f) \\ & \quad - \widehat{\beta}_1(f)\sin(2\pi f t_i) - \widehat{\beta}_2(f)\cos(2\pi f t_i)\}^2, \end{split}$$ and the maximum likelihood estimator for $f$ is $$\widehat{f} = \operatorname*{arg\,min}_{f} {\rm RSS}(f).$$ Define ${\rm RSS}_0$ as the (weighted) sum of squared residuals when fitting a model with only an intercept term $m$. The periodogram is defined as $$\label{eq:glsP} S_{\rm LS}(f) = \frac{(n-3)({\rm RSS}_0 - {\rm RSS}(f))}{2{\rm RSS}(f)}.$$ The periodogram has the property that if the light curve of the star is white noise (i.e., $y_i = m + \epsilon_i$), $S_{\rm LS}(f)$ has an $F_{2,n-3}$ distribution. Thus the periodogram may be used for controlling the “false alarm probability,” the potential that a peak in the periodogram is due to noise [@Schwarzenberg1996]. A large number of period estimation algorithms in astronomy are closely related to GLS. The LS method is identical to GLS but first normalizes magnitudes to mean $0$ and does not fit the $m$ term [@Lomb1976; @Scargle1982]. The “harmonic analysis of variance” includes an arbitrary number of harmonics in Equation [@Quinn1991; @Schwarzenberg1996]. @Bretthorst2013 incorporates Bayesian priors on the parameters $\beta_1$ and $\beta_2$. The method is similar to performing a discrete Fourier transform and selecting the frequency which maximizes the @Deeming1975 periodogram. However, @Reimann1994 showed that GLS has better consistency properties than the Deeming periodogram. ![Light curve of a Mira in the LMC observed by OGLE (black points), decomposed following Eqn. \[eqn:basicDecomposition\]. Top panel: fitted light curve; middle panel: periodic signal, $m+q(t)$; bottom panel: stochastic variations, $m+h(t)$.[]{data-label="fig:onemira"}](fig02.eps){width="49.00000%"} It is also possible to use non-sinusoidal models but compute and minimize the residual sum of squares as above. For example, @Hall2000 consider the Nadaraya-Watson estimator and @Reimann1994 uses the Supersmoother algorithm. @Wang2012 used Gaussian processes with a periodic kernel and found the period with maximum likelihood or minimum leave-one-out cross-validation error. None of the above methods account for the non-periodic variation present in Miras. While these methods are adequate for densely sampled Mira light curves (where the quantity of data overwhelms model inadequacy), their performance deteriorates in the sparsely sampled regime. In Section \[sec.evaluation\], we compare our proposed model with the LS method. The SP model {#sec.model} ============ Suppose the data $\{(t_i,y_i,\sigma_i)\}_{i=1}^n$ are modeled by $$y_i = g(t_i) + \sigma_i\epsilon_i\, ,$$ where $g(t_i)$ is the light curve signal and the $\epsilon_i\sim N(0,1)$ is independent of other $\epsilon_j$s. The signal of the light curve is further decomposed into three parts, $$\begin{split} g(t) & = m + q(t) + h(t)\\ & = m + \beta_1\cos(2\pi ft) + \beta_2\sin(2\pi ft) + h(t)\, , \label{eqn:basicDecomposition} \end{split}$$ where $m$ is the long-run average magnitude, $q(t) = \beta_1\cos(2\pi ft) + \beta_2\sin(2\pi ft)$ with frequency $f$ is the exactly periodic signal, and $h(t)$ is the stochastic deviation from a constant mean magnitude, caused by the formation and destruction of dust in the cool atmospheres of Miras. Fig. \[fig:onemira\] provides an example of the decomposition for a Mira light curve. The first two terms $m + q(t)$ in Eqn. \[eqn:basicDecomposition\] are exactly the same as the GLS model of Eqn. \[eq:gls2\]. To simplify notation, we define ${\mathbf{b}}_f(t) =(\cos(2\pi ft),\sin(2\pi ft))^T$, so that $q(t) = {\mathbf{b}}_f(t)^T{\boldsymbol{\beta}}$. The subscript in ${\mathbf{b}}_f(t)$ emphasizes that the basis is parameterized by the frequency $f$. An SP statistical model is constructed in Eqn. \[eqn:basicDecomposition\] if we assume $h(t)$ is a smooth function that belongs to a reproducing kernel Hilbert space $\mathcal{H}$ with norm $\Vert\cdot\Vert_{\mathcal{H}}$ and a reproducing kernel $K(\cdot,\cdot)$. For this model, if the frequency $f$ is known, we obtain a least squares kernel machine considered in @Liu2007. Because the frequency is unknown, the response function is nonlinear in $f$. This nonlinearity and the multimodality in $f$ of the residual sum of squares provide additional challenges that require a novel solution. Besides the additive formulation in Eqn. \[eqn:basicDecomposition\], another possible solution to account for the quasi-periodicity is a multiplicative model such as $g(t) = m + h(t) q(t)$, where the amplitude of the strictly periodic term $q(t)$ is modified by a smooth function $h(t)$. However, the multiplicative model is more computationally intensive in nature and requires imposing a positive constraint on $h(t)$. As we will show in the following subsections, the $h(t)$ term in the additive model can be easily absorbed into the likelihood function. Nevertheless, the multiplicative approach is an interesting alternative approach to model formulation and is open to future study. Equivalent formulations ----------------------- Following §5.2 of @Rasmussen2005, for fixed $f$, the parameters $m,\beta_1,\beta_2$ and $h(t)$ in Eqn. \[eqn:basicDecomposition\] are jointly estimated by minimizing $$\begin{split} & \sum_{i=1}^n \frac{1}{\sigma_i^2} [ y_i - m - \beta_1\cos(2\pi ft_i) \\ & \qquad - \beta_2\sin(2\pi ft_i) - h(t_i) ]^2 + \lambda \Vert h(\cdot)\Vert_{\mathcal{H}}^2, \label{eqn:generalFormulation} \end{split}$$ where $\lambda$ is a regularization parameter. A smoothing/penalized spline model for $h(t)$ is a special case of the general formulation of Eqn. \[eqn:generalFormulation\] with a specifically defined kernel; see §6.3 of @Rasmussen2005. For fixed $\lambda$, the solution of $h(t)$ is a linear combination of $n$ basis functions $K(t_i,t)$, $i=1,2,\cdots,n$, by the representer theorem [@Kimeldorf1971; @OSullivan1986]. It is still left for us to choose the regularization parameter $\lambda$ to balance data fitting and the smoothness of the function $h(t)$. An equivalent point of view to the above regularization approach is to impose a Gaussian process prior on the function $h(t)$; see §5.2.3 of @Rasmussen2005. The benefit of this view is that it provides an automatic method for selecting the regularization parameter $\lambda$. In particular, we can absorb $\lambda$ into the definition of the norm $\Vert \cdot\Vert_{\mathcal{H}}$ and assume the term $h(t)$ in Eqn. \[eqn:basicDecomposition\] follows a Gaussian process, $h(t) \sim {\mathcal{GP}}(0,k_{{\boldsymbol{\theta}}}(t,t'))$, with the squared exponential kernel $k_{{\boldsymbol{\theta}}} (t,t') = \theta_1^2\exp \left(-\frac{(t-t')^2}{2\theta_2^2}\right),$ and parameters ${\boldsymbol{\theta}}= (\theta_1,\theta_2)$. The Gaussian process assumption implies that at any finite number of time points $t_1,t_2,\cdots,t_s$, the vector $(h(t_1),\cdots,h(t_s))$ is multivariate normally distributed, with zero mean and covariance matrix $\mathbf{K} = (k(t_i,t_j))$. This imposes a prior on the function space of $h(t)$. We also impose priors on $m$ and ${\boldsymbol{\beta}}$ in Eqn. \[eqn:basicDecomposition\]. In particular, we assume $m \sim {\mathcal{N}}(m_0, \sigma_m^2)$ and $ {\boldsymbol{\beta}}\sim {\mathcal{N}}({\mathbf{0}},\sigma_b^2{\mathbf{I}})$. The prior mean $m_0$ can be interpreted as the average magnitude of Miras in a certain galaxy, and $\sigma_m^2$ is the variance of Miras in that galaxy; the prior variance $\sigma_b^2$ is the variance of the light curve amplitude. These prior parameters can be determined using previous studies. For example, in §\[sec.evaluation\], we use well-sampled light curves of LMC Miras [@Soszynski2009] to obtain values of these parameters. It is advisable to check the sensitivity of these prior specifications. The benefit of using priors on $m$ and ${\boldsymbol{\beta}}$ is three-fold: first, they introduce regularization by using information from early studies; second, they provide a natural device for separating the estimation of frequency and the light curve signal component using Bayesian integration when the parameter of interest is the frequency; lastly, the regularization parameter ${\boldsymbol{\theta}}$ of the non-parametric function is allowed to be chosen by the maximum likelihood, without resorting to the computationally expensive cross-validation method. In summary, we have built the following hierarchical model for a Mira light curve: $$\label{eqn:hier} \begin{split} & y_i | m,{\boldsymbol{\beta}}, g(t_i) \sim {\mathcal{N}}(g(t_i), \sigma_i^2), \\ & g(t) = m + {\mathbf{b}}_f(t)^T{\boldsymbol{\beta}}+ h(t), \\ & m \sim {\mathcal{N}}(m_0, \sigma_m^2), {\boldsymbol{\beta}}\sim {\mathcal{N}}({\mathbf{0}},\sigma_b^2{\mathbf{I}}),\\ & h(t)| {\boldsymbol{\theta}}\sim {\mathcal{GP}}(0,k_{{\boldsymbol{\theta}}} (t,t')), \end{split}$$ where ${\boldsymbol{\theta}}$ and $f$ are fixed parameters. In this model, the frequency parameter $f$ is of key interest to our study. We do not perform a fully Bayesian inference by imposing a prior distribution on $f$ because the likelihood function of $f$ is highly irregular, with numerous local maxima, and Monte Carlo computation of the posterior is expensive and intractable for large astronomical surveys. Previously, @Baluev2013 applied a Gaussian process model to study the impact of red noise in radial velocity planet searches. While his maximum likelihood method is a classical frequentist approach in statistics, our approach can be considered as a hybrid of Bayesian and frequentist approaches. We treat the parameter of interest $f$, and the parameters for the kernel ${\boldsymbol{\theta}}$ of the Gaussian process as fixed, and impose a prior distribution on other parameters. This is similar to the type-II maximum likelihood estimation of parameters of a Gaussian process or regularization parameters in function estimation; see §5.2 of @Rasmussen2005. From the Bayesian point of view, ${\boldsymbol{\theta}}$ and $f$ are treated as hyper-parameters that in turn are estimated by the empirical Bayes method. Because the Gaussian process plays a critical role in modeling departure of light curves from periodicity, we may also refer to our model more precisely as the nonlinear SP Gaussian process model. Estimation of the frequency and the periodogram ----------------------------------------------- Let ${\mathbf{y}}= (y_1,y_2,\cdots, y_n)$ be the observation vector of the magnitudes of a light curve. By integrating out $m,{\boldsymbol{\beta}}$ and ${\mathbf{h}}$ from the joint distribution given by Eqn. \[eqn:hier\], we get the marginal distribution of ${\mathbf{y}}$, $p({\mathbf{y}}|{\boldsymbol{\theta}}, f)$, which is a multivariate normal with mean ${\boldsymbol{\mu}}= m_0{\mathbf{1}}$ and covariance matrix $${\mathbf{K}}_y = \left( \sigma_m^2 + \sigma_b^2 {\mathbf{b}}_f(t_i)^T {\mathbf{b}}_f(t_j) + k_{{\boldsymbol{\theta}}}(t_i, t_j) +\sigma^2_i\delta_{ij} \right)_{n\times n},$$ where $\delta_{ij} = 1$ if $i=j$ and $\delta_{ij} = 0$ if $i\neq j$. Therefore, the log likelihood of ${\boldsymbol{\theta}}$ and $f$ is $$\begin{split} Q({\boldsymbol{\theta}},f) = &\log(p({\mathbf{y}}| {\boldsymbol{\theta}}, f))\\ = &-\frac{1}{2} ({\mathbf{y}}-m_0{\mathbf{1}})^T{\mathbf{K}}_y^{-1}({\mathbf{y}}-m_0{\mathbf{1}}) \\ &\qquad -\frac{1}{2}\log\det {\mathbf{K}}_y -\frac{n}{2}\log(2\pi) \label{eqn:mainObj}. \end{split}$$ The maximum likelihood estimator of ${\boldsymbol{\theta}}$ and $f$ is obtained by maximizing $Q({\boldsymbol{\theta}},f)$. Since the likelihood function is differentiable with respect to ${\boldsymbol{\theta}}$ but highly multimodal in the parameter $f$, standard optimization methods cannot be directly used to jointly maximize over ${\boldsymbol{\theta}}$ and $f$. We adopt a profile likelihood method as follows. For each frequency $f$ over a dense grid, we compute the maximum likelihood estimator $\widehat{{\boldsymbol{\theta}}}_f = \operatorname*{arg\,max}_{{\boldsymbol{\theta}}} Q({\boldsymbol{\theta}},f)$. This can be done using the quasi-Newton method. Then we perform a grid search to find the maximum profile likelihood estimator of $f$, i.e., $$\label{eqn:fhat} \hat{f} = \operatorname*{arg\,max}_f Q(\widehat{{\boldsymbol{\theta}}}_f, f)\, ,$$ the estimated period is $\hat{P}=1/\hat{f}$. The details of the algorithm are given in §\[sec.quasi\]. The profile log-likelihood as a function of the frequency $f$ is adopted as the *periodogram* for our model, $$S_{SP}(f) = Q(\widehat{{\boldsymbol{\theta}}}_f, f)\, . \label{eqn:periodogram}$$ It contains the spectral information of the signal. The frequency of the dominant harmonic component is expected to be the location of the peak of this profile likelihood. Computation of the periodogram {#sec.quasi} ------------------------------ Now we present the details of computing the profile likelihood. Because $Q({\boldsymbol{\theta}},f)$ is highly multimodal in the frequency parameter $f$, we follow the commonly used strategy of optimization through grid search. On the other hand, since $Q({\boldsymbol{\theta}},f)$ is differentiable in parameter ${\boldsymbol{\theta}}$, the quasi-Newton method can be employed to optimize over ${\boldsymbol{\theta}}$ for fixed $f$, and obtain the profile likelihood (Eqn. \[eqn:periodogram\]). The gradient of the log likelihood (Eqn. \[eqn:mainObj\]) with respect to $\theta_j (j=1,2)$ is $$\frac{\partial}{\partial \theta_j} Q({\boldsymbol{\theta}},f)\!=\! \frac{1}{2}\mathrm{tr}\left(({\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^T\!-\!{\mathbf{K}}_y^{-1})\frac{\partial {\mathbf{K}}_y}{\partial \theta_j}\right)$$ where ${\boldsymbol{\alpha}}\!=\!{\mathbf{K}}^{-1}_y({\mathbf{y}}-m_0{\mathbf{1}})$. In general, the objective function for the Gaussian process model is not convex in its kernel parameters ${\boldsymbol{\theta}}$ and global optimization cannot be guaranteed. Fig. \[fig:lc.surface\] shows a surface plot of $Q({\boldsymbol{\theta}},f)$ as a function of ${\boldsymbol{\theta}}$ for one simulated light curve, with $f$ fixed at the true frequency. The surface exhibits unimodality in this case, although it is not convex. The computation involved in calculating the profile likelihood through the quasi-Newton method can be intensive. Since the objective function (Eqn. \[eqn:mainObj\]) is non-convex in ${\boldsymbol{\theta}}$, generally multiple starting points should be attempted to find the global optimizer when applying the quasi-Newton method. In addition, evaluating the objective function and the gradient function requires inversion of the covariance matrix whose computation cost is of the order $O(n^3)$. During each quasi-Newton iteration, these evaluations could be repeated several times because multiple step size might be attempted. To make the computation more challenging, all of the above needs to be repeated at hundreds or even thousands of densely gridded $f$s per light curve. Furthermore, the method may need to be applied to hundreds of thousands or millions of light curves from large astronomical surveys. In order to speed up computation over the dense grid of frequency values, we use the result of applying the quasi-Newton method at one frequency value as a warm start for the subsequent frequency value. Specifically, the optimizer $\widehat{{\boldsymbol{\theta}}}_f$ and its approximate inverse Hessian matrix are provided as quantities to start the quasi-Newton iterations for the next frequency value on the dense grid. When the initial point is near the local minimizer and the inverse Hessian matrix is a good approximation to the true Hessian matrix, the quasi-Newton algorithm will converge at superlinear rate; the step size of $\alpha=1$ will be accepted by the Wolfe descent condition, avoiding evaluation of the objective function multiple times to determine the appropriate step size during each iteration [see Ch. 6 of @Nocedal2006 for a more rigorous mathematical discussion]. We find that a warm start can speed up the computation significantly but sometimes we need to restart with random initial values to ensure convergence to the global optimum. The pseudocode provided in the Appendix describes our algorithm. Estimation of the signal and its components ------------------------------------------- After the parameters $f$ and ${\boldsymbol{\theta}}$ are fixed at their maximum likelihood estimates $\widehat{f}$ and $\widehat{{\boldsymbol{\theta}}}_{\hat{f}}$, we can perform the inference of the light curve signal $g(t)$ and its components in the standard Bayesian framework. Interested readers may consult Ch. 2 of @Rasmussen2005 for a detailed discussion of this topic. Firstly, we could obtain the posterior distribution of ${\boldsymbol{\gamma}}= (m,{\boldsymbol{\beta}}^T)$, the parameters for the long run average magnitude and the exactly periodic term. The prior of ${\boldsymbol{\gamma}}$ is ${\mathcal{N}}({\boldsymbol{\gamma}}_0,{\boldsymbol{\Sigma}}_\gamma)$ with ${\boldsymbol{\gamma}}_0=(m_0,0,0)^T$ and ${\boldsymbol{\Sigma}}_\gamma = \mathrm{diag}(\sigma_m^2,\sigma_b^2,\sigma_b^2)$. Its posterior distribution is ${\boldsymbol{\gamma}}| {\mathbf{y}}\sim {\mathcal{N}}(\bar{{\boldsymbol{\gamma}}}, \bar{{\boldsymbol{\Sigma}}}_\gamma)$ with $$\label{equ:gammapost} \begin{split} \bar{{\boldsymbol{\gamma}}} = & \left({\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{H}}+{\boldsymbol{\Sigma}}_\gamma^{-1}\right)^{-1} \\ & \left({\boldsymbol{\Sigma}}_\gamma^{-1}{\boldsymbol{\gamma}}_0+{\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{y}}\right)\, , \\ \bar{{\boldsymbol{\Sigma}}}_\gamma = & \left({\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{H}}+ {\boldsymbol{\Sigma}}_\gamma^{-1}\right)^{-1}, \end{split}$$ [where]{} $${\mathbf{h}}(t)\!=\!(1,{\mathbf{b}}_{\hat{f}}(t)^T)^T, {\mathbf{H}}\!=\!({\mathbf{h}}(t_1),{\mathbf{h}}(t_2),\cdots,{\mathbf{h}}(t_n))^T,$$ and$\,{\mathbf{K}}_c\!=\!\big(k_{\widehat{{\boldsymbol{\theta}}}_{\hat{f}}}(t_i,\!t_j)\!+\!\sigma_i^2\delta_{ij}\!\big)\!_{{\tiny\textit{n}}\times\!{\tiny\textit{n}}}\,$with$\,\widehat{f}\,$and$\,\widehat{{\boldsymbol{\theta}}}_{\hat{f}}\,$plugged in. Consider the prediction of light curve magnitude at a specific time point $t^*$. Define the vector ${\mathbf{k}}^* = (k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t_1),\ \cdots,\ k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t_n))^T$. Conditional on $({\mathbf{y}}, {\boldsymbol{\gamma}})$, the distribution of $g(t^*)| {\mathbf{y}}, {\boldsymbol{\gamma}}$ is a multivariate normal with mean $ {\mathbf{h}}(t^*)^T \gamma + {\mathbf{k}}_{{\boldsymbol{\theta}}}(t^*,{\mathbf{t}}) {\mathbf{K}}_c^{-1} ({\mathbf{y}}- {\mathbf{H}}\gamma)$ and variance $k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t^*) - ({\mathbf{k}}^*)^T {\mathbf{K}}_c^{-1} {\mathbf{k}}^*$. With the posterior distribution of ${\boldsymbol{\gamma}}$ given in Eqn. \[equ:gammapost\], we are able the remove ${\boldsymbol{\gamma}}$ from the above conditional distribution of $g(t^*)$. Finally, we get the posterior distribution of the signal at $t^*$ as $g(t^*)| {\mathbf{y}}\sim {\mathcal{N}}(\bar{g}^*, \bar{\sigma}_{g^*}^2)$ with $$\begin{split} \bar{g}^* = & {\mathbf{h}}(t^*)^T \bar{\gamma} + {\mathbf{k}}(t^*,{\mathbf{t}}) {\mathbf{K}}_c^{-1} ({\mathbf{y}}- {\mathbf{H}}\bar{\gamma})\, ,\\ \bar{\sigma}^2_{g^*} = & k_{\widehat{{\boldsymbol{\theta}}}}(t^*,t^*) - ({\mathbf{k}}^*)^T {\mathbf{K}}_c^{-1} {\mathbf{k}}^*+ {\mathbf{r}}^T\bar{{\boldsymbol{\Sigma}}}_\gamma{\mathbf{r}}\, , \end{split} \label{equ:gpredict}$$ where ${\mathbf{r}}={\mathbf{h}}(t^*)-{\mathbf{H}}^T{\mathbf{K}}_c^{-1}{\mathbf{k}}^*$. Simulation of M33 light curves {#sec.construct.test} ============================== It is not possible to evaluate the period estimation accuracy of our method directly on the M33 data because the “ground truth” is unknown. Instead, we construct a test data set by smoothing the well-sampled OGLE light curves to infer continuous functions, then resample these functions to match the observational patterns of the M33 data, and at last add noise to the light curves. This data set can serve as a testbed for future studies of comparing different period estimation methods. We will now describe the M33 observations and the construction of the test data set. As the whole simulation procedure is a complicated process, we will discuss its components in detail from §4.1 to §4.4. The whole simulation procedure will be summarized in §4.5. Characteristics of the M33 observations --------------------------------------- Most of the disk of M33 was observed by the DIRECT [@Macri2001] and M33SSS [@Pellerin2011] projects in the $BVI$ bands, with a combined baseline of $7-9$ years and a sampling pattern that depends on the exact location within the disk (see Fig. \[fig:obs.gaps\]). The large area of coverage and long baseline of these observations make them suitable for Mira searches. We use the $I$-band observations to carry out the simulations, as this is the wavelength range where Miras are brightest (out of the three bands used by these projects). Detailed descriptions of the M33 observations can be found in the above referenced papers. We use the data products from a new reduction that will be presented in a companion paper (W. Yuan et al. 2016, in prep.). $I$-band light curves are available for $\sim 2.5\times 10^5$ stars, with a median of 44 measurements and a maximum of 170. . We model the relation between a magnitude measurement $m$ and its uncertainty $\sigma$ as $$\sigma = a(t_i',F)^{[m-b(t_i',F)]} + c(t_i',F)\, , \label{equ.sigma.mag}$$ for each observation field $F$ and each observation night $t_i'$, where $a(t_i',F)$, $b(t_i',F)$ and $c(t_i',F)$ are field- and night-specific constants. There are 31 different fields in total, $F=0,1,\cdots, 9, a,b,\cdots, u$. The parameters are determined via least-squares fitting using all the measurements for the specific field $F$ and night $t_i'$. Fig. \[fig:simu.sigmag\] shows the $m-\sigma$ relation for a typical field. ![$m-\sigma$ relation for a given night and field within M33. The solid red line is the best-fit relation using the empirical function $\sigma = a^{(m-b)} + c$, with $a=2.666,\,b=23.117,\,c=0.008$.[]{data-label="fig:simu.sigmag"}](fig05.eps){width="49.00000%"} In order to test the SP periodogram we need sparsely sampled, moderately noisy Mira light curves with known periods. Thus, we characterize the sampling patterns and noise levels of the M33 observations and simulated Mira light curves of known periods using the OGLE observations of these objects in the LMC. Matching the M33 observation pattern ------------------------------------ The first step in simulating a Mira light curve is to randomly select a sampling pattern based on the light curve of an actual star in some field $F$, $\{t_i'\}_{i=1}^n$ with $n\in [10,170]$. A random time shift $s$ is added, $t_i = t_i' + s$ for $i=1,2,\cdots, n$. The random shift $s$ follows a uniform distribution over the interval $[0, P_0]$, where $P_0$ is the true period of the LMC Mira selected during the artificial light curve generation process. This helps to simulate a large number of unique light curves sampled at random phases using the limited number of template light curves. The Mira template light curves ------------------------------ The template Mira light curves are obtained by using our SP model to fit the Mira light curves in the LMC, collected by the OGLE project [@Soszynski2009]. A total number of 1663 Miras have been observed in $I$ with very high accuracy, excellent phase coverage, and a long baseline (the median and mean number of observations are 466 and 602, respectively, with a baseline of $\sim 7.5$ years for most fields). Because the LMC light curves are densely sampled with high quality, we can adopt a more complicated model to provide a higher fidelity fit. Following §5.4.3 of @Rasmussen2005, instead of Eqn. \[eqn:basicDecomposition\], the signal light curve $g(t)$ is decomposed into $$\label{eqn:fullgpmodel} g(t) = m + l(t) + q(t) + h(t),$$ where $m$ is the long run average magnitude, $l(t)$ is the long-term (low-frequency) trend across different cycles, $q(t)$ is the periodic term, and $h(t)$ is small-scale (high-frequency) variability within each cycle. The latter three terms are modeled by the Gaussian process with different kernels. In particular, we use the squared exponential kernel $k_l(t_1,t_2) = \theta_1^2 \exp(-\frac{1}{2}\frac{(t_1-t_2)^2}{\theta_2^2})$ for $l(t)$, another squared exponential kernel $k_h(t_1,t_2) = \theta_6^2 \exp(-\frac{1}{2}\frac{(t_1-t_2)^2}{\theta_7^2})$ for $h(t)$, and lastly a periodic kernel $$\begin{split} k_q(t_1,t_2) = \theta_3^2 \exp \bigg ( -\frac{1}{2}&\frac{(t_1-t_2)^2}{\theta_4^2} \\ &- \frac{2\sin^2(2\pi f(t_1-t_2))}{\theta_5^2} \bigg ) \end{split}$$ for $q(t)$. Note the periodic kernel allows the light curve amplitude to change across cycles. The maximum likelihood method is applied to fit each LMC light curve, fixing $f$ to the OGLE value and solving for the unknown parameters $(\theta_1,\theta_2, \cdots, \theta_7)$. Fig. \[fig:complexDecomposition\] is an illustration of the model fitting result using Eqn \[eqn:fullgpmodel\] based on the same light curve as in Fig. \[fig:onemira\]. Notice that the more complex model in Fig. \[fig:complexDecomposition\] is only suitable for a densely sampled light curve. Once the sampling pattern is chosen, one of the template light curves will be selected according to the luminosity function described in the next subsection. With the selected template, the magnitude of the simulated light curve signal at $t_i'$ with shift $s$ is $g(t_i'+s)$, which is computed with Eqn. \[eqn:fullgpmodel\] in a similar way as Eqn. \[equ:gpredict\]. ![Light curve of a Mira in the LMC observed by OGLE (black points), decomposed following Eqn \[eqn:fullgpmodel\]. Top panel: the fitted light curve; second panel: long-term signal, $m+l(t)$; third panel: periodic term, $m + q(t)$; bottom panel: stochastic variations, $m+h(t)$.[]{data-label="fig:complexDecomposition"}](fig06.eps){width="49.00000%"} Matching the luminosity function\ to the M33 observations --------------------------------- While the OGLE observations of LMC Miras are deep enough to detect these objects over their entire range of luminosities, the M33 observations become progressively more incomplete for fainter and redder objects. We derived an empirical completeness function for the M33 observations as follows. We fitted the observed luminosity function $\mathcal{F}_0(I)$ using an exponential for $I \in [18.5,20]$ mag and extrapolated to fainter magnitudes, obtaining $\mathcal{F}_1(I)$. The empirical completeness function is then $\mathcal{C}(I) = \mathcal{F}_1(I) / \mathcal{F}_0(I)$. We randomly picked $\{t_i'\}_{i=1}^n$ from the M33 light curves. For each $\{t_i'\}_{i=1}^n$, we selected a (LMC-based) template using $\mathcal{C}(I+6.2)$ as the probability distribution. The value of $+6.2$ mag accounts for the approximate difference in distance modulus between the LMC and M33. In this way the resulting luminosity function of the simulated light curves is statistically the same as that of the real M33 observations. The simulation procedure ------------------------ With all the components discussed above, we are able to present the whole simulation procedure here. In order to generate one simulated Mira light curve matching the sampling characteristics of the M33 observations, the first step is to randomly select a sampling pattern $\{t_i'\}_{i=1}^n$, and then add a random shift $s$, $t_i = t_i' + s$, $i=1,2,\cdots, n$. The second step is to randomly select a template light curve according to the luminosity function, then compute the light curve signal $g(t_i'+s)$ for the selected sampling pattern $\{t_i'\}_{i=1}^n$. The third step is to use the best-fit relations (Eqn. \[equ.sigma.mag\]) to add photometric noise via $$y_i = g(t_i' + s)+6.2+ \sigma_i\epsilon_i\, ,$$ where $+6.2$ mag is the approximate relative distance modulus, $\epsilon_i$ is drawn from $\mathcal{N}(0,1)$, and $\sigma_i$ is computed from $$\sigma_i = a(t_i',F)^{[g(t_i)+6.2-b(t_i',F)]} + c(t_i',F).$$ for the selected observation pattern $t_i'$ and field $F$. Following this procedure, we generate one simulated light curve $\{t_i',y_i,\sigma_i\}_{i=1}^n$. The procedure is repeated until $10^5$ suitable light curves are generated, excluding any with $<10$ data points or sampling on $<7$ nights. Performance evaluation {#sec.evaluation} ====================== Having generated the test data set, we evaluate the performance of the SP model and compare it with the GLS model. We choose prior parameters for the SP model of $m_0 = 15.62 + 6.2$, $\sigma_m = 10$ and $\sigma_b =1$. The adopted value of $m_0$ is the average $I$ magnitude of Miras in the LMC and once again $+6.2$ is the approximate relative distance modulus between M33 and the LMC. The values of $\sigma_m$ and $\sigma_b$ are larger than those derived from the LMC samples in order to make those priors non-informative. Although fitting the SP model is computationally slower than the LS model, we find that our model gives an overall improvement in various metrics. For both methods, the periodograms are computed on a dense frequency grid from $1/2000$ to $1/100$ with a spacing of the order of $10^{-5}$. For the GLS method, we chose a spacing of (0.05/time span) or $\sim2.5\times 10^{-5}$, which results in optimal performance for this simulation. For our SP method, we chose a slightly smaller value of $10^{-5}$ to facilitate the warm start mechanism in our algorithm (see Appendix) given that small changes in frequency result in tiny changes of the objective function. The aliasing effect ------------------- We fit the entire simulated data set using the SP model. Fig. \[fig:lc.and.spec\] gives an example of a simulated light curve and its SP periodogram (Eqn. \[eqn:periodogram\]). In this example, the true frequency (labeled by the blue dotted line) is successfully recovered. Aliasing frequencies at $f\pm 1/365$ d affect most periodograms when dealing with sparsely observed astronomical data. The red dashed line in Fig. \[fig:lc.and.spec\] indicates the aliasing frequency at $f+1/365$ where a strong peak exists. This is not a rare case, and for some light curves the one-year beat aliasing frequencies have higher log likelihoods than the true frequencies. Fig. \[fig:fvsf\] compares the recovered and true frequencies for all simulated light curves. Two secondary strips parallel to the main one and offset by $\pm0.00274$ represent $\hat{f} = f \pm 1/365$, respectively. Other aliasing frequencies, such as $2f$, $3f$, $0.5f$, etc., are also noticeable. Lastly, due to the sampling pattern of some light curves, the side lobes of the main peak can be higher than the central value. These manifest as close parallel strips to the aforementioned features. Accuracy assessment ------------------- The estimated frequency is considered as correct if $\Delta f=|\hat{f}-f_0|<C_f$ for each light curve. The estimation accuracies for the two methods are summarized in Table \[tbl:accuracy\] for several different values of $C_f$. We choose $C_f = 2.7\times 10^{-4}$ to stringently bind the one-to-one strip in Fig. \[fig:fvsf\]. Overall, the SP correctly estimates the period for 69.4% of the light curves, while the LS model has an accuracy of 63.6%. The improvement of SP over LS is more evident for C-rich Miras, with about 10% higher accuracy, while the improvement for O-rich Miras is smaller, with about 3% higher accuracy. The difference in performance arises because C-rich Miras often exhibit larger stochastic deviations that can be better captured by the SP model, while O-rich Miras have more stable light curves that can be modeled reasonably well with the LS method. We also compute the estimation accuracy of each method by grouping the light curves according to the number of observations, as shown in the left panels of Fig. \[fig:accuracy\]. The top and bottom rows show results for C- and O-rich Miras, respectively. The performance difference is once again more evident in the C-rich category. Note that accuracy is not a monotonic function of the number of observations, implying this is not a good indicator [*per se*]{} of the information content of the light curves for frequency (period) estimation. Thus, we define another metric, called [*phase coverage*]{}. Recall that the times of observation for a given light curve are $t_1,t_2,\cdots,t_n$. Given a period of $P$, these are converted into corresponding phases by $s_i = (t_i\ \mathrm{ mod }\ P)/P,\ i=1,2,\cdots, n\,$ in the closed interval $[0,1]$. Now, define $$J = \Big(\bigcup_i (s_i-l,s_i+l)\Big)\cap [0,1]\, ,$$ for a specific $l>0$, the phase coverage can be measured by $\lambda(J)$ where $\lambda(\cdot)$ is the Lebesgue measure (we choose $l=0.02$). $\lambda(J)$ describes the “length” of the union of the intervals $J$. For example, $\lambda(J) = 0.1$ for $J = (0.1,0.2)$, and $\lambda(J) = 0.2$ for $J = (0.1,0.2)\cup (0.5, 0.6)$. We divide the light curves into 100 groups such that their $\lambda(J)$ is in one of the intervals $(k/100,(k+1)/100]$ for $k=0,1,\cdots,99$ and compute the estimation accuracy for each subset. The results for the two models are plotted in the middle column of Fig. \[fig:accuracy\]. Now the estimation accuracy is monotonically increasing as a function of phase coverage. The accuracy improvement of our method is highest when the phase coverage is around 0.5 for C-rich Miras. As the phase coverage approaches the extremes (0 or 1), the performance difference between the two methods diminishes. At $\lambda(J) \approx 0$, both methods will fail because this is a hopeless situation. At the other extreme, when $\lambda(J)\approx 1$ and abundant information is available for frequency estimation, both methods have an accuracy close to 1. The periodogram $S_{\rm SP}(f)$ of our model defined in Eqn. \[eqn:periodogram\] provides more information than just the optimal frequency. Suppose $f_1$ is the largest local maximal (global maximum) of $S_{\rm SP}(f)$, and $f_2$ is the second largest local maximal of $S_{\rm SP}(f)$. Now define [conf]{} $=S_{\rm SP}(f_1)-S_{\rm SP}(f_2)\ge 0$. The value of [conf]{} serves as a confidence measurement of the global optimal estimate in Eqn. \[eqn:fhat\]. Larger values of [conf]{} indicate smaller uncertainty in our estimate, and thereby the estimate is more reliable. Now, let $c_0$ be the smallest value, and let $c_1, \dots, c_{100}$ be the $1$st–$100$th percentiles of all the [conf]{} values computed for all the light curves. Each light curve can be assigned to a percentile group if its [conf]{} is in $(c_{k-1},c_{k}]$ for some $k\in\{1,2,\cdots,100\}$. After assigning all light curves by [conf]{} to their corresponding percentile groups, the estimation accuracy in each group can be computed. The same procedure is applied to the GLS model, with the $p$-value of the F-statistics given in Eqn. \[eq:glsP\] for the top peak being used as its [conf]{} measurement. The result is plotted in the right column of Fig. \[fig:accuracy\]. The accuracy of our SP method is much higher than the LS model in the top 40 groups. In particular, the accuracy of our method is higher than 90% in the top 20 groups for both C- and O-rich Miras. Light curves with high values of [conf]{} are particularly reliable for constructing Period-Luminosity relations (hereafter, PLRs) based on the “Wesenheit” function [@Madore1982]. This function enables a simultaneous correction for the effects of dust attenuation and finite width of the instability strip by defining a new magnitude $W_I\!=\!I\!-\!1.55(V\!-\!I)$, where $V$ and $I$ are the mean magnitudes in those filters. Figure \[fig:pl.compare\] compares PLRs based on $W_I$ magnitudes and periods determined by OGLE and estimated with each of the two models. The top and bottom rows display the PLRs for C- and [ccrrr]{} & SP & 58.1 & 55.3 & 56.5\ & LS & 49.4 & 51.6 & 50.6\ & SP & 69.6 & 63.8 & 66.3\ & LS & 60.1 & 60.6 & 60.4\ & SP & 73.5 & 66.2 & 69.4\ & LS & 63.7 & 63.5 & 63.6\ & 43,116 & 56,884 & [O-rich Miras, respectively. The leftmost column shows the PLRs based on the actual OGLE periods, while the next two sets of columns show the corresponding relations based on SP or LS periods for the simulated light curves with the top 10% and 40% values of [conf]{}.]{} In order to provide a quantitative comparison of the improvement obtained with our SP method, we calculated the dispersion of the actual $W_I$ PLRs and their recovered counterparts as a function of [conf]{} value as follows, separately for C- and O-rich Miras. First, we selected all objects of a given class with $2\!<\!\log P\!<\!3$. If the @Soszynski2009 catalog did not provide a $V$ measurement for a given variable, the missing value was estimated through linear interpolation of the $(I,V\!-\!I)$ relation for objects of the same class within $|\Delta\log P|<0.05$ dex. We fitted a quadratic PLR $$m=a+b(\log P-2.3)+c(\log P-2.3)^2$$ with iterative $3\sigma$ clipping (removing $\sim 5$% of the data). We then computed the dispersion of the initially selected OGLE sample about the best-fit relation, including outliers. This yielded “benchmark” dispersions of 0.45 & 0.54 mag for C- & O-rich variables, respectively. Keeping the best-fit relation fixed, we computed the dispersion of recovered PLRs using all artificial light curves within a certain range of [conf]{} (top 10%, top 20%, $\dots$), using the periods and [conf]{} values derived by the SP or the LS method. As in the case of the OGLE samples, we only considered objects with $2\!<\!\log P\!<\!3$. The results are plotted in Fig. \[fig:pl.sig\]. The SP subsamples exhibit lower [llrrrr]{} 00082 & O & 14.241 & 16.509 & 164.84 &\ 00094 & C & 15.120 & 18.885 & 332.30 &\ 00098 & C & 15.159 & 17.921 & 323.10 &\ 00115 & C & 14.932 & 16.947 & 176.13 &\ 00355 & O & 14.199 & 16.219 & &\ [(or at worst, equal) dispersions than their LS counterparts for all percentiles and for both subtypes. As discussed previously, the improvement provided by our method is strongest for C-rich Miras and diminishes in significance as one includes light curves with progressively lower confidence values.]{} Summary {#sec.discussion} ======= In this paper, we developed a nonlinear SP Gaussian process model for estimating the periods of sparsely sampled quasi-periodic light curves, motivated by the desire to detect Miras in an existing set of observations of M33. We conducted a large-scale high-fidelity simulation of Mira light curves as observed by the DIRECT/M33SSS surveys to compare our model with the GLS method. Our model shows improved accuracy under various metrics. The simulation data set is provided as a testbed for future comparison with other methods. The SP model will be used in a companion paper to search for Miras in M33, estimate their periods, and study the resulting PLRs.   SH was partially supported by Texas A&M University-NSFC Joint Research Program. WY & LMM acknowledge financial support from the NSF through AST grant \#1211603 and from the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University. JZH was partially supported by NSF grant DMS-1208952. The authors acknowledge the Texas A&M University Brazos HPC cluster that contributed to the research reported here. [[*Simulated light curves:*]{} A tarfile, containing $10^5$ simulated light curves are generated following the procedure of §\[sec.construct.test\]. Each light curve is stored in one file with three columns: MJD, $I$ magnitude, and uncertainty. The file name, e.g., lc006788.dat is generated sequentially and is only meant for bookkeeping purposes. A mapping between simulated light curve ID and the original OGLE object is given in the file “lc.dat”, which can also be found in the tarfile.]{} [[*Mira variables:*]{} Table\[tbl:oglemira\] summarizes the relevant properties of OGLE LMC Miras from @Soszynski2009 that were used to simulate the light curves: OGLE ID, main period and mean $I$ & $V$ magnitudes. It includes some extrapolated values of $V$ for objects with missing data (suitably identified with a “\*”). This table can be used to compare true versus derived periods and to generate Period-Luminosity relations.]{} [[*Software:*]{} The related software package, `varStar`, has been released under a GPL3 license [@He2016]. The active software development repository can be found at [github.com/shiyuanhe/varStar](github.com/shiyuanhe/varStar).]{} * *Quasi-Newton’s Method with Grid Search**\ **Input:** Maximal and minimal trial frequencies $f_M\!>\!f_m\!>\!0$; frequency step $\Delta f$; $n$ observations $\{t_i,y_i,\sigma_i\}$.\ **Output:** Periodogram $S(f)$ evaluated at the trial frequencies.\ Initialize ${\boldsymbol{\theta}}^{(0)}$ and $\mathbf{H}^{(0)}$, and $f\gets f_m$; $p \gets 0$; $\mathbf{t}_p \gets -\mathbf{H}^{(p)} \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p)}, f)$; ${\boldsymbol{\theta}}^{(p+1)}\gets {\boldsymbol{\theta}}^{(p)} +\alpha_p \mathbf{t}_p$ and the step size $\alpha_p$ satisfying the Wolfe condition; $\mathbf{d}_p \gets {\boldsymbol{\theta}}^{(p+1)}-{\boldsymbol{\theta}}^{(p)}$, $\mathbf{e}_p \gets \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p+1)}, f)- \frac{\partial}{\partial {\boldsymbol{\theta}}} Q({\boldsymbol{\theta}}^{(p)}, f)$; $\rho_p= 1/\mathbf{d}_p^T\mathbf{e}_p$; $\mathbf{H}^{(p+1)}\gets (\mathbf{I}-\rho_p \mathbf{d}_p\mathbf{e}_p^T) \mathbf{H}^{(p)} (\mathbf{I}-\rho_p \mathbf{e}_p\mathbf{d}_p^T) +\rho_p\mathbf{d}_p\mathbf{d}_p^T$; $p \gets p + 1$; $\widehat{{\boldsymbol{\theta}}}_{f} \gets {\boldsymbol{\theta}}^{(p)}$, and $S(f)\gets Q(\widehat{{\boldsymbol{\theta}}}_{f},f)$; $\mathbf{H}^{(0)} \gets \mathbf{H}^{(p)}$ and ${\boldsymbol{\theta}}^{(0)}\gets {\boldsymbol{\theta}}^{(p)}$;

ArXiv

--- abstract: 'Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation.' author: - | Alicia M. Sintes\ [Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), ]{}\ [Schlaatzweg 1, 14473 Potsdam, Germany]{}\ \ Alan A. Coley and Des J. McManus\ [Department of Mathematics, Statistics and Computing Science.]{}\ [Dalhousie University. Halifax, NS. Canada B3H 3J5]{} title: ' On space-times admitting shear-free, irrotational, geodesic null congruences' --- 22.0 true cm 15.24 true cm 1true cm 1 true cm -0.8cm Ł[[L]{}]{} Introduction ============ In this article we wish to extend earlier work on shear-free, irrotational and geodesic (SIG) timelike and spacelike congruences [@des; @des2] to SIG [*null*]{} congruences. The fact that we are dealing with null congruences means that we have to approach the problem in a completely different way; we must make extensive use of the Newman-Penrose formalism. Thus, we wish to study a congruence of curves whose tangent vector ${\bf k}$ is null and geodesic. Hence, we have a family of null geodesics $x^a=x^a(y^{\alpha},v)$, where $y^{\alpha}$ distinguishes the different geodesics, and $v$ is the affine parameter along a fixed geodesic. The null tangent vector is $k^a={\partial x^a \over \partial v} $, and satisfies ${k^a}_{;b}k^b=0$. The spin coefficients are defined in [@Kramer], where $\rho=-(\theta + i\omega)$ is called the complex divergence and $\sigma$ is the complex shear. The geodesic condition implies that the spin coefficient $\kappa$ vanishes and $\epsilon +\bar\epsilon =0$ follows from the choice of an affine parameter along the congruence. The congruence is said to be shear-free if $\sigma=0$. Also, from the relation $k_{[a;b}k_{c]}=(\bar \rho-\rho)\bar m_{[a}m_bk_{c]}$ [@Pirani], it follows that $w=0$ (i.e., zero twist) is a necessary and sufficient condition for ${\bf k}$ to be hypersurface orthogonal. First we shall briefly review some of the results of relevance to this work. Goldberg and Sachs [@Gold] proved that if a gravitational field contains a shear-free, geodesic, null congruence ${\bf k}$, then $\kappa=\sigma=0$, and if R\_[ab]{}k\^ak\^b=R\_[ab]{}k\^am\^b=R\_[ab]{}m\^am\^b=0  ,\[s1\] then the field is algebraically special (i.e., $\Psi_0=\Psi_1=0$), and ${\bf k}$ is a degenerate eigendirection. In addition, a vacuum metric is algebraically special if and only if it contains a shear-free geodesic null congruence. A space-time admits a geodesic, shear-free, twist-free ($\kappa=\sigma=\omega=0$) and diverging ($\rho=\bar\rho=\theta=-1/r$) null congruence ${\bf k}$, and satisfies (\[s1\]), if and only if the metric can be written in the form ds\^2=2r\^2 P\^[-2]{}(z,|z,u)dzd|z -2dudr -2H(z,|z,r,u)du\^2  . Robinson-Trautman models [@Robi] with this metric have been found for vacuum, Einstein-Maxwell and pure radiation fields with or without a cosmological constant [@Kramer]. For geodesic null vector fields we have that $(\theta +i\omega)_{,a}k^a+ (\theta +i\omega)^2+\sigma\bar\sigma= -R_{ab}k^ak^b/2$. Therefore, in the non-diverging case (i.e., $\rho=-(\theta +i\omega)=0$), if the energy condition $T_{ab}k^ak^b\ge 0$ is satisfied, it follows that $\sigma=0=R_{ab}k^ak^b$. Thus, non-twisting (and therefore geodesic) and non-expanding null congruences must be shear-free. Hence, the space-time is algebraically special, and it corresponds to vacuum, Einstein-Maxwell, and pure radiation field. Perfect fluid solutions violate $R_{ab}k^ak^b=0$ unless $\mu+p=0$. This class of solutions has been studied by Kundt [@Kundt]. Another important case corresponds to the Kerr-Schild metric, which is given by $g_{ab}=\eta_{ab}-2\phi k_ak_b$. The null vector ${\bf k}$ of a Kerr-Schild metric is geodesic if and only if the energy-momentum tensor obeys the condition $T_{ab}k^ak^b=0$, and then ${\bf k}$ is a multiple principal null direction of the Weyl tensor and the space-time is algebraically special. The general properties of the Kerr-Schild metrics and their applications to vacuum, Einstein-Maxwell, and pure radiation space-times can be found in [@Kramer]. Finally, we note the algebraically special perfect fluid space-times corresponding to the generalized Robinson-Trautman solutions investigated by Wainwright [@Wain]. They are characterized by a multiple null eigenvector ${\bf k}$ of the Weyl tensor which is geodesic, shear-free, and twist-free but expanding (i.e., $\Psi_o=\Psi_1=0$, $\kappa=\sigma=\omega=0$, $\rho=\bar\rho\not= 0$), and the four-velocity obeys $u_{[a;b}u_{c]}=0$, $k_{[c}k_{a];b}u^b=0$. The line-element of the space-time can be written in the form ds\^2= -[12]{}\^2(r,u)P\^[-2]{}(z,|z, u)dzd|z +2du(dr-Udu)  , \[11\] with U=r(P)\_[,u]{}+ U\^0(z,|z, u)+ S(r,u)  , \_[,r]{}=0  ,0  . In this case no dust solutions nor solutions of Petrov types $III$ and $N$ are possible. Analysis ======== Let us consider space-times admitting a shear-free, irrotational, geodesic null congruence in which the source of the gravitational field is a [*combination of a perfect fluid and null radiation*]{}, so that the energy-momentum tensor has the form T\_[ab]{}=(+p)u\_au\_b -p g\_[ab]{} +\^2k\_ak\_b  , \[13\] where $u^a$ is the four-velocity of the fluid, $\mu$ and $p$ are the density and the pressure of the fluid, respectively, and ${\bf k}$ is a null vector. The null radiation is geodesic, twist-free, and shear-free, and defines the null congruence. Wainwright [@Wain] proved that for a space-time in which there exists a SIG null congruence, coordinates can be chosen so that the metric takes on the simplified form (\[11\]) with $u=x^1$, $r=x^2$, $z=x^3+i x^4$, the tangent field of the null congruence is given by $k^a=\delta^a_2$, $k_a=\delta^1_a$, and we can introduce the null tetrad k\^a= \^a\_r  , & l\^a= \^a\_u+ U\^a\_r  , & m\^a=P\^[-1]{}( \^a\_3+ i\^a\_4 )  ,\ k\_a=\^u\_a  , & l\_a= -U \^u\_a + \^r\_a  ,& m\_a=P\^[-1]{}(\^3\_a + i \^4\_a)/2  . With the sign convention used here we have that $u^au_a=k^al_a=1=-m^a\bar m_a$. Note that the null radiation is everywhere tangent to the repeated null congruence of the space-time. First, since $\Phi_{01}\equiv-{1\over 2}R_{ab}k^am^b=0$, we conclude that the four-velocity satisfies $u^am_a=0$, and hence it can be expressed in terms of the null tetrad by u\^a=[1 B]{}(B\^2 k\^a +l\^a) u\_a=[1 B]{}\[(B\^2-U)\^u\_a + \^r\_a\]  , \[36\] for some function $B$. The conditions $\Phi_{02}\equiv-{1\over 2}R_{ab}m^am^b=0$ and $\Phi_{12}\equiv-{1\over 2}R_{ab}m^al^b=0$ are satisfied identically. The non-zero components of the Ricci tensor are & &\_[00]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})k\^ak\^b= [12]{}(+p)([**ku**]{})\^2  ,\ & &\_[11]{}-[14]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})(k\^al\^b+m\^a|m\^b)= [14]{}(+p)([**ku**]{}) ([**lu**]{}) ,\ & &\_[22]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})l\^al\^b= [12]{}(+p)([**lu**]{})\^2 +[12]{}\^2  . In addition, since ${\bf k\cdot u}={1\over \sqrt{2} B}$ and ${\bf l\cdot u}={1\over \sqrt{2}} B$ implies ${\bf l\cdot u}=B^2({\bf k\cdot u})$, we obtain B\^2\_[00]{}&=& 2\_[11]{}   , \[34\]\ B\^4\_[00]{}&=&\_[22]{}- [12]{}\^2  . \[35\] If we now assume that the fluid is non-rotating, then $ B^2=U+F(r,u)$, and the compatibility condition (\[34\]) can be written as (U+F)\_[00]{}=2\_[11]{} .\[39\] On differentiating this equation successively with respect to $z$ and $r$, we obtain the restriction (\^2)\_[,rrr]{}\[[U\^0]{}\_[,z]{}+r(P)\_[,uz]{}\]=0  . There are consequently two different cases to consider. In the first case ${U^0}_{,z}+r(\ln P)_{,uz}=0$, which is equivalent to ${U^0}_{,z}=(\ln P)_{,uz}=0$, so that $P=P(z,\bar z)$ and $U^0=U^0(u)$. Obviously, the solutions admit a multiply transitive group of motions, $G_3$, acting on the 2-spaces $r=$const, $u=$const, of constant curvature, and belong to class $II$ of Stewart and Ellis [@Stew]. The metric (\[11\]) can then be rewritten as ds\^2=-\^2(r,u)[2dzd|z(1+[k2]{}z|z)\^2 ]{}+ 2du(dr-U(r,u)du)  . \[44\] The non-zero Ricci components are given by & & \_[00]{}=-[\_[,rr]{}]{}  ,\ & & \_[11]{}=[\_[,r]{}\_[,u]{}2\^2]{} + [(\_[,r]{})\^2U2\^2]{}- [U\_[,rr]{}4]{} + [k4 \^2]{}  ,\ & & \_[22]{}=[\_[,u]{}U\_[,r]{}]{}- [\_[,uu]{}]{} -2[\_[,ur]{}U ]{}-[\_[,r]{}U\_[,u]{}]{} -[\_[,rr]{}U\^2 ]{}  , and the Ricci scalar is given by =12= 4[\_[,r]{}U\_[,r]{}]{}+ 2[\_[,r]{}\_[,u]{}\^2]{} +2 [(\_[,r]{})\^2U\^2]{} + 4 [\_[,ur]{}]{}+ U\_[,rr]{} +4 [\_[,rr]{}U ]{} +[k\^2]{}  . Hence, the metric (\[44\]) can be interpreted as pure radiation plus a perfect fluid where $\mu$ and $p$ are given by =[R 4]{} +6\_[11]{} , p=-[R 4]{}+2\_[11]{} , \[45\] $u_a$ is determined by (\[36\]) with $B^2=2 \Phi_{11}/\Phi_{00}$, and $\phi^2$ is given by \^2=2( \_[22]{}-4[\_[11]{}\^2\_[00]{}]{})  . \[47\] In the second case (i.e., ${\chi^2}_{,rrr}=0$) two possibilities arise: (i) & \^2=r,& = 1 \[52\]\ (ii) & \^2=(r\^2-k\^2), & k=const  .\[53\] In both subcases $\chi=\chi(r)$, and they can be written together as $\chi^2=ar^2+2br+c $, with $a$, $b$, $c$ taken to be appropriate constants. From equation (\[39\]) we obtain aU\^0 -b(P)\_[,u]{} + K =G(u)  , \[55\] and \[\^2 S\_[,r]{} -S(\^2)\_[,r]{}\]\_[,r]{} + [F \^2]{}= G(u)  , \[56\] where $K\equiv 4P^2 (\ln P)_{z\bar z}$, $\Sigma\equiv b^2-ac$, and $G(u)$ is an arbitrary function of $u$. Subcase $(i)$: $a=c=0$, $b=\epsilon/2$. Integrating equation (\[56\]) we see that $S$ can be written in the form S=rh(u)+2G(u)rr -f(u) -[12]{}r\^r [drr]{}F(r,u)  , \[58\] where $h(u)$ and $f(u)$ are arbitrary functions of $u$. Subcase $(ii)$: $a=\epsilon$, $b=0$, $c=-\epsilon k^2$, $\Sigma=k^2$. We obtain S=-G(u) + f(u) \^2 + h(u)\^2 -2k\^2\^2\^r [dr\^2(r)]{}F(r,u)  .\[59\] Therefore, the metric (\[11\]) with $\chi(r)$ given by (\[52\]) or (\[53\]), $S(r,u)$ given by (\[58\]) or (\[59\]), and $P(z, \bar z, u)$ satisfying (\[55\]) can be interpreted as pure radiation plus a perfect fluid, in which the four-velocity is determined by (\[36\]) and $\phi^2$, $\mu$ and $p$ are determined by (\[45\]) and (\[47\]), respectively. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the European Union, TMR Contract No. ERBFMBICT961479 (AMS), the Natural Sciences and Engineering Research Council of Canada (AAC) and the Canadian Institute for Theoretical Astrophysics (DJM). [99]{} A.A. Coley and D.J. McManus, Class. Quantum Grav., [**11**]{}(1994)1261. D.J. McManus and A.A. Coley, Class. Quantum Grav., [**11**]{}(1994)2045. D. Kramer, H. Stephani, M.A.H. MacCallum and E. Herlt, [*Exact Solutions of Einstein’s Field Equations*]{}, Deutscher Verlag der Wissenschaften, Berlin (1980). F.A.E. Pirani, in [*Lectures on General Relativity, 1964 Brandeis Summer Institute in Theoretical Physics*]{}, Vol. 1. Prentice-Hall, Englewood Cliffs, NJ (1965). J.N. Goldberg and R.K. Sachs, Acta. Phys. Polon., Suppl. [**22**]{}(1962)13. I. Robinson and A. Trautman, Proc. Roy. Soc. Lond., [**A265**]{}(1962)463. W. Kundt, Z. Phys., [**163**]{}(1961)77. J. Wainwright, Int. J. Theor. Phys., [**10**]{}(1974)39. J.M. Stewart and G.F.R. Ellis, J. Math. Phys., [**9**]{}(1968)1072.

ArXiv

= 10000 Recently there has been much interest in the search of unconventional electron behavior deviating from the Fermi liquid picture[@unconv]. Besides this, the other paradigm that is well-established on theoretical grounds is the Luttinger liquid behavior of one-dimensional (1D) electron systems[@sol; @hal]. There have been suggestions that this behavior could be extended to two-dimensional (2D) systems, in the hope that it may explain some of the features of the copper-oxide materials[@and]. However, at least for the Luttinger model, the analytic continuation in the number $D$ of dimensions has shown that the Luttinger liquid behavior is lost as soon as one departs from $D = 1$[@ccm; @arri]. Several authors have also analyzed the possibility that singular interactions could lead to the breakdown of the Fermi liquid picture[@sing]. With regard to real low-dimensional systems, such as carbon nanotubes, the main electron interaction comes actually from the long-range Coulomb potential $V(|{\bf r}|) \sim 1/|{\bf r}| $. This is also the case of the 2D layers in graphite, which have a vanishing density of states at the Fermi level. Quite remarkably, a quasiparticle decay rate linear in energy has been measured experimentally in graphite[@exp], pointing at the marginal Fermi liquid behavior in such 2D layers. Due to the singular Coulomb interaction, the imaginary part of the electron self-energy in the 2D system behaves at weak $g$ coupling like $g^2 \omega $[@expl]. It is crucial, though, the fact that the effective coupling scales at low energy as $g \sim 1/\log (\omega )$. This prevents the logarithmic suppression of the quasiparticle weight, which gets corrected by terms of order $g^2 \log (\omega ) \sim 1/\log (\omega )$[@marg]. In this letter we investigate whether the long-range Coulomb interaction may lead to the breakdown of the Fermi liquid behavior at any dimension between $D = 1$ and 2. The issue is significant for the purpose of comparing with recent experimental observations of power-law behavior of the tunneling conductance in multi-walled nanotubes[@mwnt]. These are systems whose description lies between that of a pure 1D system and the 2D graphite layer. It turns out, for instance, that the critical exponent measured for tunneling into the bulk of the multi-walled nanotubes is $\alpha \approx 0.3$. This value is close to the exponent found for the single-walled nanotubes[@bock; @yao]. However, it is much larger than expected by taking into account the reduction due to screening ($\sim 1/\sqrt{N}$) in a wire with a large number $N$ of subbands, what points towards sensible effects of the long-range Coulomb interaction in the system. We develop the analytic continuation in the number of dimensions having in mind the low-energy modes of metallic nanotubes, which have linear branches crossing at the Fermi level. From this picture, we build at general dimension $D$ a manifold of linear branches in momentum space crossing at a given Fermi point. We consider the hamiltonian $$\begin{aligned} H & = & v_F \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \; \Psi^{+} ({\bf p}) \; \mbox{\boldmath $\sigma \cdot $} {\bf p} \; \Psi ({\bf p}) \nonumber \\ \lefteqn{ + e^2 \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \; \rho ({\bf p}) \; \frac{c(D)}{|{\bf p}|^{D-1}} \; \rho (-{\bf p}) \;\;\;\;\;\; } \label{ham}\end{aligned}$$ where the $\sigma_i $ matrices are defined formally by $ \{ \sigma_i , \sigma_j \} = 2\delta_{ij}$. Here $\rho ({\bf p})$ are density operators made of the electron modes $\Psi ({\bf p})$, and $ c(D)/|{\bf p}|^{D-1} $ corresponds to the Fourier transform of the Coulomb potential in dimension $D$. Its usual logarithmic dependence on $|{\bf p}|$ at $D = 1$ is obtained by taking the 1D limit with $ c(D) = \Gamma ((D-1)/2)/(2\sqrt{\pi})^{3-D}$. The dispersion relation $\varepsilon ({\bf p}) = \pm |{\bf p}|$ is that of Dirac fermions, with a vanishing density of states at the Fermi level above $D = 1$. This ensures that the Coulomb interaction remains unscreened in the analytic continuation. At $D = 2$ we recover the low-energy description of the electronic properties of a graphite layer, dominated by the presence of isolated Fermi points with conical dispersion relation at the corners of the Brillouin zone[@graph]. In the above picture, we are neglecting interactions that mix the two inequivalent Fermi points common to the low-energy spectra of graphite layers and metallic nanotubes. In the latter, such interactions have been considered in Refs. and , with the result that they have smaller relative strength ($\sim 0.1/N$, in terms of the number $N$ of subbands) and remain small down to extremely low energies. More recently, the question has been addressed regarding the interactions in the graphite layer, and it also turns out that phases with broken symmetry cannot be realized, unless the system is doped about half-filling[@nos] or it is in a strong coupling regime[@khves]. We will accomplish a self-consistent solution of the model by looking for fixed-points of the renormalization group transformations implemented by the reduction of the cutoff $\Lambda $[@sh]. As usual, the integration of high-energy modes at that scale leads to the cutoff dependence of the parameters in the low-energy effective theory. We will see that the Fermi velocity $v_F$ grows in general as the cutoff is reduced towards the Fermi point. On the other hand, the electron charge $e$ stays constant as $\Lambda \rightarrow 0$. This comes from the fact that the polarizability $\Pi $ does not show any singular dependence on the high-energy cutoff $\Lambda $ for $D < 3$. The polarizability is then given by $$\Pi ({\bf k}, \omega_k) = b(D) \frac{v_F^{2-D} {\bf k}^2} { | v_F^2 {\bf k}^2 - \omega_k^2 |^{(3-D)/2} }\; ,\;$$ where $b(D) = \frac{2}{ \sqrt{\pi} } \frac{ \Gamma ( (D+1)/2 )^2 \Gamma ( (3-D)/2 ) }{ (2\sqrt{\pi})^D \Gamma (D+1) }$. The dependence of $v_F$ on the cutoff $\Lambda $ implies an incomplete cancellation between self-energy and vertex corrections to the polarizability. The dressed polarizability depends therefore on the effective Fermi velocity $v_F (\Lambda )$. The renormalized value of $v_F$ is determined by fixing it self-consistently to the value obtained in the electron propagator $G$ corrected by the self-energy contribution $$\begin{aligned} \Sigma ({\bf k}, \omega_k) & = & - e^2 \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \int \frac{d \omega_p}{2\pi } \nonumber \\ \lefteqn{ G ({\bf k} - {\bf p}, \omega_k - \omega_p) \frac{-i}{ \frac{|{\bf p}|^{D-1}}{c(D)} + e^2 \Pi ({\bf p}, \omega_p) } . } \label{selfe}\end{aligned}$$ The fixed-points of the renormalization group in the limit $\Lambda \rightarrow 0$ determine the universality class to which the model belongs. At $D = 2$, we are bound to obtain the low-energy fixed-point at vanishing coupling of the model of Dirac fermions with Coulomb interaction[@marg]. On the other hand, at $D = 1$ there has to be presumably a fixed-point corresponding to Luttinger liquid behavior. We note, however, that no solution of the model has been obtained yet without carrying dependence on the transverse scale needed to define the 1D logarithmic potential. Our dimensional regularization overcomes the problem of introducing such external parameter, which prevents a proper scaling behavior of the model[@wang]. At general $D$, the self-energy (\[selfe\]) shows a logarithmic dependence on the cutoff at small frequency $\omega_k$ and small momentum ${\bf k}$. This is the signature of the renormalization of the electron field scale and the Fermi velocity. In the low-energy theory with high-energy modes integrated out, the electron propagator becomes $$\begin{aligned} \frac{1}{G} & = & \frac{1}{G_0} - \Sigma \approx Z^{-1} ( \omega_k - v_F \mbox{\boldmath $\sigma \cdot$}{\bf k}) \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ - Z^{-1} f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \left( \frac{n(3-D)}{n(3-D)+2} \omega_k \right. } \nonumber \\ \lefteqn{ + \left. \left(1 - \frac{2}{D} \frac{n(3-D)+1}{n(3-D)+2} \right) v_F \mbox{\boldmath $\sigma \cdot$} {\bf k} \right) h_n (D) \log (\Lambda ) , } \label{prop}\end{aligned}$$ where $g = b(D) c(D) e^2 / v_F $, $f(D) = \frac{1}{ 2^D \pi^{(D+1)/2} \Gamma (D/2) b(D) }$, $h_n (D) = \frac{ \Gamma (n(3-D)/2 + 1/2) } { \Gamma (n(3-D)/2 + 1) }$ . The quantity $Z^{1/2}$ represents the scale of the bare electron field compared to that of the renormalized electron field for which $G$ is computed. The renormalized propagator $G$ must be cutoff-independent, as it leads to observable quantities in the quantum theory. This condition is enforced by fixing the dependence of the effective parameters $Z$ and $v_F$ on $\Lambda $ as more states are integrated out from high-energy shells. We get the differential renormalization group equations $$\begin{aligned} \Lambda \frac{d}{d \Lambda} \log Z (\Lambda ) & = & - f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \frac{n(3-D)}{n(3-D)+2} h_n (D) , } \label{zflow} \end{aligned}$$ $$\begin{aligned} \Lambda \frac{d}{d \Lambda} v_F (\Lambda ) & = & - v_F f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \left( 1 - \frac{1}{D} \frac{n(3-D)(2-D) + 2}{n(3-D) + 2} \right) h_n (D) . } \label{vflow}\end{aligned}$$ At $D = 2$, the right-hand-side of these equations can be summed up to the functions that have been found previously in the renormalization of the graphite layer[@marg]. Furthermore, they also provide meaningful expressions in the 1D limit. At $D = 1$, the right-hand-side of Eq. (\[vflow\]) vanishes identically as a function of the variable $g $. Therefore, the 1D model has formally a line of fixed-points, as it happens in the case of a short-range interaction. The scaling of the electron wavefunction can be read from the right-hand-side of Eq. (\[zflow\]), which becomes $(2 + g)/(2\sqrt{1 + g}) - 1$ at $D = 1$. This coincides with the anomalous dimension that is found in the solution of the Luttinger model, what provides an independent check of the renormalization group approach to the 1D system. We have therefore a model that interpolates between marginal Fermi liquid behavior, that is known to characterize the 2D model, and non-Fermi liquid behavior at $D = 1$. As the electron charge $e$ is not renormalized for $D < 3$, the scaling of the effective coupling $g = b(D) c(D) e^2 / v_F $ is given after Eq. (\[vflow\]) by $$\begin{aligned} \Lambda \frac{d}{d \Lambda} g (\Lambda ) & = & f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+2} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \left( 1 - \frac{1}{D} \frac{n(3-D)(2-D) + 2}{n(3-D) + 2} \right) h_n (D) . } \label{aflow}\end{aligned}$$ The right-hand-side of Eq. (\[aflow\]) is a monotonous increasing function of $g $, for any dimension between 1 and 2, as observed in Fig. \[one\] . = 6cm = 6cm We find that, away from $D = 1$, there is only one fixed-point of the renormalization group at $g = 0$. The scale dependence of the effective coupling $e^2 /v_F$ is displayed for different values of $D$ in Fig. \[two\], where the flow to the fixed-point is seen. Consequently, the scale $Z$ of the wavefunction is not renormalized to zero in the low-energy limit, and the quasiparticle weight remains finite above $D = 1$. We conclude then that, even in a model that keeps the Coulomb interaction unscreened, the breakdown of the Fermi liquid behavior only takes place formally at $D = 1$. = 8cm = 8cm The subtlety concerning the long-range Coulomb interaction is that the function $c(D)$ diverges in the limit $D \rightarrow 1$. This is actually what transforms the power-law dependence of the potential into a logarithmic dependence at $D = 1$. We observe that the 1D limit and the low-energy limit $\Lambda \rightarrow 0$ do not commute. If we stick to $D = 1$, we obtain a divergent coupling $g $ for the Coulomb interaction as well as a divergent electron scaling dimension. At any dimension slightly above $D = 1$, however, the fixed-point is at $g = 0$, with its corresponding vanishing anomalous dimension. In order to understand whether the 1D model has any stable fixed-point for finite values of $e^2 /v_F$, one can study the model by performing an expansion in powers of $g^{-1}$. The Fermi velocity is renormalized by terms that are analytic near the point $g = \infty$, and that lead to the scaling equation $$\Lambda \frac{d}{d \Lambda} v_F (\Lambda ) = \\ v_F f(D) \left( 3 - D - \frac{2}{D} \right) \frac{ \Gamma (D/2 - 1) }{ \Gamma ((D+1)/2) } \label{inf}$$ up to terms of order $O(g^{-1})$. In the limit $D \rightarrow 1$, $g \rightarrow \infty$, the right-hand-side of Eq. (\[inf\]) vanishes identically. This confirms, on nonperturbative grounds, that the 1D model with the Coulomb interaction has a line of fixed-points covering all values of $e^2 /v_F$. In the vicinity of $D = 1$, the presence of such critical line becomes sensible, and a crossover takes place to a behavior with a sharp reduction of the quasiparticle weight. This can be seen in the renormalization of the electron field scale $Z$, displayed in Fig. \[three\]. For values of $D$ above $\approx 1.2$, we have a clear signature of quasiparticles in the value of $Z$ at low energies. For lower values of $D$, the picture cannot be distinguished from that of a vanishing quasiparticle weight for all practical purposes. The drastic suppression of the electron field scale $Z$ takes place over a variation of only two orders of magnitude in the energy scale. = 6cm = 6cm The above picture allows us to make contact with the experiments carried out in multi-walled nanotubes[@mwnt]. In the proximity of the $D = 1$ fixed-point, the density of states displays an effective power-law behavior, with an increasingly large exponent. Moving to the other side of the crossover, the density of states approaches the well-known behavior of the graphite layer, $n( \varepsilon ) \sim |\varepsilon |$. In Fig. \[four\] we give the representation of the density of states $$n( \varepsilon ) \sim Z( \varepsilon ) |\varepsilon |^{D-1}$$ for several dimensions approaching $D = 1$. = 6cm = 6cm A value $e^2 /(\pi^2 v_F) = 0.5 $ for the bare coupling is appropriate for typical multi-walled nanotubes, as it takes into account the reduction due to the interaction with the inner metallic cylinders[@egger]. We observe that the exponents of $n( \varepsilon )$ at different dimensions are always larger than a lower bound $\alpha \approx 0.26$. This is in agreement with the values measured experimentally. Our analysis stresses the need of an appropriate description of the dimensional crossover between one and two dimensions, showing that the picture of a thick nanotube as an aggregate of 1D channels does not allow to obtain the correct values of the critical exponents. To summarize, we have studied the renormalization of the Coulomb interaction in graphene-based structures. We have made a rigorous characterization of the different behaviors, as we have proceeded by identifying the fixed-points of the theory. We have seen that the Fermi liquid behavior persists formally for any dimension above $D = 1$, as it also happens in the case of a short-range interaction[@ccm]. On the other hand, the proximity to the 1D fixed-point influences strongly the phenomenology of real quasi-onedimensional systems, giving rise to an effective power-law behavior of observables like the tunneling density of states. This is the case of the multi-walled nanotubes, for which we predict a lower bound for the corresponding exponent that turns out to be very close to the value measured experimentally. Financial support from CICyT (Spain) and CAM (Madrid, Spain) through grants PB96/0875 and 07N/0045/98 is gratefully acknowledged. J. Gan and E. Wong, Phys. Rev. Lett. [**71**]{}, 4226 (1993). C. Nayak and F. Wilczek, Nucl. Phys. B [**417**]{}, 359 (1994). S. Chakravarty, R. E. Norton and O. F. Syljuasen, Phys. Rev. Lett. [**74**]{}, 1423 (1995). J. González, F. Guinea and M. A. H. Vozmediano, Nucl. Phys. B [**485**]{}, 694 (1997). J. Sólyom, Adv. Phys. [**28**]{}, 201 (1979). F. D. M. Haldane, J. Phys. C [**14**]{}, 2585 (1981). P. W. Anderson, [*The Theory of Superconductivity in the High-$T_c$ Cuprates*]{} (Princeton Univ. Press, Princeton, 1997). C. Castellani, C. Di Castro and W. Metzner, Phys. Rev. Lett. [**72**]{}, 316 (1994). This result is reinforced by the fact that the coupling of Luttinger liquids by the slightest hopping in the transverse direction leads also to Fermi liquid behavior of the system, as shown by X. G. Wen, Phys. Rev. B [**42**]{}, 6623 (1990), and E. Arrigoni, Phys. Rev. Lett. [**83**]{}, 128 (1999). P.-A. Bares and X. G. Wen, Phys. Rev. B [**48**]{}, 8636 (1993). A. Houghton, H.-J. Kwon, J. B. Marston and R. Shankar, J. Phys. Condens. Matter [**6**]{}, 4909 (1994). C. Castellani, C. Di Castro and A. Maccarone, Phys. Rev. B [**55**]{}, 2676 (1997). S. Yu [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 483 (1996). J. González, F. Guinea and M. A. H. Vozmediano, Phys. Rev. Lett. [**77**]{}, 3589 (1996). J. González, F. Guinea and M. A. H. Vozmediano, Phys. Rev. B [**59**]{}, R2474 (1999). C. Schönenberger [*et al.*]{}, Appl. Phys. A [**69**]{}, 283 (1999). A. Bachtold [*et al.*]{}, report cond-mat/0012262. M. Bockrath [*et al.*]{}, Nature [**397**]{}, 598 (1999). Z. Yao [*et al.*]{}, Nature [**402**]{}, 273 (1999). J. González, F. Guinea and M. A. H. Vozmediano, Nucl. Phys. B [**406**]{}, 771 (1993). L. Balents and M. P. A. Fisher, Phys. Rev. B [**55**]{}, R11973 (1997). R. Egger and A. O. Gogolin, Phys. Rev. Lett. [**79**]{}, 5082 (1997); Eur. Phys. J. B [**3**]{}, 281 (1998). J. González, F. Guinea and M. A. H. Vozmediano, Phys. Rev. B [**63**]{}, (2001) to appear. D. V. Khveshchenko, report cond-mat/0101306. R. Shankar, Rev. Mod. Phys. [**66**]{}, 129 (1994). On phenomenological grounds, one can however make sense of such dependence on the transverse dimension, which leads to scale-dependent critical exponents as reported by C. Kane, L. Balents and M. P. A. Fisher, Phys. Rev. Lett. [**79**]{}, 5086 (1997), S. Bellucci and J. González, Eur. Phys. J. B [**18**]{}, 3 (2000), and D. W. Wang, A. J. Millis and S. Das Sarma, report cond-mat/0010241. R. Egger, Phys. Rev. Lett. [**83**]{}, 5547 (1999).

ArXiv

--- abstract: 'Bio-medical ontologies can contain a large number of concepts. Often many of these concepts are very similar to each other, and similar or identical to concepts found in other bio-medical databases. This presents both a challenge and opportunity: maintaining many similar concepts is tedious and fastidious work, which could be substantially reduced if the data could be derived from pre-existing knowledge sources. In this paper, we describe how we have achieved this for an ontology of the mitochondria using our novel ontology development environment, the [Tawny-OWL]{}library.' address: 'School of Computing Science, Newcastle University, Newcastle-upon-Tyne, UK' author: - 'Jennifer D. Warrender and Phillip Lord[^1]' bibliography: - '2015\_scaffolding\_pwl\_jw.bib' title: Scaffolding the Mitochondrial Disease Ontology from extant knowledge sources --- Introduction ============ Bio-medical ontologies vary in size, with largest containing millions of concepts. Building ontologies of this size is complex, time-consuming and expensive and just as challenging to maintain and update. Ontologies are only one of many mechanisms for the computational representation of knowledge. In some cases, ontologies are created where many of the needed concepts will be available elsewhere as terms in different structured representations. Being able to reuse these representations as a *scaffold* for the rest of an ontology might be able to reduce the cost and work-load of producing ontologies. This is evidenced by, for instance, SIO [@sio] which contains a list of all the chemical elements. Or the Gene Ontology (GO) [@go], which contains many terms related to chemical homeostasis, each of which need to relate to a specific chemical described in ChEBI [@chebi]. In addition to being described elsewhere, these concepts are often highly similar to each other. In extreme cases such as the amino acid ontology [@greycite9379], ontologies can consist of only related concepts, and “support” concepts that are used to describe them. One solution to this is the use of patterns. A pattern is an abstract specification of an ontology axiomatisation with a number of “variables”. The pattern is instantiated by providing values for these variables, which are then expanded into the full axiomatisation providing one or more concepts. Patterns have been implemented by a number of different tools, which differ in how the patterns are specified, and how and when the values are provided for the variables. For example, *termgenie* is a website which allows submission to GO (and others) [@Dietze_2014]. Variable values are entered through a form which then generates axioms, definitions and cross-references. For instance, this is the axiomatisation from termgenie when defining the term “cytosine homeostasis” is_a: GO:0048878 {is_inferred="true"} ! chemical homeostasis intersection_of: GO:0048878 ! chemical homeostasis intersection_of: regulates_levels_of CHEBI:16040 ! cytosine relationship: regulates_levels_of CHEBI:16040 {is_inferred="true"} ! cytosine As well as the axiomatisation, termgenie also generates a number of different annotations including a definition, submitter information, and status. With termgenie, patterns are specified through the use of JavaScript functions. In addition to termgenie, other systems also allow patterns. For example, both the desktop and web version of [Protégé]{}contain forms, which grant users the ability to customise the GUI and specify several axioms at once. In this case, patterns are declaratively defined (implicitly, with a GUI design) in XML [@tudorach_icd_webprotege]. Applications like Populous [@Jupp_Wolstencroft_Stevens_2011] and Rightfield [@rightfield] use spreadsheets or spreadsheet-like interfaces to enter data, which is then transformed into a set of OWL axioms based on a pattern. In the case of these two, the patterns are specified in OPPL, a pattern language for OWL which can also be used independently [@aranguren_Stevens_Antezana_2009]. Finally, the Brain API allows programmatic construction of ontologies in an easy to use manner using Java [@croset2013]. While these systems are all aimed at somewhat different use-cases, they all address the same problem; how to produce a large number of concepts all of which are similar, and to do so with a high-degree of repeatability. However, the use of this form of patternised ontology tool presents a number of problems. These tools provide a mechanism for adding many axioms at once, but not removing them again[^2]. If the knowledge changes, then this is a problem as the axioms added from a given pattern need to be removed or updated. Furthermore, if the knowledge engineering changes i.e. the pattern is updated, then all axioms added from any use of the pattern must also be updated. In this paper, we describe how we have addressed these problems with the Mitochondrial Disease Ontology (MDO), through the use of the [Tawny-OWL]{}environment, which is a fully programmatic environment for ontology development. With [Tawny-OWL]{}, we can use a *pattern-first* ontology development process, building with patterns and data from extant knowledge sources from the start. This has allowed us to generate a *scaffold* which we can then populate further with hand-crafted links between parts of this scaffold where the knowledge exists. As a result, it is possible to update both the knowledge and the patterns by simply regenerating the ontology. This process promises to aid in both the construction and maintenance of ontologies. The MDO is available from <https://github.com/jaydchan/tawny-mitochondria>. [Tawny-OWL]{}is available from <https://github.com/phillord/tawny-owl>. The Mitochondria Disease Ontology (MDO) {#sec:mitoch-dise-ontol} ======================================= Mitochondria are complex organelles found in most eukaryotic cells. Their key function is to enable the production of ATP through oxidative phosphorylation, providing usable energy for the rest of the cell. The mitochondria carry their own small genome containing 37 genes in human. Many other genes are involved in producing proteins involved in mitochondrial function, but these are encoded in the nuclear genome. A number of mitochondrial genes are associated with diseases; the first identified of these is the MELAS [@melas], which is most commonly caused by a point mutation in a tRNA found in the mitochondrial genome. As with many areas of biology, mitochondrial research is a large, knowledge-rich discipline. Our purpose with the MDO is to attempt to formalise this knowledge, using an incremental or “pay-as-you-go” data integration approach. The ontology here serves as a tool for reasoning and knowledge exploration, rather than to form as a reference ontology [@handbook2]. This is an approach we have previously found useful in classifying phosphatases [@wolstencroftetal2006]. The hope is that we can incorporate new knowledge as it is released, checking it for consistency and cross-linking it with existing knowledge. [Tawny-OWL]{} {#sec:tawny} ============= In this section, we give a brief description of [Tawny-OWL]{} [@tawny] and how it supports pattern-first development. [Tawny-OWL]{}is a library written in Clojure, a dialect of lisp. It wraps the OWL API [@owlapi] and allows the fully programmatic constructions of ontologies. It has a simple syntax which was modelled on the Manchester Syntax [@ms2], modified to integrate well with Clojure. It can be used to make simple statements in OWL: (defclass A :super (some r B)) which makes defines a new class $A$ such that $A\sqsubseteq~\exists~r~B$. Although this is similar to the equivalent Manchester Syntax statements, [Tawny-OWL]{}provides a feature called “broadcasting” which is, essentially a form of pattern. So this following statement: (some r B C) is equivalent to the two statements $\exists~r~B$ and $\exists~r~C$. We apply the first two arguments (|some| and |r|) to the remaining ones consecutively. It also provides simple patterns, such as the covering axiom, so: (some-only r B C) is equivalent to three statements $\exists~r~B$, $\exists~r~C$ and $\forall~r~(B~\sqcup~C)$. While the patterns shown here are provided by [Tawny-OWL]{}, end ontology developers are using the same programmatic environment. Patterns are encoded as functions and instantiated with function calls. For instance, we could define |some-only| as follows: (defn some-only \[property & classes\] (list (some property classes) (only property (or classes)))) Here |defn| introduces a new function, |property & classes| are the arguments, and |list| packages the return values as a list. |some|, |only| and |or| are defined by [Tawny-OWL]{}as the appropriate OWL class constructors. It is, therefore, possible to build *localised patterns* — custom patterns for use predominately with the current ontology [@warrender_thesis_2015]. Patterns can call each other and can be of arbitrary complexity. The use of [Tawny-OWL]{}, therefore, inverts the usual style of ontology development. Non-patternised classes are just trivial instantiations of patterns. Building a Mitochondrial Scaffold {#sec:build-mitoch-scaff} ================================= Following a requirements gathering phase for MDO, it was clear from our competency questions (for example “What are all the genes/proteins that are associated with a specific syndrome?”) that we needed many concepts which were heavily repetitive, and further which have comprehensive and curated lists available. We describe these parts of the domain knowledge as the *scaffold*. For example, there are around 761 genes whose products are involved in mitochondrial function. Classes representing these genes do not, in the first instance, require complex descriptions, and are defined within MDO as follows: (defclass Gene) (defn gene-class \[name\] (owl-class name :label name :super Gene)) This pattern is then populated using a simple text file, with the 761 gene names present. The gene pattern is an extremely simple pattern, as these concepts are self-standing. Other parts of the ontology are even simpler; for instance, for describing mitochondrial anatomy, the classes have similar complexity to the genes, but there are only 15. In this case, classes are defined with a pattern and a list “hard-coded” into the MDO source code, rather than using an external text file. Other patterns are more complex. For instance, the subclasses of |Disease| are defined as follows: (defn disease-class \[name omim lname\] (let \[disease (owl-class name :label name :super Disease)\] (if-not (nil? omim) (refine disease :annotation (see-also (str “OMIMID:” omim)))) (if-not (nil? lname) (refine disease :label (str “Long name:” lname))))) This function adds two annotations to each disease class, if they are available. This function also demonstrates the use of conditionals (|if|), predicates (|nil?|) and string concatenation (|str|); these are not provided by [Tawny-OWL]{}, but by Clojure and demonstrate the value of building [Tawny-OWL]{}inside a fully programmatic environment. Fitting out the Scaffold {#sec:what-we-have} ======================== The top-level of the MDO is shown in Figure \[fig:mtoplevel\]. Of these classes, “Paper” and “Term” are described later. ![The top-level structure of Mitochondrial Disease Ontology. Classes that are a part of the scaffold are coloured in orange, while classes that are built on top of the scaffold are coloured in green.[]{data-label="fig:mtoplevel"}](mstructure.png){width="\columnwidth"} The remaining classes define the scaffold, which now has a total of 1357 classes; a break-down of these classes and their sources is shown in Table \[tab:genericStats\]. For the next stage of the process, we are now building on top of this scaffold, using hand-crafted and bespoke knowledge. This is being achieved by manual extraction of knowledge from papers about mitochondria. Our initial process is to find references in papers to the terms that are represented by classes we have built in the scaffold, and draw explicit relationships between these papers and the scaffolded knowledge that they describe. Currently, these classes also use a patternised approach; the raw data is held in a bespoke (but human readable) syntax[^3], which is then parsed and used to instantiate patterns. In total, there are now 2174 classes created from this approach from around 30 papers. These terms currently are not defined beyond their name and the source paper from which they were identified. We do not consider them directly as part of the scaffold, as they are not from an extant knowledge source, but one that we have created; they are the first layer build on top of our scaffold. We expect future layers to use the [Tawny-OWL]{}syntax directly, as the knowledge increases in complexity and decreases in regularity. Resiliance to Change {#sec:resiliance-change} ==================== One key feature of our development process is that the OWL which defines the MDO is no longer *source code* but generated. Rather it is generated from patterns defined in [Tawny-OWL]{}and text files which are used to instantiate these patterns. The in-memory OWL classes and associated OWL files are generated on-demand, by *evaluating* the patterns. Effectively, we regenerate the ontology every time we restart the environment. In this section, we consider the types of changes that can happen, and how these changes impact on MDO. The scaffold of MDO is sensitive to changes in its dependency knowledge sources. First, new terms can be entered into extant sources, which will necessitate the addition of new classes. For the MDO, this simply necessitates re-importing the knowledge. The addition of equivalent new classes will then happen automatically according to the patterns already defined; no other changes should be necessary for the MDO, although we may wish to refer to the new classes in other parts of the ontology. Second, terms may be removed from dependencies; so, for example, a disease may be redefined by the UMDF. In many cases, for the MDO, this is not problematic – the equivalent classes will simply disappear from the ontology. [Tawny-OWL]{}provides two features to help with changes to terms in the scaffold when these terms are also referred to outside of the scaffold. [Tawny-OWL]{}uses a “declare-before-use” semantics, so removal of classes from the scaffold will cause fail-fast behaviour when they are used elsewhere. The Brain environment uses the same semantics for similar reasons [@croset2013]. In addition, [Tawny-OWL]{}provides a “deprecation” facility which allows the developer to continue refer to terms from the scaffold which have been removed, but to receive warnings about this use; this is rather like obsolescence, but happens automatically[^4]. Third, the MDO scaffold can also cope straight-forwardly with changes to patterns. As with the addition or removal of terms from dependencies, pattern changes will simply take place by re-evaluating the ontology. Finally, the MDO is resilient to changes in ontology engineering conventions. For example, MDO does not use OBO style numeric identifiers, nor provide stable IRIs for integration with linked data sources since these are not critical at the current time[^5]. They, however, could be added easily to all existing (and future) terms in a few lines of code, using an existing facility within [Tawny-OWL]{}for minting and persisting numeric identifiers in an automatic, yet managed, way. This change would just alter IRIs and would have no impact on references between concepts inside or outside of the scaffold. In conclusion, as well as enabling rapid construction of the MDO, we believe that the pattern-first scaffolding approach should also allow easy maintenance of the ontology. Discussion {#sec:discussion} ========== In this paper, we have described how we have used a number of extant knowledge sources, combined with patterns defined using the [Tawny-OWL]{}library to rapidly, reliably and repeatedly construct a scaffold for MDO. We have previously used a related patternised methodology to construct a complex ontology describing human chromosome rearrangements (i.e. The Karyotype Ontology (KO) [@warrender-karyotype]). However, unlike KO, the mitochondrial knowledge we want to encapsulate is found in numerous independent sources (e.g. published papers and online databases) and in a variety of formats (e.g. “free text” and CSV); the use of several patterns to form a scaffold is unique to MDO. Conversely, the axiomatisation of MDO from these sources is simple; this cannot be said for KO, most of which is generated from a single large pattern [@warrender-pattern]. In addition, while our knowledge of the karyotype is constrained and is essentially finished, the community’s understanding of mitochondria and mitochondrial disease is incomplete and will grow in response to the demands of changing knowledge. This methodology is extremely attractive for a number of reasons. First of all, it allows a very rapid way of scaffolding an ontology for a complex area of knowledge. At this stage, most of the classes created are simple and self-standing, although in some cases do have relationships to other entities in the scaffold. At this point, we have built the ontological equivalent of a data warehouse: terms have been taken from elsewhere and have undergone a form of schema reconciliation into ontological classes. One key feature of the MDO is that it has been built using tools designed for software development; these tools are relatively advanced and well-maintained[^6] [@tawny]. Moreover, recreating the MDO ontology from our original [Tawny-OWL]{}source code is an intrinsic part of the development process; there is no complex release process and any ontology developer can recreate the OWL file with a single command. While, the system as it stands has a high-degree of replicability, the design decisions implicit in the source code are not necessarily apparent. For the basic scaffold this is, perhaps, not a major issue, however as MDO is developed outside of its scaffold , we expect to integrate more documentation into the source code itself, using *lentic*, a recently developed tool for literate programming [@greycite23590]. We believe that the engineering process that we have used to build the scaffold is resilient to change, as described in Section \[sec:resiliance-change\]. Despite this resilience, our use of external sources of knowledge does bring with it new dependencies, with all the issues that this entails for change management. We believe that we can manage this by borrowing best practice from software engineering. Importing knowledge into the scaffold can, in many cases, happens entirely automatically from our extant knowledge sources. Considering just the gene lists, we can either import from a local, fixed copy of this list, or take the current version live from the NCBI portal. In software engineering terms, the former is a *release dependency* and provides stability, while the latter is a *snapshot dependency* which will fail-fast, allowing rapid incorporation of new knowledge. The latter is particularly useful within a continuous integration environment which are used with other ontologies [@greycite2899], and are also fully supported by [Tawny-OWL]{}[@tawny]. Although we have not described its usage here, with the MDO we are not forced to use [Tawny-OWL]{}for all development. It would be possible to combine predominately hand-crafted development using [Protégé]{}, for instance, with some patternised classes; for example, the OBI uses this approach [@2041-1480-1-s1-s7]. For, the MDO, in fact almost all terms other than the top-level has been created from other syntaxes, generally a flat-file. For larger projects, we envisage that most ontology developers would not need to use the programmatic nature of [Tawny-OWL]{}. While we appreciate the value of a single environment, a tool should not force all users into it. In this paper, we have described our approach to building the MDO using a patternised scaffold based around existing knowledge sources. While the work described in this paper allows us to integrate structured data into an ontology, we are now investigating new ways of integrating unstructured literate-based knowledge into our ontology; while we have started the process of formalising, this new knowledge is far from finished. As described in this paper, though, a pattern-first, scaffolded approach to ontology development has enabled us to make significant advances with the MDO. We believe that this approach is likely to be applicable to many other domains also. [^1]: To whom correspondence should be addressed: [email protected] [^2]: OPPL can remove axioms as well as add them but this is not automatic. [^3]: In this case *EDN* which is a text representation of Clojure data structures; it looks rather like JSON. [^4]: [Tawny-OWL]{}is implemented in a Lisp and so is homoiconic; this makes it particularly straight-forward to automate code updates if we choose. [^5]: Our initial intention was to use PURLS from [www.purl.org](www.purl.org) but have found practical problems with generating these. [^6]: And, usefully, not dependent on academic developers for future maintenance.

ArXiv

--- abstract: 'A lack of code-switching data complicates the training of code-switching (CS) language models. We propose an approach to train such CS language models on monolingual data only. By constraining and normalizing the output projection matrix in RNN-based language models, we bring embeddings of different languages closer to each other. Numerical and visualization results show that the proposed approaches remarkably improve the performance of CS language models trained on monolingual data. The proposed approaches are comparable or even better than training CS language models with artificially generated CS data. We additionally use unsupervised bilingual word translation to analyze whether semantically equivalent words in different languages are mapped together.' address: | Graduate Institute of Communication Engineering, National Taiwan University\ `{f04942141, b03902042, hungyilee}@ntu.edu.tw` bibliography: - 'Template.bib' title: 'Training a Code-Switching Language Model with Monolingual Data' --- Code-Switching, Language Model Introduction {#sec:intro} ============ Code-switching (CS), which occurs when two or more languages are used within a document or a sentence, is widely observed in multicultural areas. Related research is characterized by a lack of data; the application of prior knowledge [@zeng2017improving; @6639306] or additional constraints [@li2013improved; @ying2014language] would alleviate this issue. Because it is easier to collect monolingual data than CS data, efficiently utilizing a large amount of monolingual data would be a solution to the lack of CS data [@hamed2017building]. Recent work [@gonen2018language] attempts to train a CS language model using fine-tuning. Similar work [@garg2017dual] integrates two monolingual language models (LMs) by introducing a special “switch” token in both languages when training the LM, and further incorporating this within automatic speech recognition (ASR). Other works synthesize additional CS text using the modeled distribution from the data [@winata2018learn; @yilmaz2018acoustic]. Generative adversarial neural networks [@goodfellow2014generative; @arjovsky2017wasserstein] learn the CS point distribution from CS text [@chang2018code]. In this paper, we propose utilizing constraints to bring word embeddings of different languages closer together in the same latent space, and to normalize each word vector to generally improve the CS LM. Similar constraints are used in end-to-end ASR [@khassanov2019constrained], but have not yet been reported for CS language modeling. Related prior work [@audhkhasi2017direct; @settle2019acoustically] attempts to initialize the word embedding with unit-normalized vectors in ASR but does not keep the unit norm during training. Initial experiments on CS data showed that constraining and normalizing the output projection matrix helps LMs trained on monolingual data to better handle CS data. Code-Switching Language Modeling {#sec:proposed} ================================ In our approach, we use monolingual data only for training; CS data is for validation and testing only. RNN-based Language Model {#ssec:rnn} ------------------------ We adopt a recurrent neural network (RNN) based language model [@Mikolov2010RecurrentNN]. Given a sequence of words $[w_1, w_2, \dots, w_T]$, we obtain predictions $y_i$ by applying transformation $W$ on RNN hidden states $h_i$ with softmax computation: $$\label{eq:rnn} \begin{aligned} y_i &= \mathrm{softmax}(W h_i) \end{aligned} $$ where $i = 1, 2, \dots, T$ and $h_0$ is a zero vector. Specifically, the output projection matrix is denoted by $W \in \mathbb{R}^{V \times z}$, where $V$ is the vocabulary size and $z$ is the hidden layer size of the RNN. Gradient descent is then used to update the parameters with a cross entropy loss function. Consider two languages [$\mathit{L1}$]{} and [$\mathit{L2}$]{} in CS language modeling: the output projection matrix $W$ is partitioned into $W_1$ and $W_2$, with each row indicating the latent representations of each word in [$\mathit{L1}$]{} and [$\mathit{L2}$]{} respectively. With careful organization, the output projection matrix $W$ can be written as $\begin{bmatrix}W_1 \\ W_2\end{bmatrix}$. Constraints on Output Projection Matrix {#ssec:constraint} --------------------------------------- By optimizing the LM with [$\mathit{L1}$]{} and [$\mathit{L2}$]{} monolingual data, it is possible to improve the perplexity on both sides. Word embedding distributions have arbitrary shapes based on their language characteristics. Without seeing bilingual word pairs, however, the two distributions may converge into their own shape without correlating to each other. It is difficult to train an LM to switch between languages. To train an LM with only monolingual data, we assume that overlapping embeddings benefit CS language modeling. To this end, we attempt to bring word embeddings of [$\mathit{L1}$]{} and [$\mathit{L2}$]{}, that is $W_1$ and $W_2$, closer to each other. We constrain $W_1$ and $W_2$ in the two ways; Fig. \[fig:method\] shows an overview of the proposed approach. ### Symmetric Kullback–Leibler Divergence {#ssec:skld} Kullback–Leibler divergence (KLD) is a well-known measurement of the distance between two distributions. Minimizing the KLD between language distributions overlaps the embedding space semantically. We assume that both $W_1$ and $W_2$ follow a $z$-dimensional multivariate Gaussian distribution, that is, $$\begin{aligned} & W_1 \sim N(\mu_1, \Sigma_1), & W_2 \sim N(\mu_2, \Sigma_2) \end{aligned}$$ where $\mu_1, \mu_2 \in \mathbb{R}^z$ and $\Sigma_1, \Sigma_2 \in \mathbb{R}^{z \times z}$ are the mean vector and co-variance matrix for $W_1$ and $W_2$ respectively. Based on the assumption of Gaussian distribution, we can easily compute KLD between $W_1$ and $W_2$. Due to the asymmetric characteristic of KLD, we adopt the symmetric form of KLD (SKLD), that is, the sum of KLD between $W_1$ and $W_2$ and that between $W_2$ and $W_1$: $$\begin{aligned} L_{\mathit{SKLD}} &= \frac{1}{2}\left[tr(\Sigma^{-1}_1\Sigma_2 + \Sigma^{-1}_2\Sigma_1)\right. \\ &\:\:\:\:\left.+ (\mu_1 - \mu_2)^T (\Sigma^{-1}_1 + \Sigma^{-1}_2) (\mu_1 - \mu_2) - 2z\right].\end{aligned}$$ ### Cosine Distance {#ssec:cd} Cosine distance (CD) is a common measurement for semantic evaluation. By minimizing CD, we are attempting to bring the semantic latent space of languages closer. Similar to SKLD, we compute the mean vector $\mu_1$ and $\mu_2$ of $W_1$ and $W_2$ respectively, and CD between two mean vectors is obtained as $$L_{\mathit{CD}} = 1 - \frac{\mu_1 \cdot \mu_2}{\|\mu_1\| \|\mu_2\|},$$ where $\|\cdot\|$ denotes the $\ell^2$ norm. We hypothesize the latent representation of each word in [$\mathit{L1}$]{} and [$\mathit{L2}$]{} is distributed in the same semantic space and overlaps by minimizing SKLD or CD. Output Projection Matrix Normalization {#sec:normalizing} -------------------------------------- Apart from the constraints from Section \[ssec:constraint\], we propose normalizing the output projection matrix, that is, each word representation is divided by its $\ell^2$ norm to possess unit norm. Note that normalization is independent of constraints, and can be applied together. In normalization, we consider semantically equivalent words $w_j$ and $w_k$: the cosine similarity between their latent representation $v_j$ and $v_k$ should be 1, implying the angle between them is 0, that is, they have the same orientation. By Eq. (\[eq:rnn\]), we observe that the probabilities $y_{i,j}=\frac{\exp(v_j \cdot h_i)}{\sum_{m=1}^V \exp(v_m \cdot h_i)}$ and $y_{i,k}=\frac{\exp(v_k \cdot h_i)}{\sum_{m=1}^V \exp(v_m \cdot h_i)}$ are not necessarily equal because the magnitude of $v_j$ and $v_k$ might not be the same. However, being a unit vector, normalization guarantees that given the same history, the probabilities of two semantically equivalent words generated by the LM will be equal. Thus normalization is helpful for clustering semantically equivalent words in the embedding space, which improves language modeling in general. Experimental Setup {#sec:exp_setup} ================== Corpus {#ssec:corpus} ------ The South East Asia Mandarin-English (SEAME) corpus [@Lyu2010SEAMEAM] was used for the following experiments. It can be simply separated into two parts by its literal language. The first part is monolingual, containing pure Mandarin and pure English transcriptions, the two main languages in this corpus. The second part is code-switching (CS) sentences, where the transcriptions are a mix of words from the two languages. The original data consists of *train*, *dev\_man*, and *dev\_sgn*.[^1] Each split contains monolingual and CS sentences, but *dev\_man* and *dev\_sgn* are dominated by Mandarin and English respectively. We held out 1000 Mandarin, 1000 English, and all CS sentences (because we needed only monolingual data to train the LM) from *train* as the validation set. The remaining monolingual sentences were for the training set. Similar to prior work [@khassanov2019constrained], we used *dev\_man* and *dev\_sgn* for testing, but to balance the Mandarin-to-English ratio, we combined them together as the testing set. Pseudo Code-switching Training Data {#ssec:training_method} ----------------------------------- To compare the performance of the constraints and normalization with an LM trained on CS data, we also introduce pseudo-CS data training, in which we use monolingual data to generate artificial CS sentences. Two approaches are used to generate pseudo-CS data: **Word substitution** Given only monolingual data, we randomly replace a word in monolingual sentences with its corresponding word in the other language based on the substitution probability to produce CS data. However, this requires a vocabulary mapping between the two languages. We thus use the bilingual translated pair mapping provided by MUSE [@conneau2017word].[^2] Note that not all translated words are in our vocabulary set. **Sentence concatenation**: We randomly sample sentences from different languages from the original corpus and concatenate them to construct a pseudo-CS sentence which we add to the original monolingual corpus. Evaluation Metrics {#ssec:evaluation} ------------------ Perplexity (PPL) is a common measurement of language modeling. Lower perplexity indicates higher confidence in the predicted target. To better observe the effects of the techniques proposed above, we computed five kinds of perplexity on the corpus: 1) **ZH**: PPL of monolingual Mandarin sentences; 2) **EN**: PPL of monolingual English sentences; 3) **CS-PPL**: PPL of CS sentences; 4) **CSP-PPL**: the PPL of CS points, which occur when the language of the next word is different from current word; 5) **Overall**: the PPL of the whole corpus, including monolingual and CS sentences. Due to the difference between CS-PPL and CSP-PPL, these perplexities are separately measured. Clearly, improvements in CS-PPL do not necessarily translate to improvements in CSP-PPL; as CS sentences often contain a majority of non-CS points, CS-PPL is likely to benefit more from improving monolingual perplexity than from improving CSP-PPL. Implementation {#ssec:implementation} -------------- Due to the limited amount of training data, we adopted only a single recurrent layer with long short-term memory (LSTM) cells for language modeling [@sundermeyer2012lstm]. The hidden size for both the input projection and the LSTM cells was set to 300. We used a dropout of 0.3 for better generalization, and trained the models using Adam with an initial learning rate of 0.001. In order to obtain better results, the training procedure was stopped when the overall perplexity on the validation set did not decrease for 10 epochs. All reported results are the average of 3 runs. Results {#sec:results} ======= Language Modeling {#ssec:ppl_results} ----------------- The results are in Table \[table:ppl\], which contains results for (A) the language model trained with monolingual data only; (B) word substitution with substitution probability;[^3] and (C) sentence concatenation as mentioned in Section \[ssec:training\_method\]. (D), (E), and (F) are the results after applying the normalization from Section \[sec:normalizing\] on (A), (B), and (C) respectively. Baselines in rows (a)(d)(g) represent the language model trained without constraints or normalization.[^4] Observing rows (a)(d)(g), we observe that learning with pseudo-CS sentences indeed helps considerably in CS perplexity, which is reasonable because the LM has seen CS cases during training even though the training data is synthetic. However, comparing rows (b)(c) with (d) and (g) reveals that after applying additional constraints, the LM trained on monolingual data only is comparable or even better in terms of both monolingual (ZH and EN columns) and CS (CS-PPL and CSP-PPL columns) perplexity than LMs trained with pseudo-CS data. Whether using monolingual or pseudo-CS data for training, normalizing the output projection matrix generally improves language modeling. Even trained with monolingual data only, normalization also improves CSP-PPL, as shown in rows (a) and (j). Thus we conclude that the monolingual data in our corpus has a similar sentence structure, and normalization yields a similar latent space, aiding in switching between languages. After applying SKLD and normalization together, the CSP-PPL improves, yielding the best results in the monolingual data training case. The perplexity of CS points is reduced significantly when constraints are applied on the output projection matrix by minimizing SKLD or CD without degrading the performance on monolingual data. Rows (k)(n)(q) also show that combining the SKLD constraint with normalization results in the best performance on each kind of perplexity over only monolingual and pseudo-CS data. Visualization {#ssec:pca_eval} ------------- In addition to numerical analysis, we seek to determine if the overlapping level of embedding space is aligned with the perplexity results. We applied principal component analysis (PCA) on the output projection matrix, and then visualized the results on a 2-D plane. Fig. \[fig:pca\] shows the visualized results of different approaches. Fig. \[sfig:pca:baseline\] shows that embeddings of two languages are linear separable with monolingual data only and without applying any proposed approach. After synthesizing pseudo-CS data for training as shown in Fig. \[sfig:pca:ws\], the embeddings of the two languages are closer than Fig. \[sfig:pca:baseline\] but without excessive overlap. In Fig. \[sfig:pca:skld\], they totally overlap. This corresponds to the numerical results in Table \[table:ppl\]: the closer the embeddings are, the lower the perplexity is.[^5] Unsupervised Bilingual Word Translation {#ssec:bilingual_word_translation} --------------------------------------- To analyze whether words with equivalent semantics in different languages are mapped together with the proposed approaches, we conducted experiments on unsupervised bilingual word translation. Given a word $w$ existing in the same bilingual pair mapping mentioned in Section \[ssec:training\_method\], each word in the other language is ranked according to the cosine similarity of their embeddings. If the translated word of $w$ is ranked as the $r$-th candidate, then the reciprocal rank is $\frac{1}{r}$. The mean reciprocal rank (MRR) is used as an evaluation metric, which is the average of the reciprocal ranks; thus the MRR should be less than 1.0, and the closer to 1.0 the better. The proportion of correct translations that are in the top 10 candidate list ($r \leq 10$) is also reported as “P@10” [@Xing2015NormalizedWE]. In order to mitigate the degradation in performance caused by low-frequency words, we selected words only with a frequency greater than 80, resulting in about 200 vocabulary words in Mandarin and English respectively, and 55 bilingual pairs used for unsupervised bilingual word translation. The results of bilingual word translation are in Table \[table:mrr\]. We see performance for Mandarin-English translation (column (A)) in both MRR and P@10 that is worse than that in the reverse direction (column (B)). Row (i) demonstrates that the unconstrained baseline performs poorly, whereas additional constraints and normalization in rows (ii) and (iii) yield significantly improved MRR and P@10 compared with row (i). This suggests that constraints and normalization for CS language modeling indeed enhance semantic mapping. Sentence Generation {#ssec:setence_gen} ------------------- We further evaluated the sentence generation ability of language models trained only with monolingual data. Given part of a sentence, we used the language model to complete the sentence. Two generated sentences and their given inputs are shown in Table \[table:sentence\_generateion\]. Our best approach with SKLD constraint and normalization, listed in column (C), switches languages either from English to Mandarin (row (i)) or from Mandarin to English (row (ii)). However, the baseline model in column (B) fails to code-switch from either side. Conclusions {#sec:conclusions} =========== In this work, we train a code-switching language model with monolingual data by constraining and normalizing the output projection matrix, yielding improved performance. We also present an analysis of selected results, which shows our approaches help monolingual embedding space overlap and improves the measurements on bilingual word translation evaluation. [^1]: https://github.com/zengzp0912/SEAME-dev-set [^2]: https://github.com/facebookresearch/MUSE [^3]: We performed grid search on the substitution probability and 0.2 achieved the lowest perplexity. [^4]: A smoothed 5-gram model was also evaluated, but it yielded worse performance than the baseline. Due to limited space, we omit the results here. [^5]: Due to limited space, we do not show the visualization results of sentence concatenation/CD which is quite similar to Fig. \[sfig:pca:ws\]/\[sfig:pca:skld\].

ArXiv

--- abstract: 'A phenomenological QCD quasiparticle model provides a means to map lattice QCD results to regions relevant for a variety of heavy-ion collision experiments at larger baryon density. We report on effects of collectives modes and damping on the equation of state.' author: - | R. Schulze[^1], B. Kämpfer\ \ Forschungszentrum Dresden-Rossendorf, PF 510119, 01314 Dresden, Germany\ Institut für Theoretische Physik, TU Dresden, 01062 Dresden, Germany date:   title: Equation of state for QCD matter in a quasiparticle model --- Strongly interacting matter is governed by the fundamental theory of QCD, which can be solved numerically using Monte-Carlo calculations on the lattice. However, reliable results are still limited to rather small net baryon densities [@Eji06]. As an alternative approach to obtain thermodynamic gross properties of the quark-gluon plasma, a thermodynamic quasiparticle model (QPM) incorporating 1-loop QCD in hard thermal loop (HTL) approximation can be utilized [@Pes00; @BIR01; @BKS07a; @Sch08]. Employing [@Sch08] the Cornwall-Jackiw-Tomboulis formalism, the entropy density assumes the simple form of a sum $s=s_{g,\text{T}}+s_{g,\text{L}}+s_{q,\text{Pt.}}+s_{q,\text{Pl.}}+s'$ over partial entropy density contributions from four quasiparticle excitations (transverse and longitudinal gluons, quarks and plasminos). The residual interaction term $s'$ vanishes at 2-loop order for the generating functional. Individual contributions read $$s_{i}\sim d_{i}\int_{{\mathrm{d}^{4}k}()}\{\pi\varepsilon({\text{Im}}D_{i}^{-1})\Theta\!\left(\xi_{i}{\text{Re}}D_{i}^{-1}\right)-\arctan\frac{{\text{Im}}\Pi_{i}}{{\text{Re}}D_{i}^{-1}}+\mbox{Re}D_{i}\mbox{Im}\Pi_{i}\Big\},$$ where $\int_{{\mathrm{d}^{4}k}()}$ represents the convolution of the parentheses $\{\}$ with the derivatives of the distribution functions with respect to the temperature $T$, i.e. $\int{\mathrm{d}^{3}k}\int_{-\infty}^{\infty}{\mathrm{d}\omega}/(2\pi)^{4}(\partial{n_\text{B}}/\partial T)$ for the gluons and $\int{\mathrm{d}^{3}k}\int_{0}^{\infty}{\mathrm{d}\omega}/(2\pi)^{4}(\partial{n_\text{F}}/\partial T+\partial{n_\text{F}}^{\text{A}}/\partial T)$ for quarks and plasminos (superscript A for antiparticles). The sign constant $\xi_{i}$ is $-1$ for quasiparticles with real particle interpretation (transverse gluons and quarks) and $+1$ for the collective modes (longitudinal gluons and plasminos). $D_{i}$ ($\Pi_{i}$) stands for the propagators (self-energies) of species $i$. From the entropy density, the remaining state variables can be constructed in a self-consistent manner. To describe results of lattice QCD calculations at zero chemical potential, a temperature shift is introduced into the running coupling $g^{2}$ changing it to an effective coupling $G^{2}$; the parameters of the coupling (a scale parameter and the temperature shift) are then adjusted to the lattice data. To obtain the coupling $G^{2}$ at nonzero chemical potential $\mu$, the self-consistency of the model and the stationarity of the thermodynamic potential are employed, leading to a quasilinear partial differential equation for the coupling (dubbed flow equation)$$a_{T}\frac{\partial G^{2}}{\partial T}+a_{\mu}\frac{\partial G^{2}}{\partial\mu}=b$$ with coefficients $a_{T,\mu}$ and $b$ listed in [@Sch08]. This is the HTL QPM, as the HTL approximation is used for dispersion relations. ![Scaled entropy density $s/T^{3}$ (left) and pressure $p/T^{4}$ (right) of $2+1$ quark flavors as functions of the scaled temperature $T/T_{c}$ for several values of the quark chemical potential $\mu$. Lattice data (symbols) for $\mu=0$ from [@Kar07]. The termination of the curves at $T\leq T_{c}$ is at the conjectured transition line to a confined state, cf. [@Sch08; @Sch07].\[fig:cuts\]](muTc_cuts_sdT3.eps "fig:")![Scaled entropy density $s/T^{3}$ (left) and pressure $p/T^{4}$ (right) of $2+1$ quark flavors as functions of the scaled temperature $T/T_{c}$ for several values of the quark chemical potential $\mu$. Lattice data (symbols) for $\mu=0$ from [@Kar07]. The termination of the curves at $T\leq T_{c}$ is at the conjectured transition line to a confined state, cf. [@Sch08; @Sch07].\[fig:cuts\]](muTc_cuts_pdT4.eps "fig:") For simplified versions of the HTL QPM, e.g. neglecting collective modes, the solution of the flow equation leads to ambiguities. It was shown that collective modes and Landau damping as well as the use of the momentum-dependent HTL dispersion relations are essential to preserve the self-consistency of the model [@Sch07]. Utilizing the full model, thermodynamic gross properties of the quark-gluon plasma can be obtained. As an example, the entropy density and pressure along lines of constant chemical potential are exhibited in Figure \[fig:cuts\]. From these state quantities, it is possible to provide an equation of state for present and upcoming heavy-ion experiments such as at RHIC, LHC [@BKS07b], SPS and FAIR. In particular at FAIR the baryon density effects covered by our model become severe. One author (RS) thanks the organizers of the conference for support and the opportunity to present his results. [1]{} S. Ejiri, F. Karsch, E. Laermann, and C. Schmidt, *Phys. Rev.* D 73 (2006) 054506 A. Peshier, B. Kämpfer, and G. Soff, *Phys. Rev.* C 61 (2000) 045203 J.-P. Blaizot, E. Iancu, and A. Rebhan, *Phys. Rev.* D 63 (2001) 065003 M. Bluhm, B. Kämpfer, R. Schulze, and D. Seipt, *Eur. Phys. J.* C 49 (2007) 205 R. Schulze, M. Bluhm, and B. Kämpfer, *Eur. Phys. J.* ST 155 (2008) 177 F. Karsch, *J. Phys.* G 34 (2007) S627 R. Schulze, *Quasiparticle description of QCD thermodynamics: effects of finite widths, Landau damping and collective excitations*, Diploma thesis, Technical University Dresden (2007) M. Bluhm, B. Kämpfer, R. Schulze, D. Seipt, and U. Heinz, *Phys. Rev.* C 76 (2007) 034901 [^1]: [email protected]

ArXiv

--- abstract: 'Schrödinger cat states are crucial for exploration of fundamental issues of quantum mechanics and have important applications in quantum information processing. Here, we propose and experimentally demonstrate a method for manipulating cat states in a cavity with the Aharonov-Anandan phase acquired by a superconducting qubit, which is dispersively coupled to the cavity. Based on this dispersive coupling, the qubit can be forced to trace out a circuit in the projective Hilbert space conditional on one coherent state. By preparing the cavity in a superposition of two coherent states, the geometric phase associated with this transport is encoded to the relative probability amplitude of these two coherent states. We demonstrate the photon-number parity of a cat state in a cavity can be controlled by adjusting this geometric phase, which offers the possibility for protecting its quantum coherence from single-photon loss. Based on this geometric effect, we realize phase gates for one and two photonic qubits whose logical basis states are encoded in two quasi-orthogonal coherent states. We further demonstrate two-cavity gates with symmetric and asymmetric Fock state encoding schemes. Our method can be directly extended to implementation of controlled-phase gates between error-correctable logical qubits.' author: - 'Y. Xu' - 'W. Cai' - 'Y. Ma' - 'X. Mu' - 'W. Dai' - 'W. Wang' - 'L. Hu' - 'X. Li' - 'J. Han' - 'H. Wang' - 'Y. P. Song' - 'Zhen-Biao Yang' - 'Shi-Biao Zheng' - 'L. Sun' title: Geometrically manipulating photonic Schrödinger cat states and realizing cavity phase gates --- 0.5cm When a quantum system is parallel-transported along a circuit in its quantum state space, it collects information about the geometry of this path, acquiring a “memory" of its motion in the form of a phase. This phase is referred to as the geometric phase and has close relations to many physical phenomena [@Anandan1992The; @Wilczek1989Geometric]. This effect was first discovered by Berry in the context of adiabatic passage [@berry1984quantal]. One remarkable feature of Berry phase is that it is robust against fast parameter fluctuations whose effect on the enclosed parameter-space area averages out [@Chiara2003Berry]. As such, Berry phase has been considered as a choice for fault-tolerant quantum computation [@duan2001geometric; @jones2000geometric]. So far, observation of this phase and demonstration of its noise-resilient feature have been reported in various physical systems [@jones2000geometric; @Tycko1987Adiabatic; @Leek1889Observation; @Filipp2009Experimental; @Pechal2012Geometric; @gasparinetti2016measurement]. Berry’s discovery has triggered considerable interest in quantum-mechanical geometric effects, leading to important generalizations in various directions [@aharonov1987phase; @Samuel1988General]. In particular, Aharonov and Anandan defined geometric phase in the projective Hilbert space, instead of in parameter space [@aharonov1987phase], removing the adiabatic condition. The geometric nature of Aharonov-Anandan (AA) phase lies in the fact that it is related to the area enclosed by the circuit traversed by the state vector. When two or more quantum systems are coupled, the geometric phase acquired by one system can be employed to manipulate the quantum state of the others [@Zheng2004Unconventional; @Pechal2012Geometric]. The geometric phase of a harmonic vibrational mode of trapped ions has been utilized for implementing high-fidelity entangling gates for the ionic qubits [@Leibfried2003Experimental]. In a recent experiment [@Song2017Geometric], the geometric phase of a continuous-variable field mode was observed through Ramsey interference and used for realizing controlled phase gates with up to four qubits in a superconducting circuit. On the other hand, it has been shown that the geometric phase of a superconducting qubit can be used for realizing Selective Number-dependent Arbitrary Phase (SNAP) gates on a cavity [@Krastanov2015]. This kind of gates has been experimentally demonstrated and used to produce a single-photon state [@Heeres2015]. Recently, a quantum controlled-NOT (CNOT) gate between two cavity systems has been demonstrated by use of both the dynamical and AA phases produced by controllably coupling these cavities to a qubit  [@Rosenblum2018]. This gate requires the logic states of the control qubit to be respectively encoded on the vacuum state and a nonzero photon-number state, which renders it incompatible with quantum error correction schemes; on the occurrence of single-photon loss the control qubit will collapse to a Fock state, leading to complete loss of the stored information. ![Geometric manipulation of a photonic cat state. (a) Schematic of the nonadiabatic AA phase of a qubit. Two successive $\pi$ rotations of the qubit produce a geometric phase $\gamma = \pi + \varphi$, where $\varphi$ is the angle between the two rotation axes. (b) Experimental sequence to manipulate the cat state. A cavity is dispersively coupled to the qubit and initialized in a cat state $\left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\sqrt{2}\right\rangle}}\right)/\sqrt{2}$ with the help of an ancillary qubit $Q_2$. The AA phase produced by the rotations of $Q_1$ conditional on the cavity’s vacuum state is encoded in the probability amplitude of ${\ensuremath{\left|0\right\rangle}}$, resulting in a phase gate. (c) Measured Wigner function of the cavity state before the phase gate, corresponding to fidelity of 0.980 to the ideal cat state. (d) Wigner function of the cavity state after the gate with $\varphi=0$. The slight rotation and deformation of the Wigner function is due to the self-Kerr effect of the cavity. (e) Measured parity of the cavity state as a function of $\varphi$ after a displacement $D(-\sqrt{2} e^{i\delta} )$ for different values of $\delta$. Symbols are experimental data, in excellent agreement with numerical simulations (solid lines).[]{data-label="fig:fig1"}](Figure1_final.pdf) We here propose and experimentally demonstrate a scheme for manipulating the parity of a cat state in a cavity with the AA phase of a qubit dispersively coupled to the cavity in a superconducting circuit. Cat states are of fundamental interest [@Deleglise] and can be used to encode error-correctable logical qubits [@LeghtasPRL2013; @Mirrahimi2014; @Ofek2016; @heeres2017implementing]. Thus, manipulating these states and protecting them from decoherence is a subject of great importance. In our experiment, the qubit is parallel-transported along a closed loop on the Bloch sphere, picking up a geometric phase, conditional on one of the two quasiclassical components forming the cat state. We demonstrate the photon-number parity of the cat state can be manipulated by this geometric operation. This manipulation technique, in combination with the parity jump tracking method [@SunNature], allows for the protection of the quantum coherence of cat states from single-photon loss. We then employ this phase to realize logic gates for a cat-encoded qubit, and generalize our method to implementation of two-cavity controlled-phase gates with different encoding schemes and two-cavity SNAP gates for entangling two cavities. Our procedure can be directly generalized to implement gates between logic qubits with inherent error correction function. ![Quantum process tomography of single-cavity geometric phase gates. (a) Experimental sequence. (b) The Pauli transfer process $R$ matrix fidelity as a function of $m$, the number of the Z gate on the cavity state. The inserts show the measured $R$ matrices after one and nine Z gates, respectively. A linear fit of the process fidelity decay gives the Z gate fidelity $F_\mathrm{Z} = 0.987\pm0.001$. (c) The measured and ideal Pauli transfer $R$ matrices of the S gate and T gate with fidelities $F_{S} = 0.968$ and $F_{T} = 0.964$.[]{data-label="fig:fig2"}](Figure2_final.pdf) ![Two-cavity geometric phase gate. (a) A 3D view of Device B. A superconducting transmon qubit $Q_3$ at the center couples to two coaxial cavities $S_1$ and $S_2$, which couple to two other individual ancillary transmon qubits $Q_1$ and $Q_2$, respectively. Each of these transmon qubits independently couples to a stripline readout resonator used to perform simultaneous single-shot readout. (b) Schematic of the experimental sequence. (c) Ideal (left) and measured Pauli transfer $R$ matrices of two-cavity CZ gates with coherent state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|2\sqrt{2}\right\rangle}}$} (middle) and Fock state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} (right) for both cavities. The process fidelities, $F_\mathrm{CZ\_ED}$ ($F_\mathrm{ED}$), for these two encodings are 0.727 (0.869) and 0.862 (0.905), respectively. (d) Ideal (left) and measured (right) Pauli transfer $R$ matrices of the two-cavity CNOT gate with the encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} for cavity $S_1$ and {${\ensuremath{\left|0\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$, ${\ensuremath{\left|1\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$} for cavity $S_2$. The corresponding process fidelity $F_\mathrm{CNOT\_ED}$ ($F_\mathrm{ED}$) is 0.829 (0.857).[]{data-label="fig:fig3"}](Figure3_final.pdf) The experiments presented in this work are based on two circuit quantum electrodynamics (QED) devices [@Wallraff; @Clarke2008Superconducting; @You2011Atomic; @Schoelkopf2013; @Gu2017Microwave]. Device A, on which single-cavity geometric phase gates are performed, consists of two transmon qubits simultaneously dispersively coupled to two three-dimensional cavities [@Paik; @Kirchmair; @Vlastakis; @Liu2017; @Wang2017]. The parameters and architecture setup are described in Ref. [@Xu2018]. Device B, on which two-cavity geometric phase gates are performed, consists of three transmon qubits dispersively coupled to two cylindrical cavities [@Reagor2016] and three stripline readout cavities [@axline2016an]. The device parameters are described in Ref. [@Supplement]. In Device A, the coupling between the qubit ($Q_1$) used to produce the geometric phase and the cavity used to encode this phase is described by the Hamiltonian $$H=-\hbar \chi _{\mathrm{qs}}a^{\dagger }a{\ensuremath{\left|e\right\rangle}}{\ensuremath{\left\langlee\right|}},$$ where $\chi_{\mathrm{qs}}$ denotes the qubit frequency shift induced by per photon, $a^{\dagger }$ and $a$ are the creation and annihilation operators for the particular cavity field respectively, and ${\ensuremath{\left|e\right\rangle}}$ $({\ensuremath{\left|g\right\rangle}})$ is the excited (ground) state of the qubit. In Device B, the qubit, commonly coupled to two cavities used to store the photonic qubits, undergoes a frequency shift dependent on the photon numbers of both cavities. The geometric manipulation technique is well exemplified with the even cat state $\left({\ensuremath{\left|\alpha\right\rangle}} + {\ensuremath{\left|-\alpha\right\rangle}} \right)/\sqrt{2}$, where ${\ensuremath{\left|\alpha\right\rangle}}$ and ${\ensuremath{\left|-\alpha\right\rangle}}$ are coherent states with $\left\langle \alpha | -\alpha \right\rangle \approx 0$. To realize conditional qubit rotations, a phase-space displacement, $D(\alpha)$, is applied to the cavity, transforming its state to $\left({\ensuremath{\left|2\alpha\right\rangle}} + {\ensuremath{\left|0\right\rangle}} \right)/\sqrt{2}$. The qubit, initially in the ground state $\left| g\right\rangle $, is then driven by a classical field on resonance with the qubit frequency conditioned on the cavity’s vacuum state $\left| 0\right\rangle $. We here assume that the Rabi frequency $\varepsilon $ of the drive is much smaller than $\bar{n}\chi _\mathrm{qs}$, where $\bar{n}$ is the average photon number of the state ${\ensuremath{\left|2\alpha\right\rangle}}$. In this case, the qubit’s state is not changed by the drive when the cavity is in ${\ensuremath{\left|2\alpha\right\rangle}}$ due to the large detuning, and the system dynamics is described by the effective Hamiltonian $$H_{\mathrm{eff}}=\frac 12\hbar \varepsilon e^{i\phi }\left| e\right\rangle \left\langle g\right| \otimes \left| 0\right\rangle \left\langle 0\right| +h.c.,$$ where $\phi $ is the phase of the drive. This Hamiltonian produces a qubit rotation $R_{{\bf n}}^\theta $ conditional on the cavity’s vacuum state, where $R_{{\bf n}}^\theta $ represents the operation that rotates the qubit’s state by an angle $\theta =\int_0^\tau \varepsilon dt$ around the axis ${\bf n}$ with an angle $\phi $ to $x$ axis on the equatorial plane of the Bloch sphere, with $\tau $ being the pulse duration. After two successive conditional $\pi $ rotations $R_{{\bf n}_1}^{\pi ,0}=R_{ {\bf n}_1}^\pi \otimes \left| 0\right\rangle \left\langle 0\right| $ and $R_{ {\bf n}_2}^{\pi ,0}=R_{{\bf n}_2}^\pi \otimes \left| 0\right\rangle \left\langle 0\right| $, the qubit makes a cyclic evolution, returning to the initial state $\left| g\right\rangle $ but acquiring a phase $\gamma =\pi+\Delta \phi = \Omega/2$, where $\Delta \phi =\phi _1-\phi _2$ represents the angle between the two rotation axes, and $\Omega$ is the solid angle substended by the trajectory traversed by the qubit on the Bloch sphere, as shown in Fig. \[fig:fig1\](a). This conditional phase shift is encoded in the probability amplitude of the state component $\left| 0\right\rangle $, leading to the cavity state $\left( {\ensuremath{\left|2\alpha\right\rangle}} + e^{i\gamma}{\ensuremath{\left|0\right\rangle}} \right)/\sqrt{2}$. A subsequent displacement $D(-\alpha )$ transforms the cavity to the state $(\left| \alpha \right\rangle +e^{i\gamma }\left| -\alpha \right\rangle )/\sqrt{2}$. Due to the quantum interference of the two superposed coherent state components $\left| \alpha \right\rangle $ and $\left| -\alpha \right\rangle $, the cavity parity $P$ exhibits a periodical oscillation when the geometric phase $\gamma $ is varied: $P = \cos{\gamma}$. The procedure allows for manipulation of the parity of the cat state; when $\gamma=\pi$, the parity is reversed. This procedure is equivalent to a phase gate operation for the cavity qubit with ${\ensuremath{\left|\alpha\right\rangle}}$ and ${\ensuremath{\left|-\alpha\right\rangle}}$ acting as the logic basis states, and can be used to correct for the parity jump caused by single-photon loss. To simplify the operation, in our experiment the cavity displacement before the conditional qubit rotation is incorporated with the preparation of the initial cavity state; ${\ensuremath{\left|0\right\rangle}}$ and ${\ensuremath{\left|2\alpha\right\rangle}}$ act as the two logic basis states for the gate demonstration. The experimental sequence to manipulate a cat state with Device A is shown in Fig. \[fig:fig1\](b). The cavity is initialized in the cat state $\left({\ensuremath{\left|0\right\rangle}} + {\ensuremath{\left|2\sqrt{2}\right\rangle}}\right)/\sqrt{2}$ \[the measured Wigner function is shown in Fig. \[fig:fig1\](c)\] with the help of ancillary qubit $Q_2$ following the gradient ascent pulse engineering (GRAPE) technique [@Khaneja2005; @deFouquieres2011]. The two subsequent conditional $\pi$ rotations on $Q_1$, with the first one around the axis with an angle $\varphi$ to the $x$ axis and the second one around the $x$ axis, yield a geometric phase $\gamma = \pi + \varphi$ conditional on the cavity’s vacuum state. The Wigner function of the cavity state after this single-cavity geometric phase gate is shown in Fig. \[fig:fig1\](d) with $\varphi=0$, which demonstrates the phase-space inteference fringes are reversed by the geometric manipulation. After a displacement $D(-\sqrt{2} e^{i\delta} )$, the parity of the cavity state as a function of $\varphi$ is measured and shown in Fig. \[fig:fig1\](e), in excellent agreement with numerical simulations. Process tomography is used to benchmark the cavity geometric phase gate performance, with the experimental sequence shown in Fig. \[fig:fig2\](a). Since trusted operations and measurements necessary for quantum process tomography are unavailable in the cat-encoded subspace, we characterize the gate by decoding the quantum information on the cavity back to the transmon qubit $Q_2$. We use the so-called Pauli transfer process $R$ matrix as a measure of our gate [@Chow2012Universal], which connects the input and output Pauli operators with $P_\mathrm{out} = R\cdot P_\mathrm{in}$. Figure \[fig:fig2\](b) shows the $R$ matrix fidelity decay as a function of $m$, the number of the $\pi$-phase (Z) gate. The fidelity at $m=0$ quantifies the “round trip" process fidelity $F_{\mathrm{ED}}=0.969$ of the encoding and decoding processes only. A linear fit of the process fidelity decay gives the Z gate fidelity $F_Z = 0.987$, also consistent with the fidelity calculated from $F_Z = 1 - \left( F_\mathrm{ED} - F_{Z\_\mathrm{ED}} \right)$, where $F_\mathrm{Z\_ED}=0.957$ is the measured fidelity including the encoding and decoding processes. The measured and the ideal Pauli transfer $R$ matrices of the S gate and T gate are shown in Fig. \[fig:fig2\](c), where $\mathrm{S} = {\ensuremath{\left|0\right\rangle}}_L {\ensuremath{\left\langle0\right|}} + i {\ensuremath{\left|1\right\rangle}}_L {\ensuremath{\left\langle1\right|}}$ and $\mathrm{T} = {\ensuremath{\left|0\right\rangle}}_L{\ensuremath{\left\langle0\right|}} + \exp{(i\pi/4)} {\ensuremath{\left|1\right\rangle}}_L {\ensuremath{\left\langle1\right|}}$. Our method can be directly generalized to implementation of controlled-phase gates between two photonic qubits encoded in two cavities that are dispersively coupled to one common superconducting qubit [@Wang2016Schrodinger; @Gao2018]. The photon-number-dependent qubit frequency shift allows for the qubit $2\pi$ rotation conditional on both cavities being in the vacuum state ${\ensuremath{\left|00\right\rangle}}$ with a drive at the corresponding qubit frequency. With the encoding ${\ensuremath{\left|0\right\rangle}}_L = {\ensuremath{\left|0\right\rangle}}$, this rotation produces a $\pi$-phase shift if and only if both photonic qubits are in ${\ensuremath{\left|0\right\rangle}}_L$; the other logic basis state of each qubit can be any nonzero photon-number state or a coherent state with a sufficiently large amplitude. If the two photonic basis states in each cavity correspond to two coherent states, the controlled-Z (CZ) gate can be achieved by sandwiching this conditional qubit rotation between two pairs of displacement operations: the first pair of displacements transform the coherent state of each cavity used as the logic basis state ${\ensuremath{\left|0\right\rangle}}_L $ to the vacuum state, and the second pair restore the original coherent states. Figure \[fig:fig3\] shows the two-cavity geometric phase gates based on Device B, whose schematic is shown in Fig. \[fig:fig3\](a). Besides the transmon qubit simultaneously coupled to both cavities, each cavity also dispersively couples to another ancillary transmon qubit for encoding/decoding and measurement purposes. Figure \[fig:fig3\](b) shows the experimental sequence for the process tomography to benchmark the gate performance. In our experiment, the CZ gate is implemented with two different encoding schemes: coherent state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|2\alpha\right\rangle}}$} and Fock state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} for both cavities. The measured Pauli transfer $R$ matrices of the CZ gates, together with the $R$ matrix for the ideal CZ gate, are shown in Fig. \[fig:fig3\](c). The process $R$ matrix fidelities, $F_{\mathrm{CZ\_ED}}$ ($F_\mathrm{ED}$), for these two encodings are 0.727 (0.869) and 0.862 (0.905), respectively. We note that for the Fock state encoding the infidelity dominantly comes from the encoding-decoding process; for the coherent state encoding, besides the encoding-decoding error the fidelity loss mainly comes from the Kerr effects that deform the coherent states during the gate operation. Our approach can also be used to realize a CNOT gate between two cavities. As a particular example, the control cavity $S_1$ is encoded with Fock states {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} and the target cavity $S_2$ is encoded with {${\ensuremath{\left|0\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$, ${\ensuremath{\left|1\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$}. We note that this gate is equivalent to the CZ gate with the logic basis states of the target qubit encoded in the zero- and two-photon states. The measured Pauli transfer $R$ matrix of the two-cavity CNOT gate is shown in Fig. \[fig:fig3\](d), corresponding to a process $R$ matrix fidelity, $F_{\mathrm{CNOT\_ED}}$ ($F_\mathrm{ED}$), of 0.829 (0.857). , respectively. Solid black outlines are for the ideal density matrices. Measured imaginary parts for both states are smaller than 0.04 and not shown. The fidelities for ${\ensuremath{\left|\Phi_{\pm}\right\rangle}}$ are 0.957 and 0.930, respectively.[]{data-label="fig:fig4"}](Figure4_final.pdf) We finally show that our conditional dynamics can be used to deterministically create high-fidelity single-photon Bell states ${\ensuremath{\left|\Phi_{\pm}\right\rangle}} = \left({\ensuremath{\left|01\right\rangle}}\pm{\ensuremath{\left|10\right\rangle}}\right)/\sqrt{2}$. The approach is an extension of the previously reported SNAP operation for universal control of one cavity [@Krastanov2015; @Heeres2015] to two cavities. When combined with the single-cavity SNAP gates, our method can be used to realize arbitrary universal multi-cavity control. The experimental sequence is shown in Fig. \[fig:fig4\](a), where a conditional $2\pi$ rotation on qubit $Q_3$ is sandwiched in between two pairs of phase-space displacements of the cavities. With help of the two ancillary qubits, joint Wigner tomography of the two cavities is performed and two slice cuts of the measured two-mode Wigner function are shown in Figs. \[fig:fig4\](b-c). The density matrices of ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$, reconstructed by mapping the state of the two cavities to qubits $Q_1$ and $Q_2$ and then jointly measuring the state of these qubits \[as in the tomography measurement of Fig. \[fig:fig3\](b)\], are displayed in Figs. \[fig:fig4\](d-e), with state fidelities of 0.957 and 0.930, respectively. One distinct feature of our gate dynamics is that it is compatible with error correction schemes. For logic qubits whose basis states are encoded in even cat states, the photon-number parity can be used as an error syndrome of the single-photon loss [@LeghtasPRL2013; @Mirrahimi2014; @Ofek2016]. With this encoding, each of the two-qubit logic basis states is composed of four two-mode coherent state components, and a CZ gate can be realized by subsequently performing four conditional phase operations. Each operation inverts the phase of one of the four components forming a specific logic basis state, and can be realized by sandwiching a qubit $2\pi$ rotation conditional on the cavities’ vacuum state between two pairs of phase-space displacements of the cavities. For another kind of error-correctable logic qubits binomially encoded as $\{{\ensuremath{\left|0\right\rangle}}_L=({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|4\right\rangle}})/\sqrt{2}, {\ensuremath{\left|1\right\rangle}}_L=~{\ensuremath{\left|2\right\rangle}}\}$ [@Michael2016; @Hu2018], the CZ gate can be realized by a qubit $2\pi$ rotation conditional on the joint cavities’ state ${\ensuremath{\left|2\right\rangle}}{\ensuremath{\left|2\right\rangle}}$, enabled with a drive at the qubit’s frequency associated with this cavities’ state. For our device, the error due to Kerr effects is larger than that caused by single-photon loss. With the improvement of the device design, the Kerr strengths can be significantly mitigated [@Rosenblum2018]. We plan to design and fabricate a device with improved performance, and demonstrate gates with error-correctable logic qubits. For a setup with three or more cavities coupled to one common qubit, the qubit $2\pi $ rotation conditional on all cavities being in the vacuum state directly leads to a phase gate acting on these cavities if the vacuum state in each cavity acts as the logic basis state ${\ensuremath{\left|0\right\rangle}}_L$. This kind of gates is useful for quantum error correction [@Reed2012Realization] and serves as a central element for implementation of the quantum search algorithm [@Nielsen]. In addition to implementation of quantum gates, our method can be used for stabilizing the parity of a cat state: When an environmentally-induced parity jump occurs, one can correct for it through the combination of a conditional qubit $2\pi$ rotation and two displacement operations. We thank valuable discussions with Chen Wang. This work was supported by National Key Research and Development Program of China No.2017YFA0304303 and the National Natural Science Foundation of China under Grants No.11474177, No. 11874114, No. 11674060, and No. 11875108. [48]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ** (, ). , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , ****, (). , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , ****, (). . , , , , , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , (). , , , , , , , ****, (). , , , , , , , , , , , (). , , , , , , , ****, (). , ** (, ).

ArXiv

--- author: - 'Massimo Meneghetti, Rodolfo Argazzi, Francesco Pace, Lauro Moscardini, Klaus Dolag, Matthias Bartelmann, Guoliang Li, Masamune Oguri' bibliography: - '../TeXMacro/master.bib' date: '*Astronomy & Astrophysics, submitted*' title: 'Arc sensitivity to cluster ellipticity, asymmetries and substructures' --- Introduction ============ Thanks to the improvements in the quality and in the depth of astronomical observations, in particular from space, an increasing number of gravitational arcs has recently been discovered near the centres of many galaxy clusters [see e.g. @BR05.1]. Since the appearance of these images reflects the shape of the gravitational potential which is responsible for their large distortions, strong lensing is, in principle, a very powerful tool for investigating how the matter, in particular the dark component, is distributed in the inner regions of cluster lenses. Determining the inner structure of galaxy clusters is one of the major goals in cosmology, because it should allow us to set important constraints on the growth of the cosmic structures in the Universe. Moreover, constraining the mass distribution in the centre of dark matter halos has become increasingly important in the recent years, since observations of the dynamics of stars in galaxy-sized systems revealed the presence of a potential problem within the Cold-Dark-Matter (CDM) scenario. While numerical simulations in this cosmological framework predict that dark matter halos in a large range of masses should develop density profiles characterised by an inner cusp, observations of the rotation curves of dwarf and low-surface-brightness galaxies suggest that these objects rather have flat density profiles [@FL94.1; @MO94.1; @BU95.1; @BU97.1; @MG98.1; @DA00.1; @FI01.1]. While the centres of galaxies are dominated by stars, which renders it extremely complicated to derive constraints on the distribution of their dark matter, galaxy clusters are an alternative and, in many respects, preferable class of objects for testing the predictions of the CDM model. In fact, several authors already tried to investigate the inner structure of these large systems, using and often combining several kinds of observations. Apart from lensing, the gravitational potential of galaxy clusters can be traced with several other methods, for example through the emission in the X-ray band by the hot intra-cluster gas. However, while gravitational lensing directly probes the matter content of these objects, the other techniques usually rely on some strong assumptions about their dynamical state and the interaction between their baryonic and dark matter. For example, it must be often assumed that the gas is in hydrostatic equilibrium within the dark matter potential well and that the system is spherically symmetric. Some ambiguous results were found when comparing the constraints on the inner structure of clusters as obtained from X-ray and lensing observations. First, masses estimated from strong lensing are usually larger by a factor of 2-3 than the masses obtained from X-ray observations [@CH03.1; @OT04.1]. Deviations from axial symmetry and substructures are known to be important factors in strong lensing mass estimates [see e.g. @BA95.2; @BA96.2; @ME03.1; @OG05.1; @GA05.1]. Second, the constraints on the inner slope of the density profiles seem to be compatible with a wide range of inner slopes [@ET02.1; @LE03.1; @AR02.1; @SA03.1; @BA04.1; @GA05.1]. Apart from the above-mentioned uncertainties affecting the X-ray measurements, strong lensing observations also have several potential weaknesses. First of all, arcs are relatively rare events. Frequently, all the constraints which can be set on the inner structure of clusters via strong lensing depend on a single or on a small number of arcs and arclets observed near the cluster core. Second, arcs are the result of highly non-linear effects. This implies that their occurrence and their morphological properties are very sensitive to ellipticity, asymmetries and substructures of the cluster matter distribution. Reversing the problem, this means that, in order to reliably describe the strong lensing properties of galaxy clusters, all of these effects must be taken into account. Fitting the positions and the morphology of gravitational arcs to derive the underlying mass distributions of the lensing clusters, usually requires to build models with multiple mass components, each of which is characterised by its ellipticity and orientation [see e.g. @KN93.1; @CO05.1; @BR05.1]. Even describing the cluster lens population in a statistical way requires to use realistic cluster models [@ME00.1; @ME03.1; @ME03.2; @OG02.1; @OG03.1; @DA04.1; @HE05.1]. Despite the fact that the importance of ellipticity, asymmetries and substructures for strong lensing appears clearly in many previous studies, many questions still remain. For example, what is the typical scale of substructures which contribute significantly to the strong lensing ability of a cluster? Where are they located within the clusters? What is the relative importance of asymmetries compared to ellipticity? Moreover, how do substructures influence the appearance of giant arcs? All of these open problems are important for those studies aiming at constraining cosmological parameters from statistical lensing, or at determining the inner structure of galaxy clusters through gravitational arcs. This paper aims at answering to these questions. To do so, we quantify the impact of ellipticity, asymmetries and substructures by creating differently smoothed models of the projected mass distributions of some numerical clusters. We gradually move from one smoothed model to another through a sequence of intermediate steps. The plan of the paper is as follows. In Sect. \[sect:nummod\], we discuss the characteristics of the numerically simulated clusters that we use in this study; in Sect. \[sect:raytr\], we explain how ray-tracing simulations are carried out; Sect. \[sect:smooth\] illustrates how we obtain smoothed versions of the numerical clusters; in Sect. \[sect:power\], we suggest a method to quantify the amount of substructures, asymmetry and ellipticity of the cluster lenses, based on multipole expansions of their surface density fields; Sect. \[sect:resu\] is dedicated to the discussion of the results of our analysis. Finally, we summarise our conclusions in Sect.\[sect:conclu\]. Numerical models {#sect:nummod} ================ The cluster sample used in this paper is made of five massive dark matter halos. One of them, labelled $g8_{\rm hr}$, was simulated with very high mass resolution, but contains only dark-matter. The others, the clusters $g1$, $g8$, $g51$ and $g72$ have lower mass resolution but are obtained from hydro-dynamical simulations which also include gas. The halos we use here are massive objects with masses $8.1\times 10^{14}\:h^{-1}M_\odot$ ($g72$), $8.6\times 10^{14}\:h^{-1}M_\odot$ ($g51$), $1.4\times 10^{15}\:h^{-1}M_\odot$ ($g1$) and $1.8\times 10^{15}\:h^{-1}M_\odot$ ($g8$ and $g8_{\rm hr}$) at $z=0.3$. We have chosen this redshift because it is close to where the strong lensing efficiency of clusters is the largest for sources at $z_{\rm s} \gtrsim 1$ [@LI05.1]. The clusters were extracted from a cosmological simulation with a box-size of $479\,h^{-1}\,{\rm Mpc}$ of a flat $\Lambda$CDM model with $\Omega_0=0.3$, $h=0.7$, $\sigma_8=0.9$, and $\Omega_{\rm b}=0.04$ (see @YO01.1). Using the “Zoomed Initial Conditions” (ZIC) technique [@TO97.2], they were re-simulated with higher mass and force resolution by populating their Lagrangian volumes in the initial domain with more particles, appropriately adding small-scale power. The initial displacements are generated using a “glass” distribution [@WH96.1] for the Lagrangian particles. The re-simulations were carried out with the Tree-SPH code GADGET-2 [@SP01.1; @SP05.1]. For the low resolution clusters, the simulations started with a gravitational softening length fixed at $\epsilon=30.0\,h^{-1}\,\mathrm{kpc}$ comoving (Plummer-equivalent) and switch to a physical softening length of $\epsilon=5.0\,h^{-1}\,\mathrm{kpc}$ at $1+z=6$. The particle masses are $m_{\rm DM}=1.13\times 10^9\:h^{-1}M_\odot$ and $m_{\rm GAS}=1.7\times 10^8\:h^{-1}M_\odot$ for the dark matter and gas particles, respectively. For the high-resolution cluster $g8_{\rm hr}$ the particle mass is $m_{\rm DM}=2.0\times 10^8\:h^{-1}M_\odot$ and the softening was set to half of the value used for the low resolution runs. Its virial region at $z=0.3$ contains more than nine million particles, which allow us to well resolve substructures on scales down to those of galaxies. Despite the lower mass resolution with respect to $g8_{\rm hr}$, the other low resolution clusters also contain several million particles within their virial radii. To introduce gas into the high-resolution regions of the low-resolution clusters, each particle in a control run containing only dark matter was split into a gas and a dark matter particle. These were displaced by half the original mean inter-particle distance, so that the centre-of-mass and the momentum were conserved. Selection of the initial region was done with an iterative process involving several low-resolution, dissipation-less re-simulations to optimise the simulated volume. The iterative cleaning process ensures that all these haloes are free of contaminating boundary effects up to at least 3 to 5 times the virial radius. The simulations including gas particles follow only the non radiative evolution of the intra-cluster medium. More sophisticated versions of these clusters, where radiative cooling, heating by a UV background, and a treatment of the star formation and feedback processes were included exist and their lensing properties have been studied in detail by [@PU05.1]. The cluster $g8_{\rm hr}$ is in principle a higher-resolution, dark-matter only version of the cluster $g8$, which was simulated with non-radiative gas physics. Nevertheless the two objects can only be compared statistically. Indeed, the introduction of the gas component as well as the increment of the mass resolution introduce small perturbations to the initial conditions, which lead to slightly different time evolutions of the simulated halos. Furthermore, also the presence of gas and its drag due to pressure lead to significant changes in the assembly of the halo. A detailed discussion of such differences can be found in [@PU05.1]. For this reason, the lensing properties of $g8$ and $g8_{hr}$ at $z=0.3$ are not directly comparable. Lensing simulations {#sect:raytr} =================== Ray-tracing simulations are carried out using the technique described in detail in several earlier papers (e.g. @BA98.2 [@ME00.1]). We select a cube of $6\,h^{-1}$Mpc comoving side length, centred on the halo centre and containing the high-density region of the cluster. The particles in this cube are used for producing a three-dimensional density field, by interpolating their position on a grid of $1024^3$ cells using the [*Triangular Shaped Cloud*]{} method [@HO88.1]. Then, we project the three-dimensional density field along the coordinate axes, obtaining three surface density maps $\Sigma_{i,j}$, used as lens planes in the following lensing simulations. The lensing simulations are performed by tracing a bundle of $2048 \times 2048$ light rays through a regular grid, covering the central sixteenth of the lens plane. This choice is driven by the necessity to study in detail the central region of the clusters, where critical curves form, taking into account the contribution from the surrounding mass distribution to the deflection angle of each ray. Deflection angles on the ray grid are computed using the method described in @ME00.1. We first define a grid of $256\times256$ “test” rays, for each of which the deflection angle is calculated by directly summing the contributions from all cells on the surface density map $\Sigma_{i,j}$, $$\vec \alpha_{h,k}=\frac{4G}{c^2}\sum_{i,j} \Sigma_{i,j} A \frac{\vec x_{h,k}-\vec x_{i,j}}{|\vec x_{h,k}-\vec x_{i,j}|^2}\;,$$ where $A$ is the area of one pixel on the surface density map and $\vec x_{h,k}$ and $\vec x_{i,j}$ are the positions on the lens plane of the “test” ray ($h,k$) and of the surface density element ($i,j$). Following [@WA98.2], we avoid the divergence when the distance between a light ray and the density grid-point is zero by shifting the “test” ray grid by half-cells in both directions with respect to the grid on which the surface density is given. We then determine the deflection angle of each of the $2048\times2048$ light rays by bi-cubic interpolation between the four nearest test rays. The position $\vec y$ of each ray on the source plane is calculated by applying the lens equation. If $\vec y$ and $\vec x$ are the angular positions of source and image from an arbitrarily defined optical axis passing through the observer and perpendicular to the lens and source planes, this is written as $$\vec y = \vec x -\frac{D_{\rm ls}}{D_{\rm s}}\vec \alpha(\vec x)\;,$$ where $D_{\rm ls}$ and $D_{\rm s}$ are the angular diameter distances between the lens and the source planes and between the observer and the source plane, respectively. Then, a large number of sources is distributed on the source plane. We place this plane at redshift $z_\mathrm{s}=2$. Keeping all sources at the same redshift is an approximation justified for the purposes of the present case study, but the recent detections of arcs in high-redshift clusters [@ZA03.1; @GL03.1] indicate that more realistic simulations will have to account for a wide source redshift distribution. The sources are elliptical with axis ratios randomly drawn from $[0.5,1]$. Their equivalent diameter (the diameter of the circle enclosing the same area of the source) is $r_\mathrm{e}=1''$. They are distributed on a region on the source plane corresponding to one quarter of the field of view where rays are traced. As in our earlier studies, we adopt an adaptive refinement technique when placing sources on their plane. We first start with a coarse distribution of $32\times32$ sources and then increase the source number density towards the high-magnification regions of the source plane by adding sources on sub-grids whose resolution is increased towards the lens caustics. This increases the probability of producing long arcs and thus the numerical efficiency of the method. In order to compensate for this artificial source-density enhancement, we assign a statistical weight to each image for the following statistical analysis which is proportional to the area of the sub-grid cell on which the source was placed. By collecting rays whose positions on the source plane lie within any single source, we reconstruct the images of background galaxies and measure their length and width. Our technique for image detection and classification was described in detail by [@BA94.1] and used by [@ME00.1; @ME01.1; @ME03.2; @ME03.1; @TO04.1] and [@ME05.1]. The modifications recently suggested by [@PU05.1] for increasing the accuracy of the measurements of the arc properties have been included in our code. The simulation process ends in a catalogue of images which is subsequently analysed. Smoothed representations of the cluster {#sect:smooth} ======================================= We aim here at separating the effects of substructures, ellipticity and asymmetries on the strong lensing efficiency of our numerical clusters. We work in two dimensions, starting from the projected two-dimensional mass distribution of each lens. Since the lens’ surface density profile plays a crucial role for strong lensing, we keep it fixed and vary only the shape of the density contours. As a first step, we construct a fiducial model for the smooth mass distribution of the lens. This is done by measuring the ellipticity and the position angle of the surface density contours as function of radius in the projected mass map. The projected density is measured in circles of increasing radii $x$. We determine the quadrupole moments of the density distribution in each aperture, $$S_{ij}(x) = \frac{\int {\mathrm{d}}^2 x \Sigma(\vec x) (x_i-x_{c,i})(x_j-x_{c,j})}{\int {\mathrm{d}}^2 x \Sigma(\vec x)} \, , \, i,j \in (1,2) ,$$ where $\vec x_c$ denotes the position of the cluster centre and the integrals are extended to the area enclosed by the aperture. Both the ellipticity $\epsilon$ and the position angle $\phi$ of the iso-density contour corresponding to the chosen aperture radius are derived from the elements of the tensor $S_{ij}(r)$: $$\begin{aligned} \epsilon(x) & = &\sqrt{\frac{(S_{11}-S_{22})^2+4S_{12}^2}{(S_{11}+S_{22})^2}} \, , \\ \phi(x) & = & \frac{1}{2} \arctan{\frac{2S_{12}}{S_{11}-S_{22}}} \, .\end{aligned}$$ {width="\hsize"} The ellipticity and position angle profiles are used in combination with the mean surface density profile $\overline{\Sigma}(x)$ of the lens for constructing a smoothed surface density map, $$\Sigma^{\rm sm}(\vec x)=\overline{\Sigma}(x_{\epsilon,\phi})\;, \label{eq:smooths}$$ where the equivalent radius $x_{\epsilon,\phi}$ is given by $$\begin{aligned} x_{\epsilon,\phi} & = & \left(\frac{[x_1\cos\phi(x)+x_2\sin\phi(x)]^2}{[1-e(x)]} \right. \nonumber \\ &+& \left. [-x_1\sin\phi(x)+x_2\cos\phi(x)]^2[1+e(x)]\right)^{1/2}\;, \label{eq:eqrad}\end{aligned}$$ and $e=1-\sqrt{1-\epsilon^2}/(1+\epsilon)=1-\sqrt{1-\epsilon}/\sqrt{1+\epsilon} \approx \epsilon$ (for $\epsilon \ll 1$). The resulting map conserves the mean surface density profile of the cluster and reproduces well the twist of its iso-density contours, i.e. the asymmetries of the projected mass distribution. This is shown in Fig. \[fig:smooth\]. The panel on the left column shows the original surface density map for one projection of the high-resolution cluster $g8_{\rm hr}$. Contour levels start at $\sim 3.6\times10^{15}\,h\,M_\odot\,$Mpc$^{-2}$ and are spaced by $5\times10^{15}\,h\,M_\odot\,$Mpc$^{-2}$. The top-middle panel shows the smoothed map obtained from Eq. (\[eq:smooths\]). In the rest of the paper, we will call this smoothed model the “asymmetric” model. The same colour scale and spacing of the contour levels as in the first panel are used. In the smoothed map, the substructures on all scales are removed and redistributed in elliptical shells around the cluster. Comparing the strong lensing properties of the original and of the smoothed map, we can therefore quantify [*the net effect of substructures*]{} on the cluster strong lensing efficiency. By subtracting the smoothed from the original map, we obtain a residual map showing which substructures will not contribute to lensing after smoothing. We plot this residual map in the right panel in the upper row of Fig. \[fig:smooth\]. Similarly, the effects of other cluster properties can be separated. For example, we can remove asymmetries and deviations from a purely elliptical projected mass distribution by disabling the twist of the iso-density contours in our smoothing procedure. For doing so, we fix the ellipticity and the position angle to a constant value, $\epsilon=\epsilon_{\rm crit}$ and $\phi=\phi_{\rm crit}$. We choose $\epsilon_{\rm crit}$ and $\phi_{\rm crit}$ to be those measured in the smallest aperture containing the cluster critical curves. A smoothed map of the cluster is created as explained earlier. The results are shown in the middle panels of Fig. \[fig:smooth\]. A comparison of the lensing properties of this new “elliptical” model with those of the previously smoothed map allows us to quantify [*the effect of asymmetries*]{}, which are large-scale deviations from elliptical two-dimensional mass distributions. Finally, even the cluster ellipticity can be removed, still preserving the same mean density profile. This is easily done by inserting $\epsilon=0$ in Eq. (\[eq:smooths\]). The resulting smoothed map and the residuals obtained by subtracting it from the original projected mass map are shown in the bottom panels of Fig.\[fig:smooth\]. If we compare the lensing efficiency of such an “axially symmetric” model to that of the previously defined elliptical model, we quantify [*the effect of ellipticity*]{} on the cluster strong lensing properties. For each smoothing method, we simulate lensing of background galaxies not only for the extreme cases of the totally smoothed maps but also for partially smoothed mass distributions. Adding the residuals $R$ to the smoothed map, the original surface density map of the cluster is indeed re-obtained, $$\Sigma(\vec x)=\Sigma^{\rm sm}(\vec x)+R(\vec x) \,.$$ Substructures of different sizes can be gradually removed from the cluster mass distribution by filtering the residual map with some filter function of varying width before re-adding it to the totally smoothed map. This can be done, for example, by convolving the residual map $R$ with a Gaussian $G$ of width $\sigma$: $$\begin{aligned} \widetilde{R}(\vec x,\sigma)&=&R(\vec x)*G(\vec x,\sigma) \, , \\ G(\vec x,\sigma)&=&\frac{1}{2\pi \sigma^2} \exp{\left(-\frac{\vec{x}^2}{2\sigma^2}\right)} \, .\end{aligned}$$ The width $\sigma$ defines the scale of the substructures which will be filtered out of the mass map. Surface density maps with intermediate levels of substructures are finally obtained by adding the filtered residuals to the smoothed map, $$\widetilde{\Sigma}(\vec x,\sigma)=\Sigma^{\rm sm}(\vec x)+\widetilde{R}(\vec x,\sigma) \,.$$ ![Surface density maps of the same cluster projection as shown in Fig. \[fig:smooth\], smoothed with increasing smoothing length from the top left to the bottom right panels. The background smoothed model is the one shown in the upper right panel of Fig. \[fig:smooth\]. The smoothing lengths in the four panels are $0,47,141$ and $470\,h^{-1}$kpc comoving, respectively. The horizontal side length of each panel is $6\,h^{-1}$Mpc comoving.[]{data-label="fig:incsmooth"}](figures/xy_3to20_lin_smoothing.eps){width="\hsize"} This procedure allows us to investigate what is [*the characteristic scale of substructures which are important for strong lensing*]{}. Moreover, comparing how the lensing properties of different clusters react to smoothing, we can quantify [*the impact of substructures in halos with different degrees of asymmetry and ellipticity*]{}. A sequence of smoothed versions of the same cluster model is shown in Fig \[fig:incsmooth\]. The smoothing length $\sigma$ is $0,47,141$ and $470\,h^{-1}$ comoving kpc from the top left to the bottom right panel, respectively. Another important issue is to understand [*where the substructures must be located in order to have a significant impact*]{} on the strong lensing properties of clusters. To address this problem, we remove from the clusters the substructures located outside apertures of decreasing equivalent radius $x_{\epsilon,\phi}$. Again, this is done by modifying the residuals of the smoothed maps. We multiply the residual map with the function, $$T(\vec x_{\epsilon,\phi},l)=\left\{ \begin{array}{r@{\quad \quad}l} 1 & (x_{\epsilon,\phi}<l) \\ \exp{\left(-\frac{\vec x_{\epsilon,\phi}^2}{2\sigma_{\rm cut}^2}\right)} & (x_{\epsilon,\phi} \ge l) \end{array}\right.\;, \label{eq:wf}$$ where $\sigma_{\rm cut}=100\,h^{-1}$kpc comoving and $l$ is the equivalent distance beyond which the substructures are suppressed. The Gaussian tail of the window function was applied to avoid sudden discontinuities in the surface density maps. ![Surface density maps of the same cluster projection shown in Fig. \[fig:smooth\], suppressing the substructures in shells of decreasing equivalent radius from the top left to the bottom right panels. The background smoothed model is the one shown in the upper right panel of Fig. \[fig:smooth\]. The equivalent radii beyond which the substructures are removed are $1174,704,352$ and $235\,h^{-1}$kpc comoving, respectively. The horizontal side length of each panel is $6\,h^{-1}$Mpc comoving.[]{data-label="fig:subrad"}](figures/xy_3to20_lin_sub_frames10_6_3_2.eps){width="\hsize"} Results of the removal of cluster substructures at different radii are shown in Fig. \[fig:subrad\]. From the top left to the bottom right panel, the cut-off scales $l$ are $1174,704,352$ and $235\,h^{-1}$kpc comoving, respectively. The residual maps filtered with the window function (\[eq:wf\]) were re-added to the totally smoothed asymmetric maps for obtaining surface density distributions with the desired level of substructures within a given equivalent radius. Quantifying the amount of substructures, asymmetry and ellipticity {#sect:power} ================================================================== The variations of ellipticity and position angle of the iso-density contours are given by the functions $\epsilon(x)$ and $\phi(x)$, which were defined in the previous section. These are shown for the three projections of cluster $g8_{\rm hr}$ in Fig. \[fig:ellphi\]. They illustrate that the projection along the $x$-axis of this cluster, shown in Fig. \[fig:smooth\], is the most elliptical at the relevant radii, with an ellipticity which grows from $\epsilon\sim 0.4$ to $\sim0.58$ within the inner $\sim 1 \,h^{-1}$Mpc. The iso-density contours have almost constant orientation in this projection. The projection along the $y$-axis exhibits the largest variations of ellipticity in the central region of the cluster, with $\epsilon$ growing from $\sim 0.25$ to $\sim 0.52$. It is also characterised by a large twist of the iso-density contours, whose orientations change by up to $\sim 20$ degrees. When projected along the $z$-axis, the cluster appears more circular and with fairly constant ellipticity ranging between $\sim 0.22$ and $\sim 0.32$. The twist of the iso-density contours is moderate within the inner $1\,h^{-1}$Mpc. ![Variations of ellipticity (top panel) and position angle (bottom panel) of the iso-density contours of the three projections of cluster $g8_{\rm hr}$ as function of the distance from the cluster centre. The cluster has a virial radius of $\sim 2.7h^{-1}$Mpc.[]{data-label="fig:ellphi"}](figures/ellprof.eps "fig:"){width="\hsize"} ![Variations of ellipticity (top panel) and position angle (bottom panel) of the iso-density contours of the three projections of cluster $g8_{\rm hr}$ as function of the distance from the cluster centre. The cluster has a virial radius of $\sim 2.7h^{-1}$Mpc.[]{data-label="fig:ellphi"}](figures/phiprof.eps "fig:"){width="\hsize"} We now quantify the amount of substructure within the numerical clusters by means of multipole expansions of their surface density maps [@ME03.1]. Briefly, starting from the particle positions in the numerical simulations, we compute the surface density $\Sigma$ at discrete radii $x_n$ and position angles $\theta_k$ taken from $[0,1.5]\,h^{-1}\,\mbox{Mpc}$ and $[0,2\pi]$, respectively. For any $x_n$, each discrete sample of data $\Sigma(x_n,\theta_k)$ is expanded into a Fourier series in the position angle, $$\Sigma(x_n,\theta_k)=\sum_{l=0}^\infty\,S_l(x_n) \mathrm{e}^{-\mathrm{i}l\theta_k}\;, \label{eq:expansion}$$ where the coefficients $S_l(x_n)$ $$S_l(x_n)=\sum_{k=0}^\infty\,\Sigma(x_n,\theta_k) \mathrm{e}^{\mathrm{i}l\theta_k}\;,$$ can be computed using fast-Fourier techniques. We define the power spectrum $P_l(x_n)$ of the multipole expansion $l$ as $P_l(x_n)=|S_l(x_n)|^2$. As discussed by @ME03.1, the amount of substructure, asymmetry and ellipticity in the mass distributions of the numerically simulated cluster at any distance $x_n$ from the main clump can be quantified by defining an integrated power $P_\mathrm{circ}(x_n)$ as the sum of the power spectra over all multipoles, from which the monopole is subtracted in order to remove the axially symmetric contribution, $$P_\mathrm{circ}(x_n)=\sum_{l=0}^\infty\,P_l(x_n) - P_0(x_n)\;.$$ This quantity measures the deviation from a circular distribution of the surface mass density at a given distance $x_n$ from the cluster centre. In a fully analogous way, we can quantify the deviation from a purely elliptical surface mass density by subtracting from $P_\mathrm{circ}$ the quadrupole term $P_2(x_n)$: $$P_\mathrm{ell}(x_n)=P_\mathrm{circ}(x_n) - P_2(x_n)\;.$$ Separating the effects of asymmetries and substructures by means of contributions to the multipoles is not an easy task. The two terms are mixed together in the multipole expansion. In this paper, asymmetries are assumed to be large-scale deviations from a purely elliptical mass distribution, which result in variations of ellipticity and position angle of the iso-density contours as functions of radius. Their contribution to the azimuthal multipole expansion is mostly contained in the low-[*l*]{} tail. On the other hand, for small $x_n$ even relatively small substructures subtend large angles with respect to the cluster centre. The dipole term itself contains a contribution from substructures. Assuming that substructures are localised lumps of matter whose angular extent is $\lesssim \pi/2$ and then contribute to all multipoles of order larger than $l\sim 2$, we quantify the amount of substructures at radius $x_n$ in the cluster by means of the quantity $$P_\mathrm{sub}(x_n)=P_\mathrm{ell}(x_n)-P_1(x_n)\;,$$ where the power corresponding to the dipole, $P_1(x_n)$ has been subtracted from $P_\mathrm{ell}(x_n)$. ![Power in substructures as a function of distance from the cluster centre for the three projections along the $x$-, $y$- and $z$-axes of cluster $g8_{\rm hr}$.[]{data-label="fig:powers"}](figures/pwsrprof.eps){width="\hsize"} Fig.\[fig:powers\] shows the radial profiles of $P_{\rm sub}$ for the three projections along the $x$, $y$ and $z$-axes of cluster $g8_{\rm hr}$. Peaks along the curves indicate the presence of substructures. The amplitude of the peaks is a growing function of the mass of the substructures. Clearly, in the projection along the $x$-axis massive lumps of matter are located at distances of $\sim 550$ and $\sim 700\,h^{-1}$kpc from the cluster centre. Substructures are abundant also at radii of $\sim 1$ and $\sim 1.25\,h^{-1}$Mpc. This is visible in the left panel of Fig.\[fig:smooth\]. Instead, the most dominant substructures in the projection along the $y$-axis are located outside the region of radius $1\,h^{-1}$Mpc. Only a relatively small peak is observed at $\sim 650\,h^{-1}$kpc from the centre. Finally, when projected along the $z$-axis, the cluster contains a large amount of substructures at the distance of $\sim 800\,h^{-1}$kpc from the centre. Other peaks are located at radii $>1\,h^{-1}$Mpc. Results {#sect:resu} ======= In this section, we describe the lensing properties of the numerical clusters in the sample we have analysed and quantify the impact of ellipticity, asymmetries and substructures on their ability to produce arcs. Magnification and caustics {#sect:mag} -------------------------- The lensing properties of the two-dimensional mass distributions generated using the previously explained methods can be easily determined using the standard ray-tracing technique described in Sect. \[sect:raytr\]. The ability of galaxy clusters to produce strong lensing events is expected to reflect both the presence of substructures embedded into their halos and the degree of ellipticity and asymmetry of their mass distributions. Indeed, all of these factors contribute to increase the shear field of the clusters. This was shown for example by [@TO04.1] and later confirmed by [@ME04.1] and [@FE05.1], who found that the passage of substructures through the cluster cores can enhance the lensing cross section for the formation of arcs with large length-to-width ratios by orders of magnitude. [@ME03.1] show that elliptical models with realistic density profiles produce a number of arcs larger by a factor of ten compared to axially-symmetric lenses with the same mass. Analogous results were obtained by [@OG03.1], who compared the lensing efficiency of triaxial and spherically symmetric halos, and more recently by [@HE05.1]. ![Maps of the tangential-to-radial magnification ratio for the same cluster projection showed in the previous figures. From the top left to the bottom right panel, we show the maps corresponding to the original cluster and to three smoothed versions of it: using the asymmetric, the elliptical and the axially symmetric background models.[]{data-label="fig:lw_maps"}](figures/lw_maps.eps){width="\hsize"} ![Cumulative distributions of tangential-to-radial magnifications in the maps showed in Fig.\[fig:lw\_maps\].[]{data-label="fig:lw_cum"}](figures/lw.eps){width="\hsize"} {width="0.33\hsize"} {width="0.33\hsize"} {width="0.33\hsize"} By smoothing the two-dimensional mass distribution of the clusters, both the levels of substructures and asymmetries are decreased. Thus, we expect their ability to produce highly distorted arcs to be somewhat reduced. This expectation is supported by the fact that the regions of the lens plane where the tangential-to-radial magnification ratio exceeds a given threshold shrink significantly when the smoothing is applied. This is shown in Fig. \[fig:lw\_maps\]. The map of the tangential-to-radial magnification ratio of the original cluster (top right panel) is compared to those obtained by smoothing its surface density map using the asymmetric (top left panel), the elliptical (bottom left panel) and the axially symmetric (bottom right panel) background models. The cumulative distributions of the pixel values in these maps are displayed in Fig.\[fig:lw\_cum\]. The probability of having pixels where the tangential-to-radial magnification ratio exceeds the minimal value decreases at least by a factor of two, due to removal of substructures, asymmetries and ellipticity. This does not imply that the cross section for arcs with large length-to-width ratio decreases by the same amount, since the excess of pixels with large tangential-to-radial magnification ratio in the unsmoothed map is in part due to isolated lumps of matter whose angular scale is similar or smaller than the angular scale of the sources. By definition, the lensing cross section, which measures a cluster’s ability to produce arcs, is an area encompassing the lens’ caustics. Thus, the more extended the caustics are, the larger is generally the lensing cross section. In Fig. \[fig:cau\_smooth\] we show how the caustic shape changes as the smoothing length is varied. Results are shown for each of the smoothing schemes applied. As expected, the caustics shrink as the smoothing length increases. Comparable trends are found for the asymmetric and elliptical background models, for which the change of the caustic length is not dramatic. On the other hand, if the cluster surface density is gradually smoothed towards an axially symmetric distribution, the evolution of the lens’ caustics is much stronger. Similar reductions of the caustic sizes are found when suppressing the substructures outside a given radius. This is shown in Fig. \[fig:cau\_sub\]. Clearly, substructures at distances of the order of $1\,h^{-1}$Mpc already play a significant role for strong lensing. Although they are located far away from the cluster critical region, the external shear they produce is remarkable and determines an expansion of the lens’ caustics. ![Lens’ caustics obtained after removing substructures from the region outside a given radius, as given by the labels in the figure. The side length is $500\,h^{-1}$kpc comoving, corresponding to $\sim 1'$. The figure refers to the cluster projection along the $x$ axis.[]{data-label="fig:cau_sub"}](figures/caustics_sub_0.5Mpc_1arcmin.eps){width="\hsize"} {width="0.33\hsize"} {width="0.33\hsize"} {width="0.33\hsize"} Lensing cross sections ---------------------- Aiming at quantifying the differences between the strong-lensing efficiency of clusters with different amounts of ellipticity, asymmetries and substructures, we focus on the statistical distributions of the arc length-to-width ratios. Indeed, the distortion of the images of background galaxies lensed by foreground clusters is commonly expressed in terms of these ratios. The efficiency of a galaxy cluster for producing arcs with a given property can be quantified by means of its lensing cross section. This is the area on the source plane where a source must be placed in order to be imaged as an arc with that property. The lensing cross sections for large and thin arcs are computed as described in detail in several previous papers [@ME05.1 see e.g]. We consider here the cross sections for arcs whose length-to-width ratio exceeds a threshold $(L/W)_{\rm min}=7.5$, and refer to these arcs as [*giant*]{} arcs. ### A particular case: the cluster $g8_{\rm hr}$ The impact of ellipticity, asymmetry and substructures on the lensing cross section for giant arcs obviously depends on the particular projected mass distribution of the lens. Large differences can be found even between different projections of the same cluster. As an example, we show in Fig. \[fig:avcs\_smooth\] the lensing cross sections for giant arcs as a function of the smoothing length for the three projections of cluster $g8_{\rm hr}$. Results are shown for all the smoothing methods described earlier. The cross sections have been normalised to that of the unsmoothed lens. The horizontal lines in each plot indicate the limiting values reached when the surface density maps are completely smoothed. Three different realizations of background source distributions were used to calculate the errorbars. As expected, the lensing cross sections decrease as the smoothing scale increases. The decrement is generally rapid for small smoothing lengths, then becomes shallower. The differences between the three projections are large. When smoothing with an elliptical background model, maximal variations of the cross section of the order of $40\%$ are found for the projections along the $x$- and $y$-axes. For these two projections, circularising the surface-mass distributions reduces the cross section by $\sim 85-90\%$. However, while for the projection along the $x$-axis smoothing using the asymmetric background model reduces the lensing cross section by $35\%$, for the projection along the $y$-axis the cross section becomes only $\sim 20 \%$ smaller. The differences between these two projections can be explained as follows. First, as discussed earlier in the paper, when projected along the $x$-axis, the cluster has important substructures close to its centre. This is evident in Fig. \[fig:powers\]: substructures are significant at radii between $\sim 400 - 800 \,h^{-1}$kpc. On the other hand, when projected along the $y$-axis the cluster exhibits significant substructures only at larger radii, $>1 \,h^{-1}$Mpc. Since strong lensing occurs in the very inner region of the cluster, the impact of substructures close to the centre is larger than that of substructures farther away. Second, in the projection along the $y$-axis, the twist of the iso-density contours and the variations of their ellipticity are significantly larger than for the projection along the $x$-axis (see Fig.\[fig:ellphi\]). Therefore, while for the projection along the $x$-axis the deviation from a purely elliptical mass distribution is mostly due to the effects of substructures, in the projection along the $y$-axis it is due to both substructures ($\sim 20\%$) and asymmetries ($\sim 25\%$). Asymmetries which are due to the presence of large-scale density fluctuations distort the isodensity contours which are elongated in some particular direction, varying their ellipticity and position angle. Such large-scale modes contribute to the shear field of the cluster, pushing the critical lines towards regions of lower surface density and increasing their size. Consequently, the strong lensing cross section also increases. As shown in Sect. \[sect:power\], when projected along the $z$-axis, the cluster appears rounder. Consequently, a smooth axially symmetric representation of this lens which conserves its surface density profile has a lensing cross section for giant arcs which is only $50\%$ smaller than that of the original cluster. Smoothing using the asymmetric or the elliptical background models is equivalent and leads to a reduction of the lensing cross section by $\sim 30\%$. The absence of significant differences between these two smoothing schemes indicates that asymmetries play little role in this projection, while the large substructure observed at $\sim 800\,h^{-1}$kpc from the centre has a significant impact on the lensing properties of this lens, even being at relatively large distance from the region where strong lensing occurs. ![Comparison between the low and the high resolution version of cluster g8. The lensing cross section for arcs with length-to-width ratio larger than $7.5$ averaged over three orthogonal projections of the same cluster are plotted versus the smoothing function.[]{data-label="fig:avcs_g8hl"}](figures/normcs_g8_hl.eps){width="\hsize"} The smoothing length for which the curves converge to the values for the completely smoothed maps tell us the characteristic scale of cluster substructures which is important for lensing. In those projections where localised substructures play an important role, i.e. in the projections along the $x$- and the $z$-axes, the relevant scales are smaller ($\lesssim 100 - 300\,h^{-1}$kpc), while for the projection where asymmetries are more relevant they are larger ($\lesssim 400\,h^{-1}$kpc). Converting these spatial scales into the corresponding mass scales in not an easy task, especially because we are dealing with substructures in two dimensions. Tentatively, we can assume that the substructures are spherical and their mean density corresponds to the virial overdensity $\Delta_v(z)$. For $z=0.3$, in the cosmological framework where our simulations are carried out, $\Delta_v \sim 123$. Then, the above mentioned spatial scales correspond to masses between $\sim 4\times 10^{11}$ and $\sim 2\times 10^{13} h^{-1}\,M_\odot$. ![Mean lensing cross section for arcs with length-to-width ratio larger than $7.5$ of four low-resolution clusters as a function of the smoothing length. Solid, dashed and dotted lines refer to smoothing adopting the asymmetric, the elliptical and the axially symmetric background models, respectively. The critical regions of the lenses have maximal radii in the range $\sim 100 - 250 h^{-1}$kpc.[]{data-label="fig:avcs_low_all"}](figures/normcs_low_all.eps){width="\hsize"} The three cluster projections whose lensing properties were discussed above were carried out along the three orthogonal axes of the simulation box. In general, these axes do not coincide with the cluster’s principal axes because it is randomly oriented with respect to the simulation box. Thus, the roundest and the most elliptical cluster projections that we have studied are not necessarily the roundest and the most elliptical possible projections, respectively. In the case of $g8_{hr}$, however, the principal axes do not differ substantially from those of the simulation box. The cluster turns out to be prolate with axis ratios $I_1/I_2 \sim 1.9$ and $I_2/I_3 \sim 1.1$. When projected along the major principal axis, i.e. in its roundest projection, the ellipticity in the central region is slightly smaller than in the projection along the $z$-axis, varying between $0.1$ and $0.2$. When projected along the two other principal axes, the cluster has ellipticity and twist profiles very similar to those for the projections along the $x$- and the $y$-axes. For these reasons, the differences between the strong lensing cross sections of the purely elliptical and of the axially-symmetric smoothed models are modest in the roundest projection, even smaller than for the previously discussed projection along the $z$-axis. Indeed, we find that the ellipticity accounts for only $10\%$ of the lensing cross section in this case. When projected along the other two principal axes, the impact of the ellipticity is similar to that for the projections along the $x$- and $y$-axes. ### Mean lensing cross sections The example of cluster $g8_{\rm hr}$ shows that, depending on the particular configuration of a lens, the impact of ellipticity, asymmetry and substructures can be substantially different in different clusters. Nevertheless, we can try to estimate what is the statistical impact of all these factors. For doing that, we repeat the analysis shown for the cluster $g8_{\rm hr}$ on our sample of clusters simulated with lower mass resolution. Among them, we analyse the lensing properties of the low-resolution analogue of cluster $g8_{\rm hr}$. When we compare the sensitivity to smoothing of the low- and the high-resolution versions of the same cluster, we do not find significant differences between them. Fig. \[fig:avcs\_g8hl\] shows how the lensing cross section for giant arcs changes as a function of the smoothing length for all three smoothing schemes applied. Each curve is the average over the three independent projections of the clusters. The thick and thin lines refer to the high- and low-resolution runs, respectively. Considering that, as discussed in Sect. \[sect:nummod\], the two simulations have quite different mass distributions, the differences shown here, which are still within the error bars, have little significance. This suggests that our results are not affected by problems of mass resolution of the numerical simulations. ![Lensing cross section for arcs with length-to-width ratio larger than $7.5$ as a function of the minimal equivalent radius containing substructures. Results are shown for all three cluster projections. The curves are normalised to the cross section of the cluster containing all its substructures, corresponding to $l=\infty$. The critical regions of the lenses have maximal radii in the range $\sim 100 - 250 h^{-1}$kpc.[]{data-label="fig:avcs_sub"}](figures/normcs_low_all_dist.eps){width="\hsize"} In Fig. \[fig:avcs\_low\_all\] we show the variation of the lensing cross section vs. smoothing length averaged over all the low-resolution clusters which we have analysed. For each cluster, we use the three projections along the $x$, $y$ and $z$ axes, i.e. 12 lens planes in total. The results are shown for all three smoothing schemes adopted. The curves show that on average removing the substructures from the clusters reduces their lensing cross section by about $30\sim 35 \%$. Removing asymmetries, i.e. transforming the cluster mass distributions to purely elliptical, further reduces the lensing cross section for giant arcs by $\sim 10 \%$. If also ellipticity is removed, the mean lensing cross section becomes $\sim 20\%$ of that of the unsmoothed lenses. The typical scales for the substructures which mostly affect the lensing properties of their host halos are $\lesssim 150\,h^{-1}$kpc ($M \lesssim 10^{12}h^{-1}\,M_\odot$). Note that this does not mean that larger substructures do not affect the lensing cross sections: simply, they are less abundant. We verified that only one of the clusters in our sample ($g72$) is undergoing a major merger with a massive substructure ($M_{\rm sub}\sim 4\times 10^{14}h^{-1}\,M_\odot$) at $z=0.3$. Note also that the largest scale sub-halos contribute also to the asymmetry of the projected mass distributions. This means that smoothing further using the asymmetric background model does not remove these large substructures completely. When smoothing using the background elliptical and axially-symmetric models the smoothing length at which the lensing cross section approximates that of the completely smoothed lenses is slightly larger, because larger-scale contributions to the surface density fields must be removed. ### Location of important substructures We now investigate what is the typical location of substructures which are important for lensing. By ray-tracing through the mass distributions obtained after removing substructures from outside a given equivalent radius, as discussed at the end of Sect.\[sect:smooth\], we find that the strong lensing efficiency of clusters is sensitive to substructures located within a quite large region around the cluster centre. For demonstrating this, we show in Fig. \[fig:avcs\_sub\] how the lensing cross section changes as a function of the minimal radius containing substructures. The cross sections are again normalised to those of the unsmoothed lenses. In the projection along the $y$-axis of cluster $g8_{\rm hr}$ (short dashed line), we note that the lensing cross section decreases quickly when removing substructures outside an equivalent radius of $\sim 1\,h^{-1}$Mpc. The lensing cross section for giant arcs is already reduced by $\sim 10\%$ when the minimal equivalent radius containing substructures is $\sim 800\,h^{-1}$kpc. In fact, in this projection there are two large substructures at distances between $1$ and $1.2\,h^{-1}$Mpc from the cluster centre, which seem to influence the strong lensing efficiency of this lens. Note that the critical lines in this projection of the cluster extend up to $\sim 200 h^{-1}$kpc from the cluster centre. In the other two projections of the same cluster (long dashed and dotted lines), where large substructures are located closer to the centre, a similar decrement of the lensing cross section is observed at much smaller equivalent radii, between $300$ and $450\,h^{-1}$kpc. When averaging over the low-resolution cluster sample, we still find that the lensing cross sections start to decrease when substructures outside a region of equivalent radius $\sim 1\,h^{-1}$Mpc are removed from the clusters. While the minimal radius containing substructures is further reduced, the cross sections continue to become smaller. The evolution is initially shallow. A reduction of $\sim 15\%$ is observed at a minimal equivalent radius $\sim 300\,h^{-1}$kpc. If substructures are removed from an even smaller region around the centre of the clusters, the decrement of the lensing cross sections becomes faster. The critical regions of the lenses in our sample have maximal radii in the range $\sim 100 - 250 h^{-1}$kpc. This shows that substructures close to the cluster centre are the most relevant for strong lensing, but substructures located far away from the cluster critical region for lensing also have a significant impact on the cluster lensing cross sections. Arc shapes, locations and fluxes -------------------------------- ![Effects of substructures on scales $\lesssim 50\,h^{-1}$kpc on the morphology of gravitational arcs. In the left panels shown are the arcs formed out of two different sources lensed by the unsmoothed mass distribution displayed in the top left panel of Fig.\[fig:incsmooth\]. In the right panels, shown are the corresponding images obtained by using as lens the mass distribution given in the top right panel of Fig.\[fig:incsmooth\], which has been obtained by smoothing with a smoothing length of $47\,h^{-1}$kpc. In all panels, the critical lines are drawn for comparison. The top and the bottom panels have sizes of $38''\times 57''$ and $36''\times 56''$, respectively. []{data-label="fig:splitjoinarcs"}](figures/3079_3488.eps){width="\hsize"} Small changes in the positions of the caustics and therefore in the positions of the critical lines can have huge consequences on the appearance and location of gravitational arcs. In order to describe these effects we compare here the characteristics of the images of the same population of background sources lensed both with the unsmoothed projected mass distributions of the numerical clusters and with weakly smoothed versions of them. We smooth using the asymmetric model and using a smoothing length of $47\,h^{-1}$kpc ($M\sim 5\times 10^{10} h^{-1}\,M_\odot$), not significantly exceeding the scale of galaxies in clusters. The aim of this discussion is to show that even relatively small substructures may play a crucial role in determining the appearance of gravitational arcs. Sources are first distributed around the caustics of the unsmoothed lens following the method discussed in Sect. \[sect:raytr\]. Then, the same sources are used when ray-tracing through the surface density maps from which substructures are removed. Each source conserves its position, luminosity, ellipticity and orientation, allowing to directly measure the effects that removing substructures and asymmetries has on several properties of the same arcs. For this experiment, we use an extended version of our ray-tracing code which includes several observational effects, like sky brightness and photon noise, allowing to mimic observations in several photometric bands. We assume that the sources have exponential luminosity profiles and shine with a luminosity in the $B$-band $L_B=10^{10}\,L_{\odot}$. We simulate exposures of $3$ksec with a telescope with diameter of $8.2$m (VLT-like). The throughput of the telescope has been assumed to be $0.25$. The surface brightness of the sky in the $B$-band has been fixed at $22.7$mag per square arcsec. In this ideal situation, no seeing is simulated. The effects of all these observational effects on the morphological properties of arcs will be discussed in a forthcoming paper. ![Example of gravitational arc shifted by substructures. The size of the each frame is $27''\times 34''$. Left panel: simulation including substructures. Right panel: simulation performed after smoothing with a smoothing length of $47\,h^{-1}$kpc.[]{data-label="fig:arcshift"}](figures/1965.eps){width="\hsize"} In earlier studies, [@ME00.1] and [@FL00.1] showed that the impact of galaxy-sized cluster subhalos on the statistical properties of gravitational arcs with large length-to-width ratios is very modest. These results are confirmed in the present study. As shown in the previous section, the lensing cross sections for long and thin arcs decrease by $\sim 20\%$ when smoothing the cluster surface densities on scales $\sim 50\,h^{-1}$kpc. On smaller scales the decrement is only of a few percent. However, the morphology of individual arcs is strongly affected in several cases. Arcs can become longer or shorter, thinner or thicker. In other cases, more dramatic morphological changes are found. For example, cluster galaxies locally perturb the cluster potential such as to break long arcs, while in other cases the opposite effect occurs. Two examples are shown in Fig.\[fig:splitjoinarcs\]. In the upper panels, the same source is imaged as two short arcs or as a single long arc when the unsmoothed (left panel) and the smoothed lens (right panel) are used, respectively. The lenses displayed in the top panels of Fig. \[fig:incsmooth\] have been used for these simulations. In the bottom panels, a single long arc becomes a smaller arclet in absence of substructures. Super-imposed on each graph are the critical lines. They tend to wiggle around individual substructures in the left panels, while they are more regular in the right panels. Substructures slightly shift the high-magnification regions of a cluster relative to the background sources, inducing remarkable changes in the shape of their images and in their multiplicity. For the cluster projection used in this example, the image multiplicity is increased for $\sim 21\%$ of the sources producing arcs longer than $5''$, when smoothing is applied, indicating that long arcs break up. On the other hand, for $\sim 10\%$ of them the image multiplicity decreases, showing that the caustics shrink and sources move outside of them, Consequently the number of their images decreases. In several cases, substructures are also responsible for significant shifts of the positions of gravitational arcs. An example is shown in Fig. \[fig:arcshift\]. The size of each frame is $27''\times 34''$. The morphological properties of the arc in the two simulations are almost identical. The arc length is $\sim 11''$, the arc width is $\sim0.6''$. The luminosity peak of the arc, which we use for measuring the shift, is moved towards the bottom left corner of the frame by $\sim 8.5''$, when substructures on scales smaller than $47\,h^{-1}$kpc are smoothed away. Similar cases are frequent. For the lens used in this example, $\sim 27\%$ of the long arcs (length $>5''$) found in the simulation including substructures are shifted by more than $5''$ after smoothing. About $\sim 4\%$ of them is shifted by more than $10''$. From this analysis, long arcs which split into smaller arclets are excluded. Finally, substructures affect the fluxes received from the lensed sources. The histogram in Fig. \[fig:fluxhist\] (solid line) shows the probability distribution function of the differences $\Delta B=B-B_{sm}$ between magnitudes of arcs with length $>5''$ measured in the simulations where the unsmoothed and smoothed lens projected mass maps were used as lens planes, respectively. The analysis is restricted to arcs whose length exceeds $5''$ in the simulation containing all substructures. Some arcs are magnified, some others demagnified by the substructures. The maximal variations in luminosity correspond to $\Delta B \sim -2.3 \div +2.4$. The distribution is slightly skewed towards the negative values, indicating that in absence of substructures arcs tend to be less luminous. In fact, substructures contribute to magnify the sources, as discussed in Sect. \[sect:mag\]. Sources of different size are expected to be differently susceptible to the substructures. The dashed and the dotted lines in Fig. \[fig:fluxhist\] respectively show how the probability distribution function of $\Delta B$ changes when the source size is increased or decreased by a factor of two compared to the original source size used in the simulations. As expected, larger sources are less sensitive to perturbations by small substructures in the lenses. ![Probability distribution function of the differences between arc magnitudes in simulation including and excluding substructures on scales $<47\,h^{-1}$kpc. Results are shown for three different source sizes. See text for more details.[]{data-label="fig:fluxhist"}](figures/fluxhist_new.eps){width="\hsize"} Similar results were found for some other cluster models. For other lenses, the impact of the substructures on the properties of individual arcs is even stronger. The observed arc shifts have tangential and radial components. Generally, the tangential shifts are larger than the radial shifts. However, when large substructures located close to the critical regions of clusters are smoothed away, significant radial shifts are possible, given that the relative size of the critical lines changes dramatically. In Fig. \[fig:radtanshifts\], the radial shifts of long arcs (length $>5''$) is plotted versus the tangential shifts. Different symbols are used to identify arcs produced by different numerical clusters. As anticipated, for the majority of the arcs produced by the clusters $g1$, $g8$, $g8_{hr}$ and $g51$ the radial shifts are within few arcseconds, while tangential shifts of $10''$ and more are frequent. On the other hand, the arcs produced by the cluster $g72$ have significantly larger radial shifts. As mentioned above, $g72$ is undergoing a major merger and a secondary lump of matter occurs near the cluster centre. The cluster critical lines, along which arcs form, are elongated towards it. When moderate smoothing is applied, the impact of the merging substructure is attenuated and the critical line shrinks substantially. Thus, the arcs move towards the centre of the cluster and their morphology and flux are also strongly affected. ![Distribution of long arcs (length $>5''$) in the plane radial ($\Delta R$) vs. tangential ($R\Delta \phi$) shift. Different symbols identify arcs produced by different clusters. The arcs produced by the cluster $g72$, which is experiencing a major merger, are given by the small filled squares.[]{data-label="fig:radtanshifts"}](figures/radtanshifts.eps){width="\hsize"} Some results for all the cluster models we analysed are summarised in Tab.\[tab:subarcs\]. All of these effects might have an enormous impact in lensing analysis of clusters, in particular when modelling a lens by fitting gravitational arcs. These results show that any substructure on scales comparable to those of galaxies should be included in the model in order to avoid systematic errors. This problem will be addressed in detail in a following paper, in particular regarding the possible biases in strong lensing mass determinations. However, by making the wrong assumption of axial symmetry, we can approximately estimate the errors due to the radial shifts of the arcs. For axially symmetric lenses, the mean convergence within the critical line is $\overline\kappa(<x_c)=1$. The mass within $x_c$ is then $$M(<x_c)=\pi \Sigma_{\rm cr} x_c^2 \;,$$ where $$\Sigma_{\rm cr}= \frac{c^2}{4 \pi G} \frac{D_{\rm s}}{D_{\rm l} D_{\rm ls}}$$ is the critical surface mass density and $x_c$ is in physical units. We assume that the position of an arc traces the position of the critical line. Then, if an arc distance from the centre changes from $R$ to $R'$, the relative variation of the mass inferred from strong lensing is $$\frac{\Delta M}{M}=\frac{M-M'}{M}=\frac{R^2-R'^2}{R^2}\;.$$ The distribution of such $\Delta M/M$, as derived from the radial shifts displayed in Fig.\[fig:radtanshifts\], is shown in Fig. \[fig:mbias\]. Without suitably modelling the effects of substructures, the typical errors in mass determinations are within a factor of two, but larger errors are also possible. Since substructures generally contribute to enlarge the critical lines, a larger mass within the critical line would be required in order to have an arc at the observed distance from the cluster centre. ![Distribution of the relative variations of mass determinations from strong lensing, assuming axial symmetry and that the arc position trace the location of the critical lines.[]{data-label="fig:mbias"}](figures/mbias.eps){width="\hsize"} Note that even substructures far away from the cluster centre are important. For example, keeping the inner structure of the projection along the $y$-axis of cluster $g8_{\rm hr}$ unchanged, while removing the big substructures at distances $> 1\,h^{-1}$Mpc, we find that more than $\sim 50\%$ of the long arcs are shifted by at least $5''$. Moreover, image multiplicity increases for $26\%$ and decreases for $8\%$ of the sources respectively. ----------- ------- ------------ ------------ -------------- ------------ Cluster proj. inc. mult. dec. mult. shift $>5''$ $\Delta B$ \[$\%$\] \[$\%$\] \[$\%$\] $g8_{hr}$ x 21.2 9.8 26.6 -2.3/+2.4 y 19.7 3.9 11.1 -1.7/+0.8 z 23.7 10.1 15.5 -2.5/+1.6 $g1$ x 47.0 1.0 24.0 -1.9/+1.1 y 26.1 0.0 20.0 -1.7/+0.7 z 24.5 0.0 29.5 -2.5/+0.8 $g8$ x 24.0 4.0 27.0 -1.7/+1.0 y 35.4 9.5 51.1 -2.1/+2.1 z 31.2 6.1 28.2 -2.5/+0.5 $g51$ x 33.1 6.2 44.7 -2.3/+1.7 y 35.6 5.1 65.7 -2.5/+1.3 z 39.1 6.2 54.3 -1.5/+1.3 $g72$ x 36.0 4.0 79.4 -2.5/+2.1 y 62.5 0.0 57.1 -2.4/+0.0 z 22.5 0.0 70.9 -2.4/+1.7 mean 32.1 4.4 40.3 -2.1/+1.3 ----------- ------- ------------ ------------ -------------- ------------ : Effects of substructures on gravitational arcs. Column 1: cluster name; column 2: projection; column 3: percentage of sources whose image multiplicity increases; column 4: percentage of sources whose image multiplicity decreases; column 5: percentage of long arcs ($l>5''$), whose positions result to be shifted by more than $5''$ when substructures are smoothed away; column 6: maximal variations of magnitudes of long arcs. \[tab:subarcs\] If relatively small substructures can alter many of the properties of gravitational arcs, even asymmetries may be relevant. As noted earlier the projection along the $y$-axis is the most asymmetric of $g8_{hr}$. Comparing the properties of arcs lensed by the smoothed asymmetric and elliptical models of this lens, we find significant shifts in the location of about $45\%$ of the long arcs. For $\sim 20\%$ of the sources producing long arcs, the multiplicity is changed. Conclusions {#sect:conclu} =========== In this paper we have quantified the impact of several properties of realistic cluster lenses on their strong lensing ability. In particular, our goal was to separate the effects of substructures, asymmetries and ellipticity. For doing that, we analysed the lensing properties of one numerical cluster simulated with very high mass resolution. In addition, we studied four other clusters obtained from N-body simulation with a lower mass resolution. Each cluster was projected along three independent directions. For each projection, we constructed three completely smoothed versions. Each of them conserves the mean surface density profile of the mass distribution of the cluster. However, the first reproduces the variations of the ellipticity and of the position angle of the isodensity contours as functions of the distance from the centre; the second has elliptical isodensity contours with fixed ellipticity and orientation; the third is an axially symmetric model. The lensing properties of the numerical clusters, of their smoothed analogues and of several intermediate versions were investigated using standard ray-tracing techniques. Our main results can be summarised as follows: - Substructures, asymmetries and ellipticity contribute to increase the ability of clusters to produce strong lensing events. Substructured, asymmetric and highly elliptical clusters produce more extended high magnification regions in the lens plane where long and thin arcs can form. Indeed, substructures, asymmetries and ellipticity determine the location and the shape of the lens caustics around which sources must be located in order to be strongly lensed by the clusters. - The impact of substructures, asymmetries and ellipticity on the lensing cross section for producing giant arcs is different for different lenses. The lensing properties of the most symmetric clusters appear to be particularly influenced by the substructures. On the contrary, substructures are less important in asymmetric lenses. - On average, we quantify that substructures account for $\sim 30\%$ of the total cluster cross section, asymmetries for $\sim 10\%$ and ellipticity for $\sim 40\%$. - The substructures that typically contribute to lensing are on scales $\lesssim 150 - 200 \,h^{-1}$kpc. Assuming a virial overdensity of $\sim 123$ for $z=0.3$, this corresponds to mass scales of the order of $\sim 10^{12} h^{-1}\,M_\odot$. Substructures on larger scales are not as frequent in our cluster sample, but, if present, they can boost significantly the lensing cross section [see e.g. @TO04.1; @ME05.1]. - Substructures play a more important role when they are located close to the cluster centre. However, the lensing cross section for giant arcs is sensitive to substructures within a wide region around the cluster core. In particular, our simulations show that the sensitivity to substructures far from the centre is particularly high in those clusters whose inner regions are unperturbed. In these cases, the loss of strong lensing efficiency due to removing the substructures from the clusters is correlated with substructures within a region of $\sim 1 \, h^{-1}$Mpc in radius; on the contrary, clusters containing substructures in the inner regions are “screened” against external perturbers. - Even small substructures ($l\lesssim 50$kpc, $M\lesssim 5\times 10^{10}h^{-1}\,M_\odot$) influence the appearance and the location of gravitational arcs. The perturbations to the projected gravitational potential of the cluster induced by the substructures alter the multiplicity of the images of individual sources. Moreover, they change the morphology and the flux of the images themselves. Finally, they can shift the position of arcs with significant length to width ratios by several arcseconds on the sky. These results highlight several important aspects of strong lensing by clusters. First, any model for cluster lenses cannot neglect the effects of asymmetries, ellipticity and substructures. Clusters which may appear as relaxed and symmetric, for example in the X-rays, are potentially those which are most sensitive to the smallest substructures, located even at large distances from the inner cluster regions, critical for strong lensing. Even subhalos on the scales of galaxies can influence the strong lensing properties of their hosts and alter the shape and the fluxes of gravitational arcs. Therefore, if the lens modelling is not carried out at a very high level of detail, it may result in being totally incorrect. Second, the high sensitivity of gravitational arcs to deviations from regular, smooth and symmetric mass distributions suggests that strong gravitational lensing is potentially a powerful tool to measure the level of substructures and asymmetries in clusters. Since, as we said, the sensitivity to substructures is higher in the case of more symmetric lenses, we conclude that dynamically active clusters, like those undergoing major merger events, should be quite insensitive to “corrugations” in the projected mass distribution but highly sensitive to asymmetries. Arcs could then be used to diagnose mergers in clusters. Conversely, substructures should become increasingly important for the arc morphology as clusters relax. Then the level of substructures in clusters should be quantified by measuring their effect on the arc morphology. This is particularly intriguing since measuring the fine structures of gravitational arcs has become feasible thanks to the high spatial resolution reached in observations from space. Third, the strong impact of asymmetries and substructures on the lensing properties of clusters and the wide region in the cluster where these last can be located in order to produce a significant effect further support the picture that mergers might have a huge impact on the cluster optical depth for strong lensing, as suggested in several previous studies [@TO04.1; @ME04.1; @FE05.1].

ArXiv

--- abstract: 'For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_\lambda\}_{\lambda\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ for all $\lambda\in J$, under natural technical conditions we prove that the sum $K+C_\lambda$ has positive Lebesgue measure for almost all values of the parameter $\lambda$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.' address: - 'Anton Gorodetski University of California, Irvine' - 'Scott Northrup University of California, Irvine' author: - '[A. Gorodetski]{}' - '[S. Northrup]{}' title: On Sums of Nearly Affine Cantor Sets --- [^1] Introduction and Main results {#sec:intro} ============================= Questions on the structure and properties of sums of Cantor sets appear naturally in dynamical systems [@n1; @n2; @n3; @PaTa], number theory [@CF; @Mal; @Moreira], harmonic analysis [@BM; @BKMP], and spectral theory [@EL06; @EL07; @EL08; @Y]. J.Palis asked whether it is true (at least generically) that the arithmetic sum of dynamically defined Cantor sets either has measure zero, or contains an interval (see [@PaTa]). This claim is currently known as the “Palis’ Conjecture”. The conjecture was answered affirmatively in [@MY] for generic dynamically defined Cantor sets. For sums of generic [*affine*]{} Cantor sets Palis’ Conjecture is still open. Even for the simplest case of middle-$\alpha$ Cantor sets these questions are non-trivial and not completely settled. By a middle-$\alpha$ Cantor set we mean the Cantor set $$\label{e.ddcs} C=\cap_{n=0}^{\infty}I_n, \ I_{n+1}=\cup_{i=1}^{m}\varphi_i(I_n), \ \varphi_i(I_0)\cap \varphi_j(I_0)=\emptyset \ \text{for}\ i\ne j,$$ where $I_0=[0,1]$, $m=2$, $\varphi_1(x)=ax$, $\varphi_2(x)=(1-a)+ax$, $a=\frac{1}{2}(1-\alpha)$. Let us denote this Cantor set by $C_a$. It is easy to show (using dimensional arguments, e.g. see Proposition 1 in Section 4 from [@PaTa]) that if $\frac{\log 2}{\log 1/a}+\frac{\log 2}{\log 1/b}<1$ then $C_a+C_b$ is a Cantor set. On the other hand, Newhouse’s Gap Lemma (e.g. see Section 4.2 from [@PaTa], or [@n1]) implies that if $\frac{a}{1-2a}\frac{b}{1-2b}>1$ then $C_a+C_b$ is an interval. This still leaves a “mysterious region” $R$ in the space of parameters, see Figure \[f.1\], and Solomyak [@So97] showed that for Lebesgue a.e. $(a, b)\in R$ one has $Leb(C_a+C_b)>0$. \[f.1\] ![The region $R$ studied by B.Solomyak in [@So97]](mysterious.png){height="2in"} A description of possible topological types of $C_a+C_b$ was provided in [@MO]. It is still an open question whether $C_a+C_b$ contains an interval for a.e. $(a,b)\in R$. Solomyak’s result was generalized to families of homogeneous self-similar Cantor sets (i.e. Cantor sets given by (\[e.ddcs\]) where all contractions $\{\varphi_i\}_{i=1, \ldots, m}$ are linear with the same contraction coefficient) by Peres and Solomyak [@PeSo]. They showed that for a fixed compact set $K\subseteq \mathbb{R}$ and a family $\{C_\lambda\}$ of homogeneous Cantor sets parameterized by a contraction rate $\lambda$ (i.e. all contractions have the form $\varphi_i(x)=\lambda x +D_i(\lambda)$, $D_i\in C^1$) one has $$\begin{gathered} \label{e.hdsone} \text{dim}_H(C_\lambda+K)= \text{dim}_HC_\lambda + \text{dim}_HK \ \text{for a.e.}\ \lambda\in (\lambda_0, \lambda_1) \\ \text{if} \ \ \ \text{dim}_HC_\lambda + \text{dim}_HK<1\ \ \text{for all}\ \lambda\in (\lambda_0, \lambda_1), \ \text{and} \end{gathered}$$ $$\begin{gathered} \label{e.hdlone} Leb(C_\lambda+K)>0 \ \text{for a.e.}\ \lambda\in (\lambda_0, \lambda_1) \\ \text{if} \ \ \ \text{dim}_HC_\lambda + \text{dim}_HK>1\ \ \text{for all}\ \lambda\in (\lambda_0, \lambda_1). \end{gathered}$$ In the case when $K$ is a non-linear $C^{1+\varepsilon}$-dynamically defined Cantor set, the set of exceptional parameters in (\[e.hdlone\]) in fact has zero Hausdorff dimension, see [@Shm Theorem 1.4]. For a more general case of sums of dynamically defined Cantor sets $C$ and $K$ on the first glance the mentioned above results by Moreira and Yoccoz [@MY] provide the complete answer. But in practice in many cases one has to deal with a finite parameter families of Cantor sets, or even with a specific fixed Cantor sets $C$ and $K$, and [@MY] does not provide specific genericity assumptions that could be verified in a particular given setting. Specific conditions that would allow to claim that $$\label{e.hs} \text{dim}_H(C+K)=\min\left(\text{dim}_HC + \text{dim}_HK, 1\right)$$ are currently known [@HS; @NPS; @PeShm], but the case $\text{dim}_HC + \text{dim}_HK>1$ turned out to be more subtle. In this paper we address this question in the case of affine (all $\varphi_i$ in (\[e.ddcs\]) are affine contractions, not necessarily with the same contraction coefficients) and close to affine dynamically defined Cantor sets. \[t.1\] Suppose $J\subseteq \mathbb{R}$ is an interval and $\lbrace C_\l \rbrace_{\lambda\in J}$ is a family of dynamically defined Cantor sets generated by contracting maps $$\label{e.start}\lbrace f_{i, \lambda}(x)=c_{i}(\lambda)x+b_{i}(\lambda)+g_{i}(x, \lambda)\rbrace_{i=1}^m$$ such that the following holds: $$c_i(\lambda), b_i(\lambda)\ \text{are $C^1$-functions of}\ \lambda;$$ $$\label{e.c} \frac{d|c_i|}{d\lambda} \le -\delta<0\ \text{for all $\lambda\in J$ and some uniform $\delta>0$};$$ $$\label{e.g} g_i(x,\lambda)\in C^2, \ \text{with small (based on $\{c_i(\lambda)\}, \{b_i(\lambda)\}$) $C^2$-norm}.$$ Then for any compact $K\subset \mathbb{R}$ with $$\label{e.condition} \dim_H(K)+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J,$$ the sumset $K+C_\lambda$ has positive Lebesgue measure for a.e. $\l\in J$. Theorem \[t.1\] can be generalized in a straightforward way to a larger class of nearly affine Cantor sets where topological Markov chains are allowed instead of the full Bernoulli shift in the symbolic representation (see [@MY] or [@PaTa] for detailed definitions). We restrict ourselves to the case of the full shift only to keep the exposition more transparent. We strongly believe that the assumption on $C_\lambda$ being close to affine is an artefact of the proof, and that a similar statement should hold in a more general setting, for a family of non-linear dynamically defined Cantor sets without any smallness assumptions on non-linearity. We plan to address this question in a future publication. Consider now the non-homogeneous affine case, that is a Cantor set generated by (\[e.ddcs\]), where $\varphi_k(x)=\lambda_kx+d_k$. Moreover, let us include it into a family $\{K_{\Lambda}\}$, where $$\Lambda=(\lambda_1, \ldots, \lambda_m), \ \lambda_k\in J_k\subset (-1, 0)\cup (0,1), \ \text{and} \ d_k=d_k(\Lambda)\ \text{is $C^1$}.$$ The last condition in (\[e.ddcs\]) implies that $$[d_i(\Lambda)+\l_i K_\Lambda]\cap [d_j(\Lambda)+\l_j K_\Lambda] = \emptyset\ \text{ for}\ i\neq j,$$ which is sometimes called [*strong separation condition*]{}, e.g. see [@PeSo]. Fubini’s theorem together with Theorem \[t.1\] gives the following statement. \[c.1\] Suppose $K$ is a compact subset of the real line, and a family $\{K_\Lambda\}$ of affine Cantor sets as above is given such that $$\dim_H K + \dim_H K_\Lambda > 1 \ \text{ for all}\ (\l_1,\dots,\l_m)\in J_1\times\dots\times J_m.$$ Then for a.e. $(\l_1,\dots,\l_m)\in J_1\times\dots\times J_m$ the set $K+K_\Lambda$ has positive Lebesgue measure. Notice that in this setting any affine Cantor set is completely determined by $2m$ parameters, namely $(\lambda_1, \ldots, \lambda_m)\in \Lambda$ and $(d_1, \ldots, d_m)$. Admissible $2m$-tuples of the parameters (i.e. such that $\varphi_k([0,1])\subseteq [0,1]$ for each $k=1, 2, \ldots, m$, and the strong separation condition holds) form a subset in $\mathbb{R}^{2m}$. As an immediate consequence of Corollary \[c.1\] we have \[c.2\] Generically (for Lebesgue almost all admissible tuples of the parameters) the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one. It is interesting to compare these results with Theorem E from [@ShmS] that claims that for any two affine Cantor sets $C_1$ and $C_2$ with sum of dimensions greater than one, $\text{dim}_H\,\{u\in \mathbb{R}\ |\ Leb(C_1+uC_2)=0\}=0$. The idea of proof of Theorem \[t.1\] is to find some measures supported on $K$ and $C_\lambda$ whose convolution is absolutely continuous with respect to the Lebesgue measure. Since support of a convolution of two measures is the sum of their supports, this would prove that $Leb(K+C_\lambda)>0$. In Section \[s.acc\] we provide the statement of a result from [@DGS] on absolute continuity of convolutions of singular measures under certain conditions. Then in Section \[sec:main\] we verify those conditions for some specific measures supported on $K$ and $C_\lambda$. Absolute continuity of convolutions {#s.acc} =================================== Let $\Omega=\mathcal{A}^{\Z_+}$ with $|\mathcal{A}| = m \geq 2$ be the standard symbolic space, equipped with the product topology. Let $\mu$ be a Borel probability measure on $\Omega$. Let $J$ be a compact interval and assume we are given a family of continuous maps $\Pi_\l:\Omega\to\R$, for $\l \in J$, such that $C_\l = \Pi_\l(\Omega)$ are the Cantor sets, and let $\nu_\l = \Pi_\l (\mu)$. For a word $u\in\mathcal{A}^{n}$, $n\geq 0$, denote by $|u| = n$ its length and by $[u]$ the cylinder set of elements of $\Omega$ that have $u$ as a prefix. For $\omega,\tau\in\Omega$ we write $\omega\wedge\tau$ for the maximum common subword in the beginning of $\omega$ and $\tau$ (empty if $\omega_0\neq\tau_0$; we set the length of the empty word to be zero). For $\omega,\tau\in \Omega$, let $\pwt(\l):=\Pi_\l(\omega)-\Pi_\l(\tau)$. We will need the following statement. \[BlackBox\] Let $\eta$ be a compactly supported Borel probability measure on $\R$ of exact local dimension $d_\eta$. Suppose that for any $\varepsilon > 0$ there exists a subset $\Omega_\varepsilon \subset \Omega$ such that $\mu(\Omega_\varepsilon) > 1 -\varepsilon$ and the following holds; there exist constants $C_1,C_2,C_3,\alpha,\beta,\gamma>0$ and $k_0\in\Z_+$ such that $$\label{BlackBox0} d_\eta+\dfrac{\gamma}{\beta} > 1 \text{ and } d_\eta > \dfrac{\beta -\gamma}{\alpha},$$ $$\label{BlackBox1} \max_{\l\in J} |\pwt(\l)| \leq C_1 m^{-\alpha|\omega\wedge\tau|} \text{ for all }\omega,\tau\in\Omega_\varepsilon, \omega\neq\tau,$$ $$\label{BlackBox2} \sup_{v\in\R} Leb(\lbrace \l\in J: |v+\pwt(\l)|\leq r\rbrace )\leq C_2 m^{|\omega\wedge\tau|\beta}r$$ for all $\omega,\tau\in\Omega_\varepsilon, \omega\neq\tau$ such that $|\omega\wedge\tau| \geq k_0$, and $$\label{BlackBox3} \max_{u\in \mathcal{A}^{n}, [u]\cap \Omega_\varepsilon \neq \emptyset} \mu([u])\leq C_3m^{-\gamma n} \text{ for all } n\geq 1.$$ Then the convolution $\eta\ast\nu_\l$ is absolutely continuous with respect to the Lebesgue measure for a.e. $\l\in J$. In fact, in Proposition \[BlackBox\] the condition on exact dimensionality of the measure $\eta$ can be replaced by the following condition (and this is the only consequence of exact dimensionality of $\eta$ that was used in the proof of Proposition \[BlackBox\] in [@DGS]): $\eta$ is a compactly supported Borel probability measure on the real line, such that $$\label{e.exdim} \eta[B_r(x)] \leq Cr^{d_\eta}, \text{ for all }x\in\R \text{ and } r>0.$$ Proofs {#sec:main} ====== Here we construct the measure $\eta$ supported on $K$ and a family of measures $\nu_\lambda$ with $supp\, \nu_\l=C_\l$ such that Proposition \[BlackBox\] can be applied. Since absolute continuity of the convolution $\eta * \nu_\l$ implies that $Leb(C_\l+K)>0$, this will prove Theorem \[t.1\]. Let us start with construction of the measure $\eta$. The compact set $K\subset \mathbb{R}$ satisfies the condition (\[e.condition\]), i.e. $\dim_H(K)+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J$. Take any constant $d\in (0, \dim_H(K))$ that is sufficiently close to $\dim_H(K)$ to guarantee that $d+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J$. By Frostman’s Lemma (see, e.g., [@Mattila Theorem 8.8]), there exists a Borel measure $\eta$ supported on $K$ such that (\[e.exdim\]) holds with $d_\eta=d$. Let $\{C_\lambda\}_{\l\in J}$ be a family of dynamically defined Cantor sets generated by contractions $f_{k, \lambda}:[0,1]\to [0,1]$, $k=1, \ldots, m$, given by (\[e.start\]). Define the map $\xi: C_\lambda\to\mathbb{R}$ by $$\xi(x)=\log|f_{k, \lambda}'(f_{k, \lambda}^{-1}(x))|\ \ \text{if}\ \ x\in f_{k, \lambda}([0,1]).$$ Due to [@Man], there is an ergodic Borel probability measure $\mu_\l$ on $C_\l$ (namely, the equilibrium measure for the potential $(\text{dim}_H\,C_{\lambda})\xi(x)$) that satisfies the condition $-h_{\mu_\l}/\mu_\l (\xi) = \dim_H (C_\l)$. This is also a measure on $C_\l$ such that $\dim_H(\mu_\l) = \dim_H (C_\l)$ (i.e. [*the measure of maximal dimension*]{}). Sometimes it is convenient to consider one expanding map $$\Phi_\lambda:\cup_{k=1}^m f_{k, \lambda}([0,1])\mapsto [0,1], \ \text{where}\ \Phi_\lambda(x)=f^{-1}_{k,\lambda}(x)\ \text{ for}\ x\in f_{k, \lambda}([0,1]),$$ instead of the collection of contractions $\{f_{1, \lambda}, \ldots, f_{m,\lambda}\}$. Notice that $\Phi_\lambda(C_\lambda)=C_\lambda$, and the Lyapunov exponent of $\Phi_\lambda$ with respect to the invariant measure $\mu_\lambda$ is equal to $-\mu_\l (\xi)$. We will denote this Lyapunov exponent by $Lyap^u(\mu_\lambda)$. Since $\mu_\l$ is a measure of maximal dimension, we have $$\dim_H(\mu_\l) = \frac{h_{\mu_\l}(\Phi_\l)}{Lyap^u(\mu_\l)}=\dim_H(C_\l).$$ For each $\omega\in\Omega$ let $F_{\l}^n(\omega) = f_{\omega_0,\l}\circ\dots\circ f_{\omega_n,\l}(x)$, for a fixed $x\in [0,1]$. Then we can define the map $\Pi_\lambda : \Omega \to \R$ given by $$\Pi_\lambda (\omega) = \lim_{n\to\infty} F_{\l}^n(\omega),$$ where in fact the limit does not depend on the initial point $x\in [0,1]$. For any $\l_1, \l_2\in J$ the map $h_{\l_1, \l_2}:C_{\l_2}\to C_{\l_1}$ defined by $h_{\l_1, \l_2}=\Pi_{\l_1}\circ \Pi^{-1}_{\l_2}$ is a homeomorphism. It is well known (e.g. see Section 19 in [@KH]) that this homeomorphism must be Hölder continuous. Moreover, due to [@PV] the following statement holds. \[l.help1\] For any $\lambda_0\in J$ and any $\tau\in (0,1)$ there exists a neighborhood $V\subseteq J$, $\lambda_0\in V$, such that for any $\l\in V$ the conjugacy $h_{\l, \l_0}:C_{\l_0}\to C_{\l}$ as well as its inverse $h_{\l_0, \l}:C_{\l}\to C_{\l_0}$ are Hölder continuous with Hölder exponent $\tau$. Define the measure $\mu$ on $\Omega$ by $\mu:=\Pi_{\l_0}^{-1}(\mu_{\l_0})$, and set $$\nu_\lambda:=\Pi_{\l}(\mu)=\Pi_{\l}(\Pi^{-1}_{\l_0}(\mu_{\l_0}))=h_{\l, \l_0}(\mu_{\l_0})=h_{\l, \l_0}(\nu_{\l_0}).$$ If both $h_{\l, \l_0}$ and $h_{\l_0, \l}$ are Hölder continuous with Hölder exponent $\tau$ then $$\tau\dim_HC_{\l_0}=\tau\dim_H\nu_{\l_0}\le \dim_H\nu_\l\le \frac{1}{\tau}\dim_H\nu_{\l_0}=\frac{1}{\tau}\dim_HC_{\l_0}.$$ Since in Lemma \[l.help1\] the value of $\tau$ can be taken arbitrarily close to one, we get the following statement. \[l.help2\] For any $\lambda_0\in J$ there exists a neighborhood $W\subseteq J$, $\lambda_0\in W$, such that for any $\l\in W$ we have $$d_\eta + \dim_H \nu_\l > 1.$$ It is clear that in order to prove Theorem \[t.1\] it is enough to prove that for each $\l_0\in J$ there exists a neighborhood $W, \l_0\in W$, such that the sum $K+C_{\l}$ has positive Lebesgue measure for a.e. $\l$ from $W$. For a given $\l_0\in J$ we can choose positive $\varepsilon,\alpha, \beta$, and $\gamma$ in such a way that $$d_\eta + \dim_H \nu_{\l_0} > 1+\varepsilon,$$ $$\alpha < \frac{Lyap^u(\nu_{\l_0})}{\log m} < \beta,$$ $$\gamma < \frac{h_{\nu_{\l_0}}(\Phi_{\l_0})}{\log m}.$$ If $\alpha, \beta$ are sufficiently close to $\frac{Lyap^u(\nu_{\l_0})}{\log m}$, and $\gamma$ is sufficiently close to $\frac{h_{\nu_{\l_0}}(\Phi_{\l_0})}{\log m}$, then we also have $$d_\eta + \frac{\gamma}{\beta} > 1,$$ which is one of the conditions (\[BlackBox0\]) of Proposition \[BlackBox\], and also $$\frac{\beta}{\alpha} < 1+\frac{\varepsilon}{2}.$$ Decreasing if needed the neighborhood $W$ given by Lemma \[l.help2\] we can guarantee that for all $\l\in W$ the following property holds: $$\frac{h_{\nu_\l}(\Phi_\l)}{Lyap^u(\nu_\l)} -\frac{\gamma}{\alpha} < \frac{\varepsilon}{2}.$$ Therefore $$d_\eta > 1+\varepsilon - \dim_H(\nu_\l) > \frac{\beta}{\alpha} - \frac{\gamma}{\alpha}$$ for $\l\in W$, which implies another part of the condition (\[BlackBox0\]) of Proposition \[BlackBox\], namely, $$d_\eta > \frac{\beta - \gamma}{\alpha}.$$ Finally let us notice that if $W$ is small, then we have $$\alpha < \frac{Lyap^u(\nu_\l)}{\log m} < \beta,$$ $$\gamma < \frac{h_{\nu_\l}(\Phi_\l)}{\log m}$$ for all $\l\in W$. In order to verify the conditions (\[BlackBox1\]), (\[BlackBox2\]), and (\[BlackBox3\]) of Proposition \[BlackBox\], we will try to mimic the proof of Theorem 3.7 from [@DGS]. We will show that for a given small $\varepsilon>0$ there are subsets $\Omega_1$ and $\Omega_2$ in $\Omega$ such that $\mu(\Omega_i)>1-\frac{\varepsilon}{2}$, $i=1,2$, and properties (\[BlackBox1\]) and (\[BlackBox2\]) hold for all $\omega,\tau\in\Omega_1$, and (\[BlackBox3\]) holds for all $\omega,\tau\in\Omega_2$. This will imply that all these conditions hold for all $\omega,\tau\in\Omega_{\varepsilon}=\Omega_1\cap\Omega_2$ with $\mu(\Omega_\varepsilon)>1-\varepsilon$, i.e. justify application of Proposition \[BlackBox\], and therefore prove Theorem \[t.1\]. For $\omega\in \Omega$, $\omega=\omega_0\omega_1\ldots\omega_n\ldots$, set $p(\lambda)=\Pi_\lambda(\omega)$ and $$\label{e.ls} l^{(s)}=\frac{df_{\omega_{s-1}, \lambda}}{dx}(\Phi^s_{\lambda}(p(\lambda))).$$ We will also write $l^{(s)}(\lambda)$ or $l^{(s)}_\omega$ if we need to emphasize the dependence of $l^{(s)}$ on $\lambda$ or $\omega$. Notice that $\{l^{(s)}\}$ is a sequence of multipliers of the contractions along the orbit of point $p(\lambda)$ under the map $\Phi_\lambda$, and if Lyapunov exponent at $p(\lambda)$ exists then $$Lyap^u(p(\lambda))=-\lim_{n\to \infty} \frac{1}{n}\sum_{s=1}^n\log\left|l^{(s)}\right|.$$ \[Egorov\] Given $\epsilon > 0$, there exists a set $\Omega_1\subset \Omega$ with $\mu(\Omega_1) > 1-\frac{\epsilon}{2}$ and $N\in\N$ such that $$\label{e.alphabeta}\alpha\log m < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|<\beta\log m$$ for every $\l\in W$, $n\ge N$, and all $p\in \Pi_\l(\Omega_1)$. Let us start with the first part of the inequality (\[e.alphabeta\]). First we will show that for a fixed $\l\in W$ and a given $\varepsilon'>0$, there exists $\Omega'$ with $\mu (\Omega') > 1-\varepsilon'$ and $N\in \N$ such that $$\alpha\log m+\xi < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|,$$ where $0 < \xi < Lyap^u(\mu_\l) - \alpha\log m$, for all $n\ge N$ and all $p\in \Pi_\l(\Omega')$. By the Birkhoff Ergodic Theorem, $$\begin{aligned} Lyap^u(\mu_\l) & =\\ & = \int \log \| D\Phi_\l (\Pi_\l(\omega)\|\,d\mu(\omega) \\ & = \lim_{n\to\infty} \frac{1}{n}\sum_{s=1}^n \log \| D\Phi_\l(\Phi_\l^s(\Pi_\l(\omega))\| \\ & = \lim_{n\to\infty}-\frac{1}{n}\sum_{s=1}^n \log \left|l_\omega^{(s)}(\l)\right|\end{aligned}$$ for $\mu$-a.e. $\omega\in\Omega$. Thus by Egorov’s theorem, there exists $\Omega' \subset \Omega$ with $\mu(\Omega') > 1-\varepsilon'$ such that the convergence is uniform on $\Omega'$. Thus there exists $N\in \N$ such that $\alpha\log m+\xi < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|$ for all $n\ge N$ and all $p\in \Pi_\l(\Omega')$. Next we will show that $N$ can be chosen uniformly in $\l \in W$. Let $\varepsilon >0$ be given. Consider the family of functions $$L_\omega (\l) = -\log \|D\Phi_\l(\Pi_\l(\omega)\|.$$ We can treat the elements of this family as functions of $\l$ with parameter $\omega$. Then $\lbrace L_\omega (\l)\rbrace_{\omega\in\Omega}$ is an equicontinuous family of functions and there exists $t>0$ such that if $|\l_1 - \l_2 | \leq t$, then $|L_\omega(\l_1) - L_\omega(\l_2)| < \frac{\xi}{100}$ for any $\omega\in\Omega.$ Consider a finite $t$-net $\lbrace y_1,\dots,y_M\rbrace$ in $W$, containing $M=M(W,t)$ points. For each point $y_j$ we can find a set $\Omega^{(j)}\subset \Omega$, $\mu(\Omega^{(j)}) > 1 - \frac{\epsilon}{4M}$, and $N_j\in \N$ such that for every $n \geq N_j$ and every $\omega\in \Omega^{(j)}$, we have $$\frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) = -\frac{1}{n}\sum_{s=1}^{n} \log \left|l_\omega^{(s)}(y_j)\right| > \alpha\log m +\xi.$$ Take $\Omega_1 = \cap_{j=1}^M \Omega^{(j)}$. We have $$\mu (\Omega_1) > 1 - M\frac{\varepsilon}{4M} = 1-\frac{\varepsilon}{4},$$ and for every $\l\in W$ there exists $y_j$ with $|y_j-\l| \leq t$. So for every $\omega\in\Omega_1\subseteq\Omega^{(j)}$ and every $n \ge N =\max \lbrace N_1,\dots,N_M\rbrace$, we have $$\begin{aligned} -\frac{1}{n}\sum_{s=1}^{n} \log \left|l_\omega^{(s)}(\l)\right| & = \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(\l)\\ & \geq \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) - \left|\frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) - \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(\l)\right| \\ & \geq \alpha\log m + \xi -\frac{\xi}{100} \\ & > \alpha\log m +\frac{\xi}{2} \\ & > \alpha\log m,\end{aligned}$$ which proofs the first part of the inequality (\[e.alphabeta\]). The proof of the second part is analogous. This concludes the proof of Lemma \[Egorov\]. Notice that Lemma \[Egorov\] directly implies that for $p\in \Pi_\l(\Omega_1)$ and $n\ge N$ we have $$\label{e.abproduct} m^{-\beta n} < \left|\prod_{s=1}^n l^{(s)}\right| < m^{-\alpha n}.$$ The next statement is a simple partial case of Lemma 3.12 from [@DGS]. \[l.333simple\] There is a constant $C >0$ such that for any word $\omega_0\omega_1\ldots\omega_n\in\mathcal{A}^{n+1}$, any $\lambda\in J$, and any $x,y\in I_0=[0,1]$ the following holds. Set $$p=f_{\omega_0, \lambda}\circ f_{\omega_1, \lambda}\circ\ldots \circ f_{\omega_n, \lambda}(x),$$ and define $\{l^{(s)}\}$ by (\[e.ls\]). Denote $$q=f_{\omega_0, \lambda}\circ f_{\omega_1, \lambda}\circ\ldots \circ f_{\omega_n, \lambda}(y).$$ Then $$\frac{1}{C}\left|\prod_{s=1}^n l^{(s)}\right| \leq \frac{|p-q|}{|x-y|} \leq C\left|\prod_{s=1}^n l^{(s)}\right|.$$ The property (\[BlackBox1\]) for all $\omega, \tau\in \Omega_1$ follows now from (\[e.abproduct\]) and Lemma \[l.333simple\]. In order to check (\[BlackBox2\]) for some $\omega, \tau\in \Omega$ it is enough to show that $$\label{e.ineq} \left|\frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|>C' m^{-\beta |\omega \wedge \tau|}$$ for some uniform (independent of $\omega, \tau\in \Omega$) constant $C'>0$. Let us consider some $\omega, \tau\in \Omega$, and set $n=|\omega \wedge \tau|$. Let us denote $$P_0(\lambda)=\Pi_{\lambda}(\omega), \ Q_0(\lambda)=\Pi_{\lambda}(\tau), \ \$$ and $$\ P_s=\Phi^s_{\lambda}(P_0), \ Q_s=\Phi_{\lambda}^s(Q_0), \ s=0, \ldots, n.$$ Notice that the distance between $P_n$ and $Q_n$ is uniformly bounded away from zero. Indeed, since $n=|\omega \wedge \tau|$, $P_n$ and $Q_n$ belong to different elements of Markov partition of $C_\lambda$. Let us also denote $$k^{(s)}_\lambda=f_{\omega_{s-1}} \ \ \ \text{and}\ \ \ l^{(s)}(\lambda)=\frac{\partial k^{(s)}_\lambda}{\partial x}(P_{s}(\lambda))\ \ \text{for}\ \ s=1, 2, \ldots, n.$$ We have $$\label{e.ineqnext} k^{(s)}_\lambda(x)=k^{(s)}_\lambda(P_s(\lambda))+l^{(s)}\cdot (x-P_s(\lambda)) + O((x-P_s(\lambda))^2)$$ and $$\label{e.ineqnext} \frac{\partial k^{(s)}_\lambda}{\partial x}(x)=l^{(s)}+ O(x-P_s(\lambda)).$$ Notice that $P_0=k^{(1)}_\lambda\circ \ldots \circ k^{(n)}_\lambda(P_n)$, and $Q_0=k^{(1)}_\lambda\circ \ldots \circ k^{(n)}_\lambda(Q_n)$. To prove (\[e.ineq\]) we need to find a bound on $$\begin{aligned} \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)=\frac{d}{d\l}(P_0(\l)-Q_0(\l))=\end{aligned}$$ $$\begin{aligned} & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l)) + \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial P_n}{\partial\l}(\l) \\ & - \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l)) - \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial Q_n(\l)}{\partial\l} \\ & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\left(\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l))-\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l))\right) \\ & + \sum_{i=1}^n \frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right) \\ & + \left(\left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial P_n}{\partial\l}(\l) - \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial Q_n(\l)}{\partial\l}\right)\\ & = S_1 + S_2 + S_3 \end{aligned}$$ Let us estimate $S_1$. We have $$\begin{aligned} S_1 = & \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\left(\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l))-\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l))\right)\\ & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1} l^{(s)}\right)\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}(W_{i}(\l))(P_{i}(\l) - Q_{i}(\l))\end{aligned}$$ where $W_{i}(\l)$ is a point between $P_{i}(\l)$ and $Q_{i}(\l)$. Since we have $$\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}=\frac{\partial c_{\omega_{i-1}}}{\partial\l}+\frac{\partial^2 g_{\omega_{i-1}}(x,\l)}{\partial x\partial\l},$$ the assumption (\[e.c\]) implies that $\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}$ has sign opposite to the sign of $l^{(i)}$. Also, it is easy to see that $P_{i}(\l) - Q_{i}(\l)$ has the same sign as $$\left(\prod_{s=i+1}^n l^{(s)}\right)(P_{n}(\l) - Q_{n}(\l)).$$ Therefore all terms in the sum $S_1$ have the same sign as $$-\left(\prod_{s=1}^n l^{(s)}\right)(P_{n}(\l) - Q_{n}(\l)).$$ Using Lemma \[l.333simple\], assumption (\[e.c\]), and the fact that $|P_{n}(\l) - Q_{n}(\l)|$ is bounded away from zero, this implies that $$\begin{aligned} \label{e.last} |S_1| = & \sum_{i=1}^n \left|\prod_{s=1}^{i-1} l^{(s)}\right|\left|\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}(W_{i}(\l))\right|\left|P_{i}(\l) - Q_{i}(\l)\right|\ge nC^{*}\left|\prod_{s=1}^{n} l^{(s)}\right|\end{aligned}$$ for some constant $C^*>0$. Let us now estimate $S_2$. Let us remind that $k_{\l}^{(s)}(x) = f_{\omega_{s-1}, \l} (x)= c_{\omega_{s-1}}({\l})x+b_{\omega_{s-1}}{(\l)} + g_{\omega_{s-1}}{(x, \l)},$ where the $C^2$-norm of $g_{\omega_{s-1}}{(x,\l)}$ is small. $$\begin{aligned} |S_2| &= \left|\sum_{i=1}^n \frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & \leq \sum_{i=1}^n \left|\frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\right|\cdot\left|\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & \leq \sum_{i=1}^n C\left|\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & = C \sum_{i=1}^n \left|\prod_{s=1}^{i-1} l^{(s)}\right| \left|1-\frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right|\end{aligned}$$ \[l.33\] $$\left|1-\frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right| \leq C^{\prime}\left|\prod_{s=i}^{n} l^{(s)}\right|\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}$$ for some $C^{\prime} > 0$. Note that if $A$ is near 1 and $B$ is much smaller than 1, we have that $$|\log A| < B \text{ implies }|A-1| \leq 2B.$$ Indeed, $$\begin{aligned} |\log A| < B & \Rightarrow e^{-B}-1 < A-1 < e^B-1 \\ & \Rightarrow -B+O(B^2) < A-1 < B+O(B^2)\\ & \Rightarrow |A-1| < 2B\end{aligned}$$ for small $B$. To prove Lemma \[l.33\], we will show that $\left| \log \frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right|$ is small. By the mean value theorem and using Lemma \[l.333simple\] we get $$\begin{aligned} \left| \log \frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right| & = \left|\sum_{s=1}^{i-1} \log \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l)) - \sum_{s=1}^{i-1} \log \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right| \\ & \leq C \sum_{s=1}^{i-1} \left|\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))-\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right| \\ & = C\sum_{s=1}^{i-1} \left| \frac{\partial g_{\omega_{s-1}}}{\partial x}(Q_{s}(\l)) - \frac{\partial g_{\omega_{s-1}}}{\partial x}(P_{s}(\l))\right| \\ & = C\sum_{s=1}^{i-1} \left|\frac{\partial^2 g_{\omega_{s-1}}}{\partial x^2}(V_{s+1})\right|\left|Q_{s}(\l) - P_{s}(\l)\right| \\ & \leq \widetilde{C}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2} \sum_{s=1}^{i-1} \left|\prod_{j=s+1}^{n} l^{(s)}\right| \\ & \leq \widetilde{\widetilde{C}}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2} \left| \prod_{s=i}^n l^{(s)}\right|\end{aligned}$$ since the terms of the last sum are bounded by a geometrical progression. This proves Lemma \[l.33\]. Therefore we have $$\label{e.eqnew1} |S_2| \leq n{C^{\prime\prime}}\left|\prod_{s=1}^n l^{(s)}\right|\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}.$$ Notice that Lemma \[l.33\] implies also that for come constant $\hat{C}>0$ we have $$\label{e.eqnew2} |S_3|\le \hat{C}\left|\prod_{s=1}^n l^{(s)}\right|.$$ Now combining (\[e.last\]), (\[e.eqnew1\]), and (\[e.eqnew2\]) we get $$\begin{aligned} \left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|=|S_1+S_2+S_3|\ge \left(nC^{*}-n{C^{\prime\prime}}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}-\hat{C}\right)\left|\prod_{s=1}^{n} l^{(s)}\right|.\end{aligned}$$ Therefore one can choose smallness of the $C^2$ norms of $\{g_i\}_{i=1, \ldots, m}$ in (\[e.g\]) so that for some $\delta^*>0$ and all large enough values of $n\in \mathbb{N}$ we have $$\begin{aligned} \label{e.estfinal} \left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|\ge n\delta^*\left|\prod_{s=1}^{n} l^{(s)}\right|\end{aligned}$$ for any $\omega, \tau\in \Omega$ with $|\omega\wedge\tau|=n$. In particular, if $\omega, \tau\in \Omega_1$ then (\[e.estfinal\]) together with (\[e.abproduct\]) imply that $$\left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|\ge n\delta^*m^{-\beta n}=n\delta^*m^{-\beta |\omega\wedge\tau|},$$ which implies (\[e.ineq\]) and hence verifies the assumption (\[BlackBox2\]). Finally, the Shannon-McMillan-Breiman Theorem implies that $$-\frac{1}{n}\log \mu([\omega]_n)\to h_{\mu}(\sigma)$$ for $\mu$-a.e. $\omega \in \Omega$. By Egorov’s theorem, there exists a set $\Omega_2\subset \Omega$ with $\mu(\Omega_2) > 1-\varepsilon/2$ such that this convergence is uniform in $\omega\in \Omega_2$. Thus we have $$-\frac{1}{n}\log \mu([\omega]_n)\to h_{\mu}(\sigma) > \gamma\log m$$ uniformly for $\omega\in\Omega_2$. So for $n$ sufficiently large, we have that $$\mu([\omega]_n) < m^{-\gamma n}.$$ Hence if $C>0$ is sufficiently large then for all $n\geq 1$ we have $$\mu([\omega]_n) < Cm^{-\gamma n}.$$ Now let $\Omega_\varepsilon = \Omega_1 \cap \Omega_2$, then $\mu(\Omega_\varepsilon) > 1-\varepsilon$ and all conditions of Proposition \[BlackBox\] hold on $\Omega_\varepsilon$. This concludes the proof of Theorem \[t.1\]. [00]{} B. Bárány, M. Pollicott, K. Simon, Stationary measures for projective transformations: the Blackwell and Fürstenberg measures, *J. Stat. Phys.* **148** (2012), 393–421. G. Brown, W. Moran, Raikov systems and radicals in convolution measure algebras, [*J. London Math. Soc. (2)*]{} [**28**]{} (1983), no. 3, pp. 531–542. G. Brown, M. Keane, W. Moran, C. Pearce, An inequality, with applications to Cantor measures and normal numbers, [*Mathematika*]{} [**35**]{} (1988), no. 1, pp. 87–94. T. Cusick, M. Flahive, The Markoff and Lagrange spectra,[*Mathematical Surveys and Monographs*]{}, [**30**]{}, American Mathematical Society, Providence, RI, 1989. D. Damanik, Dynamical upper bounds for one-dimensional quasicrystals, *J. Math. Anal. Appl.* **303** (2005), 327–341. D. Damanik, M. Embree, A. Gorodetski, Spectral properties of Schrödinger operators arising in the study of quasicrystals, chapter in *Mathematics of Aperiodic Order* (editors Johannes Kellendonk, Daniel Lenz, Jean Savinien), series *Progress in Mathematics, Birkhäeuser*, **309** (2015), 307–370. D. Damanik, A. Gorodetski, Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian, *Nonlinearity* **22** (2009), 123–143. D. Damanik, A. Gorodetski, Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, *Commun. Math. Phys.* **305** (2011), 221–-277. D. Damanik, A. Gorodetski, The density of states measure of the weakly coupled Fibonacci Hamiltonian, *Geom. Funct. Anal.* **22** (2012), 976–-989. D. Damanik, A. Gorodetski, B. Solomyak, Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian, [*Duke Mathematical Journal*]{} **164** (2015), 1603–1640. D. Damanik, A. Gorodetski, W. Yessen, The Fibonacci Hamiltonian, preprint (arXiv:1403.7823). S. Even-Dar Mandel, R. Lifshitz, Electronic energy spectra and wave functions on the square Fibonacci tiling, *Phil. Mag.* **86** (2006), 759–764. S. Even-Dar Mandel, R. Lifshitz, Electronic energy spectra of square and cubic Fibonacci quasicrystals, *Phil. Mag.* **88** (2008), 2261–2273. S. Even-Dar Mandel, R. Lifshitz, Bloch-like electronic wave functions in two-dimensional quasicrystals, preprint (arXiv:0808.3659). I. Garcia, A family of smooth Cantor sets, *Ann. Acad. Sci. Fenn. Math.* **36** (2011), 21–45. M. Hall, On the sum and product of continued fractions, [*Ann. of Math. (2)*]{}, [**48**]{} (1947), pp. 966–993. M. Hochman, P. Shmerkin, Local entropy averages and projections of fractal measures, *Ann. of Math.* **175** (2012), 1001-–1059. B. Honary, C. Moreira, M. Pourbarat, Stable intersections of affine Cantor sets, *Bull. Braz. Math. Soc.* **36** (2005), 363–378. R. Ilan, E. Liberty, S. Even-Dar Mandel, R. Lifshitz, Electrons and phonons on the square Fibonacci tilings, *Ferroelectrics* **305** (2004), 15–19. A. Katok, B. Hasselblatt, *Introduction to the Modern Theory of Dynamical Systems*, Cambridge University Press, 1995. R. Lifshitz, The square Fibonacci tiling, *J. of Alloys and Compounds*, **342** (2002), 186–190. A. Malyshev, Markov and Lagrange spectra (a survey of the literature), [*Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.*]{} , [**67**]{} (1977) pp. 5–38 (in Russian). A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, *Ergodic Theory Dynam. Systems* **1** (1981), 451–-459. P. Mattila, [*Geometry of Sets and Measures in Euclidean Spaces*]{}, Cambridge University Press, Cambridge, 1995. H. McCluskey, A. Manning, Hausdorff dimension for horseshoes, *Ergodic Theory Dynam. Systems* **3** (1983), 251-–260. M. Mei, Spectra of discrete Schrödinger operators with primitive invertible substitution potentials, *J. Math. Phys.* **55** (2014), no. 8, 082701, 22pp. P. Mendes, F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, *Nonlinearity* **7** (1994), 329–343. C. Moreira, Sums of regular Cantor sets, dynamics and applications to number theory, International Conference on Dimension and Dynamics (Miskolc, 1998), *Period. Math. Hungar.* **37** (1998), 55-–63. C. Moreira, J.-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, *Ann. of Math.* **154** (2001), 45–96. F. Nazarov, Y. Peres, P. Shmerkin, Convolutions of Cantor measures without resonance, *Israel J. Math.* **187** (2012), 93–116. J. Neunhäuserer, Properties of some overlapping self-similar and some self-affine measures, *Acta Math. Hungar.* **92** (2001), 143–161. S. Newhouse, Non-density of Axiom A(a) on $S^2$, [*Proc. A.M.S. Symp. Pure Math.*]{}, [**14**]{}, (1970), 191–202. S. Newhouse, Diffeomorphisms with infinitely many sinks. [*Topology*]{} [**13**]{}, (1974), 9–18. S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, [*Publ. Math. I.H.E.S.*]{}, [**50**]{}, (1979), 101–151. S.-M. Ngai, Y. Wang, Self-similar measures associated to IFS with non-uniform contraction ratios, *Asian J. Math.* **9** (2005), 227–244. J. Palis, F. Takens, [*Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations*]{}, Cambridge University Press, 1993. Palis J., Viana M., On the continuity of Hausdorff dimension and limit capacity for horseshoes. [*Lecture Notes in Math., 1331, Springer, Berlin,*]{} 1988. Y. Peres, W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, *Duke Math. J.* **102** (2000), 193–251. Y. Peres, P. Shmerkin, Resonance between Cantor sets, *Ergodic Theory Dynam. Systems* **29** (2009), 201-–221. Y. Peres, B. Solomyak, Self-similar measures and intersections of Cantor sets, *Trans. Amer. Math. Soc.* **350** (1998), 4065–4087. Y. Peres, B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, *Math. Res. Lett.* **3** (1996), 231–239. M. Pollicott, Analyticity of dimensions for hyperbolic surface diffeomorphisms, *Proceedings of the American Mathematical Society* **143** (2015), 3465–3474. M. Pollicott, K. Simon, The Hausdorff dimension of $\lambda$-expansions with deleted digits, *Trans. Amer. Math. Soc.* **347** (1995), 967–983. S. Saeki, On convolution squares of singular measures, *Illinois J. Math.* **24** (1980), 225-–232. A. Sannami, An example of a regular Cantor set whose difference set is a Cantor set with positive measure, [*Hokkaido Math. J.*]{} [**21**]{} (1992), no. 1, 7–24. P. Shmerkin, On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions, [*Geometric and Functional Analysis*]{}, [**24**]{} (2014), 946–958. P. Shmerkin, B. Solomyak, Absolute continuity of self-similar measures, their projections and convolutions, preprint (arXiv:1406.0204). K. Simon, B. Solomyak, Hausdorff dimension for horseshoes in $\R^3$, *Ergodic Theory Dynam. Systems* **19** (1999), 1343–1363. K. Simon, B. Solomyak, M. Urbanski, Invariant measures for parabolic IFS with overlaps and random continued fractions, *Trans. Amer. Math. Soc.* **353** (2001), 5145–5164. B. Solomyak, On the random series $\sum\pm \lambda^n$ (an Erdős problem), *Ann. of Math.* **142** (1995), 611–625. B. Solomyak, On the measure of arithmetic sums of Cantor sets, *Indag. Math.* (*N.S.*) **8** (1997), 133–141. B. Solomyak, Measure and dimension for some fractal families, *Math. Proc. Cambridge Philos. Soc.* **124** (1998), 531–546. N. Wiener, A. Wintner, Fourier-Stieltjes transforms and singular infinite convolutions, *Amer. J. Math.* **60** (1938), 513–522. W. Yessen, Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian, preprint (arXiv:1410.3102). [^1]: A. G.  and S. N. were supported in part by NSF grants DMS–1301515.

ArXiv

--- abstract: 'We derive the nested Bethe Ansatz solution of the fully packed O($n$) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behaviour. In the $n=0$ limit we obtain the exact compact exponents $\gamma=1$ and $\nu=1/2$ for Hamiltonian walks, along with the exact value $\kappa^2 = 3 \sqrt 3 /4$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.' address: - '$^a$Department of Mathematics, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia' - '$^b$Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba, Meguroku, Tokyo 153, Japan' author: - 'M. T. Batchelor$^a$, J. Suzuki$^b$ and C. M. Yung$^a$' date: 1 June 1994 title: Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice --- The configurational statistics of polymer chains have long been modelled by self-avoiding walks. In the low-temperature limit the enumeration of a single self-attracting polymer in dilute solution reduces to that of compact self-avoiding walks. A closely related problem isi that of Hamiltonian walks in which the self-avoiding walk visits each site of a given lattice and thus completely fills the available space. Hamiltonian walks are directly related to the Gibbs-DiMarzio theory for the glass transition of polymer melts[@gkm]. More than thirty years ago now Kasteleyn obtained the exact number of Hamiltonian walks on the Manhattan oriented square lattice[@k]. More recently this work has been significantly extended to yield exactly solved models of polymer melts[@dd]. The critical behaviour of Hamiltonian walks on the Manhattan lattice has also been obtained from the $Q=0$ limit of the $Q$-state Potts model [@d]. In particular this Hamiltonian walk problem has been shown[@dd; @d] to lie in the same universality class as dense self-avoiding walks, which follow from the $n=0$ limit in the low-temperature or densely packed phase of the honeycomb O($n$) model[@n; @dense]. As exact results for Hamiltonian walks are confined to the Manhattan lattice, the behaviour of Hamiltonian walks on non-oriented lattices, and the precise scaling of compact two-dimensional polymers, remains unclear[@opb; @c1; @c2]. The exact value of the (Hamiltonian) geometric exponent $\gamma^H$ was conjectured to be $\gamma^H=\gamma^D$, where $\gamma^D=\frac{19}{16}$ was extracted via the Coulmb gas method for dense self-avoiding walks[@dense; @c1]. However, recent numerical investigations of the collapsed and compact problems are more suggestive of the value $\gamma^H=1$[@c2; @ct; @bbgop]. More recently, Bl[ö]{}te and Nienhuis[@bn2] have argued that a universality class different to dense walks governs the O($n$) model in the zero temperature limit (the fully packed loop model). Based on numerical evidence obtained via finite-size scaling and transfer matrix techniques, along with a graphical mapping at $n=1$, they argued that the model lies in a new universality class characterized by the superposition of a low-temperature O($n$) phase and a solid-on-solid model at a temperature independent of $n$. This model is identical to the Hamiltonian walk problem in the limit $n=0$. In this Letter we present exact results for Hamiltonian walks on the honeycomb lattice from an exact solution of this fully packed loop model. We derive the physical quantities which characterize Hamiltonian walks on the honeycomb lattice. These include a closed form expression for the connectivity, or entropy, and an exact infinite set of geometric scaling dimensions which include a conjectured value by Bl[ö]{}te and Nienhuis[@bn2]. Our results settle the abovementioned controversy in favour of the universal value $\gamma^H=1$. In general the partition function of the O($n$) loop model can be written as $${\cal Z}_{{\rm O}(n)} = \sum t^{{\cal N}-{\cal N}_b} n^{{\cal N}_L},$$ where the sum is over all configurations of closed and nonintersecting loops covering ${\cal N}_b$ bonds of the honeycomb lattice and ${\cal N}$ is the total number of lattice sites (vertices). Here the variable $t$ plays the role of the O($n$) temperature, $n$ is the fugacity of a closed loop and ${\cal N}_L$ is the total number of loops in a given configuration. For the particular choice $t = t_c$, where[@n] $$t_c^2 = 2 \pm \sqrt{2-n},$$ the related vertex model is exactly solvable with a Bethe Ansatz type solution for both periodic[@b1; @bb; @s1] and open[@bs] boundary conditions. This critical line is depicted as a function of $n$ in Fig. 1. Here we extend the exact solution curve along the line $t=0$, where the only nonzero contributions in the partition sum (1) are for configurations in which each lattice site is visited by a loop, i.e. with ${\cal N}={\cal N}_b$[@noteb]. This is the fully packed model recently investigated by Blöte and Nienhuis[@bn2]. We consider a lattice of ${\cal N} = 2 M N$ sites as depicted in Fig. 2, i.e. with periodic boundaries across a finite strip of width $N$. The allowed arrow configurations and the corresponding weights of the related vertex model are shown in Fig. 3. Here the parameter $n = s + s^{-1} = 2 \cos \lambda$. In Fig. 2 we also show a seam to ensure that loops which wrap around the strip pick up the correct weight $n$ in the partition function. The corresponding weights along the seam are also given in Fig. 3[@note1]. We find that the eigenvalues of the row-to-row transfer matrix of the vertex model are given by $$\Lambda = \prod_{\alpha=1}^{r_1} - {\sinh (\theta_{\alpha} - {\rm i} \frac{\lambda}{2}) \over \sinh (\theta_{\alpha} + {\rm i} \frac{\lambda}{2})} \prod_{\mu=1}^{r_2} - {\sinh (\phi_{\mu} + {\rm i} \lambda) \over \sinh \phi_{\mu}} + {\rm e}^{{\rm i} \epsilon} \prod_{\mu=1}^{r_2} - {\sinh (\phi_{\mu} - {\rm i} \lambda) \over \sinh \phi_{\mu}}$$ where the roots $\theta_{\alpha}$ and $\phi_{\mu}$ follow from $${\rm e}^{{\rm i} \epsilon} \left[ - {\sinh (\theta_{\alpha} - {\rm i} \frac{\lambda}{2}) \over \sinh (\theta_{\alpha} + {\rm i} \frac{\lambda}{2})} \right]^N = - \prod_{\mu=1}^{r_2} - {\sinh (\theta_{\alpha} - \phi_{\mu}+{\rm i} \frac{\lambda}{2}) \over \sinh (\theta_{\alpha} - \phi_{\mu} - {\rm i} \frac{\lambda}{2})} \prod_{\beta=1}^{r_1} {\sinh (\theta_{\alpha} - \theta_{\beta}-{\rm i} \lambda) \over \sinh (\theta_{\alpha} - \theta_{\beta}+{\rm i} \lambda)}, \quad \alpha=1,\ldots,r_1.$$ $${\rm e}^{{\rm i} \epsilon} \prod_{\alpha=1}^{r_1} - {\sinh (\phi_{\mu}-\theta_{\alpha} - {\rm i} \frac{\lambda}{2}) \over \sinh (\phi_{\mu}-\theta_{\alpha} + {\rm i} \frac{\lambda}{2})} = - \prod_{\nu=1}^{r_2} {\sinh (\phi_{\mu} - \phi_{\nu}-{\rm i} \lambda) \over \sinh (\phi_{\mu} - \phi_{\nu}+{\rm i} \lambda)}, \quad \mu=1,\ldots,r_2.$$ Here the seam parameter $\epsilon=\lambda$ for the largest sector and $\epsilon=0$ otherwise. Apart from the seam, this exact solution on the honeycomb lattice follows from earlier work by Baxter on the colourings of the hexagonal lattice [@b2]. Baxter derived the Bethe Ansatz solution and evaluated the bulk partition function per site in the region $n\ge2$. The corresponding vertex model was later considered in the region $n<2$ with regard to the polymer melting transition at $n=0$[@si]. More generally, the fully packed loop model can be seen to follow from the honeycomb limit of the solvable square lattice $A_2^{(1)}$ loop model [@wn; @r]. Equivalently, the related vertex model on the honeycomb lattice is obtained in the appropriate limit of the $A_2^{(1)}$ vertex model on the square lattice in the ferromagnetic regime. This latter model is the $su(3)$ vertex model[@su3]. One can verify that the above results follow from the honeycomb limit of the Algebraic Bethe Ansatz solution of the $su(3)$ model[@bvv] with appropriate seam. It should be noted that Reshetikhin[@r] has performed similar calculations to those presented here, although in the absence of the seam, which plays a crucial role in the underlying critical behaviour. Defining the finite-size free energy as $f_N = N^{-1} \log \Lambda_0$, we derive the bulk value to be $$f_\infty = \int_{-\infty}^{\infty} {\sinh^2\! \lambda x \, \sinh (\pi -\lambda)x \over x\, \sinh \pi x \, \sinh 3 \lambda x } dx . \label{fbulk}$$ This result is valid in the region $0 < \lambda \le \pi/2$, where the Bethe Ansatz roots defining the largest eigenvalue $\Lambda_0$ are all real. We note that the most natural system size $N$ is a multiple of 3, for which the largest eigenvalue occurs with $r_1=2N/3$ and $r_2=N/3$ roots. In the limit $\lambda \rightarrow 0$ $f_\infty$ reduces to the known $n=2$ value[@b2; @b1], $$f_\infty = \log \left[ \frac{3 \Gamma^2(1/3)}{4 \pi^2}\right]. \label{flim}$$ There is however, a cusp in the free energy at $\lambda = \pi/2$. For $\lambda > \pi/2$ the largest eigenvalue has roots $\theta_{\alpha}$ shifted by i$\pi/2$. The result for $f_\infty$ is that obtained from (\[fbulk\]) under the interchange $\lambda \leftrightarrow \pi - \lambda$, reflecting a symmetry between the regions $-2 \le n \le 0$ and $0 \le n \le 2$. Thus the value (\[flim\]) holds also at $n = -2$, in agreement with the $t_c \rightarrow 0$ limiting value[@b1]. As our interest here lies primarily in the point $n = 0$ ($\lambda = \pi/2$), we confine our attention to the region $0 \le n \le 2$. At $n = 0$, we find that the above result for $f_\infty$ can be evaluated exactly to give the partition sum per site, $\kappa$, as $$\kappa^2 = 3 \sqrt 3/4,$$ and thus $\kappa = 1.13975 \ldots$ follows as the exact value for the entropy or connective constant of Hamiltonian walks on the honeycomb lattice. This numerical value has been obtained previously via the same route in terms of an infinite sum [@s2]. Our exact result (8) is to be compared with the open self-avoiding walk, for which $\mu^2 = 2 + \sqrt 2$[@n], and so $\mu = 1.84775 \ldots$ It follows that for self-avoiding walks on the honeycomb lattice the entropy loss per step due to compactness, relative to the freedom of open configurations, is exactly given by $$\frac{1}{2} \log \left[ \frac{3 \sqrt 3}{4 (2+\sqrt 2)} \right] = - 0.483161 \ldots$$ The central charge $c$ and scaling dimensions $X_i$ defining the critical behaviour of the model follow from the dominant finite-size corrections to the transfer matrix eigenvalues[@cx]. For the central charge, $$f_N \simeq f_{\infty} + \frac{\pi \zeta c}{6 N^2}.$$ The scaling dimensions are related to the inverse correlation lengths via $$\xi_i^{-1} = \log ( \Lambda_0/\Lambda_i) \simeq 2 \pi \zeta X_i/N.$$ Here $\zeta=\sqrt 3 /2$ is a lattice-dependent scale factor. The derivation of the dominant finite-size corrections via the Bethe Ansatz solution of the vertex model follows that given for the $su(3)$ model in the antiferromagnetic regime[@devega] (see, also [@am]). The derivation is straightforward though tedious and we omit the details. In the absence of the seam, we find that the central charge is $c=2$ with scaling dimensions $X = \Delta^{(+)} + \Delta^{(-)}$, where $$\Delta^{(\pm)} = \frac{1}{8}\, g \, \mbox{\boldmath $n$}^T C\, \mbox{\boldmath $n$} + \frac{1}{8\, g}\, (\mbox{\boldmath $h$}^{\pm})^{T} C^{-1} \mbox{\boldmath $h$}^{\pm} - \frac{1}{4} \mbox{\boldmath $n$}\cdot\mbox{\boldmath $h$}^{\pm},$$ $C$ is the $su(3)$ Cartan matrix and $\mbox{\boldmath $n$}=(n_1,n_2)$ with $n_1$ and $n_2$ related to the number of Bethe Ansatz roots via $r_1 = 2N/3 - n_1$ and $r_2 = N/3 - n_2$[@note2]. The remaining parameters $\mbox{\boldmath $h$}^{\pm} = (h_1^{\pm},h_2^{\pm})$ define the number of holes in the root distribution in the usual way[@devega]. We have further defined the variable $g = 1 - \lambda/\pi$. With the introduction of the seam $\epsilon = \lambda$, we find that the central charge of the fully packed O($n$) model is exactly given by $$c = 2 - 6(1-g)^2/g.$$ This is the identification made by Blöte and Nienhuis[@bn2]. At $n=0$ we have $g=1/2$, and thus $c = -1$. On the other hand, both Hamiltonian walks on the Manhattan lattice[@dd; @d] and dense self-avoiding walks[@dense] lie in a different universality class with $c=-2$. However, as we shall see below, they do share common sets of scaling dimensions and thus critical exponents. This sharing of exponents between the fully packed and densely packed loop models has already been anticipated by Blöte and Nienhuis in their identification of the leading thermal and magnetic exponents[@bn2]. Here we derive an exact infinite set of scaling dimensions. Of most interest is the so-called watermelon correlator, which measures the geometric correlation between $L$ nonintersecting self-avoiding walks tied together at their extremities $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$. It has a critical algebraic decay, $$\langle \phi_L(\mbox{\boldmath $x$}) \phi_L (\mbox{\boldmath $y$}) \rangle_c \sim |\mbox{\boldmath $x$}-\mbox{\boldmath $y$}|^{-2 X_L},$$ where $X_L$ is the scaling dimension of the conformal source operator $\phi_L(\mbox{\boldmath $x$})$[@dense]. As along the line $t=t_c$, these scaling dimensions are associated with the largest eigenvalue in each sector of the transfer matrix. The pertinent scaling dimensions follow from the more general result $$X = \frac{1}{2}\,g\left(n_1^2+n_2^2-n_1 \, n_2 \right) - \frac{(1-g)^2}{2\,g}.$$ The sectors of the transfer matrix are labelled by the Bethe Ansatz roots via $L=n_1+n_2$. The minimum scaling dimension in a given sector are given by $n_1=n_2=k$ for $L=2k$ and $n_1=k-1, n_2=k$ or $n_1=k, n_2=k-1$ for $L=2k-1$. Thus we have the set of geometric scaling dimensions $X_L$ corresponding to the operators $\phi_L$ for the loop model, $$\begin{aligned} X_{2 k-1} &=& \frac{1}{2}\,g \left(k^2-k+1\right) - \frac{(1-g)^2}{2\,g},\\ X_{2 k} &=& \frac{1}{2}\,g\, k^2 - \frac{(1-g)^2}{2\,g},\end{aligned}$$ where $k=1,2,\ldots$ The magnetic scaling dimension is given by $X_{\sigma}=X_1$ which agrees with the identification made in [@bn2]. The eigenvalue related to $X_2$ appears in the $n_d=2$ sector of the loop model, i.e. with two dangling bonds[@bn2]. At $n=0$ this more general set of dimensions reduces to $$\begin{aligned} X_{2 k-1} &= \frac{1}{4} \left(k^2-k\right), \\ X_{2 k} &= \frac{1}{4} \left(k^2-1\right).\end{aligned}$$ In comparison, the scaling dimensions for dense self-avoiding walks are[@dense] $$X^{\rm DSAW}_L = {\mbox{\small $\frac{1}{16}$}}\left(L^2 - 4 \right).$$ Thus we have the relations $$\begin{aligned} X_{2 k-1} &=& X^{\rm DSAW}_{2 k -1} + {\mbox{\small $\frac{3}{16}$}},\\ X_{2 k} &=& X^{\rm DSAW}_{2 k}.\end{aligned}$$ Note that $X_1=X_2=0$ and $X_L > 0$ for $L>2$. Identifying $X_{\epsilon}=X_2$ as in [@dense], then the exponents $\gamma=1$ and $\nu = 1/2$ follow in the usual way[@note3]. These are indeed the exponents to be expected for compact or space filling two-dimensional polymers. The corresponding scaling dimensions for Hamiltonian walks on the Manhattan lattice are as given in (19)[@d]. Exact Bethe Ansatz results on this model indicate that the scaling dimensions $X_{\sigma}=X_{\epsilon}=0$, from which one can also deduce that $\gamma=1$ and $\nu = 1/2$[@bosy]. We also expect these results to hold for Hamiltonian walks on the square lattice. Extending the finite-size scaling analysis of the correlation lengths for self-avoiding walks on the square lattice [@dense] down to the zero-temperature limit $t=0$, we see a clear convergence of the central charge and leading scaling dimension to the values $c=-1$ and $X_1=0$ for even system sizes, with $X_2=0$ exactly. These results are the analog of the present study on the honeycomb lattice where $N = 3k$ is most natural in terms of the Bethe Ansatz solution. Our results indicate that fully packed self-avoiding walks on the honeycomb lattice have the same degree of “solvability" as self-avoiding walks on the honeycomb lattice. The fully packed loop model with open boundaries is also exactly solvable[@yb]. The derivation of the surface critical behaviour of Hamiltonian walks is currently in progress. It is a pleasure to thank A. L. Owczarek, R. J. Baxter, H. W. J. Blöte, B. Nienhuis and K. A. Seaton for helpful discussions and correspondence. This work has been supported by the Australian Research Council. See, e.g., M. Gordon, P. Kapadia and A. Malakis, J. Phys. A [**9**]{}, 751 (1976); J. F. Nagle, P. D. Gujrati and M. Goldstein, J. Phys. Chem. [**74**]{}, 2596 (1984); T. G. Schmalz, G. E. Hite and D. J. Klein, J. Phys. A [**17**]{}, 445 (1984); H. S. Chan and K. A. Dill, Macromolecules [**22**]{}, 4559 (1989) and references therein. P. W. Kasteleyn, Physica [**29**]{}, 1329 (1963). B. Duplantier and F. David, J. Stat. Phys. [**51**]{}, 327 (1988). B. Duplantier, J. Stat. Phys. [**49**]{}, 411 (1987). B. Nienhuis, Phys. Rev. Lett. [**49**]{}, 1062 (1982). B. Duplantier, J. Phys. A [**19**]{}, L1009 (1986); H. Saleur, Phys. Rev. B [**35**]{}, 3657 (1987); B. Duplantier and H. Saleur, Nucl. Phys. B [**290**]{}, 291 (1987). A. L. Owczarek, T. Prellberg and R. Brak, Phys. Rev. Lett. [**70**]{} 951 (1993). B. Duplantier, Phys. Rev. Lett. [**71**]{} 4274 (1993). A. L. Owczarek, T. Prellberg and R. Brak, Phys. Rev. Lett. [**71**]{} 4275 (1993). C. J. Camacho and D. Thirumalai, Phys. Rev. Lett. [**71**]{} 2505 (1993). D. Bennett-Wood, R. Brak, A. J. Guttmann, A. L. Owczarek and T. Prellberg, J. Phys. A [**27**]{}, L1 (1994). H. W. J. Blöte and B. Nienhuis, Phys. Rev. Lett. [**72**]{}, 1372 (1994). R. J. Baxter, J. Phys. A [**19**]{}, 2821 (1986). M. T. Batchelor and H. W. J. Blöte, Phys. Rev. Lett. [**61**]{}, 138 (1988); Phys. Rev. B. [**39**]{}, 2391 (1989). J. Suzuki, J. Phys. Soc. Jpn. [**57**]{}, 2966 (1988). M. T. Batchelor and J. Suzuki, J. Phys. A [**26**]{}, L729 (1993). Exact information can also be obtained along the lines $n=0$ and $n=1$. We are indebted to R. J. Baxter for this remark. There are several ways to define the seam. This particular choice is consistent with the vertex weight gauge factors and the seam used in the corresponding solution of the vertex model along the line $t=t_c$: see, Refs. [@b1; @bb; @s1]. R. J. Baxter, J. Math. Phys. [**11**]{}, 784 (1970). J. Suzuki and T. Izuyama, J. Phys. Soc. Jpn. [**57**]{}, 818 (1988). S. O. Warnaar and B. Nienhuis, J. Phys. A [**26**]{}, 2301 (1993). N. Yu. Reshetikhin, J. Phys. A [**24**]{}, 2387 (1991). I. V. Cherednik, Theor. Math. Phys. [**47**]{}, 225 (1981); O. Babelon, H. J. de Vega and C. M. Viallet, Nucl. Phys. B [**190**]{} 542 (1981). O. Babelon, H. J. de Vega and C. M. Viallet, Nucl. Phys. B [**200**]{} 266 (1982). J. Suzuki, J. Phys. Soc. Jpn. [**57**]{}, 687 (1988). H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. [**56**]{}, 742 (1986); I. Affleck, Phys. Rev. Lett. [**56**]{}, 746 (1986); J. L. Cardy, Nucl. Phys. B [**270**]{}, 186 (1986). H. J. de Vega, J. Phys. A [**21**]{}, L1089 (1988). F. C. Alcaraz and M. J. Martins, J. Phys. A [**23**]{}, L1079 (1990). The parameters $n_1$ and $n_2$ are related to de Vega’s parameters $S_1$ and $S_2$ via the action of the Cartan matrix: $S_1=2n_1-n_2$ and $S_2=2n_2-n_1$. Using $\eta=2 X_{\sigma}$, $1/\nu = 2 - X_{\epsilon}$ and $\gamma = (2-\eta)\nu$. M. T. Batchelor, A. L. Owczarek, K. A. Seaton and C. M. Yung, “Surface critical behaviour of an O($n$) loop model related to two Manhattan lattice walk problems", A.N.U preprint MRR051-94. C. M. Yung and M. T. Batchelor, “Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices", A.N.U preprint MRR042-94.

ArXiv

--- abstract: 'The observation of isolated positive and negative charges, but not isolated magnetic north and south poles, is an old puzzle. Instead, evidence of effective magnetic monopoles has been found in the abstract momentum space. Apart from Hall-related effects, few observable consequences of these abstract monopoles are known. Here, we show that it is possible to manipulate the monopoles by external magnetic fields and probe them by universal conductance fluctuation (UCF) measurements in ferromagnets with strong spin-orbit coupling. The observed fluctuations are not noise, but reproducible quasiperiodic oscillations as a function of magnetisation direction, a novel Berry phase fingerprint of the magnetic monopoles.' author: - 'Kjetil M.D. Hals$^{1}$, Anh Kiet Nguyen$^1$, Xavier Waintal$^2$ and Arne Brataas$^1$' title: Effective Magnetic Monopoles and Universal Conductance Fluctuations --- Quantum states in solids are classified by a crystal momentum vector and a band index. The space spanned by the momentum vectors is known as the momentum space. Each band index defines an energy-band of allowed electronic energy levels in the momentum space. Momentum-space magnetic monopoles arise from energy-band crossings  [@Bohm:book03]. Each band crossing point produces a magnetic monopole with a quantised topological magnetic charge, characterised by a Chern number [@Bohm:book03]. An electric particle traversing a closed curve in momentum space accumulates a geometric phase from the monopole fields [@Berry:prca84; @Sundaram:prb99]. So far, these abstract monopoles have revealed themselves only through Hall-related effects [@Fang:science03; @Nagaosa:RMF10], but we show that they can also be manipulated and probed by UCF measurements. UCFs are observed experimentally as reproducible fluctuations in the conductance in response to an applied external magnetic field $\rm{B}$ [@Lee:prb86]. The fluctuation pattern is known as the magneto-fingerprint of the sample [@Lee:prb86]. Recent experiments on the ferromagnetic semiconductor (Ga, Mn)As report two different $\rm{B}_c$ periods in the conductance fluctuations [@Vila:prl07; @Neumaier:prl07]: a slow, conventional oscillation for high magnetic fields and a much faster oscillation for low fields when the magnetisation rotates. The present work reinterprets these recent experimental results and shows that the fast oscillations are caused by the relocation of momentum-space magnetic monopoles. Rotation of the magnetisation relocates the monopoles, which leads to a geometric phase change of the closed momentum-space curves. The numerical results demonstrate, in good agreement with the experiments, that a geometric phase change is observed with fast UCF oscillations, implying a novel Berry phase fingerprint of the monopoles. The underlying physics of UCFs is quantum interference between different paths across the sample [@Lee:prb86]. Let $\rm{A}_c$ denote the quantum mechanical probability amplitude for propagating along the classical path $\mathbf{x}_c (t)$. The amplitude can be expressed as $\rm{A}_c= \left| \rm{A}_c\right| \exp(i\rm{S}[\mathbf{x}_c(t)]/\hbar)$ in terms of the action $\rm{S}[\mathbf{x} (t)]= \int \rm{dt L }(\mathbf{x},\mathbf{\dot{x}})$, where $\rm{L}(\mathbf{x},\mathbf{\dot{x}})$ is the Lagrangian, $\mathbf{\dot{x}}\equiv \rm{d}\mathbf{x}/\rm{dt}$, and $\left| \rm{A}_c\right|^2$ is the probability to follow the path $\mathbf{x}_c(t)$ [@Rammer:book07]. When an external magnetic field $\mathbf{B}$ is applied, the term $-e\int \rm{dt}\mathbf{\dot{x}} \cdot \mathbf{A}$ should be added to the action [@Rammer:book07], where $e$ is minus the electron charge, and $\mathbf{A}$ is the vector potential corresponding to $\mathbf{B}=\boldsymbol{ \nabla } \times\mathbf{A}$. Let us separate out the magnetic field-dependent phase and rewrite the amplitude as $\rm{A}_c = \tilde{\rm{A} }_c\exp (-i e /\hbar \int \rm{dt}\mathbf{\dot{x}}_c\cdot \mathbf{A} )$. The conductance $\rm{G}$ is proportional to the total probability of propagating across the sample, $\rm{G(B)}\propto \left|\sum_c \rm{A}_c \right|^2$. Reformulating the line integral associated with the vector potential as a surface integral using Green’s theorem, one finds $\rm{G(B)}\propto \sum_{c c^{'}} \tilde{\rm{A}}_c^{*} \tilde{\rm{A}}_{c^{'}} \exp(i 2\pi \Phi_{c c^{'}}(B)/\Phi_0 ) $, where $\Phi_{c c^{'}}(B)$ is the magnetic flux enclosed by the loop formed by the paths $\mathbf{x}_{c}(t)$ and $\mathbf{x}_{c^{'}}(t)$ and $\Phi_0 \equiv h/e$. Changing the magnetic field randomises the phase difference between different pairs of paths, causing the conductance to fluctuate. A typical period $\rm{B}_c$ of these quasiperiodic oscillations is when the dominant paths experience a relative phase shift of $2\pi$. Assuming that typical paths approximately enclose the sample area $\mathcal{A}$, leads to $\rm{B}_c = \Phi_0 / \mathcal{A}$ [@Lee:prb86]. A closed loop in real space also corresponds to a closed loop in momentum space. In systems with either broken inversion or time-reversal symmetry, there is also a phase associated with paths in momentum space [@Sundaram:prb99; @Bohm:book03]. Semiclassically, this Berry phase effect is included in the Lagrangian as $\hbar\mathbf{A}^{(n)}\cdot \mathbf{\dot{k}}$, where $\mathbf{A}^{(n)} (\mathbf{k})=i \left\langle u_n \left| \boldsymbol{ \nabla }_{\mathbf{k}} \right| u_n \right\rangle$ is the Berry connection, $\left| u_n \right\rangle$ is the periodic part of the Bloch function, and $n$ is the band index [@Sundaram:prb99]. The propagation amplitudes accumulate a geometric phase factor $\exp(i\int \rm{d}\mathbf{k}\cdot \mathbf{A}^{(n)}(\mathbf{k}))$ along a path in momentum space. A closed momentum-space curve acquires a phase equal to the flux of the effective field $\mathbf{\Omega}^{(n)} (\mathbf{k})= \boldsymbol{ \nabla }_{\mathbf{k}}\times \mathbf{A}^{(n)}(\mathbf{k})$ that the loop encloses [@Sundaram:prb99; @Bohm:book03]. The effective field is known as the Berry curvature [@Berry:prca84; @Bohm:book03]: $$\begin{aligned} \mathbf{\Omega}^{(n)} (\mathbf{k}) & = & i\sum_{m\neq n} \frac{ \left\langle u_n \left| \boldsymbol{ \nabla }_{\mathbf{k}} H \right| u_m \right\rangle \times \left\langle u_m \left| \boldsymbol{ \nabla }_{\mathbf{k}} H \right| u_n \right\rangle }{ \left( \rm{E}_{n}(\mathbf{k}) - \rm{E}_{m}(\mathbf{k}) \right) ^2 }, \label{BerryCurvature} \end{aligned}$$ where $H$ is the Hamiltonian of the system and $\rm{E}_n(\mathbf{k})$ is the dispersion relation of the $n$th band. Momentum-space magnetic monopoles are singularities in the Berry curvature where energy bands cross at isolated points [@Fang:science03; @Nagaosa:RMF10]. In ferromagnets with strong spin-orbit coupling, the Hamiltonian is not invariant under rotation of the magnetisation [@Jungwirth:RMP06]. Changing the magnetisation direction relocates the magnetic monopoles, inducing a geometric phase change in the propagation amplitudes. The external magnetic field can rotate the magnetisation in UCF experiments on ferromagnets. The phase change of a closed real-space curve then also acquires important contributions from the geometric phase change of the corresponding closed momentum-space curve. We demonstrate that the magnetic monopoles give rise to fast conductance oscillations at low magnetic fields. This novel and large magnetic monopole effect is qualitatively different from the studies of Berry phase effects in two-dimensional electron gases with Rashba spin-orbit coupling since these systems exhibit no effective momentum-space monopoles [@Engel:prb00]. Also, the effect we compute quantitively differs from the weak peak splitting effects seen therein by 1 order of magnitude. In the following discussion, the Berry phase effect on UCFs will be investigated for the ferromagnetic semiconductor (Ga, Mn)As. The system is modeled by the Hamiltonian [@Jungwirth:RMP06] $$H = (\gamma_1 + \frac{5}{2} \gamma_2) \frac{\mathbf{P}^2}{2 m_e} - \frac{\gamma_2}{m_e} (\mathbf{P} \cdot \mathbf{J})^2 + \mathbf{h} \cdot \mathbf{J} + V(\mathbf{r}). \label{Hamiltonian}$$ The band structure of the host compound is described by the two first terms in Eq. , characterised by the Luttinger parameters $\gamma_1$ and $\gamma_2$. $\mathbf{J}$ is a vector of $4 \! \times \! 4$ spin matrices for 3/2 spins, and $\mathbf{P}= \mathbf{p} - e \mathbf{A}$ is the canonical momentum operator in the presence of an external magnetic field $\mathbf{B}=\boldsymbol{ \nabla } \times\mathbf{A}$. $m_e$ denotes the electron mass. The third term describes the exchange interaction between the holes and the local magnetic moments, modeled by a homogenous exchange field $\mathbf{h}$. To model disorder, we used the impurity potential $V(\mathbf{r}) = \sum_{i} V_i \delta(\mathbf{r}-\mathbf{R}_i)$, where $V_i$ and $\mathbf{R}_i$ are the strength and position of impurity number $i$ and $\delta(\mathbf{r})$ is the delta function. The magnetocrystalline anisotropy in (Ga, Mn)As is complicated and depends on several material parameters such as doping, strain and shape: see Ref. [@Jungwirth:RMP06] and references therein. We consider two cases: 1) a perpendicular easy magnetisation axis that is valid, for example, for (Ga, Mn)As grown on (Ga, In)As and 2) a uniaxial in-plane easy magnetisation axis that is valid for (Ga, Mn)As bars grown on a GaAs substrate [@Jungwirth:RMP06]. Here, the magnetisation and hence $\mathbf{h}$ are assumed to be governed by the following magnetic free energy $\varepsilon = K_u \sin^2(\phi_M) - M B \cos(\phi_M - \phi_B)$, where $\phi_M$ ($\phi_B$) is the angle between the exchange field $\mathbf{h}$ (applied magnetic field $\mathbf{B}$) and the current. For the numerical UCF calculation, we considered a discrete rectangular conductor sandwiched between two clean reservoirs with rectangular cross sections defined by $L_x = 30~nm$ and $L_z = 14~nm$. The spacing between the lattice points is $a_x = a_y = a_z = 1~nm$, significantly smaller than the typical Fermi wavelengths used at $\lambda_F \sim 5~nm$. We assumed one impurity at each lattice site in the conductor. The current direction is $[010]$, and the crystal growth direction and applied magnetic field are along $[001]$. We used the Landau gauge $\mathbf{A} = B x \hat{y}$. For direct comparisons with experimental findings, we used parameters appropriate for (Ga, Mn)As: $\gamma_1 = 7.0$, $\gamma_2 = 2.5$, Fermi energy $E_F = 78~meV$ and $|\mathbf{h}| = 31~meV$. The impurity strengths $V_i$ are uniformly distributed between $-V_0/2$ and $V_0/2$. $V_0 = 0.75~eV$, which leads to a mean free path of $\sim 6~nm$. We assume that $M = 2 \times 10^{4}~A/m$ and the uniaxial anisotropy constant $K_u = 5 \times 10^{3}~J/m^3$, giving the anisotropy field $B_u=2 K_u / M = 0.5~T$, which is similar to the experimental value found in Ref. [@Vila:prl07]. The Landauer-B[" u]{}tikker formula is used to calculate the conductance from a stable transfer matrix method [@Usuki:prb95]. More details about the numerical calculation method can be found in Ref. [@Nguyen:prl08]. ![ (**a**) Dispersion curves along the $[001]$ axis when the exchange field $\mathbf{h}$ is pointing along $[001]$. Each band crossing point $\pm \mathbf{k}_{1,2}$ gives rise to a momentum-space monopole. The black dotted line is the Fermi level used in the numerical UCF simulation. (**b**) Energy surfaces in the $k_x k_y$ plane near the $\mathbf{k}_1$ band crossing point in figure (a). (**c**) Energy surfaces in the $k_x k_y$ plane near the $\mathbf{k}_2$ band crossing point in figure (a).[]{data-label="Fig1"}](Fig1.pdf) Let us first analyse and classify the monopoles and then use simple semiclassical considerations to estimate the Berry phase-induced UCF oscillation period. The geometric phase-induced conductance oscillations appear for weak external magnetic fields, and we can therefore neglect the real-space magnetic field in the analysis of the Berry curvature. Without magnetic fields and disorder, the Hamiltonian in Eq.  has four band crossing points located at $\pm | \mathbf{k}_1 | = \pm \sqrt{ ( \left| \mathbf{h} \right| m_e ) / ( 2\gamma_2 \hbar^2 )}$ and $\pm | \mathbf{k}_2 | = \pm \sqrt{ ( \left| \mathbf{h} \right| m_e ) / (\gamma_2 \hbar^2 )}$ along the momentum-space axis parallel to the exchange field, as shown in Fig. \[Fig1\]a. Each crossing point gives rise to a magnetic monopole. The eigenfunctions of the Hamiltonian are of the form $\exp(i\mathbf{k\cdot r})\, \boldsymbol{\chi}_{n,\mathbf{k}}$ where $\boldsymbol{\chi}_{n,\mathbf{k}}$ is a four-component spinor. Because the helicity operator $\hat{\Sigma} \equiv \mathbf{k}\cdot \mathbf{J} / \left| \mathbf{k} \right|$ commutes with the Hamiltonian when $\mathbf{k}$ is parallel to $\mathbf{h}$, the eigenspinors along this axis in momentum space are $\boldsymbol{\chi}_{m_z, \mathbf{k || h }} = \hat{U}(\theta,\phi) \left| m_z \right\rangle$ where $\left| m_z \right\rangle$ are eigenvectors of $\hat{J}_z$ and $\hat{U}(\theta,\phi)$ is the unitary rotation operator that rotates the quantisation axis parallel to $\mathbf{h}$. At the point $\mathbf{k}_1$, $\boldsymbol{\chi}_{3/2}$ and $\boldsymbol{\chi}_{1/2}$ are degenerate eigenspinors. Close to this point, the Hamiltonian couples these two states only weakly to the $\boldsymbol{\chi}_{-3/2}$ and $\boldsymbol{\chi}_{-1/2}$ states, and it can therefore be written as a $2\times 2$ matrix in the basis of $ \left( 1 \ 0 \right)^T\equiv \boldsymbol{\chi}_{1/2} $ and $ \left( 0 \ 1 \right)^T \equiv \boldsymbol{\chi}_{3/2} $. Expanding the Hamiltonian around the degenerate point, $H= H(\mathbf{k}_1) + \delta \mathbf{k}\cdot \boldsymbol{\nabla}_{\mathbf{k}_1} H$ ($\delta \mathbf{k} \equiv \mathbf{k} - \mathbf{k}_1$), treating the last term as a perturbation and considering the case when $\mathbf{h} = \left| \mathbf{h} \right| \hat{\mathbf{z}}$, we obtain the local $2\times 2$ Hamiltonian: $$\rm{H} = \rm{E_0} \left( \delta \mathbf{k} \right)\hat{I} + \frac{1}{2}\mathbf{x}\cdot\boldsymbol{\sigma}, \label{LocalHamiltonian}$$ where $\rm{E_0} \left( \delta \mathbf{k} \right)$ is an energy shift of the two energy bands away from the source point, $\boldsymbol{\sigma}$ is a vector of Pauli matrices, $\rm{\hat{I}}$ is the identity matrix, and $\mathbf{x}\equiv 2\gamma_2 \hbar^2 \left| \mathbf{k}_1 \right| / m_e \left( -\sqrt{3} \delta k_x,\: \sqrt{3} \delta k_y, \: 2\delta k_z \right)$. When the crystal momentum $\mathbf{k}$ varies in time, the effective Hamiltonian in Eq.  describes a spin in a time-varying magnetic field and the electron accumulates a well-known geometric phase from the Berry curvature field [@Berry:prca84] $$\boldsymbol{\Omega}^{( \pm )} \left( \mathbf{k} \right) = \mp \frac{\mathbf{k} }{ 2 \left| \mathbf{k} \right|^3} , \label{}$$ where $\mp$ refer to the upper and lower energy bands near $\mathbf{k}_1$. We have here reparameterised the momentum space as $\mathbf{k}\mapsto \mathbf{k}= \left( -x_1,\: x_2,\: x_3 \right)$ for clarity. The topological magnetic charge of this monopole, its Chern number, is $1/2\pi \int_{S^2} \boldsymbol{\Omega}^{( \pm )}\cdot\mathbf{\hat{n}}\, \rm{dS} = \mp 1$ [@Bohm:book03]. Because the Berry curvature is inversely proportional to $\left( E_{n}(\mathbf{k}) - E_{m}(\mathbf{k}) \right)^2$, bands that nearly cross over a larger region in momentum space produce a stronger monopole. The structures of the energy bands in Figs. \[Fig1\]b-c show that the $\mathbf{k}_2$ monopole is stronger than the $\mathbf{k}_1$ monopole. A similar simple perturbative analysis of the $\mathbf{k}_2$ monopole cannot be carried out, but numerically, we find that the curvature decays asymptotically as $\mathbf{k}^{-2}$, and have a Chern number of $\pm 2$. As can be seen from Fig. \[Fig1\]a, the Berry curvature field from the two $\pm \mathbf{k}_1$ monopoles is experienced by orbits in the third and fourth bands, whereas the $\pm \mathbf{k}_2$ monopoles give a geometric phase effect only to momentum-space curves in the second and third bands. In the third band, the $\pm \mathbf{k}_1$ and $\pm \mathbf{k}_2$ monopoles have topological charges of opposite signs and therefore counteract each other. Orbits in the lowest band do not experience any monopole field. Therefore, paths on the Fermi surface of the second band, which are experiencing the strong $\pm \mathbf{k}_2$ monopoles, dominate the Berry phase-induced conductance fluctuations. ![(**a**) The second-band Fermi surface and the Berry curvature field (black arrows) on this surface when the exchange field points along $[001]$. A typical path is shown in (a) as a yellow curve where $k_z \sim \pi/ L_z$. The curve accumulates a geometric phase of $\sim 2\pi$. (**b**) The second-band Fermi surface and the Berry curvature field (black arrows) on this surface when the exchange field is rotated by an angle of $0.13$. The yellow curve represents the same real-space curve as in (a). It accumulates a vanishing geometric phase. In both plots, the red dot is the $\mathbf{k}_2$ monopole source. (**c**) The geometric phase change of the curve as a function of the rotation angle of the exchange field. The geometric phase angle and rotation angle are in units of, respectively, $2\pi$ and $\pi / 20$ rad.[]{data-label="Fig2"}](Fig2.pdf) A typical closed momentum-space curve on the second-band Fermi surface is shown in Fig. \[Fig2\]a. The corresponding real-space curve is found from the semiclassical equation $\hbar\mathbf{\dot{r}}=\boldsymbol{\nabla}_{\mathbf{k}}\rm{E}_2(\mathbf{k})|_{\rm{E_2=E_F}}$. Because the Fermi surface is not rotationally symmetric, the momentum-space curve corresponding to this fixed real-space curve changes location on the Fermi surface when $\mathbf{h}$ is rotated, as illustrated in Fig. \[Fig2\]b. The associated geometric phase change of the relocated momentum-space curve, calculated numerically, is shown in Fig. \[Fig2\]c. We found that a rotation angle on the order of $0.13$ changes the Berry phase of this fixed real-space curve by $2\pi$. The origin of the rapid phase change occurs when the curve in Fig. \[Fig2\]a encloses the strong Berry curvature field region on the Fermi surface, shown as black arrows in Fig. \[Fig2\]a-b, whereas the curve in Fig. \[Fig2\]b is relocated outside this region. The magnetic field needed to rotate the magnetisation by $0.13$ rad is therefore an estimate of the oscillation period of the Berry phase fingerprint. Using the free energy defined above, this leads to the oscillation period $B_c^{\rm{Berry}} \sim B_u /10$, where $B_u$ is the minimal external magnetic field needed to align the magnetisation along the hard magnetic anisotropy axis. ![The conductance versus the applied magnetic field for the perpendicular magnetisation easy axis (black) and in-plane easy axis (red) for two different impurity configurations. The upper curve is shifted $1.5 e^2/h$ upwards for clarity. The grey region marks the region where the sample’s magnetisation changes from in-plane to perpendicular. The curves for the perpendicular and in-plane easy axes are coincident for $B > B_u = 0.5T $ because the same impurity configuration is used. Inset: a close-up for small magnetic fields.[]{data-label="fig:Ghy"}](UCF_simulation.pdf) Let us next confirm our semiclassical analysis by a numerical UCF simulation of the system including disorder and magnetic field in Eq. . We consider two cases: 1) a perpendicular easy magnetisation axis that is valid, for example, for (Ga, Mn)As grown on (Ga, In)As and 2) a uniaxial in-plane easy magnetisation axis that is valid for (Ga, Mn)As bars grown on a GaAs substrate [@Jungwirth:RMP06]. The external magnetic field is applied along the $[ 0 0 1]$ growth direction. First, consider the case of a perpendicular magnetisation easy axis where the magnetisation is aligned along the growth direction. We see in Fig. \[fig:Ghy\] that the conductance has a weak increasing trend for increasing $B$. Here, the magnetic field squeezes the spatial extension of the wave function [@Datta:book95], allowing more conducting channels to be open for increasing $B$. Imposed on the increasing trend, there are strong conductance fluctuations with a dominant period $B_c \sim 1T$. Because the magnetisation here is always along $[0 0 1]$, the only change in the quantal phases comes from the magnetic flux. For a wire of $\mathcal{A} = 30 \times 100nm^2$ $xy$ area, the dominant fluctuation period is $\rm{B}_c = \Phi_0 / \mathcal{A}\sim 1T$ [@Lee:prb86], consistent with the data shown in Fig. \[fig:Ghy\]. Second, consider the case of an in-plane easy axis where $\mathbf{h}$ rotates from the $[0 1 0]$ direction to the hard $[ 0 0 1]$ axis when the magnetic field increases from $0$ to the anisotropy field $B_u = 0.5T$. The decreasing trend of the conductance for increasing $B \in [0,B_u]$, shown in Fig. \[fig:Ghy\], is the standard anisotropic magnetoresistance effect [@Nguyen:prl08]. Imposed on the decreasing trend, the conductance fluctuates wildly for $B < B_u$ with a period on the order of $B_u /10$. Here, changing $B$ leads to changes in the direction of $\mathbf{h}$, which [*relocates the position of the momentum-space magnetic monopoles and thereby the geometric phase for a given real-space orbit*]{}. This gives rise to the extraordinarily fast conductance fluctuations shown in Fig. \[fig:Ghy\]. For $B > B_u$, the Berry phase is fixed and the UCF again exclusively comes from the conventional magnetic flux. Similar to what is found for the intrinsic anomalous Hall effect [@Fang:science03; @Nagaosa:RMF10], the effect is strongest for Fermi energies near the monopole sources. We expect the effect to also be present for more highly doped (Ga, Mn)As systems that require a six- or eight-band model in which more monopoles are expected to exist. In conclusion, the UCF simulation in Fig. \[fig:Ghy\] semiquantitatively reproduces the experiments in Refs. [@Vila:prl07; @Neumaier:prl07], and together with our semiclassical analysis, it reinterprets the fast oscillations as a Berry phase fingerprint. This work was supported by computing time through the Notur project. [99]{} A. Bohm [*et al.*]{}, *The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics.* (Springer-Verlag, Berlin,2003). M. V. Berry, Proc. R. Soc. A [**392**]{}, 45-57 (1984). G. Sundaram and Q. Niu, Phys. Rev. B [**59**]{}, 14915 (1999). Z. Fang [*et al.*]{}, Science [**302**]{}, 92-95 (2003). N. Nagaosa, J. Sinova, S. Onoda, and A. H. MacDonald, N. P. Ong, Rev. Mod. Phys. [**82**]{}, 1539 (2010). P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B [**35**]{}, 1039 (1987). L. Vila [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 027204 (2007). D. Neumaier [*et al.*]{}, Phys. Rev. Lett. [**99**]{}, 116803 (2007). J. Rammer, *Quantum Field Theory of Non-equilibrium States.* (Cambridge University Press, 2007). T. Jungwirth, J. Sinova, J. Mašek, J. Kučera, and A. H. MacDonald, Rev. Mod. Phys. **78**, 809864 (2006). D. Loss, H. Schoeller, P. M. Goldbart, Phys. Rev. B [**48**]{}, 15218, (1993); A. F. Morpurgo, J. P. Heida, T. M. Klapwijk, B. J. van Wees, and G. Borghs , Phys. Rev. Lett. [**80**]{}, 1050 (1998); A. G. Mal’shukov, V. V. Shlyapin, and K. A. Chao, Phys. Rev. B [**60**]{}, R2161 (1999); H. A. Engel and D. Loss, Phys. Rev. B [**62**]{}, 10238 (2000). T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N. Yokoyama, Phys. Rev. B **52**, 8244 (1995). A. K. Nguyen and A. Brataas, Phys. Rev. Lett. [**101**]{}, 016801 (2008). S. Datta, *Electronic Transport in Mesoscopic Systems.* (Cambridge University Press, Cambridge U.K., 1995).

ArXiv

--- abstract: 'We show that, by studying the arrival times of radio pulses from highly-magnetized transient beamed sources, it may be possible to detect light pseudo-scalar particles, such as axions and axion-like particles, whose existence could have considerable implications for the strong-CP problem of QCD as well as the dark matter problem in cosmology. Specifically, such light bosons may be detected with a much greater sensitivity, over a broad particle mass range, than is currently achievable by terrestrial experiments, and using indirect astrophysical considerations. The observable effect was discussed in Chelouche & Guendelman (2009), and is akin to the Stern-Gerlach experiment: the splitting of a photon beam naturally arises when finite coupling exists between the electro-magnetic field and the axion field. The splitting angle of the light beams linearly depends on the photon wavelength, the size of the magnetized region, and the magnetic field gradient in the transverse direction to the propagation direction of the photons. If radio emission in radio-loud magnetars is beamed and originates in regions with strong magnetic field gradients, then splitting of individual pulses may be detectable. We quantify the effect for a simplified model for magnetars, and search for radio beam splitting in the 2GHz radio light curves of the radio loud magnetar XTEJ1810-197.' address: - 'Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel; [email protected]' - 'Department of Physics, University of Haifa, Haifa 31905, Israel ; [email protected]' author: - 'Eduardo I. Guendelman' - Doron Chelouche title: 'Radio-loud Magnetars as Detectors for Axions and Axion-like Particles' --- Introduction ============ The Peccei-Quinn mechanism (Peccei & Quinn 1977) was devised to elegantly solve to the strong-CP problem of quantum chromodynamics (QCD). This was accomplished by postulating a new quantum field and a new class of particles associated with it. The particles are pseudo-scalars that couple very weakly to the electromagnetic (EM) field. It later became apparent that such particles, termed axions, could also provide a solution to the dark matter problem (Khlopov 1999 and references therein). Besides QCD axions there are also the putative axion-like particles (ALPs), which may be related to the quintessence field, and whose existence is predicted by many versions of string theory. To date, however, there is no evidence for the existence of such particles and it is not clear that the Peccei-Quinn solution actually works. There is a longstanding interest in determining the physical properties of axions/ALPs. At present, laboratory experiments and astrophysical bounds imply that their coupling constant to the electromagnetic field, $g<10^{-10}\,{\rm GeV}^{-1}$ (see Chelouche et al. 2009 for summary). Mass limits are less stringent: if QCD axions are concerned, then their mass is probably $>10^{-6}$eV since otherwise the Universe would over-close, in contrast to observations. These limits, however, do not apply for ALPs. Here we follow the formalism given in Chelouche & Guendelman (2009; see also Guendelman 2008a,b,c) who outlined a new effect that arises from the coupling between the electromagnetic field and the axion field. The effect has the advantage of having unique observational signatures, which can be visible down to very small values of the (unknown) coupling constant compared to those accessible by other methods. Below, we outline the effect of beam splitting and look for it in the radio light curve of a radio-loud magnetar. Splitting in in-homogenous magnetic fields ========================================== The interaction term in the Lagrangian for the electromagnetic and the axion field is of the form $$\displaystyle L_{\rm int} = \frac{1}{4} g \tilde{F}^{\mu \nu}F_{\mu \nu}a = g{\bf E}\cdot {\bf B} a$$ where $E$ is the electric field (associated with the photon), $B$ the magnetic field, and $a$ the axion field. $g$ is the unknown coupling of particles to the EM field ($F_{\mu \nu}$, and its dual, $\tilde{F}^{\mu \nu}$). The full Lagrangian for the system can be written as $$L= -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} -\frac{1}{2}m_\gamma^2A^2 + \frac{1}{2} \partial_\mu a \partial ^\mu a -\frac{1}{2} m_a^2 a^2 +L_{\rm int}$$ which is comprised of the free EM Lagrangian (including an effective photon mass, $m_\gamma$, term which takes into account potential refractive index in the medium) and the Klein-Gordon equations for free particles having a rest-mass $m_a$. In the absence of $L_{\rm int}$, photons and particles (e.g., axions/ALPs) are well defined energy states of the system. However, for finite coupling, the equation of motion for the photon-particle system takes the form (Raffelt & Stodolsky 1988)[^1], $$\left [ {\bf k}^2 -\omega^2+\left \vert \begin{array}{cc} m_\gamma^2 & -gB_\| \omega \\ -gB_\| \omega & m_a^2 \end{array} \right \vert \right ] \left ( \begin{array}{l} \gamma \\ a \end{array} \right )=0, \label{mat}$$ where $\omega$ is the photon energy and $B_\|$ the magnetic field in the direction of the photon polarization (the photon’s $E$ field). Clearly, neither pure photon nor pure axion/ALP states are eigenstates of the system but rather some combination of them. Let us now focus on the limit $$\vert m_a^2-m_\gamma(\omega)^2 \vert \ll gB_\| \omega. \label{cond2}$$ This condition is met either near resonance where $m_\gamma^2 \simeq m_a^2$ or when both masses are individually smaller than $\sqrt{gB_\| \omega}$ (which limit is actually met is immaterial). The eigenstates of equation \[mat\] are then given by $$\left \vert \psi \right >_- = \left [ \left \vert \gamma \right > + \left \vert a \right > \right ]/\sqrt{2},~~\left \vert \psi \right >_+ = \left [ \left \vert \gamma \right > - \left \vert a \right > \right ]/\sqrt{2} \label{psi}$$ where $\left \vert a \right >$ is the axion state and $\left \vert \gamma \right >$ is the photon state. The eigenvalues are $m_\pm^2= \pm gB_\| \omega$. By analogy with optics, these masses are related to effective refractive indices: $n_\pm=1+\delta n_\pm\simeq 1-{m_\pm^2}/{2\omega^2}$ (for $\vert \delta n_\pm\vert \ll1$) meaning that different paths through a refractive medium would be taken by the rays. We note that there is no dependence on the particle or photon mass so long as equation \[cond2\] is satisfied. In terms of the refractive index, and in complete analogy to mechanics, the equation of motion for a ray may be found by minimizing the action $\int d{\bf s} n({\bf s})$. It is straightforward to show (Chelouche & Guendelman 2009) that the momentum imparted on each state is $$\delta p_y^\pm = \mp (g/2) \int dz \left ( \partial B_x/\partial y \right ), \label{dpy}$$ where $B_x=B_x(y,z)$, and is taken to be parallel to the photon polarization (Fig. 1). Clearly, each of the beams will be affected in a similar way while gaining opposite momenta so that the total momentum is zero and the classical wave packet travels in a straight line (along the $z$-axis). This effect is analogous to the Stern-Gerlach experiment (Fig. 1). In the limit $n_\pm \simeq 1$, the separation angle between the beams is $$\delta \phi \simeq \frac{2\vert \delta p_y \vert}{p}, \label{theta}$$ where $p$ is the beam momentum along the propagation direction, i.e., the $z$-axis. This expression holds for small splitting angles and assumes relativistic particles. The magnitude of splitting depends on the relative angle between the photon polarization and the magnetic field, as well as on the geometry (and strength) of the magnetic field, which are poorly understood in magnetars. To gain a qualitative understanding of the magnitude of the effect, we assume $B\sim \int dz \left ( \partial B_x/\partial y \right ) $, where $B$ is the magnetic field in region through which the photon propagates. Taking current limits on the value of the coupling constant of $g\sim 10^{-10}\,{\rm GeV}^{-1}$, a photon frequency of 2GHz, and a typical magnetar magnetic field, $B\sim 10^{15}$G of we find that typical splitting phase between the photon beams is $$\delta \phi \sim 1\left ( \frac{g}{10^{-10}\,{\rm GeV}^{-1}} \right ) \left ( \frac{B}{10^{15}\,{\rm G}} \right ) \left ( \frac{\omega}{2\,{\rm GHz}} \right )^{-1}\,{\rm rad}$$ (here $\omega$ is the photon frequency in GHz, and $B$ is the magnetic field in Gauss). Provided that the intrinsic pulse emitted by the magnetar is narrower than the splitting angle, a double pulse is expected to appear due to the effect of splitting. In fact, in cases where the pulses are highly beamed, hence narrow in phase (see below), considerably smaller values of the coupling constant, $g$, may be probed. Furthermore, by choosing to work at lower frequencies, smaller coupling constants may be probed. This allows one to search for ALPs in a previously unexplored phase space using the radio light curves of radio-loud magnetars. Figure 2 shows an example for what a narrow radio pulse, typical of radio-loud magnetars (see below) would look like when split due to photon-particle coupling in a highly magnetized object ($B=10^{16}$G) observed at a radio frequency of 300MHz. Clearly, splitting may be discernible down to very low values of the coupling constant. We note that, depending on the strength of the magnetic field (and to some degree also on the poorly constrained plasma density), and the contribution of the vacuum birefringence term (Adler 1971) to the effective photon mass, two split pulses or one shifted pulse (with the shifting angle being $\delta \phi /2$) will be observed. In the latter case, light curves in two or more bands are required to detect the effect. The full treatment of such issues is beyond the scope of this contribution, and is discussed at some length in Chelouche & Guendelman (2009). Below we consider only the effect of splitting when studying the (monochromatic) light curve of a radio-loud magnetar. The case of XTEJ1810-197 ======================== We adopt a pragmatic approach when searching for the effect of photon-axion coupling-induced splitting in magnetars. Given the large uncertainties in the physics of magnetars and their radio emission mechanisms, we do not know whether the effect of splitting or shifting should be observed. In addition, it is certainly possible that not all radio pulses originate from the same region in the magnetosphere, and while in some cases splitting will be observed, in other cases beam shifting will be the relevant effect. Similarly, if different pulses are emitted from different regions having different polarizations with respect to a complicated, tangled geometry of the field, then many different splitting or shifting angles are predicted. For these reasons, we aim to study the statistics of phase differences between pulses. Should all radio pulses be emitted from the same region, the data will reflect on the typical phase difference between pulses. This phase difference, which is induced by photon-particle coupling, may be discernible from other phase difference scales, which relate to the physics of the radio-emitting regions in the magnetar. We consider the 2GHz radio observations of the radio-loud magnetar XTEJ1810-197 whose data were published by Camilo et al. (2006). A total of 40 object rotations were recorded, with the light curve of a few individual rotations shown in figure \[mag1\]. As discussed in Camilo et al. (2006), the light curves are characterized by narrow transient radio pulses and, as such, are very different than those typically observed toward pulsars. When averaging the light curves of individual rotations, evidence for quasi-periodicity appears, whereby the bulk of the radio emission is confined to certain orbital phases, akin to the better studied pulsar phenomenon. Aiming to statistically study the difference in arrival phases of radio pulses, we first need to positively identify the numerous, potentially weak, narrow transient features in the light curves of XTEJ1810-197. To this end, we devised the following “peak-finder” algorithm: for each light curve (rotation), we define the (initial) standard deviation of the light curve, $\sigma$. Only those peaks that satisfy $(f(t)-\left < f \right >)/\sigma>3$ \[$f(t)$ is the time-series and $\left < f \right >$ is its mean\], are identified as peaks, and are then removed from the observed light curve. A new standard deviation, $\sigma$, is calculated for the reduced light curve, and the peak identification algorithm is executed leading to new significant peaks being identified. The scheme iterates until $\sigma$ between successive iterations converges to better than 0.01%. Figure \[mag1\] shows the results of the peak identification algorithm for a light curve corresponding to a single stellar revolution. Clearly, all peaks lie well above the fluctuating background. The phase stamps of individual peaks are identified with their maxima. In cases were a multi-maxima ridge exists, the time stamps for individual maxima is recorded. We analyze the data from individual revolutions to be less sensitive to non-stationary effects in the light curve (e.g., a varying noise level between stellar revolutions). All peaks from all stellar revolutions were identified and their phase stamps, relative to the first revolution, logged. We then evaluate the phase difference distribution taking into account all peak pairs. The results are shown in figure \[mag2\]. A clear peak is observed, by definition, at around the stellar orbital period ($\delta \phi/2\pi=1$). Two small peaks at $\delta \phi/2\pi \sim 0.3,~0.7$ are due to the secondary pulse at $\phi/2\pi \sim 0.87$ (see the inset of Fig. \[mag2\]). A second significant time-scale is apparent at a phase difference of $\delta \phi/2\pi \lesssim 0.2$. This scale roughly corresponds to the phase width of the mean main pulse (a second mean pulse exists at $\phi/2\pi \sim 0.87$ and is not shown here). Interestingly, we cannot positively identify any particular phase difference scale for $\delta \phi/2\pi <0.2$, as might be expected due to the effect of splitting. In fact, the distribution is qualitatively consistent with the predictions from a purely random origin for the radio pulses (see dotted line in Fig. \[mag2\]). Further analysis is underway. Based on our preliminary analysis, we cannot find supporting evidence for beam splitting in the 2GHz light curve of XTEJ1810-197. There remains the open possibility that, for this object and this particular waveband, we are in the regime of beam shifting, and light curve comparison with [*simultaneous*]{} observations in other wavebands may be able to detect it. Given our limited understanding of magnetars and their radio emission processes, we do not claim to interpret our null result as a limit on the photon-ALP coupling constant, $g$, or on the existence of light bosons. Conclusions =========== We show that the effect of beam splitting due to finite coupling between the axion field and the electromagnetic field (Chelouche & Guendelman 2009) may be observable in the radio-light curves of radio-loud magnetars for a plausible range of values corresponding to the properties of magnetars and photon-to-axion coupling strength. The phase between the split pulses depends linearly on the magnetic field, the photon-particle coupling constant, and on the photon wavelength. As such, this effect can be used to detect axions and ALPs with much greater sensitivity than photon-axion/ALP oscillations. Our predictions indicate that, for narrow (beamed) radio pulses, the phase between the split pulses is likely to be $\lesssim 1$rad at 2GHz. Such a timescale will contribute to the statistics of phase differences between pulses, whose underlying form is determined by radio emission processes in the magnetar itself. Searching for discernible phase difference scales, which can be related to the beam-splitting effects, in the 2GHz light curve of XTEJ1810-197, shows no clear characteristic phase scale in the range $0.1-1$rad. Interestingly, a preliminary analysis shows that the data is qualitatively consistent with narrow pulsed emission being drawn from a random process. While this could be used to shed light on the radio emission mechanism in magnetars, we cannot draw any conclusions at this stage concerning the existence of light bosons or their coupling to the electromagnetic field. We thank Scott Ransom for providing us the 2GHz data for XTEJ1810-197 in electronic form. Adler, S. L. 1971, Annals of Physics, 67, 599 Camilo, F., Ransom, S. M., Halpern, J. P., Reynolds, J., Helfand, D. J., Zimmerman, N., & Sarkissian, J. 2006, Nature, 442, 892 Chelouche, D., Rabad[á]{}n, R., Pavlov, S. S., & Castej[ó]{}n, F. 2009, ApJS, 180, 1 Chelouche, D., & Guendelman, E. I. 2009, ApJL, 699, L5 Guendelman, E. I. 2008, Modern Physics Letters A, 23, 191 Guendelman, E. I. 2008, Physics Letters B, 662, 227 Guendelman, E. I. 2008, Physics Letters B, 662, 445 Khlopov, M. Y. 1999, Cosmoparticle Physics, World Scientific Peccei, R. D., & Quinn, H. R. 1977, Physical Review Letters, 38, 1440 Raffelt, G., & Stodolsky, L. 1988, Phys. Rev. D., 37, 1237 [^1]: Unless otherwise stated, we work in natural units so that $\hbar=c=1$.

ArXiv

--- abstract: 'Based on network analysis of hierarchical structural relations among Chinese characters, we develop an efficient learning strategy of Chinese characters. We regard a more efficient learning method if one learns the same number of useful Chinese characters in less effort or time. We construct a node-weighted network of Chinese characters, where character usage frequencies are used as node weights. Using this hierarchical node-weighted network, we propose a new learning method, the distributed node weight (DNW) strategy, which is based on a new measure of nodes’ importance that takes into account both the weight of the nodes and the hierarchical structure of the network. Chinese character learning strategies, particularly their learning order, are analyzed as dynamical processes over the network. We compare the efficiency of three theoretical learning methods and two commonly used methods from mainstream Chinese textbooks, one for Chinese elementary school students and the other for students learning Chinese as a second language. We find that the DNW method significantly outperforms the others, implying that the efficiency of current learning methods of major textbooks can be greatly improved.' author: - 'Xiaoyong Yan$^{1,2}$, Ying Fan$^{1,3}$, Zengru Di$^{1,3}$, Shlomo Havlin$^{4}$, Jinshan Wu$^{1,3,\dag}$' bibliography: - 'characters.bib' title: Efficient learning strategy of Chinese characters based on network approach --- [**[Introduction]{}**]{}. It is widely accepted that learning Chinese is much more difficult than learning western languages, and the main obstacle is learning to read and write Chinese characters. However, some students who have learned certain amount of Chinese characters and gradually understand the intrinsic coherent structure of the relations between Chinese characters, quite often find out that it is not that hard to learn Chinese [@Bellassen]. Unfortunately, such experiences are only at individual level. Until today there is no textbook that have exploited systematically the intrinsic coherent structures to form a better learning strategy. We explore here such relations between Chinese characters systematically and use this to form an efficient learning strategy. Complex networks theory has been found useful in diverse fields, ranging from social systems, economics to genetics, physiology and climate systems [@Watts; @Strogatz; @Albert; @Newman; @Wu; @Costa; @Fortunato]. An important challenge in studies of complex networks in different disciplines is how network analysis can improve our understanding of function and structure of complex systems [@Costa; @Fortunato; @Chen]. Here we address the question if and how network approach can improve the efficiency of Chinese learning. Differing from western languages such as English, Chinese characters are non-alphabetic but are rather ideographic and orthographical [@Branner]. A straightforward example is the relation among the Chinese characters ‘’, ‘’ and ‘’, representing tree, woods and forest, respectively. These characters appear as one tree, two trees and three trees. The connection between the composition forms of these characters and their meanings is obvious. Another example is ‘’ (root), which is also related to the character ‘ ’ (tree): A bar near the bottom of a tree refers to the tree root. Such relations among Chinese characters are common, though sometimes it is not easy to realize them intuitively, or, even worse, they sometimes may become fuzzy after a few thousand years of evolution of the Chinese characters. However, the overall forms and meanings of Chinese characters are still closely related [@Qiu; @Bai; @Bellassen]: Usually, combinations of simple Chinese characters are used to form complex characters. Most Chinese users and learners eventually notice such structural relations although quite often implicitly and from accumulation of knowledge and intuitions on Chinese characters [@Lam1]. Making use of such relations explicitly might be helpful in turning rote leaning into meaningful learning [@Novak:Cmap], which could improve efficiency of students’ Chinese learning. In the above example of ‘’, ‘ ’, and ‘’, instead of memorizing all three characters individually in rote learning, one just needs to memorize one simple character ‘’ and then uses the logical relation among the three characters to learn the other two. However, such structural relations among Chinese characters have not yet been fully exploited in practical Chinese teaching and learning. As far as we know from all mainstream Chinese textbooks the textbook of Bellassen et al. [@Bellassen] is the only one that has taken partially the structure information into consideration. However, considerations of such relations in teaching Chinese in their textbook are, at best, at the individual characters level and focus on the details of using such relations to teach some characters one-by-one. With the network analysis tool at hand, we are able to analyze this relation at a system level. The goal of the present manuscript is to perform such a system-level network analysis of Chinese characters and to show that it can be used to significantly improve Chinese learning. Major aspects of strategies for teaching Chinese include character set choices, the teaching order of the chosen characters, and details of how to teach every individual character. Although our investigation is potentially applicable to all three aspects, we focus here only on the teaching order question. Learning order of English words is a well studied question which has been well established [@English_Order]. However, there is almost no explicit such studies in Chinese characters. In this work, the characters choice is taken to be the set of the most frequently used characters, with $99\%$ accumulated frequency [@Frequency]. To demonstrate our main point: how network analysis can improve Chinese learning, we focus here on the issue of Chinese character learning order. Although some researchers have applied complex network theory to study the Chinese character network [@Li; @Lee], they mainly focus on the network’s structural properties and/or evolution dynamics, but not on learning strategies. A recent work studied the evolution of relative word usage frequencies and its implication on coevolution of language and culture [@Petersen]. Different from these studies, our work considers the whole structural Chinese character network, but more importantly, the value of the network for developing efficient Chinese characters learning strategies. We find, that our approach, based on both word usage and network analysis provides a valuable tool for efficient language learning. [**[Data and methods.]{}**]{} Although nearly a hundred thousand Chinese characters have been used throughout history, modern Chinese no longer uses most of them. For a common Chinese person, knowing $3,000 - 4,000$ characters will enable him or her to read modern Chinese smoothly. In this work, we thus focus only on the most used $3500$ Chinese characters, extracted from a standard character list provided by the Ministry of Education of China [@Characters]. According to statistics [@Frequency], these 3500 characters account for more than $99\%$ of the accumulated usage frequency in the modern Chinese written language. ![\[fig1\] Chinese character decomposing and network construction. The numerical values in the figure represent learning cost, which will be discussed later.](Wu_fig1.pdf){width="8.4cm"} Most Chinese characters can be decomposed into several simpler sub-characters [@Qiu; @Bai]. For instance, as illustrated in Fig. \[fig1\], character ‘’(means ‘add’) is made from ‘’(ashamed) and ‘’(water); ‘’ can then be decomposed into ‘’(head, or sky) and ‘’(heart), and ‘’ can be decomposed into ‘’ (one) and ‘’(a person standing up, or big). The characters ‘’, ‘’, ‘ ’ and ‘’ cannot be decomposed any further, as they are all radical hieroglyphic symbols in Chinese. There are general principles about how simple characters form compound characters. It is so-called “Liu Shu” (six ways of creating Chinese characters). Ideally when for example two characters are combined to form another character the compound character should be connected to its sub-characters either via their meanings or pronunciations. We have illustrated those principles using characters listed in Fig. \[fig1\]. See [**[Supporting Online Material]{}**]{} for more details. While certain decompositions are structurally meaningful and intuitive, others are not that obvious at least with the current Chinese character forms [@Bai]. In this work, we do not care about the question, to what extent Chinese character decompositions are reasonable, the so-called Chinese character rationale [@Qiu], but rather about the existing structural relations (sometimes called character-formation rationale or configuration rationale) among Chinese characters and how to extract useful information from these relations to learn Chinese. Our decompositions are based primarily on Ref. [@ShuoWen; @Qiu; @Bai]. Following the general principles shown in the above example and the information in Ref. [@ShuoWen; @Qiu; @Bai] , we decompose all 3500 characters and construct a network by connecting character $B$ to $A$ (an adjacent matrix element $a_{BA}=1$, otherwise it is zero) through a directed link if $B$ is a “direct” component of $A$. Here, “direct” means to connect characters hierarchically (see Fig. \[fig1\]): Assuming $B$ is part of $A$, if $C$ is part of $B$ and thus in principle $C$ is also part of $A$, we connect only $B$ to $A$ and $C$ to $B$, but NOT $C$ to $A$. There are other considerations on including more specific characters which are not within the list of most-used $3500$ characters but are used as radicals of characters in the list, in constructing this network. More technical details can be found in the [**[Supporting Online Material]{}**]{}. Decomposing characters and building up links in this way, the network is a Directed Acyclic Graph (DAG), which has a giant component of $3687$ nodes (see [**[Supporting Online Material]{}**]{} for details on the number of nodes) and $7024$ links, plus $15$ isolated nodes. Fig. \[fullmap\] is a skeleton illustration of the full map of the network. ![\[fullmap\] Full map of the Chinese character network. For a better visual demonstration, we plot here the minimum spanning tree of the whole network which is shown in blue while other links are presented in grey as a background. All characters can be seen when the figure magnified properly. ](Wu_fig2.pdf){width="8.4cm"} As a DAG, the Chinese character network is hierarchical. Starting from the bottom in Fig. \[fig1\], where nodes have no incoming links, we can assign a number to a character to denote its level: all components of a character should have lower levels than the character itself. Fig.\[fig2\](a) shows the hierarchical distribution of characters in the network. The figure shows that the network has a small set of radical characters ($224$ nodes at the bottom level, $1$) and nearly $94\%$ of the characters lie at higher levels. Moreover, the network has a broad heterogeneous offsprings degree distribution (a node’s offspring degree is defined as its number of outgoing edges). Notice in Fig. \[fig2\](b), the number of characters with more than one (the smallest number on the vertical axis) offspring is close to $1000$ (the largest number shown on the horizontal axis). This means that less than $1000$ of the $3687$ characters are involved in forming other characters. The other characters are simply the top ones in their paths so that no characters are formed based on them. Their distribution in the different levels is also shown in Fig. \[fig2\]a. ![\[fig2\] Topological properties of Chinese character network. (a) Hierarchical distribution: number of characters at each level. The number of characters in each level that have no offspings is shown in brown. (b) Node-offspring distribution: Zipf plot, where characters are ranked according to their number of offsprings. The number of offsprings of a character is plotted against the rank of the character.](Wu_fig3.pdf){width="8.4cm"} [**[Learning Strategy.]{}**]{} The heterogeneity of the hierarchical structure reflected in the node-offspring broad distribution in the Chinese character network suggests that learning Chinese characters in a “bottom-up" order (starting from level $1$ characters and gradually climbing along the hierarchical paths) may be an efficient approach. At the level of learning of [*[individual]{}*]{} characters, Chinese teaching has indeed used this rationale[@Bellassen; @Zhou]. Other approaches are based on character usage frequencies, learning the most used characters, those appearing as the most used words first (Ref. [@Lam2] provides a critical review of this approach and others). To assess the efficiency of different approaches, which is here limited to Chinese characters learning orders, one needs a method to measure the learning efficiency. However, measuring learning efficiency is not trivial and currently, to the best of our knowledge, does not exist. In our approach, we regard a learning strategy as more efficient if it reaches the same learning goal, a desired number of learned characters or accumulated character usage frequencies, with lower learning costs compared to other strategies. The question thus becomes how to determine the learning cost? Of all possible factors related to cost, it is reasonable to assume that a character with more sub-characters and more unlearned sub-characters is more difficult to learn. For example, the character ‘’, with 5 sub-characters, is obviously more difficult to learn than ‘’, with 2 sub-characters. Conversely, it is easier to learn a character for which all sub-characters have been learned earlier than another character with same number of sub-characters all of which are previously unknown to the learner. We thus intuitively define the cost for a student to learn a character as the sum of the number of sub-characters and the learning cost of the unlearned sub-characters at his current stage. The learning cost of the unlearned sub-characters is calculated recursively until characters at the first level are reached or until all sub-characters have been learned previously. Each unlearned character of the first level contributes cost $1$, while previously learned characters contribute cost $0$. For example, assuming that, at a given stage, a student needs to learn the character ‘’ and that the student already knows the characters in blue in Fig. \[fig1\]. We demonstrate the cost for the student to learn this character. First, the character ‘ ’ has $2$ sub-characters (‘’and ‘’), and the student does not know one character, ‘’. The total cost of learning the character ‘’ is thus equals to $2$ plus the cost of learning ‘’, which, calculated using the same principle, is $2$ ($2$ sub-characters ‘’ and ‘’ , and none of which are new to the student). The cost for the student is thus $4$. If the student somehow learned the character ‘’ before and then needs to learn ‘’, the cost of acquiring ‘’ is only $2$. Thus, to learn both characters, it is cheaper to first learn ‘ ’ and then ‘’ (total cost $2+2=4$), rather than the other way around ($4+2=6$). If we assume that learning more characters, independent of their usage frequency, is the learning goal, the optimal learning strategy is to follow the node-offspring order (NOO) from many to few, which means learning characters with more offspring first. In this way, an ancestor character is always learned before its offspring characters since the ancestor has at least one more offspring than the offspring character. From the learning cost definition, we know that using this approach we never waste effort in learning characters twice. No other strategy is thus better than this one. However, in this way we might learn many characters with low usage frequencies which are less useful. Hence, as shown in Fig. \[fig3\]b, if our aim is acquiring more accumulated usage frequency, the NOO-based strategy is indeed not a good one. Being able to achieve a high accumulated usage frequency in relatively short times is not only good for those who can not spend much time but it will also help the students to do extracurricular reading. Thus, our main objective is to develop a learning strategy that reaches the highest accumulated usage frequency with limited cost. When simply following the character usage frequency order (UFO method) from high to low, one discards topological relations among characters that could help in the learning process and save cost. In UFO one learns characters at higher levels before learning those at lower levels, which is more costly. Thus, the question comes to developing a new Chinese character centrality measure of character importance, that considers both topological relations and usage frequencies. Such a measure could help to obtain a learning order better than both NOO and UFO. One additional consideration is to learn first the characters with larger out degree in the character network since here a large out degree means the character is involved as a component in many characters. The method proposed in the following in fact takes all these three aspects into consideration. Here we develop a centrality measure that we call distributed node weight (DNW) based on both network structure and on usage frequencies which are the node weights ($W^{(m)}_j$ ). Here $j$ represents the node (character) and $m$ its level in the network. The top level is $m=5$ (no outgoing links) and the bottom level is $m=0$ (no incoming links). To measure character centrality of node $j$ at level $m$, we pick each of its predecessors (denoted as node $i$ at level $m+1$) and add its weight $W^{(m+1)}_i$ multiplied by $b$ to the weight $W^{(m)}_j$ as follow: $$\label{eq1} \tilde{W}^{(m)}_j=W^{(m)}_j+b\sum_{i}W^{(m+1)}_i a_{ji},$$ where $b\geq0$ is a parameter, $a_{ji}=1 \mbox{ or } 0$ is the adjacency matrix element from node $j$ to node $i$ (whether or not character $j$ is a direct part of character $i$). In the DNW method one learns characters in order according to their centrality from highest to lowest. Thus, when $b=0$, the DNW is equivalent to the UFO method. For $b>0$, the node’s offsprings play an important role. When $b=1$ and all $W_{j}=1$ (which means ignoring the difference in character usage frequencies), the DNW centrality order becomes the node-offspring order (NOO). In this sense, the NOO is an unweighted version of the DNW. The DNW order can thus be considered a hybrid of the NOO and UFO. ![\[fig3\] Learning efficiency comparison for different learning orders: node-offspring order (NOO), usage frequency order (UFO), distributed node weight (DNW) and two common empirical orders (EM1 for Chinese pupils and EM2 for LCSL). (a) Number of characters is set as the learning goal. (b) Accumulated usage frequency is set as the learning goal. $C_{min}$ is defined as the learning cost of $1775$ characters using the NOO method and it will be used in discussion of leaning efficiency index.](Wu_fig4a.pdf "fig:"){width="4.2cm"} ![\[fig3\] Learning efficiency comparison for different learning orders: node-offspring order (NOO), usage frequency order (UFO), distributed node weight (DNW) and two common empirical orders (EM1 for Chinese pupils and EM2 for LCSL). (a) Number of characters is set as the learning goal. (b) Accumulated usage frequency is set as the learning goal. $C_{min}$ is defined as the learning cost of $1775$ characters using the NOO method and it will be used in discussion of leaning efficiency index.](Wu_fig4b.pdf "fig:"){width="4.2cm"} Using numerical analysis, we find that the optimal $b$ value for the DNW strategy is $b\simeq 0.35$, as discussed below. With this optimal parameter $b$, we compare our strategy of DNW learning order against the NOO and the UFO in Fig.\[fig3\]. We find in Fig.\[fig3\]a that DNW is close to NOO, regarding the total number of characters vs. the learning cost. However, in Fig. \[fig3\]b, the DNW is significantly better than NOO and even better than UFO, regarding the total accumulated usage frequency vs. the learning cost. In the left panel, NOO and DWN are much better than UFO, while in the right panel the UFO and DNW are much better than NOO. Thus, only the DNW demonstrates a high efficiency in both, accumulated frequency and total number of characters. The DNW in the right figure appears to be only slightly better than the UFO, but this is a little misleading. From the left figure, we can see that with the same cost, say around $1000$, although the difference between the two is relatively small in the right figure, there is a much bigger difference in the left figure. It means that even though the DNW is only slightly better than the UFO on the accumulated usage frequency, significantly more characters are learned following the DNW than the UFO. Such a difference in number of known characters sometimes is as important as the accumulated usage frequency when estimating if an individual is literate or not. For beginners, $400-500$ characters is roughly the first barrier. Many stop there. Using the UFO, this corresponds to a cost of about $2000$ while using the DNW it is around only $1000$. Thus, it will be much easier for students to overcome this barrier when using DNW compared to UFO. We next compare the DNW against two empirical commonly used orders: one is from a set of the most used Chinese textbook [@Textbook1] for primary schools in China, which contains $2475$ different Chinese characters (EM1); the other is from a mainstream Chinese textbook [@Textbook2] for students Learning Chinese as a Second Language (LCSL), which contains $1775$ different Chinese characters (EM2). We sort the two character sets by first appearances in new character lists in the two textbooks and plot their learning results in Fig.\[fig3\]. The figure shows that compared to our developed DNW method, the empirical learning orders have relatively poor performance in both the total number of characters and accumulated usage frequency. This emphasizes the urgent need of improving the efficiency of current learning Chinese characters. [**[Optimal b.]{}**]{} To find the optimal $b$ value, we define an efficiency index for learning strategies. We first take a certain learning cost and denote it as $C_{min}$, which is here set to be the learning cost of learning the total of $N_{min}=1775$ characters using the NOO order ($C_{min}=3351$, See Fig. \[fig3\]a). We intuitively assume that the sooner a curve reaches $N_{min}$ the learning is more efficient. Thus, the larger is the area under the curves in Fig. \[fig3\]a the learning can be regarded as more efficient. The same consideration holds for the curves in Fig. \[fig3\]b. We therefore, measure the area underneath the learning efficiency curves (Fig.\[fig3\]) up to cost $C_{min}$ and denote them as $S_n$ (area under the curve of number of characters v.s. cost like the ones in Fig. \[fig3\]a) and similarly $S_f$ (area under the curve of accumulated usage frequency v.s. cost like those in Fig. \[fig3\]b), respectively. The ratio between the area underneath the curves $S_{n}$ ($S_{f}$) and the area of a rectangular region defined by $C_{min}N_{min}$ ($C_{min}F_{min}$, where $F_{min}$ is the maximum accumulated frequency of the curves at $C=C_{min}$) is defined as the learning efficiency index, $$\begin{aligned} v_n=\frac{S_n}{C_{min}N_{min}},\\ v_f=\frac{S_f}{C_{min}F_{min}}. {\label{eq:speed}}\end{aligned}$$ The sooner a curve reaches $N_{min}$ ($F_{min}$) the larger is the area and so is the ratio, the more efficient is the learning order. In this sense, the above ratios serve as indexes of efficiency of learning orders. In Fig. \[fig4\], we plot $v_n$ and $v_f$ of the hybrid strategy (DNW) as functions of $b$. We also plot two lines, for comparison, showing the learning efficiency of the NOO (blue line) and UFO (green line). As $b$ increases, $v_n$ of the hybrid strategy approaches that of the NOO. On the other hand, when $b=0.35$, $v_f$ of hybrid strategy reaches its maximum. Thus, with respect to frequency usage the DNW with $b=0.35$ is the most efficient. However, if we consider also the number of characters the range of $b\in\left[0.35, 0.7\right]$ can be regarded as very good choices. As an example, in this work we use $b=0.35$, which shows a significant improvement over commonly used methods (Fig. \[fig3\]). ![\[fig4\] Efficient index of hybrid strategies as a function of b (dots). The two horizontal lines are the efficiency of the node-offspring order (blue line) and usage frequency order (green line). (a) Efficiency when using number of characters as the learning goal. (b) Efficiency when using accumulated usage frequency as the learning goal.](Wu_fig5.pdf){width="8.4cm"} In order to compare the DNW strategy against others in more detail, we have analyzed the learning cost statistics of the characters covered by cost $C_{min}$ for all the five learning strategies in Fig. \[fig5\]. Recall that $C_{min}$ is the cost of learning first $1775$ characters using the NOO and number of characters covered by this $C_{min}$ is different for different methods. Using the measure of learning cost proposed earlier, we record the learning cost of every character before the accumulated cost reaches $C_{min}$ in each learning order and then plot a histogram of learning costs of all those characters for each learning order. From Fig. \[fig5\]a, we see that in both DNW and NOO learning orders, characters with learning cost $2$ are dominant (roughly $80\%$). In these two learning orders, few characters have learning cost higher than $3$. The other three learning orders have much smaller fraction of characters of cost-$2$ and more characters with cost higher than $3$. Most Chinese characters can be decomposed into $2$ direct parts, therefore, learning cost $2$ means that when a character is learned, its parts have been quite often learned before. This is natural in the NOO order since it is designed that way. However, as seen here it also holds in the DNW order, which is the high advantage of the DNW order. In Fig. \[fig5\]b we also plot the corresponding usage frequencies of the set of characters with the same learning cost. In DNW one learns in fact about 6$\%$ less characters compared to NOO, but the usage of the characters learned in DNW is more than 30$\%$ higher. Thus DNW is significantly better than NOO. We also find that although DNW and UFO have comparable overall usage frequencies, the DNW is concentrated on the cost-$1$ and cost-$2$ characters while the UFO is distributed widely on characters with learning cost from $1$ to $4$. This illustrates further why our DNW is an efficient learning order in both the sense of total number and total usage frequency of characters. ![\[fig5\] Up to a fixed total learning cost $C_{min}$, for all five learning orders, we count and plot the number of characters according to their individual learning costs in (a) and convert the number of characters into the corresponding usage frequency in (b). ](Wu_fig6.pdf){width="8.4cm"} [**[Conclusion and Discussion]{}**]{}. We demonstrate the potential of network approach in increasing significantly the efficiency of learning Chinese. By including character usage frequencies as node weights to the structural character network, we discover and develop an efficient learning strategy which enables to turn rote learning of Chinese characters to meaningful learning. In the [**[Supporting Online Material]{}**]{}, we present an adjacency list form of the constructed network; we also list Chinese characters order according to our DNW centrality. The constructed network might also help design a customized Chinese character learning order for students who have previously learned some Chinese and want to continue their studies at their own paces. Given the information about the student’s known characters in our network, our DNW centrality measure can be adapted to be used in finding a specific student oriented optimal learning order. This goal is completely out of reach of standard textbook-based education and it will be especially useful for Chinese learners that do not study Chinese in a formal Chinese school, or study Chinese every now and then or using private tutors. We hope that our study will lead to develop textbooks applying the DNW learning order and detailed decomposition of each character. It will also be valuable for Chinese learners to have a dictionary explaining every character and word simply from a core set of small number of basic characters. Note that we are not claiming that our decomposition is perfect or that our character choice is good enough. These questions are still debated in the Chinese character structure fields. There are possibly also other topological quantities that might be valuable for Chinese learning. Considering our node-weighted network, the concept of using the shortest path to accumulate the largest node weight in shortest steps, clearly differs from the usual shortest path. How these quantities are related to Chinese learning is an interesting question that we have not discussed in this work. Writers, reporters and citizens in China have argued that the Chinese textbooks currently used in mainland China are going in the wrong direction, and textbooks used $70$ years ago seem to be more reasonable. Influenced by English teaching, Chinese teaching indeed becomes increasingly speaking- and listening-oriented [@Lam2]. Speaking- and listening-oriented approach is a reasonable way to learn a phonetic language. However, for Chinese – an ideographic language, it results an inefficient learning order of Chinese character where structurally complicated characters are often taught before simpler ones. What we are suggesting is that in designing the speaking, listening and reading materials, one should utilize the logographic relations among Chinese characters and also respect the optimal learning order discovered from analyzing the character network of the same relation. Only using a network analysis can we capture an entire picture of a network of these structural relations. [**Acknowledgements**]{} This work was supported by NSFC Grant $61174150$ and $60974084$. [**Competing interests statement**]{} The authors declare that they have no competing financial interests. [**Correspondence**]{} should be addressed to J. Wu ([email protected]). Supporting Online Material ========================== Data and methods ---------------- ### Decomposition of Chinese characters According to “Liu Shu” (six ways of creating Chinese characters), ideally when sub-characters are combined to form a character the compound character should be connected to its sub-characters either via their meanings or pronunciations. Thus, Chinese characters are usually meaningfully and coherently connected to each other. Let us start from the bottom of Fig. 1 in the main text. The four characters are “” (one), “”(person, big), “”(heart), “” (water). These characters closely resemble the shapes or characteristics of the objects to which they refer, though their forms today might not hold as much of a resemblance as their ancient forms. One can compare the modern simplified Chinese character against their ancient Zhuanti forms in the figures. Such characters are called pictographic (Xiangxing) characters. Initially, the character “” (sky) refers to the head, the primary part of a person, by placing a bar over the character “”(person, big). The meaning later developed and became the sky, heaven and god, the primary part of everything as ancient Chinese people believed. This way of forming new characters from radical parts is called “simple” ideogram (Zhishi) or “combination character” ideogram (Huiyi). These two mechanisms are in fact slightly different in that the first is based on only one radical part, usually with only a very simple additional stroke while the second usually involves two radical parts. For a character formed by these two principles, its meaning usually can be read out intuitively from the combination. For example, the character ‘’ (forest) mentioned in the introduction of the main text follows the principle of “combined” ideogram: it is a stack of three ‘’(tree). However, in this work, we will not distinguish the two mechanisms. The character ‘’ (, ashamed) is a compound character of ‘’ and ‘’. It follows a different principle, which later became popular in forming new Chinese characters, the so-called pictophonetic formation (Xingsheng). Here, ‘’ and ‘’ have exactly the same pronunciation, and the meaning of ‘’ refers to a psychological phenomenon, which was believed to be related to ‘’ (heart). The same pictophonetic relation holds among ‘’ (add), ‘’and ‘’ (water): the first two share the pronunciation while the last part ‘’ is remotely connected to the meaning of ‘’. In Fig.1 of the main text, we also notice that the characters ‘’ and ‘’ also form the characters ‘ ’(seep). The character ‘’ follows also the pictophonetic formation. It is quite common that some basic characters are used in quite a few composed characters. Here we have demonstrated four of the six principles. The other two are phonetic loan (Jiajie) and derivative cognates (Zhuanzhu). Those two principles are more on usage of characters but not on creating new characters. It is not our focus of this work to discuss various ways of usages of Chinese characters. Following the above general principles, our decompositions of characters are based primarily on Ref.\[11,12,21\]. The first is a standard reference, where the six principles were first explicitly discussed, in Chinese etymology studies, and the last two are regarded as developments of the first, mainly due to discoveries of new materials, including Oracle characters (Jiaguwen) and Bronze characters(Jinwen). Starting from $3500$ characters, our network ends up with a giant component of $3687$ nodes and $7025$ links, plus $15$ isolated nodes. Why do we have more nodes than the total number of characters we start with? In our decomposition, we find some sub-characters beyond the set of the most used $3500$ characters. Sometimes, such sub-characters are just variations of their normal forms. The situation becomes more complicated when a radical whose corresponding normal form is not within the most-used set. In such cases, we add the “never-independent characters” as extra nodes in the network. For example, ‘’ is such a rarely used character, but we keep it in our network. See Fig. 2 in the main text for the full map of structural relations among Chinese characters. ### Additional explanation of definition of learning cost We define the learning cost of a character for a student to be the sum of the number of sub-characters and the learning cost (calculated recursively) of the unlearned sub-characters at his current stage. The recursive definition seems to imply that when a student is learning a compound character, he has to recognize first the sub-characters. However, the dynamic process is only a fictitious process used to represent the difficulty that the student faces in learning the character. It does not means the learning process is indeed as such. Recall from the main text total cost of learning ‘’ before ‘’ is $4=2+2$, which is from the fact that it has $2$ sub-characters and also from the fact that cost of learning the unknown ‘’ is $2$. Therefore, determining cost of learning ‘’ first obviously involves cost of learning ‘’. However, this does not imply that the student should have known ‘’ after acquiring ‘’. If it happens so that the next time the student must learn ‘’, then the learning cost of ‘’ is still $2$ even he had learned ‘’ before. Thus the total learning cost of the two characters following the order of ‘’ $\rightarrow$ ‘’ is $6$. Of course, if the student learned the character ‘’ meaningfully, when he learn the character ‘ ’, he indeed learn also the relation between ‘’ and ‘ ’ (also the meaning of ‘’) explicitly from his books or his instructors, then the total cost for him to learn both characters is in fact $4$ (no cost for learning ‘’), which is the same cost of learning both characters in the order of ‘’ $\rightarrow$ ‘ ’. Therefore, learning closely connected characters together at the same time and learning them meaningfully would reduce the cost. Therefore, one might conclude that our definition of learning cost does not apply to such meaningful learning. However, for this we would argue that such meaningful learning has implicitly used the optimal learning orders, learning the two characters simultaneously and meaningfully is equivalent to learning them according to the proper order. Another problem related to our definition of learning cost is that we treated the number of sub-characters and the cost of unlearned sub-characters equally. This can be questioned and should be investigated further. For example, one might introduce a parameter to rescale the number of sub-characters and then sum the two together. For simplicity, we have not yet discussed this issue. Finding the proper value of such parameters from empirical studies and then comparing performance of those learning orders again using the new definition of cost should be an interesting topic. Supplemental Results -------------------- At last, we provide the two important lists of characters as final results of our network-based analysis of Chinese characters. First is the adjacency list of the network of characters. The first character of every line is the starting point of links and all other characters in the same line are the ending point of the links, meaning the first character is a part of everyone of the other characters. Second is the order of Chinese characters listed according to the calculated DNW centrality. This list includes all $3500$ characters and $b=0.5$ is used in the calculation of DNW. In the main text, when $1775$ characters are used as the learning target, we find the optimal value of parameter $b$ is $b=0.35$. Repeating the same analysis for all $3500$ characters, we find that learning efficiency is higher when $b=0.5$ is used instead of $b=0.35$. Here the list is produced when we consider the whole set of most used characters as the learning goal. The lists can be downloaded from our own still developing website on Chinese learning <http://www.learnm.org/data/>.

ArXiv

--- abstract: | We study a crime hotspot model suggested by Short-Bertozzi-Brantingham [@sbb]. The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity. More precisely, for the one-dimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: \(i) Multiple spikes of arbitrary number having the same amplitude (symmetric spikes), \(ii) Multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). We use an approach based on Liapunov-Schmidt reduction and extend it to the quasilinear crime hotspot model. Some novel results that allow us to carry out the Liapunov-Schmidt reduction are: (i) approximation of the quasilinear crime hotspot system on the large scale by the semilinear Schnakenberg model, (ii) estimate of the spatial dependence of the second component on the small scale which is dominated by the quasilinear part of the system. The paper concludes with an extension to the anisotropic case. author: - 'Henri Berestycki [^1]' - 'Juncheng Wei [^2]' - 'Matthias Winter [^3]' title: Existence of Symmetric and Asymmetric Spikes for a Crime Hotspot Model --- [**Key words:**]{} crime model, reaction-diffusion systems, multiple spikes, symmetric and asymmetric, quasilinear chemotaxis system, Schnakenberg model, Liapunov-Schmidt reduction [**AMS subject classification:**]{} Primary 35J25, 35 B45; Secondary 36J47, 91D25 Introduction: The statement of the problem ========================================== Pattern forming reaction-diffusion systems have been and are applied to many phenomena in the natural sciences. Recent works have also started to use such systems to describe macroscopic social phenomena. In this direction, Short, Bertozzi and Brantingham [@sbb] have proposed a system of non-linear parabolic partial differential equations to describe the formation of hotspots of criminal activity. Their equations are derived from an agent-based lattice model that incorporates the movement of criminals and a given scalar field representing the “attractiveness of crime”. The system in one dimension reads as follows: $$\begin{aligned} \nonumber A_{t} & =\varepsilon^{2}A_{xx}-A+\rho A+A_{0} (x), \ \mbox{in} \ (-L, L),\\ \rho_{t} & =D (\rho_{x}-2\frac{\rho}{A}A_{x})_{x}-\rho A+\gamma (x), \ \mbox{in} \ (-L, L). \label{sysoriginal}\end{aligned}$$ Here $A$ is the “attractiveness of crime” and $\rho$ denotes the density of criminals. The rate at which crimes occur is given by $\rho A$. When this rate increases, the number of criminals is reduced while the attractiveness increases. The second feature is related to the well documented occurrence of repeat offenses. The positive function $A_0 (x)$ is the intrinsic (static) attractiveness which is stationary in time but possibly variable in space. The positive function $\gamma (x)$ is the source term representing the introduction rate of offenders (per unit area). For the precise meanings of the functions $A_0 (x)$ and $\gamma (x)$, we refer to [@sbb; @sbbt; @soptbbc] and the references therein. This paper is concerned with the mathematical analysis of the one-dimensional version of this system. Let us describe our approach. Setting $$v=\frac{\rho}{A^{2}},$$ the system is transformed into $$\begin{aligned} \nonumber A_{t} & =\varepsilon^{2}A_{xx}-A+vA^{3}+A_{0} (x) \ \mbox{in} \ (-L, L),\\ (A^2v)_{t} & =D\left( A^{2}v_{x}\right) _{x}-vA^{3}+\gamma (x) \ \mbox{in} \ (-L, L). \label{sysdyn}\end{aligned}$$ We always consider Neumann boundary conditions $$A_x(-L)=A_x(L)=\rho_x(-L)=\rho_x(L)=v_{x}(-L)=v_x(L)=0.$$ Note that $v$ is well-defined and positive if $A$ and $\rho$ are both positive. The parameter $0<{\varepsilon}^2$ represents nearest neighbor interactions in a lattice model for the attractiveness. We assume that it is very small which corresponds to the temporal dependence of attractiveness dominating its spatial dependence. This models the case of attractiveness propagating rather slowly, i.e. much slower than individual criminals. It is a realistic assumption if the criminal spatial profile remains largely unchanged, or, in other words, if the relative crime-intensity does only change very slowly. This appears to be a reasonable assumption since it typically takes decades for dangerous neighborhoods, i.e. those attracting criminals, to evolve into safe ones and vice versa. Roughly speaking, a $k$ spike solution $(A, v)$ to (\[sysdyn\]) is such that the component $A$ has exactly $k$ local maximum points. In this paper, we address the issue of existence of steady states with multiple spikes in the following two cases: Symmetric spikes (same amplitudes) or asymmetric spikes (different amplitudes). Our approach is by rigorous nonlinear analysis. We apply Liapunov-Schmidt reduction to this quasilinear system. In this approach, to establish the existence of spikes, we derive the following new results: - Approximation of the crime hotspot system on the large scale of order one by the semi-linear Schnakenberg model (see Section 3, in particular equation (\[approx\])), - Estimate of the spatial dependence of the second component on the small scale of order ${\varepsilon}$, dominated by the quasilinear part of the system (see Section 6, in particular inequalities (\[estw3\]) – (\[estw1\])). We remark that asymmetric multiple spike steady states (of $k_1$ small and $k_2$ large spikes) are an intermediate state between two different symmetric multiple spike steady states of $k_1+k_2$ spikes (for which all spikes are fully developed) and $k_2$ spikes (for which the small spikes are gone). These rigorous results shed light on the formation of hotspots for the idealized model of criminal activity introduced in [@sbb]. Let us now comment on previous works. As far as we know, there are three mathematical works related to the crime model (\[sysdyn\]). Short, Bertozzi and Brantingham [@sbb] proposed this model based on mean field considerations. They have also performed a weakly nonlinear analysis on (\[sysoriginal\]) about the constant solution $$(A, \rho)= \left(\gamma +A_0, \frac{\gamma}{\gamma+A_0} \right)$$ assuming that both $A_0 (x)$ and $\gamma (x)$ are homogeneous. Rodriguez and Bertozzi have further shown local existence and uniqueness of solutions [@rb1]. In [@ccm], Cantrell, Cosner and Manasevich have given a rigorous proof of the bifurcations from this constant steady state. On the other hand, in the isotropic case, Kolokolnikov, Ward and Wei [@kww1] have studied existence and stability of multiple symmetric and asymmetric spikes for (\[sysdyn\]) using formal matched asymptotics. They derived qualitative results on competition instabilities and Hopf bifurcation and gave some extensions to two-space dimensions. The present paper provides rigorous justification for many of the results in [@kww1] and also derives some extensions. In particular, we establish here the following three new results: first, we reduce the quasilinear chemotaxis problems to a Schnakenberg type reaction-diffusion system and prove the existence of symmetric $k$ spikes. Second, this paper gives the first rigorous proof of the existence of asymmetric spikes in the isotropic case. Third, we study the pinning effect in an inhomogeneous setting $A_0 (x)$ and $\gamma (x)$. The stability of these spikes is an interesting issue which should be addressed in the future. We should mention that another model of criminality has been proposed and analyzed by Berestycki and Nadal [@bn]. In a forthcoming paper [@bw], we shall study the existence and stability of hotspots (spikes) in this system as well. It is quite interesting to observe that both models admit hotspot (spike) solutions. The structure of this paper is as follows. We formally construct a one-spike solution in Section 2 in which we state our main results. In Section 3 we show how to approximate the crime hotspot model by the Schnakenberg model. Section 4 is devoted to the computation of the amplitudes and positions of the spikes to leading order. Nondegeneracy conditions are derived in Section 5. These are required for the existence proof, given in Sections 6–8. In Section 6 we introduce and study the approximate solutions. In Section 7 we apply Liapunov-Schmidt reduction to this problem. Lastly, we solve the reduced problem in Section 8 and conclude the existence proof. In Section 9 we extend the proof of single spike solution to the case when both $A_0 (x)$ and $\gamma (x)$ are allowed to be inhomogeneous. Finally, in Section 10 we discuss our results and their significance and mention possible future work and open problems. Steady state: Formal argument for leading order and main results ================================================================ Before stating the main results, we first construct a time-independent spike on the interval $[-L,L]$ located at some point $x_0$. The construction here is carried out using classical matched asymptotic expansions. In the inner region, we assume that $v$ is a constant $v_0$ in leading order: $$v(x)\sim v_{0},\ \ \ |x-x_0|\ll 1.$$ Then, if $0<{\varepsilon}\ll 1$, the equation for $A$ becomes $$\varepsilon^2 A^{''}-A+ v_0 A^3 + A_0 (x)=0.$$ Rescaling $$A(x)=v_{0}^{-1/2} \hat{A}(y),\ \ \ \ y=\frac{x-x_0}{\varepsilon},$$ we get $$\hat{A}_{yy}-\hat{A}+ \hat{A}^3 + A_0 (x_0+\epsilon y) v_0^{1/2}=0.$$ We assume that $v_0 \to 0$ as ${\varepsilon}\to 0$. Then, at leading order, $\hat{A} (y) \sim w(y)$, where $w$ is the unique (even) solution of the following ODE $$w_{yy}-w+w^{3}=0$$ so that$$w(y)=\sqrt{2}\operatorname*{sech}\left( y\right) .$$ In the outer region, we assume that$$vA^{3}\ll1,\ \ \ \frac{x}{\varepsilon}\gg1$$ so that$$A\sim A_{0} (x).$$ We also assume that $D=\frac{\hat{D}}{{\varepsilon}^2}$, where $\hat{D}$ is a positive constant, and we estimate $$\int_{-L}^{L}vA^{3}\,dx\sim v_{0}^{-1/2}\varepsilon\int_{-\infty}^{\infty}w^{3}dy.$$ Integrating the second equation in (\[sysdyn\]), we then have $$\begin{gathered} \nonumber v_{0}^{-1/2}\varepsilon\int_{-\infty}^{\infty}w^{3}dy\sim\int_{-L}^L \gamma (x) dx,\\ v_{0}\sim\frac{\left( \int_{-\infty}^{\infty}w^{3}dy\right) ^{2}}{ \left(\int_{-L}^L \gamma (x) dx \right)^{2}}\varepsilon^{2}.\label{sys0}$$ We remark that $\int w^{3}\,dy=\int w\,dy=\sqrt{2}\pi$ so that$$v_{0}\sim\frac{2 \pi^{2}}{\left( \int_{-L}^L \gamma (x) \right) ^{2}} \varepsilon^{2}.$$ In particular, we obtain$$A(x)\sim\left\{ \begin{array} [c]{c}A_0(x)+ \dfrac{\sqrt{2} \int_{-L}^L \gamma (x) dx }{\varepsilon\pi}\,w\left(\frac{x-x_0}{\varepsilon} \right),\ \ \ x=O\left( \varepsilon\right), \\[3mm] A_{0}(x),\ \ \ \ x\gg O(\varepsilon). \end{array} \right. \label{Aunif}$$ Now we state our main theorems on the existence of multi-spike steady states for system (\[sysdyn\]). We discuss two cases. In the case of isotropic coefficients $A_0 (x) \equiv$ Constant, $\gamma (x)\equiv$ Constant, we will consider two types of solutions: \(i) Multiple spikes of arbitrary number having the same amplitude (symmetric spikes). \(ii) Multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). In the case of anisotropic coefficients $A_0 (x)$ and $\gamma (x)$, we will consider the existence of single spike solution. Our first result concerns the existence of multiple spikes of arbitrary number having the same amplitude (symmetric spikes). \[existencesym\] Assume that $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$ and $$\label{aniso} A_0(x)\equiv A_0, \gamma (x) \equiv \bar{A}-A_0 \ \ \mbox{where}\ \bar{A}>A_0.$$ Then, provided ${\varepsilon}>0$ is small enough, problem (\[sysdyn\]) has a $K$-spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep} A_{{\varepsilon}}(x)= A_0+\frac{1}{{\varepsilon}}\sum_{j=1}^K \frac{1}{\sqrt{v_j^{\varepsilon}}} w \left(\frac{x-t_j^{\varepsilon}}{{\varepsilon}}\right)+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ $$\label{hep} v_{\varepsilon}(t_i^{\varepsilon})={\varepsilon}^2 v_i^{\varepsilon},\quad i=1,\ldots,K,$$ where $$\label{tep0} t_i^{\varepsilon}\to t_i^0,\quad i=1,\ldots,K$$ with $$\label{limpos} t_i^0=\frac{2i-1-K}{K}\,L,\,i=1,\ldots,K$$ and $$\label{vep0} v_i^{\varepsilon}= v_i^0\left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}} \right)\right) ,\quad i=1,\ldots,K$$ with $$\label{limamp} v_i^0=\frac{\pi^2 K^2}{2 (\bar{A}-A_0)^2 L^2},\quad i=1,\ldots,K.$$ [[ Note that in (\[aep\]) a two-term expansion of the solution $A_{\varepsilon}$ is given, where for each spike the term $\frac{1}{\sqrt{v_j^{\varepsilon}}} w \left(\frac{x-t_j^{\varepsilon}}{{\varepsilon}}\right)$ of order $O(\frac{1}{{\varepsilon}})$ is the leading term in the inner solution and the term $A_0$ of order $O(1)$ is the leading term in the outer solution. By using the operator $T[\hat{A}]$ defined in (3.12) this two-term expansion carries over to $\hat{v}$ as well. The same remark applies to (\[aep1\]) and (\[aep200\]). The two-term expansion agrees with that in [@kww1]. ]{} ]{} The next result is about asymmetric two-spikes. \[existenceas\] Under the same assumption as in Theorem \[existencesym\], with $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$ and suppose moreover that $$\frac{2\sqrt{\pi}(\hat{D} A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \leq 1, \label{condc}$$ and $$\frac{2\sqrt{\pi}(\hat{D} A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \not= \frac{2}{\sqrt{5}}. \label{condbc}$$ Then, for ${\varepsilon}>0$ small enough, problem (\[sysdyn\]) has an asymmetric $2$-spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep1} A_{{\varepsilon}}(x)=A_0+\frac{1}{{\varepsilon}} \left( \sum_{j=1}^2 \frac{1}{\sqrt{v_i^{\varepsilon}}} w \left(\frac{x-t_i^{\varepsilon}}{{\varepsilon}}\right) +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right) \right),$$ $$\label{hep1} v_{\varepsilon}(t_i^{\varepsilon})={\varepsilon}^2 v_i^{\varepsilon},\quad i=1,\ldots,K,$$ where $t_i^{\varepsilon}$ and $v_i^{\varepsilon}$ satisfy (\[tep0\]) and (\[vep0\]), respectively. The limiting amplitudes $v_i^0$ and positions $t_i^0$ are given as solutions of (\[amp1\]) and (\[amp6\]). Condition (\[condbc\]) is a kind of nondegeneracy condition. Note that in the case of asymmetric spikes we explicitly characterize the points of non-degeneracy. The last theorem is about the existence of single spike solution in the anisotropic case \[existenceani\] Assume that ${\varepsilon}>0$ is small enough and $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$. Then, problem (\[sysdyn\]) has a single spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep200} A_{{\varepsilon}}(x)= A_0 (x)+\frac{1}{{\varepsilon}} \frac{1}{\sqrt{v_0^{\varepsilon}}} w \left(\frac{x-t_0^{\varepsilon}}{{\varepsilon}}\right)+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ $$\label{hep200} v_{\varepsilon}(t_0^{\varepsilon})={\varepsilon}^2 v_0^{\varepsilon},$$ where $$\label{tep000} t_0^{\varepsilon}\to t_0, \ \int_{-L}^{t_0} \gamma (x) dx = \int_{t_0}^L \gamma (x) dx$$ and $$\label{vep000} v_0^{\varepsilon}= \frac{ 2\pi^2}{ (\int_{-L}^L \gamma (x) dx)^2} \left(1+ O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right).$$ We notice that in the anisotropic case, the single spike location is only determined by the function $\int_{-L}^{x} \gamma (t) dt$ and $A_0 (x)$ has no effect at all. Note also that the location $t_0$ is uniquely determined by the condition $$\int_{-L}^{t_0} \gamma (x) dx= \frac{1}{2} \int_{-L}^L \gamma (x) dx.$$ With more computations, it is possible to construct multiple asymmetric spikes in the isotropic case, and also multiple spikes in the anisotropic case. Since the statements and computations are complicated, we will not present them here. We refer to [@ww-pre] for some results in this direction. Scaling and approximation by the Schnakenberg model =================================================== We will use the following notation for the domain and the rescaled domain, respectively: $$\Omega= (-L, L),\quad {\Omega}_{{\varepsilon}}=\left(-\frac{L}{{\varepsilon}},\frac{L}{{\varepsilon}}\right).$$ This section is devoted to the reduction of the system (\[sysdyn\]) to a particular Schnakenberg type reaction diffusion equation in which no chemotaxis appears. Based on the computations in Section 2, we rescale the solution and the second diffusion coefficient as follows: $$A=A_0 (x)+\frac{1}{\epsilon} \hat{A},\quad v=\epsilon^2 \hat{v},\ \ \ D=\frac{\hat{D}}{\epsilon^2}.$$ Then the steady-state problem becomes $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\hat{A}_{xx}-\hat{A}+\hat{v} (\epsilon A_0 +\hat{A})^{3} +\varepsilon^3 A_0^{''},\ x\in{\Omega}, \\ 0 & =\hat{D} \left( \left(A_0 (x)+\frac{1}{\epsilon} \hat{A}\right)^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0(x) +\hat{A})^{3}+\gamma (x),\ x\in{\Omega}. \label{sys1}\end{aligned}$$ We will consider the case when ${\varepsilon}\ll 1$ and $\hat{D}$ is constant, with Neumann boundary conditions. A key observation of this paper is that the solutions of problem (\[sys1\]) are very close to the solutions of the Schnakenberg model $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\hat{A}_{xx}-\hat{A}+\hat{v} (\epsilon A_0 +\hat{A})^{3} +\varepsilon^3 A_0^{''},\ x\in{\Omega},\\ 0 & =\hat{D} \left( A_0^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0 +\hat{A})^{3}+\gamma (x),\ x\in{\Omega}, \label{schnak1}\end{aligned}$$ with Neumann boundary conditions. To see this, we first consider the following linear problem: $$\label{vequ} \left\{ \begin{array}{l} \hat{D} (a(x) v_x)_x=f(x),\quad -L<x<L, \\[3mm] v_x(-L)=v_x(L)=0, \end{array} \right.$$ where $a\in C^1(-L,L)$, $a(x)\geq c>0$ for all $x\in(-L,L)$ and $f\in L^1(-L,L)$. We compute $$a(x) v_x(x)=\int_{-L}^x \frac{1}{ \hat{D}} f(t)\,dt.$$ So $$\label{v1} v_x (x)=\frac{1}{ a(x)} \int_{-L}^x \frac{1}{ \hat{D}} f(t)\,dt$$ and hence $$\label{ss2} v(x)- v(-L)= \int_{-L}^x \frac{1}{ \hat{D} a(s)} \int_{-L}^s f(t) dt\,ds$$ which can be rewritten as $$\label{ss3} v(x)- v(-L)=\frac{1}{\hat{D}} \int_{-L}^x K_a (x, s) f(s)\, ds,$$ where $$K_a (x, s)=\int_s^x \frac{1}{a (t)} \,dt.$$ [We note that the kernel $K_a (x, s)$ is an even (odd) function if $a(x)$ is an odd (even) function. More precisely, if $a(x)=\pm a(-x)$, then $$\label{ksym} K_a (-x, -s)=\mp K_a (x,s).$$ ]{} [We note that $v$ is an even (odd) function if $f$ is even and $a$ is even (odd). More precisely, using $$\int_{-L}^0 f(t)\,dt=\frac{1}{2}\int_{-L}^L f(t)\,dt=0 \quad \mbox{if $f$ is even},$$ we compute $$v_x(x)=\frac{1}{a(x)}\int_0^x \frac{1}{\hat{D}}f(t)\,dt.$$ Integration yields $$v(x)-v(0)=\int_0^x \frac{1}{\hat{D}a(s)}\int_0^s f(t)\,dt\,ds$$$$= \frac{1}{\hat{D}} \int_{0}^x K_a(x,s)f(s)\,ds$$ and $$v(-x)-v(0)=\frac{1}{\hat{D}} \int_{0}^{-x} K_a(-x,s)f(s)\,ds$$ $$=-\frac{1}{\hat{D}} \int_{0}^{x} K_a(-x,-s)f(-s)\,ds$$ $$=\pm\frac{1}{\hat{D}} \int_{0}^{x} K_a(x,s)f(s)\,ds$$ $$=\pm(v(x)-v(0))$$ if $a$ is an even (odd) function using (\[ksym\]). Similarly, if $f$ is odd and $a$ is odd (even), then $v$ is an even (odd) function. ]{} Integrating (\[vequ\]), we derive the necessary condition $$\label{intconstr} \int_{-L}^L f(x)\,dx=0.$$ Note that on the other hand $v$ defined by (\[vequ\]) satisfies the boundary conditions $v_x(-L)=v_x(L)=0$ provided that (\[intconstr\]) holds. This follows from (\[v1\]). Let us now consider $a(x)= \left(A_0+ \frac{\gamma }{\epsilon} w(\frac{x}{\epsilon})\right)^2$, where $ w>0$ and $ w(y)\sim e^{-|y|}$ as $|y|\to\infty$. Then we claim that $$\label{ss4} K_{a} (x,s)= K_{A_0^2} (x,s)+ O(\epsilon |s-x|)+ O\left( \left|[s,x] \cap \left(-2\epsilon \log\frac{1}{\epsilon}, 2\epsilon \log\frac{1}{\epsilon}\right)\right|\right).$$ Note that (\[ss4\]) is an $L^\infty$ estimate for $K_a(x,s)$. In fact, we have $$\int_{s}^x \frac{1}{(A_0+\frac{1}{\epsilon} w)^2}\,dt = \int_s^x \frac{1}{A_0^2} dx+ \int_{s}^x \left[\frac{1}{(A_0+\frac{1}{\epsilon} w)^2}-\frac{1}{A_0^2} \right]\,dt,$$ where $$\int_{s}^x \left[\frac{1}{A_0^2}-\frac{1}{(A_0+\frac{1}{\epsilon} w)^2}\right]\,dt= \epsilon \int_{\frac{s}{\epsilon}}^{\frac{x}{\epsilon}} \frac{ 2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2}\,dy$$ $$= \epsilon \int_{ [s/\epsilon, x/\epsilon] \cap \left\{ |y|>2 \log \frac{1}{\epsilon}\right\} } \ldots \,dy + \epsilon \int_{ [s/\epsilon, x/\epsilon] \cap \left\{ |y|<2 \log \frac{1}{\epsilon}\right\} }\ldots \,dy.$$ The first term is $O(\epsilon |x-s|)$ since $w=O({\varepsilon}^2)$ and so $ \frac{2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2 }=O(\epsilon)$. For the second term, observing that $ \frac{2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2 }=O(1)$ we derive (\[ss4\]). All these estimates are in the $L^\infty$ norm. Thus, $v$ satisfies: $$\label{ss5} v(x)- v(-L)=\frac{1}{\hat{D}} \int_{-L}^x K_{A_0^2} (x, s) f(s) ds +O\left(\epsilon \int_{-L}^x \left(|x-s|+\log\frac{1}{{\varepsilon}}\right)\, |f(s)|\, ds\right).$$ [The estimates (\[ss4\]) and (\[ss5\]) also hold if $$a (x)= \left(A_0 +\frac{\gamma}{\epsilon} \left(w\left(\frac{x-x_0}{\epsilon}\right) + \phi\right) \right)^2,$$ where $ \phi (x)$ satisfies $ |\phi (x)|\leq C \epsilon \max ( e^{- \frac{|x-x_0|}{2\epsilon}}, \sqrt{\epsilon})$. This is the class of functions that we will work with. This is also the motivation for our choice of the norm $\| \cdot \|_{*}$ (defined in (\[normdef\])). ]{} Therefore, we can approximate steady states for the crime hotspot model by the Schnakenberg model as follows: Given $\hat{A}>0$, let $\hat{v}=T[\hat{A}]$ be the unique solution of the following linear problem: $$\left\{ \begin{array}{l} \hat{D} \left( (A_0 +\frac{1}{\epsilon} \hat{A})^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0 +\hat{A})^{3}+\gamma (x)=0, \quad -L<x<L, \\[3mm] \hat{v}_{x}(-L)=\hat{v}_{x}(L)=0. \end{array} \right. \label{ta}$$ Then, by the maximum principle, the solution $T[\hat{A}]$ is positive. By the previous computations and remarks, if $ \hat{A}= w+\phi$ with $ |\phi |\leq C \epsilon \max ( e^{-\frac{|x-x_0|}{2\epsilon}}, \sqrt{\epsilon})$, it follows that $$\label{approx} T[\hat{A}]= v^{0} +O\left(\epsilon \log \frac{1}{\epsilon}\right) \quad\mbox{ in }H^2(-L,L),$$ where $v^0$ satisfies $$\left\{ \begin{array}{l} \hat{D} \left( A_0^2 v^0_{x}\right) _{x}-\frac{1}{\epsilon} v^0 (\epsilon A_0 +\hat{A})^{3}+\gamma (x)=0, \quad -L<x<L, \\[3mm] v^0_{x}(-L)=v^0_{x}(L)=0. \end{array} \right.$$ We adapt an approach based on Liapunov-Schmidt reduction which has been applied to the semilinear Schnakenberg model in [@iww2] and extend it to the quasilinear crime hotspot model. This method has also been used to study spikes for the one-dimensional Gierer-Meinhardt system in [@ww; @ww-pre] as well as two-dimensional Schnakenberg model in [@ww13]. We refer to the survey paper [@wei-survey] and the book [@ww-book] for references. Multiple asymmetric spikes for the one-dimensional Schnakenberg model have been considered using matched asymptotics in [@iww]. Existence and stability of localized patterns for the crime hotspot model have been studied by matched asymptotics in [@kww1] and results on competition instabilities and Hopf bifurcation have been shown including some extensions to two space dimensions. We remark that another approach for studying multiple spikes in one-dimensional reaction-diffusion system is the geometric singular perturbation theory in dynamical systems. For results and methods in this direction we refer to [@dgk; @dkp] and the references therein. Computation of the amplitudes and positions of the spikes ========================================================= In this section, we study (\[sys1\]) in the isotropic case (\[aniso\]). In particular, we compute the amplitudes and positions to leading order. We consider symmetric multi-spike solutions with any number of spikes and asymmetric multi-spike solutions with one small and large spike. We first write down the system for the amplitudes in case of a general number $K$ of spikes, where we have either $K$ spikes of the same amplitude or $k_1$ small and $k_2$ large spikes with $k_1+k_2=K$. We will first solve this system in the case of symmetric spikes. Then we will choose $k_1=k_2=1$ and solve this system in this special case of asymmetric spikes. Integrating the right hand side of the second equation in (\[sys1\]), we compute for $v_j=\lim_{{\varepsilon}\to 0} \hat{v}_{\varepsilon}(t_j^{\varepsilon})$: $$\label{amp1} \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}= (\bar{A}-A_0)2L.$$ Solving the second equation in (\[sys1\]), using (\[ss2\]) in combination with the approximation (\[approx\]), we get $$\hat{v}_{\varepsilon}(x)- \hat{v}_{\varepsilon}(-L)= \int_{-L}^x \frac{1}{ \hat{D} (A_0+\frac{1}{{\varepsilon}}\hat{A}_{\varepsilon})^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}_{\varepsilon}({\varepsilon}A_0+\hat{A}_{\varepsilon})^3-\bar{A}+A_0 \right) dt\,ds$$$$=\int_{-L}^x \frac{1}{\hat{D} A_0^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}_{\varepsilon}({\varepsilon}A_0+\hat{A}_{\varepsilon})^3-\bar{A}+A_0 \right) dt\,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$$$\int_{-L}^x \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} H(s-t_j^{\varepsilon})- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} \right) \frac{s+L}{2L} \right] \,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$$$= \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} (x-t_j^{\varepsilon})- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}}\right) \frac{(x+L)^2}{4L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $$H(x)= \left\{ \begin{array}{l} 1 \quad \mbox{ if } x\geq 0, \\[3mm] 0 \quad \mbox{ if }x<0. \end{array} \right.$$ Taking the limit ${\varepsilon}\to 0$ and setting $x=t_i=\lim_{{\varepsilon}\to 0}t_i^{\varepsilon}$, we derive $$v_i= \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}(t_i-t_j)- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\right) \frac{(t_i+L)^2}{4L} \right]+C_1 $$ $$\label{amp2} =\frac{1}{\hat{D}A_0^2}\left[ \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\frac{1}{2} |t_i-t_j| -\left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\right) \frac{1}{4L}t_i^2 \right]+C_2$$ for some real constants $C_1,\,C_2$ independent of $i$, where the last identity in (\[amp2\]) uses (\[amp3\]) which we now explain. We use an assumption on the position of spikes that can be stated as follows: $$\label{fi} F_i(t_1^0,t_2^0,\ldots,t_K^0):= \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \left( \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \right) \frac{t_i+L}{2L} =0,\quad i=1,\ldots,K.$$ Note that (\[fi\]) will be derived later on in Section 8 below (see equation (\[formulaf\])). We re-write (\[fi\]) and compute $$\frac{1}{2\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{1}{\sqrt{v_j}} -\left(\sum_{j=1}^{K} \frac{1}{\sqrt{v_j}}\right) \frac{t_i+L}{2L}$$ $$\label{amp3} =\sum_{j=1}^{i-1} \frac{1}{2\sqrt{v_j}} - \sum_{j=i+1}^{K} \frac{1}{2\sqrt{v_j}} - \left( \sum_{j=1}^{K} \frac{1}{\sqrt{v_j}} \right) \frac{t_i}{2L}=0.$$ From (\[amp2\]) and (\[amp3\]), we derive $$\label{amp5} v_{i+1}-v_i=\frac{\pi}{\sqrt{2} \hat{D}A_0^2} (t_{i+1}-t_i)\frac{1}{2} \left(\frac{1}{\sqrt{v_i}}- \frac{1}{\sqrt{v_{i+1}}} \right).$$ In the next two subsection we now solve these equations for the amplitudes of the spikes in the cases of both symmetric and asymmetric spikes. Symmetric spikes ---------------- We first consider the case of symmetric spikes, where $v_i=v$ is independent of $i=1,\ldots,K$, and compute the amplitude $v$ and the positions $t_i$. From (\[amp1\]), we get $$v=\frac{\pi^2 K^2}{2(\bar{A}-A_0)^2 L^2}.$$ From (\[amp5\]), the positions are $t_i=\left(-1+\frac{2i-1}{K}\right)L,\quad i=1,\ldots,K.$ The proof of the existence of multiple symmetric spikes follows from the construction of a single spike in the interval $\left(-\frac{L}{K},\frac{L}{K}\right)$. The proof of the existence of a single spike uses the implicit function theorem in the space of even functions, for which the Liapunov-Schmidt reduction method is not needed. This proof can easily be obtained by specializing the proof given for multiple asymmetric spikes given below. In this case, the proof can thus be simplified. Therefore we omit the details. Asymmetric spikes ----------------- Combining (\[amp3\]) and (\[amp5\]), we get $$v_iv_{i+1}=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \frac{L}{2} \frac{1}{\sum_{j=1}^K \frac{1}{\sqrt{v_j}}}.$$ This implies that there are only two different amplitudes which we denote by $v_s\leq v_l$ appearing $k_1$ and $k_2$ times, respectively. Hence we get $$\label{amp6} v_sv_l=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \frac{L}{2} \frac{1}{ \frac{k_1}{\sqrt{v_s}} + \frac{k_2}{\sqrt{v_l}} }.$$ Multiplying (\[amp3\]) by $\frac{t_i}{2}$ and subtracting (\[amp2\]) from the result we get $$v_i-C=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \sum_{j=1}^k\frac{t_j}{\sqrt{v_j}}\mbox{sgn}(t_j-t_i) +\frac{1}{\hat{D}A_0^2} \left(\sum_{j=1}^k\frac{\pi}{\sqrt{2}}\frac{1}{\sqrt{v_j}}\right) \frac{t_i^2}{2L},$$ where $$\mbox{sgn}(\alpha)=\left\{ \begin{array}{ll} +1 & \mbox{ if }\alpha>0, \\[2mm] 0 & \mbox{ if }\alpha=0, \\[2mm] -1 & \mbox{ if }\alpha<0. \end{array} \right.$$ Next we determine $v_s,\,v_l$ from (\[amp1\]) and (\[amp6\]). Substituting (\[amp1\]) into (\[amp6\]), we get $$v_l=\frac{1}{v_s}\frac{\pi^2}{\hat{D}A_0^2}\frac{1}{4} \frac{1}{\bar{A}-A_0}.$$ Plugging this equation into (\[amp1\]) gives $$C\left(z+\frac{1}{z}\right)=1,$$ where $$C =\frac{\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \sqrt{k_1k_2}, \quad z= \frac{\sqrt{2v_s}(\hat{D}A_0^2)^{1/4}(\bar{A}-A_0)^{1/4}}{\sqrt{\pi}} \sqrt{\frac{k_2}{k_1}}.$$ To determine a solution, we need to satisfy the necessary condition $2C<1$ which can be summarized as $$\frac{2\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \sqrt{k_1k_2}<1.$$ The second necessary condition is given by $v_s<v_l$ which is equivalent to $z< \sqrt{\frac{k_2}{k_1}}$. This implies the following cases: [**Case (i):**]{} $k_2\leq k_1$. If $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right) < 1$$ then there exists exactly one solution with $z\leq \sqrt{\frac{k_2}{k_1}}$. On the other hand, if $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)> 1$$ then there exists no solution with $z<\sqrt{\frac{k_2}{k_1}}$. [**Case (ii):**]{} $k_2> k_1$. If $2C>1$ there is no solution. If $2C<1$ and $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)< 1$$ then there exists exactly one solution with $z\leq \sqrt{\frac{k_2}{k_1}}$. If $2C<1$ and $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)> 1$$ then there exist exactly two solutions with $z< \sqrt{\frac{k_2}{k_1}}$. [**Special case:**]{} $k_1=k_2=1$. Finally, we consider the special case $k_1=k_2=1$ which belongs to Case (i) in the previous classification and we have the following results: If $2C< 1$ then there is one solution with $z< 1. $ If $2C> 1$ then there is no solution with $z> 1. $ Existence and nondegeneracy conditions ====================================== We now describe a general scheme of Liapunov-Schmidt reduction. (We refer to the survey paper [@wei-survey] for more details.) Essentially this method divides the problem of solving nonlinear elliptic equations (and systems) into two steps. In the first step, the problem is solved up to multipliers of approximate kernels. In the second step one solves algebraic equations in terms of finding zeroes of the multipliers. In this section, we linearize (\[sys1\]) around the approximate solution and derive the linearized operator as well as its nondegeneracy conditions, i.e. conditions such that the resulting linear operator is uniformly invertible. Linearizing (\[sys1\]) around the solution, we get: $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\phi_{xx}-\phi+3\hat{v}({\varepsilon}A_0+\hat{A})^2\phi+\psi ({\varepsilon}A_0+\hat{A})^3\ \ \mbox{ in }{\Omega}, \\[3mm] \nonumber 0 & =\hat{D} \left( \left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right)^2 \psi_{x}\right)_x +\hat{D} \left(2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x \\[3mm] & -\frac{3}{\epsilon} \hat{v}({\varepsilon}A_0+\hat{A})^2\phi -\frac{1}{\epsilon} \psi ({\varepsilon}A_0+\hat{A})^3 \ \ \mbox{ in }{\Omega}\label{evp}$$ with Neumann boundary conditions $$\phi_x(-L)=\phi_x(L)=\psi_x(-L)=\psi_x(L)=0.$$ Note that for the second equation of (\[evp\]) we have the necessary condition $$\int_{-L}^L \left(-\frac{3}{\epsilon} \hat{v}({\varepsilon}A_0+\hat{A})^2\phi -\frac{1}{\epsilon} \psi ({\varepsilon}A_0+\hat{A})^3 \right)\,dx=0$$ which follows by integrating the equation and using the Neumann boundary conditions for $\psi$ and $v$. In the limit ${\varepsilon}\to 0$ we get $$\sum_{j=1}^K \left[\psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}} +3\int_{-\infty}^\infty w^2\phi_j\,dy \right]=0,$$ where $\psi_j=\psi(x_j)$. The second equation of (\[evp\]) can be solved as follows, using formula (\[ss2\]) and estimate (\[approx\]): $$\psi(x)- \psi(-L)= \int_{-L}^x \frac{1}{ \hat{D} (A_0+\frac{1}{{\varepsilon}}\hat{A})^2} \int_{-L}^s \Bigg[ -\hat{D} \left( 2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x$$$$+\frac{3}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^2\phi +\frac{1}{{\varepsilon}}\psi({\varepsilon}A_0+\hat{A})^3 \Bigg] dt\,ds$$$$=\int_{-L}^x \frac{1}{\hat{D} A_0^2} \int_{-L}^s \left( \frac{3}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^2\phi +\frac{1}{{\varepsilon}}\psi({\varepsilon}A_0+\hat{A})^3 \right) dt\,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ Note that the contributions from the term $$\hat{D} \left( 2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x$$ can be estimated by $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$ since $\phi$ vanishes in the outer expansion. In the limit ${\varepsilon}\to 0$, we get $$\psi(x)- \psi(-L)= \int_{-L}^x \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}}H(s-t_j) + 3\sum_{j=1}^{i-1} \int_{-\infty}^\infty w^2\phi_j\,dy H(s-t_j) \right] \,ds $$ $$=\frac{1}{\hat{D} A_0^2} \left[\sum_{j=1}^{i-1} \psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}}(x-t_j) + 3\sum_{j=1}^{i-1} \int_{-\infty}^\infty w^2\phi_j\,dy\, (x-t_j)\right], \label{limeig}$$ where $t_{i-1}< x\leq t_{i}$. From now on, we consider the case of two spikes having different amplitudes (asymmetric spikes). Using the notation $$\Phi= \left(\begin{array}{l}\phi_1 \\[2mm] \phi_2 \end{array} \right),\quad \Psi= \left(\begin{array}{l}\psi_1 \\[2mm] \psi_2 \end{array} \right),\quad \omega= \left(\begin{array}{l}\omega_1 \\[2mm] \psi_2 \end{array} \right) = \left(\begin{array}{l}-\frac{3}{\hat{D}A_0^2}\int w^2\phi_1\,dy \\[2mm] -\frac{3}{\hat{D}A_0^2}\int w^2\phi_2\,dy \end{array} \right),$$ we can rewrite (\[limeig\]) for $x=t_i$ as follows: $$({\cal B}+{\cal C})\Psi=\boldmath{\omega},$$ where $${\cal C}=\frac{\sqrt{2}\pi}{\hat{D}A_0^2} \left( \begin{array}{cc} \displaystyle \frac{1}{v_s^{3/2}} & 0 \\[3mm] \displaystyle 0 & \displaystyle\frac{1}{v_l^{3/2}} \end{array} \right), \quad {\cal B}=\frac{1}{d_2} \left( \begin{array}{rr} 1 & -1\\ -1 & 1 \end{array} \right), \quad d_2=t_2-t_1.$$ Using ${\cal E}={\cal C}({\cal B}+{\cal C})^{-1}$, we get the following system of nonlocal eigenvalue problems (NLEPs) $$L\Phi:=\Phi_{yy}-\Phi+3w^2\Phi-3w^3\frac{\int w^2{\cal E}\Phi\,dy}{\int w^3\,dy}. \label{vecnlep}$$ Diagonalizing the matrix $\cal E$, we know from [@wei99; @wz] that (\[vecnlep\]) has a nontrivial solution iff ${\cal E}$ has eigenvalue $\lambda_e=\frac{2}{3}$. Thus it remains to compute the matrix ${\cal E}$ and its eigenvalues. We get $${\cal E}^{-1}=({\cal B}+{\cal C}){\cal C}^{-1}= {\cal B}{\cal C}^{-1}+I$$ $$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} \left( \begin{array}{rr} v_s^{3/2} & -v_l^{3/2} \\ -v_s^{3/2} & v_l^{3/2} \end{array} \right) +I.$$ Then ${\cal E}^{-1}$ has the eigenvector $v_{m,1}=\frac{1}{\sqrt{2}}(1,-1)^T$ with eigenvalue $e_{m,1}=\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} (v_s^{3/2}+v_l^{3/2})+1$ and the eigenvector $v_{m,2}=\frac{1}{\sqrt{v_s^3+v_l^3}}(v_l^{3/2},v_s^{3/2})^T$ with eigenvalue $e_{m,2}=1\not=\frac{3}{2}$. For nondegeneracy, the condition $e_{m,1}\not=\frac{3}{2}$ has to be satisfied, which is equivalent to $$\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} (v_s^{3/2}+v_l^{3/2})\not= \frac{1}{2}.$$ Using the formulas for $d_2,\,v_s,\,v_l$, we compute $$\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2}(v_s^{3/2}+v_l^{3/2})$$$$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi L}\left((\sqrt{v_s}+\sqrt{v_l})^3-3(\sqrt{v_s}v_l+v_s\sqrt{v_l})\right)$$$$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi L}\left(\frac{(\bar{A}-A_0)^{3/2} L^3}{2\sqrt{2}(\hat{D}A_0^2)^{3/2}} -3 \frac{\pi L}{2\sqrt{2}\hat{D}A_0^2}\right)$$$$=\frac{(\bar{A}-A_0)^{3/2} L^2}{4\pi\sqrt{\hat{D}A_0^2}}-\frac{3}{4}\not=\frac{1}{2}.$$ This implies the condition $$\frac{2\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L}\not= \frac{2}{\sqrt{5}}.$$ We have to exclude this point from our existence result Theorem \[existenceas\]. This is why we impose the condition (\[condbc\]) in Theorem \[existenceas\], which amounts to a nondegeneracy condition. If this condition is violated, we expect small eigenvalues to occur and it is an open question to know whether there will be spikes in this case. Approximate solutions ===================== For simplicity, we set $L=1$. In this section and the following we consider the case of general $K=1,2,\ldots$ since it does not cause any extra difficulty here, even in the case of asymmetric spikes. Let $ -1<t_1^0 < \cdots < t_j^0 < \cdots t_K^0 <1$ be $K$ points and let $v_j>0$ be $K$ amplitudes satisfying the assumptions (\[amp1\]), (\[amp2\]) and (\[amp3\]). Let $${\bf t}^0=(t_{1}^0,\ldots,t_{K}^0).$$ We first construct an approximate solution to (\[sys1\]) which concentrates near these prescribed $K$ points. Then we will rigorously construct an exact solution which is given by a small perturbation of this approximate solution. Let $-1<t_1<\cdots<t_j<\cdots<t_K<1$ be $K$ points such that $ {\bf t} = (t_1, \ldots, t_K) \in B_{{\varepsilon}^{3/4}} ( {\bf t}^0)$. Set $$\label{app11} w_j (x)= w \left( \frac{x-t_j}{{\varepsilon}} \right),$$ and $$\label{r0} r_0 =\frac{1}{10} \left( \min \left(t_1^0 +1, 1-t_K^0, \frac{1}{2}\min_{i \not = j} |t_i^0 -t_j^0|\right)\right).$$ Let $\chi: {\mathbb{R}}\to [0, 1]$ be a smooth cut-off function such that $\chi(x)=1$ for $|x|<1$ and $\chi(x)=0$ for $|x|>2$. We now define the approximate solution as $$\label{app1} \tilde{w}_j(x)= w_j (x) \chi\left(\frac{x-t_j}{r_0}\right).$$ It is easy to see that $\tilde{w}_j (x)$ satisfies $$\label{a11} {\varepsilon}^2 \tilde{w}_j^{''} - \tilde{w}_j + \tilde{w}_j^3 =\mbox{e.s.t.}$$ in $L^2(-1,1)$, where e.s.t. denotes an exponentially small term. Let $$\label{vector} \hat{A}=w_{{\varepsilon},{\bf t}}(x)= \sum_{j=1}^K \frac{1}{\sqrt{v_j^{\varepsilon}}} \tilde{w_j}(x),\quad \mbox{ where } v_j^{\varepsilon}=T[w_{{\varepsilon},{\bf t}}](t_j^{\varepsilon}),$$ $$\label{v} \hat{v}=T[w_{{\varepsilon},{\bf t}}], $$ where $T[A]$ is defined by (\[ta\]) and ${\bf t} \in B_{{\varepsilon}^{3/4}} ( {\bf t}^0)$. Then by (\[approx\]) we have $$\label{tauiia} v_i^{\varepsilon}:= T[\hat{A}] (t_i^{\varepsilon})= \lim_{{\varepsilon}\to 0} T[w_{{\varepsilon},{\bf t}}] (t_i^{\varepsilon}) +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ \[tauii\] Now let $x=t_i+{\varepsilon}y$. We find for $\hat{A}=w_{{\varepsilon}, {\bf t}}$: $$T[\hat{A}](t_i+{\varepsilon}y)-T[\hat{A}](t_i) =$$$$=\int_{t_i}^{t_i+{\varepsilon}y} \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right)^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^3-\bar{A}+A_0 \right) dt\,ds$$$$={\varepsilon}^2 \int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \int_{0}^{\bar{s}} \frac{1}{{\varepsilon}} \frac{(w(\bar{t}))^3}{\sqrt{v_i^{\varepsilon}}} d\bar{t}\,d\bar{s}$$$$+{\varepsilon}^2 \int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \,d\bar{s} \frac{1}{{\varepsilon}} \left[ \int_{-\infty}^{0} \frac{(w(\bar{t}))^3}{\sqrt{v_i}}\,d\bar{t} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}H(t_i-t_j)- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times \left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right) $$$$= {\varepsilon}P_i^{\varepsilon}(y) +{\varepsilon}\int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \,d\bar{s} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times\left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right), \label{tax}$$ in $L^2({\Omega}_{\varepsilon})$, where $$P_i^{\varepsilon}(y)=\int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \int_{0}^{\bar{s}} \frac{(w(\bar{t}))^3}{\sqrt{v_i^{\varepsilon}}} d\bar{t}\,d\bar{s}$$ using (\[amp1\]). Note that $P_i^{\varepsilon}$ is an even function and the second term is an odd function in $y$. We now derive the following estimate for all $y\geq 0$: $$(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,w_{{\varepsilon}, {\bf t}}^3 \leq C {\varepsilon}\int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}} w(\bar{s})\right)^2}\,d\bar{s}\, w^3(y)$$$$\leq \frac{C}{\hat{D}} {\varepsilon}\int_0^y {\varepsilon}^2 \frac{1}{w^2(\bar{s})}\,d\bar{s}\,w^3(y)$$$$\leq \frac{C}{\hat{D}} {\varepsilon}^3 \int_0^y e^{2\bar{s}} \,d\bar{s} \,e^{-3y}$$$$\leq\frac{C}{\hat{D}} {\varepsilon}^3 e^{-y}.$$ For $y<0$ there is an obvious modification of this estimate. Further, it can be extended to cover both the cases when $w_{{\varepsilon}, {\bf t}}^3$ is replaced by ${\varepsilon}w_{{\varepsilon}, {\bf t}}^2$ or ${\varepsilon}^2 w_{{\varepsilon}, {\bf t}}$, respectively, giving the same upper bound in either case. Now if we define the following norm $$\| f\|_{**}= \| f \|_{L^2 (\Omega_\epsilon)} + \sup_{ - \frac{L}{\epsilon} <y <\frac{L}{\epsilon} } [\max(\min_{i} e^{-\frac{1}{2} |y-\frac{t_i}{\epsilon}| }, \sqrt{\epsilon})]^{-1} | f(y)|$$ then by the decay of $w_{{\varepsilon}, {\bf t}}$ and the definition of the norm, we infer that $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,w_{{\varepsilon}, {\bf t}}^3\|_{**}= O({\varepsilon}^{5/2}), \label{estw3}$$ $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,{\varepsilon}w_{{\varepsilon}, {\bf t}}^2\|_{**}= O({\varepsilon}^{5/2}) \label{estw2},$$ $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,{\varepsilon}^2 w_{{\varepsilon}, {\bf t}}\|_{**}= O({\varepsilon}^{5/2}). \label{estw1}$$ Let us now define $$\label{sa} S_{\varepsilon}[\hat{A}] := {\varepsilon}^{2}\hat{A}_{xx}-\hat{A}+T[\hat{A}] (\varepsilon A_0+\hat{A})^{3}$$ where $T[\hat{A}]$ is defined in (\[ta\]). Next we set $\hat{A}=w_{{\varepsilon}, {\bf t}}$ and compute $S_{\varepsilon}[w_{{\varepsilon}, {\bf t}}]$. In fact, $$S_{\varepsilon}[w_{{\varepsilon}, {\bf t}}]= {\varepsilon}^{2} ({w_{{\varepsilon}, {\bf t}}})_{xx}-w_{{\varepsilon}, {\bf t}}+T[w_{{\varepsilon}, {\bf t}}]\,({\varepsilon}A_0+ w_{{\varepsilon}, {\bf t}})^{3}$$$$={\varepsilon}^2 ({w_{{\varepsilon}, {\bf t}}})_{xx}-w_{{\varepsilon}, {\bf t}}+T[w_{{\varepsilon}, {\bf t}}](t_i)\,w_{{\varepsilon}, {\bf t}}^{3}$$$$+T[w_{{\varepsilon}, {\bf t}}]({\varepsilon}^3 A_0^3+3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$+\left(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i)\right)\, w_{{\varepsilon}, {\bf t}}^{3}$$ $$\label{defe} =:E_1+E_2+E_3.$$ We compute $$E_1= \sum_{i=1}^K \frac{1}{\sqrt{v_{i}^{\varepsilon}}} \left( \tilde{w}_i^{''} -\tilde{w}_i+ \frac{T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_i} \tilde{w}_{i}^3 \right) =\mbox{e.s.t.}$$ in $L_2({\Omega}_{\varepsilon}) $ since $v_i^{\varepsilon}=T[w_{{\varepsilon}, {\bf t}}](t_i)$. Further, we get $$E_2=O({\varepsilon})$$ in $L^2({\Omega}_{\varepsilon}) $ since $T[w_{{\varepsilon}, {\bf t}}]$ is bounded in $L^\infty({\Omega}_{\varepsilon})$ and $w_{{\varepsilon}, {\bf t}}$ is bounded in $L^2({\Omega}_{\varepsilon}) $. We also notice that actually $ E_2= O({\varepsilon}e^{-\min_i (|y-\frac{t_i}{{\varepsilon}}|)})$. Lastly, we derive $$E_3= \sum_{i=1}^K \frac{1}{(v_{i}^{{\varepsilon}})^{3/2}} \left( T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)- T[w_{{\varepsilon}, {\bf t}}](t_i) \right) \tilde{w}_{{\varepsilon}, i}^3 =O({\varepsilon}^3)$$ in $L^2({\Omega}_{\varepsilon})$ by (\[estw3\]). Combining these estimates, we conclude that $$\label{estsa} \|S [ w_{{\varepsilon}, {\bf t}}]\|_{**} = O({\varepsilon}).$$ [The estimate (\[estsa\]) shows that our choice of approximate solution given in (\[vector\]) and (\[v\]) is suitable. This will enable us in the next two sections to rigorously construct a steady state which is very close to the approximate solution. ]{} Liapunov-Schmidt Reduction ========================== In this section, we use Liapunov-Schmidt reduction to solve the problem $$\label{ls} S_{\varepsilon}[ w_{{\varepsilon}, {\bf t}} + v] = \sum_{i=1}^K \beta_i \frac{d\tilde{w}_i}{dx}$$ for real constants $ \beta_i$ and a function $ v\in H^2(-\frac{1}{{\varepsilon}}, \frac{1}{{\varepsilon}})$ which is small in the corresponding norm (to be defined later), where $ \tilde{w}_i$ is given by (\[app1\]) and $w_{{\varepsilon},{\bf t}}$ by (\[vector\]). This is the first step in the Liapunov-Schmidt reduction method. We shall follow the general procedure used in [@ww]. To this end, we need to study the linearized operator $$\tilde{L}_{{\varepsilon}, {\bf t}}: H^2 (\Omega_{\varepsilon}) \to L^2(\Omega_{\varepsilon})$$ given by $$\label{lept} \tilde{L}_{{\varepsilon}, {\bf t}} := S_{\epsilon}^{'} [w_{{\varepsilon}, {\bf t}}]\phi = {\varepsilon}^2\Delta\phi-\phi +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi +(T'[w_{{\varepsilon}, {\bf t}}]\phi) ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3},$$ where for $\hat{A}=w_{{\varepsilon}, {\bf t}}$ and a given function $\phi\in L^2({\Omega})$ we define $T^{'}[\hat{A}]\phi$ to be the unique solution of $$\label{tap} \left\{ \begin{array}{ll} \hat{D} \left( (A_0 +\frac{1}{{\varepsilon}} \hat{A})^2 (T'[\hat{A}]\phi)_{x} \right)_{x} -\frac{1}{{\varepsilon}} (T'[\hat{A}]\phi) ({\varepsilon}A_0 +\hat{A})^{3} -\frac{1}{{\varepsilon}} T[\hat{A}]3 ({\varepsilon}A_0 +\hat{A})^{2}\phi =0,\\ \null \hfill\quad\text{for}\quad -L<x<L, \\[3mm] (T'[\hat{A}]\phi)_{x}(-L)=(T'[\hat{A}]\phi)_{x}(L)=0. \end{array} \right.$$ The norm for the error function $\phi$ is defined as follows $$\label{normdef} \| \phi \|_{*}= \| \phi \|_{H^2 (\Omega_{\varepsilon})} + \sup_{ - \frac{L}{{\varepsilon}} <y <\frac{L}{{\varepsilon}} } [\max (\min_{i} e^{-\frac{1}{2}|y-\frac{t_i}{{\varepsilon}}| }, \sqrt{{\varepsilon}})]^{-1} | \phi (y)|.$$ We define the approximate kernel and co-kernel, respectively, as follows: $${\cal K}_{{\varepsilon}, {\bf t}} := \mbox{span} \ \left\{ \frac{d\tilde{w}_i}{dx} \Bigg| i=1,\ldots,K\right\} \subset H^2 ({\Omega}_{\varepsilon}),$$ $${\cal C}_{{\varepsilon}, {\bf t}} := \mbox{span} \ \left\{ \frac{d \tilde{w}_i}{dx} \Bigg| i=1,\ldots,K\right\} \subset L^2 ({\Omega}_{\varepsilon}).$$ From (\[vecnlep\]) we recall the definition of the following system of NLEPs : $$L\Phi:=\Phi_{yy}-\Phi+3w^2\Phi-3w^3\frac{\int w^2{\cal E}\Phi\,dy}{\int w^3\,dy}, \label{linop1}$$ where $$\Phi= \left( \begin{array}{l} \phi_1\\ \phi_2 \\ \vdots \\ \phi_K \end{array} \right) \in(H^2({\mathbb{R}}))^K.$$ By Lemma 3.3 of [@ww] we know that $$L: (X_0\oplus \cdots \oplus X_0)^\perp \cap (H^2({\mathbb{R}}))^K \to (X_0\oplus \cdots \oplus X_0)^\perp \cap (L^2({\mathbb{R}}))^K$$ is invertible and its inverse is bounded. We will show that this system is the limit of the operator $\tilde{L}_{{\varepsilon}, {\bf t}}$ (defined in (\[lept\])) as ${\varepsilon}\to 0$. We also introduce the projection $\pi_{{\varepsilon}, {\bf t}}^\perp: L^2({\Omega}_{\varepsilon})\to {\cal C}^\perp_{{\varepsilon}, {\bf t}}$ and study the operator $L_{{\varepsilon}, {\bf t}}:= \pi_{{\varepsilon}, {\bf t}}^\perp\circ \tilde{L}_{{\varepsilon}, {\bf t}}$. By letting ${\varepsilon}\to 0$, we will show that $L_{{\varepsilon}, {\bf t}}:\, {\cal K}_{{\varepsilon}, {\bf t}}^\perp \to {\cal C}_{{\varepsilon}, {\bf t}}^\perp$ is invertible and its inverse is uniformly bounded provided ${\varepsilon}$ is small enough. This statement is contained in the following proposition. \[A\] \[mainprop\] There exist positive constants $\bar{{\varepsilon}},\,\bar{\delta}, \lambda$ such that for all ${\varepsilon}\in(0,\bar{{\varepsilon}})$, ${\bf t} \in{\Omega}^K$ with $ \min(| 1+t_1|, |1-t_K|, \min_{i \not =j}|t_i-t_j|)>\bar{\delta}$, $$\label{normesta} \|L_{{\varepsilon}, {\bf t} } \phi \|_{**}\geq \lambda \|\phi\|_{*}.$$ Furthermore, the map $$L_{{\varepsilon},{\bf t}}= \pi_{{\varepsilon},{\bf t}}^\perp\circ \tilde{L}_{{\varepsilon},{\bf t}}:\, {\cal K}_{{\varepsilon},{\bf t}}^\perp \to {\cal C}_{{\varepsilon},{\bf t}}^\perp$$ is surjective. [**Proof of Proposition \[mainprop\]:**]{} This proof uses the method of Liapunov-Schmidt reduction following for example the approach in [@iww2], [@ww] and [@ww13]. Suppose that (\[normesta\]) is false. Then there exist sequences $\{{\varepsilon}_k\},\,\{{\bf t}^k\},\,\{\phi^k\}$ with ${\varepsilon}_k\to 0$, ${\bf t}^k\in{\Omega}^K$, $\min(| 1+t_1^k|, |1-t_K^k|, \min_{i \not =j}|t_i^k-t_j^k|)>\bar{\delta}$, $\phi^k=\phi_{{\varepsilon}_k}\in K_{{\varepsilon}_k,{\bf t}^k}^\perp$, $k=1,2,\ldots$ such that $$\begin{aligned} &&\| L_{{\varepsilon}_k,{{\bf t}^k }} \phi^k\|_{**} \to 0 \label{lpt}\qquad\mbox{as }k\to\infty,\\ &&\| \phi^k\|_{*}=1,{\hspace{1cm}}k=1,\,2,\,\ldots\,. \label{onnorm}\end{aligned}$$ We define $\phi_{{\varepsilon},i}$, $i=1,2,\ldots,K$ and $\phi_{{\varepsilon},K+1}$ as follows: $$\label{phei} \phi_{{\varepsilon},i}(x)=\phi_{\varepsilon}(x) \chi\left(\frac{x-t_i}{r_0}\right),\quad x\in{\Omega},$$ $$\phi_{{\varepsilon},K+1}(x)=\phi_{{\varepsilon}}(x)-\sum_{i=1}^K \phi_{{\varepsilon},i}(x),\quad x\in{\Omega}.$$ At first (after rescaling) the functions $\phi_{{\varepsilon},i}$ are only defined on ${\Omega}_{\varepsilon}$. However, by a standard result they can be extended to ${\mathbb{R}}$ such that their norm in $H^2({\mathbb{R}})$ is still bounded by a constant independent of ${\varepsilon}$ and $\bf t$ for ${\varepsilon}$ small enough. In the following we will deal with this extension. For simplicity of notation we keep the same notation for the extension. Since for $i=1,2,\ldots,K$ each sequence $\{\phi_i^k\}:=\{\phi_{{\varepsilon}_k,i}\}$ ($k=1,2,\ldots$) is bounded in $H^2_{loc}({\mathbb{R}})$ it converges weakly to a limit in $H^2_{loc}({\mathbb{R}})$, and therefore also strongly in $L^2_{loc}({\mathbb{R}})$ and $L^{\infty}_{loc}({\mathbb{R}})$. Denoting these limits by $\phi_i$, then $\phi=\left(\begin{array}{c}\phi_1 \\ \phi_2\\ \vdots \\ \phi_K\end{array}\right)$ solves the system $L \phi=0.$ By Lemma 3.3 of [@ww], it follows that $\phi\in \mbox{Ker}(L)= X_0\oplus \cdots \oplus X_0$. Since $\phi^k\in K_{{\varepsilon}_k,t_k}^\perp$, taking $k\to\infty$, we get $\phi\in \mbox{Ker}(L)^\perp$. Therefore, we have $\phi=0$. By elliptic estimates we get $\|\phi_{{\varepsilon}_k,i}\|_{H^2({\mathbb{R}})} \to 0$ as $k\to\infty$ for $i=1,2,\ldots,K$. Further, $\phi_{{\varepsilon},K+1}\to \phi_{K+1}$ in $H^2({\mathbb{R}})$, where $\Phi_{K+1}$ satisfies $$(\phi_{K+1})_{yy}-\phi_{K+1}=0 \quad\mbox{ in }{\mathbb{R}}.$$ Therefore, we conclude that $\phi_{K+1}=0$ and $\|\phi_{K+1}^k\|_{H^2({\mathbb{R}})} \to 0$ as $k\to \infty$. Once we have $ \| \phi_i \|_{H^2 ({\mathbb{R}})} \to 0$, the maximum principle implies that $ \|\phi_i \|_{*} \to 0$ since the operator $L_{{\varepsilon}, {\bf t}}$ essentially behaves like $ \phi_i^{''} - \phi_i$ for $ |x-t_i|>>{\varepsilon}$. This contradicts the assumption that $\|\phi^k\|_{*}=1$. To complete the proof of Proposition \[A\], we just need to show that the conjugate operator to $L_{{\varepsilon}, {\bf t}}$ (denoted by $ L_{{\varepsilon}, {\bf t}}^*$) is injective from ${\cal K}_{{\varepsilon}, {\bf t}}^\perp $ to ${\cal C}_{{\varepsilon}, {\bf t}}^\perp$. Note that $ L_{{\varepsilon},{\bf t}}^*\phi=\pi_{{\varepsilon},{\bf t}}\circ \tilde{L}_{{\varepsilon},{\bf t}}^*\phi$ with $$\tilde{L}_{{\varepsilon},{\bf t}}^*\phi:= {\varepsilon}^2\Delta\phi-\phi +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi +(T'[w_{{\varepsilon}, {\bf t}}]\phi ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3}).$$ The proof for $ L_{{\varepsilon}, {\bf t}}^*$ follows the same lines as the proof for $ L_{{\varepsilon}, {\bf t}}$ and is therefore omitted. Here also the nondegeneracy condition (\[condbc\]) is required. For further technical details we refer to [@ww]. [$\Box$]{} Now we are in the position to solve the equation $${\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ S_{{\varepsilon}} ( w_{{\varepsilon}, {\bf t}} + \phi) =0. \label{solve}$$ Since $ L_{{\varepsilon}, {\bf t}}|_{K_{{\varepsilon}, {\bf t}}^{\perp}}$ is invertible (call the inverse $L^{-1}_{{\varepsilon}, {\bf t}}$) ,we can rewrite this equation as $$\phi=-(L_{{\varepsilon}, {\bf t}}^{-1} \circ {\cal \pi}_{{\varepsilon},{\bf t}}^\perp \circ S_{{\varepsilon}} (w_{{\varepsilon}, {\bf t}}) ) - ( L_{{\varepsilon}, {\bf t}}^{-1}\circ {\cal \pi}_{{\varepsilon}, {\bf t}}^\perp\circ N_{{\varepsilon}, {\bf t}}(\phi))\equiv M_{{\varepsilon}, {\bf t}}(\phi), \label{fix}$$ where $$\label{nep} N_{{\varepsilon},{\bf t}}(\phi)=S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}} +\phi ) -S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}}) -S_{{\varepsilon}}^{'} (w_{{\varepsilon}, {\bf t}}) \phi$$ and the operator $M_{{\varepsilon}, {\bf t}}$ has been defined by (\[fix\]) for $\phi\in H^2 (\Omega_{{\varepsilon}})$. The strategy of the proof is to show that the operator $M_{{\varepsilon}, {\bf t}}$ is a contraction on $$B_{{\varepsilon},\delta}\equiv\{\phi\in H^2(\Omega_{{\varepsilon}}) \,:\, \|\phi\|_{*}<\delta\}$$ if ${\varepsilon}$ is small enough and $\delta$ is suitably chosen. By (\[estsa\]) and Proposition \[A\] we have that $$\|M_{{\varepsilon}, {\bf t}}(\phi)\|_{*} \leq\lambda^{-1} \left(\|{\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ N_{{\varepsilon}, {\bf t}}(\phi) \|_{**} +\left\|{\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ S_{\varepsilon}( w_{{\varepsilon}, {\bf t}} )\right\|_{**}\right)$$$$\leq \lambda^{-1}C_0( c(\delta)\delta +{\varepsilon}),$$ where $\lambda>0$ is independent of $\delta>0$, ${\varepsilon}>0$ and $c(\delta)\to 0$ as $\delta\to 0$. Similarly, we show that $$\|M_{{\varepsilon},{\bf t}}(\phi)-M_{{\varepsilon},{\bf t}}(\phi^{'})\|_{*} \leq \lambda^{-1}C_0( c(\delta)\delta)\|\phi-\phi^{'}\|_{*},$$ where $c(\delta)\to 0$ as $\delta\to 0$. Choosing $\delta= C_3{\varepsilon}\mbox{ for $\lambda^{-1}C_0<C_3$}$ and taking ${\varepsilon}$ small enough, then $M_{{\varepsilon}, {\bf t}}$ maps $B_{{\varepsilon},\delta}$ into $B_{{\varepsilon},\delta}$ and it is a contraction mapping in $B_{{\varepsilon}, \delta}$. The existence of a fixed point $\phi_{{\varepsilon}, {\bf t}}$ now follows from the standard contraction mapping principle and $\phi_{{\varepsilon}, {\bf t}}$ is a solution of (\[fix\]). [$\Box$]{} We have thus proved There exist $\overline{{\varepsilon}}>0$ $\overline{\delta}>0$ such that for every pair of ${\varepsilon}, {\bf t}$ with $0<{\varepsilon}<\overline{{\varepsilon}}$ and ${\bf t}\in{\Omega}^K$, $1+t_1>\overline{\delta}$, $1-t_K>\overline{\delta}$, $\frac{1}{2}|t_i-t_j|> \overline{\delta}$ there is a unique $\phi_{{\varepsilon}, {\bf t}}\in K_{{\varepsilon}, {\bf t}}^{\perp}$ satisfying $S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon},{\bf t}} ) \in {\cal C}_{{\varepsilon}, {\bf t}}$. Furthermore, the following estimate holds $$\|\phi_{{\varepsilon}, {\bf t}}\|_{*}\leq C_3{\varepsilon}. \label{estphi}$$ \[lem34\] In the next section we determine the positions of the spikes so that the resulting steady state is an exact solution of the original problem. The reduced problem =================== In this section we solve the reduced problem and complete the proof of the existence result for asymmetric spikes in Theorem \[existenceas\]. By Lemma \[lem34\], for every $ {\bf t} \in B_{{\varepsilon}^{3/4}} ({\bf t}^0)$, there exists a unique $\phi_{{\varepsilon}, {\bf t} } \in {\cal K}_{{\varepsilon}, {\bf t}}^\perp$, solution of $$\label{see} S [ w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon}, {\bf t} } ] = v_{{\varepsilon}, {\bf t}} \in {\cal C}_{{\varepsilon}, {\bf t}}.$$ The idea here is to find ${\bf t}^{\varepsilon}=(t_1^{\varepsilon},\ldots,t_K^{\varepsilon})$ near ${\bf t}^0$ such that also $$\label{see2} S [w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ] \perp {\cal C}_{{\varepsilon}, {\bf t}^{\varepsilon}}$$ (and therefore $S [w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ]=0$). To this end, we let $$W_{\epsilon,i}( {\bf t}):=\frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [w_{{\varepsilon}, {\bf t}} +\phi_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx} \, dx ,$$ $$W_{\epsilon}({\bf t}):=(W_{\epsilon,1}({\bf t}),..., W_{\epsilon,K}({\bf t}) ) : B_{ {\varepsilon}^{3/4}} ({\bf t}^0) \to {\mathbb{R}}^K.$$ Then $W_{\epsilon}({\bf t})$ is a map which is continuous in ${\bf t}$ and our problem is reduced to finding a zero of the vector field $W_{\varepsilon}({\bf t})$. Let us now calculate $W_{\epsilon}({\bf t})$ as follows: $$W_{{\varepsilon}, i} ( {\bf t})= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [ w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx}$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [ w_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx}$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S_{\varepsilon}^{'}[w_{{\varepsilon}, {\bf t}}] \phi_{{\varepsilon}, {\bf t}} \frac{d \tilde{w}_i}{dx}$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L N_{\varepsilon}(\phi_{{\varepsilon}, {\bf t}}) \frac{d \tilde{w}_i}{dx}$$ $$= :I_1 + I_2+ I_3,$$ where $I_1, I_2$ and $I_3$ are defined in an obvious way in the last equality. We will now compute these three integral terms as $\epsilon \to 0$. The result will be that $I_1$ is the leading term and $I_2 $ and $ I_3$ are $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$. For $I_1$, we have $$I_1= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (E_1+E_2+E_3) \frac{d\tilde{w}_i}{dx} \, dx = \frac{v_i}{{\varepsilon}^2} \int_{-L}^L E_3 \frac{d\tilde{w}_i}{dx} \, dx +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $E_1,\,E_2,\,E_3$ are defined in (\[defe\]). For $E_1$ this estimate is obvious. For $E_2$, we use the decomposition $$T[w_{{\varepsilon}, {\bf t}}]({\varepsilon}^3 A_0^3+3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$=T[w_{{\varepsilon}, {\bf t}}]{\varepsilon}^3 A_0^3$$$$+ T[w_{{\varepsilon}, {\bf t}}](t_i)(3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$+ (T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)- T[w_{{\varepsilon}, {\bf t}}](t_i))(3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2).$$ Then we can estimate the first part directly, the second part using the fact that it is an even function in $y$ and the third part using the estimates (\[estw2\]) and (\[estw1\]). From (\[estw3\]), we derive $$\frac{v_i}{{\varepsilon}^2} \int_{-L}^L E_3 \frac{d\tilde{w}_i}{dx} \,dx$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-L/{\varepsilon}}^{L/{\varepsilon}} P_i(y)w^3 (y) \frac{w^{'} (y)}{\sqrt{v_i}} \,dy$$$$+\frac{v_i}{{\varepsilon}^2} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times \int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(\bar{s})\right)^2}\,d\bar{s} ({\varepsilon}A_0+\hat{A})^3 \frac{w^{'} (y)}{\sqrt{v_i}} \chi_i \,dy$$$$+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$ $$= \mbox{e.s.t.}- \frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $\chi_i(x)=\chi\left(\frac{x-t_i}{r_0}\right)$. Here we have used the fact that $P_i(y)$ is an even function and have computed the following integral $$\int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(t_i+{\varepsilon}\bar{s})\right)^2}\,d\bar{s} ({\varepsilon}A_0+\hat{A})^3 \frac{w^{'} (y)}{\sqrt{v_i}} \chi_i(t_i+{\varepsilon}y) \,dy$$$$=\int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(t_i+{\varepsilon}\bar{s})\right)^2}\,d\bar{s} \frac{1}{4}\frac{d}{dy}({\varepsilon}A_0+\hat{A})^4 \chi_i(t_i+{\varepsilon}y) \,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}} \int_{-L/{\varepsilon}}^{L/{\varepsilon}} \frac{1}{4}\left({\varepsilon}A_0+\hat{A}(t_i+{\varepsilon}y)\right)^2 \,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}}\int_{{\mathbb{R}}}\frac{(w(y))^2}{4v_i}\,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}v_i} \int_{{\mathbb{R}}} \frac{1}{2\cosh^2 y}\,dy +O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{ \hat{D}v_i}+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right).$$ In summary, we have $$\label{esti1} I_1=-\frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ For $I_2$, we calculate $$I_2= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S^{'}[w_{{\varepsilon}, {\bf t}}] (\phi_{{\varepsilon}, {\bf t}}) \frac{d\tilde{w}_i}{dx}$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \left[ {\varepsilon}^2\Delta\phi_{{\varepsilon}, {\bf t}}-\phi_{{\varepsilon}, {\bf t}} +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi_{{\varepsilon}, {\bf t}} +(T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}}) ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3} \right] \frac{d\tilde{w}_i}{dx}$$$$= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \left[ {\varepsilon}^2\Delta \frac{d\tilde{w}_i}{dx} - \frac{d\tilde{w}_i}{dx} + 3\tilde{w}_{i}^{2} \frac{T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_i}d\frac{\tilde{w}_i}{dx} \right] \phi_{{\varepsilon}, {\bf t}}\,dx$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \frac{T[w_{{\varepsilon}, {\bf t}}](x)-T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_{i}} \,3\tilde{w}_{i}^{2} \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx} \,dx$$$$+\frac{v_i}{{\varepsilon}^2} \int_{-L}^L T[w_{{\varepsilon}, {\bf t}}](t_i)3\left({\varepsilon}^2 A_0^2+2{\varepsilon}A_0\frac{\tilde{w}_i}{\sqrt{v_i}}\right) \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx} \,dx$$$$+\frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T[w_{{\varepsilon}, {\bf t}}](x)-T[w_{{\varepsilon}, {\bf t}}](t_i))3\left({\varepsilon}^2 A_0^2+2{\varepsilon}A_0\frac{\tilde{w}_i}{\sqrt{v_i}}\right) \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx}$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}}) {\varepsilon}^3A_0^3 \frac{d\tilde{w}_i}{dx} \,dx$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i) \left( 3{\varepsilon}^2 A_0^2 \frac{\tilde{w}_i}{\sqrt{v_i}}+3 {\varepsilon}A_0 \frac{\tilde{w}_{i}^{2}}{v_i}+\frac{\tilde{w}_{i}^{2}}{v_i^{3/2}} \right) \frac{d\tilde{w}_i}{dx} \,dx$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L [ (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(x) - (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i) ]$$$$\times \left( 3{\varepsilon}^2 A_0^3 \frac{\tilde{w}_i}{\sqrt{v_i}}+3 {\varepsilon}A_0 \frac{\tilde{w}_{i}^{2}}{v_i}+\frac{\tilde{w}_{i}^{2}}{v_i^{3/2}} \right) \frac{d\tilde{w}_i}{dx} \,dx$$$$=I_2^1+I_2^2+I_2^3+I_2^4+I_2^5+I_2^6+I_2^7.$$ With obvious notations, we now show that each one of the seven terms is $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$ as $\epsilon \to 0$. For $I_2^1$, it follows from $T[w_{{\varepsilon}, {\bf t}}]=v_i$, while for $I_2^2$, we use (\[estw3\]) and the fact that $$\left\|\frac{\phi_{{\varepsilon},{\bf t}}}{w_{{\varepsilon},{\bf t}}}\right\|_{L^\infty({\Omega}_{\varepsilon})}\leq C{\varepsilon}. \label{estphiw}$$ For $I_2^3$, we use $\|T[w_{{\varepsilon},{\bf t}}]\|_{L^\infty({\Omega}_{\varepsilon})}=O(1)$ and the fact that $\phi_{{\varepsilon},{\bf t}}$ is an even function. For $I_2^4$, we use (\[estw2\]), (\[estw1\]) and (\[estphiw\]). For $I_2^5$, the estimate is derived from $\|T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}\|_{L^\infty({\Omega}_{\varepsilon})}=O({\varepsilon})$. For $I_2^6$, we use $(T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i)=O({\varepsilon})$ and the fact that $\tilde{w}_i$ is even. Lastly, for $I_2^7$, we use estimates similar to (\[estw3\]), (\[estw2\]), (\[estw1\]) with $T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}$ instead of $T[w_{{\varepsilon},{\bf t}}]$ and the inequality $$\left\|\frac{T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}}{T[w_{{\varepsilon},{\bf t}}]}\right\|_{L^\infty({\Omega}_{\varepsilon})}\leq C{\varepsilon}.$$ By arguments similar to the ones for $I_2$, we derive $$\label{i33} I_3 = O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right) \quad\mbox{ in } L^2({\Omega}_{\varepsilon}).$$ Combining the estimates for $I_1$, $I_2$ and $I_3$, we have $$W_{{\varepsilon},i} ({\bf t})= -\frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ $$\label{formulaf} = -\frac{1}{\hat{D}} F_i ({\bf t}) + O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $F_i ({\bf t})$ was defined in (\[fi\]). By assumption (\[amp3\]), we have $ F ({\bf t}^0) =0$. Next we show that $$\det ( \nabla_{{\bf t}^0} F ({\bf t}^0)) \not = 0$$ in the case of two spikes with amplitudes $v_1=v_s<v_l=v_2$. We compute $$\nabla_{{\bf t}} F ({\bf t})= D_{\bf t} F+(D_{v} F) (D_{\bf t} v),$$ where $$D_{\bf t} F =-\left(\sum_{j=1}^2 \frac{\sqrt{2}\pi}{\sqrt{v_j}}\frac{1}{2L}\right){\cal I},$$$$D_{v} F=\frac{\sqrt{2}\pi}{4} \left( \begin{array}{cc} 0 & v_2^{-3/2} \\[3mm] -v_1^{-3/2} & 0 \end{array} \right),$$$$D_{\bf t} v= \frac{\pi}{\sqrt{2}\hat{D} A_0^2} \, \frac{1}{\left(\left(\frac{v_2}{v_1}\right)^{3/2}+1-\frac{\pi L}{\sqrt{2}\hat{D}A_0^2 v_1^{3/2}}\right)} \left( \begin{array}{cc} v_1^{-1/2} & v_2^{-1/2} \\[3mm] -v_1^{-2} v_2^{3/2} & -v_1^{-3/2} v_2 \end{array} \right).$$ This implies, using (\[amp1\]) and (\[amp6\]), that $$\nabla_{{\bf t}^0} F ({\bf t}^0)= \frac{\pi^2}{4 \hat{D} A_0^2}\, \frac{1}{\left(\left(\frac{v_2}{v_1}\right)^{3/2}+1-\frac{\pi L}{\sqrt{2}\hat{D}A_0^2 v_1^{3/2}} \right)}$$$$\times \left( \begin{array}{cc} -v_1^{-5/2}v_2^{1/2} -v_1^{-1}v_2^{-1}+v_1^{-2}+2v_1^{-3/2}v_2^{-1/2} & -v_1^{-3/2}v_2^{-1/2} \\[3mm] -v_1^{-2} & -v_1^{-5/2}v_2^{1/2} -v_1^{-1}v_2^{-1}+2v_1^{-2}+v_1^{-3/2}v_2^{-1/2} \end{array} \right).$$ Next, we compute $$\det ( \nabla_{{\bf t}^0} F ({\bf t}^0))= \frac{\pi^4}{16 (\hat{D} A_0^2)^2}\,\frac{1}{v_1^2v_2^2}\, \frac{1}{\left(\alpha^3-2\alpha^2-2\alpha+1 \right)^2}$$$$\times \left( (\alpha^3-\alpha^2-2\alpha+1)(\alpha^3-2\alpha^2-\alpha+1)-\alpha^3 \right)$$ $$= \frac{\pi^4}{16 (\hat{D} A_0^2)^2}\, \frac{1}{v_1^2v_2^2} \,\frac{\alpha^3-\alpha^2-\alpha+1}{\alpha^3-2\alpha^2-2\alpha+1},$$ where $\alpha=\sqrt{\frac{v_2}{v_1}}$. Therefore, we have $\det ( \nabla_{{\bf t}^0} F ({\bf t}^0))\not = 0$, except for two specific positive values of $\alpha$: $\alpha=1$ (the bifurcation point of asymmetric from symmetric spikes which is not included in Theorem \[existenceas\]) and $\alpha=\frac{1+\sqrt{5}}{2}$ (corresponding to the eigenvalue $e_{m,1}=\frac{3}{2}$ in Section 5 which has been excluded from Theorem \[existenceas\]). Thus, under the conditions of Theorem \[existenceas\], we get $$W_{\varepsilon}({\bf t}) = -\frac{1}{\hat{D}} \nabla_{{\bf t}^0} F ({\bf t}^0)({\bf t}-{\bf t}^0) + O\left(|{\bf t} -{\bf t}^0|^2+{\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ Since $W_{\varepsilon}({\bf t})$ is continuous in $\bf t $, standard degree theory [@danclec] implies that for $\epsilon$ small enough and $\delta$ suitable chosen there exist ${\bf t^{\epsilon}}\in B_\delta({\bf t}^0)$ such that $W_{\epsilon}({\bf t^{\epsilon}})=0$ and ${\bf t^{\varepsilon}} \to {{\bf t}^0}$. For further technical details of the argument, we refer to [@ww13]. [$\Box$]{} Thus we have proved the following proposition. \[redprob\] For $\epsilon$ small enough, there exist points ${\bf t}^{\epsilon}$ with ${ \bf t}^{\epsilon}\to { \bf t}^0$ such that $W_{\epsilon}({\bf t}^{\varepsilon})=0$. \[conver\] Finally, we complete the proof of Theorems \[existencesym\] and \[existenceas\]. [**Proof of Theorem \[existenceas\]:**]{} By Proposition \[redprob\], there exist ${\bf t}^{\varepsilon}\to {\bf t}^0$ such that $ W_{\varepsilon}({\bf t}^{\varepsilon}) =0$. In other words, $S [w_{{\varepsilon}, {\bf t}^{\varepsilon}}+\phi_{{\varepsilon}, {\bf t}^{\varepsilon}}] =0$. Let $\hat{A}_{\varepsilon}= w_{{\varepsilon}, {\bf t}^{\varepsilon}} +\phi_{{\varepsilon}, {\bf t}^{\varepsilon}},\, \hat{v}_{\varepsilon}= T[w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ]$. By the maximum principle, we conclude that $\hat{A}_{{\varepsilon}} >0,\, \hat{v}_{\varepsilon}>0$. Moreover $(\hat{A}_{\varepsilon}, \hat{v}_{\varepsilon})$ satisfies all the properties of Theorem \[existenceas\]. [$\Box$]{} [**Proof of Theorem \[existencesym\]:**]{} To prove Theorem \[existencesym\], we first construct a single spike in the interval $\left(-\frac{L}{K},\frac{L}{K}\right)$ as above. Then we continue the single spike periodically to a function in the interval $(-L,L)$ and get a symmetric multiple spike in the interval $(-L,L)$. [$\Box$]{} Proof of Theorem \[existenceani\] ================================= The proof of Theorem \[existenceani\] goes exactly as that of Theorem \[existenceas\]. First, let us derive the location of the single spikes formally: in the first equation in (\[sys1\]) the term $\varepsilon^3 A_0^{''}$ is very small and can be omitted in the computations. Thus we may assume that $$\label{n100} \hat{A} \sim \xi^{-1/2} w\left(\frac{x-t_0}{\varepsilon}\right), \ v(t^0)= \xi$$ Substituting the above expressions into the second equation of (\[sys1\]) and noting that $ \hat{v} \frac{1}{\varepsilon} (\varepsilon A_0 +\hat{A})^3 \sim \xi^{-1/2} (\int w^3) \delta_{t_0}$, we see that $\hat{v}$ satisfies in leading order $$\hat{D} ( A_0^2 \hat{v}_x)_{x} - \xi^{-1/2} \left(\int_{\mathbb{R}}w^3\,dy\right) \delta_{t_0} + \gamma (x)=0.$$ Solving the above equation, we then obtain $$\label{eqn100} \hat{v}_x (t_0 -)= - \frac{1}{\hat{D} (A_0 (t_0))^2} \int_{-L}^{t_0} \gamma (x) dx, \ \ \hat{v}_x (t_0 +)= \frac{1}{\hat{D} (A_0 (t_0))^2} \int_{t_0}^{L} \gamma (x) dx.$$ Substituting (\[n100\]) into the first equation of (\[sys1\]) and rescaling $x= t_0+\varepsilon y$, we deduce that the error becomes $$\varepsilon ( \hat{v}_x (t_0-) y^{-} + \hat{v}_x (t_0+) y^{+}) w^3 (y) + \varepsilon A_0(t_0) 3 w^2 +O(\varepsilon^2),$$ where $y^{-}=\min(y,0)$ and $y^{+}=\max(y,0)$. from which we conclude that a necessary condition for the existence of a spike at $t_0$ is that $$\int_{\mathbb{R}}\left[\varepsilon ( \hat{v}_x (t_0-) y^{-} + \hat{v}_x (t_0+) y^{+}) w^3 (y) + \varepsilon A_0(t_0) 3w^2 +O(\varepsilon^2)\right] w^{'} (y)\,dy=0$$ whence $$\hat{v}_x (t_0-)+\hat{v}_x (t_0+)=0$$ which is equivalent to (\[tep000\]). It turns out that (\[tep000\]) is also sufficient, since the derivative of $\int_{-L}^{t_0} \gamma (x) dx -\int_{t_0}^L \gamma (x) dx$ with respect to $t_0$ is $\gamma (t_0)$ which is strictly positive. The rest of the proof goes exactly as in the proof of Theorem \[existenceas\]. We omit the details. Discussion ========== In this article we have provided a rigorous mathematical analysis of the formation of spikes in the model of Short, Bertozzi and Brantingham [@sbb]. Thus, we have shown that this model naturally leads to the formation of criminality hot-spots. The existence of such hotspots is one of the main stylized facts about criminality. It is observed for an array of criminal activity types. Hot-spots are extensively reported and discussed in the criminology literature. We refer for example to the articles [@je] and [@bb] as well as to the references therein. Now, the fact that a mathematical model yields such hotspots can be viewed as passing one benchmark of validity. The findings in our paper provide such a test for the Short, Bertozzi and Brantingham model [@sbb]. Furthermore, the rigorous analysis carried here sheds light on the mechanisms for the formation of hotspots in this model and the way it quantitatively depends on the parameters. This type of analysis can then be applied to study issues such as the reduction of hotspots by crime prevention strategies or optimal use of resources to this effect. One of the goals is to understand when policing strategies actually reduce criminal activity and when they merely displace hot-spots to new areas. In this paper we have proved three main new results. First, we showed that we can reduce the quasilinear chemotaxis problems to a Schnakenberg type reaction-diffusion system and derived the existence of symmetric $k$ spikes. Next, we established the existence of asymmetric spikes in the isotropic case. Lastly, we have studied the pinning effect by inhomogeneous media $A_0 (x)$ and $\gamma (x)$. The stability of these spikes is an interesting open issue. In [@kww1] spikes in two space dimensions are considered by formal matched asymptotics. Our approach of rigorous justification can be extended to that case in a radially symmetric setting, i.e. if the domain is a disk and we construct a single spike located at the centre. We remark that in [@kww1] it is assumed that in the outer region (away from the spikes) the system in leading order is semi-linear which allows an extension of the results for the Schnakenberg model to this case. In [@kww1], for the inner region, a numerical computation by solving a core problem yields the profile of the spike. An alternative approach to the problem in one space dimension would be to write it as a first-order semi-linear ODE system and then apply standard methods, e.g. dynamical system methods for the problem on the real line. This approach becomes cumbersome when we impose Neumann boundary conditions and we add inhomogeneity. We nevertheless refer to a recent paper [@HDK], where the dynamical systems approach is used to construct traveling wave solutions of a quasi-linear reaction-diffusion-mechanical system. We remark that there are very few results concerning the analysis of spikes in quasi-linear reaction diffusion systems. As far as we know, there are two such types of systems. The first one is the chemotaxis system of Keller-Segel type. We refer to [@HP] for the background of chemotaxis models and [@KW2] for the analysis of spikes to these systems. The other one is the Shigesada-Kawasaki-Teramoto model of species segregation ([@SKT]). For the analysis of spikes in a cross-diffusion system, we refer to [@KW1], [@LN] and [@WX]. A family of related models for the diffusion of criminality has been proposed in [@bn]. We analyze the formation of hot spots in this class of models in our forthcoming work [@bw]. The equations in [@bn] also envision the possibility of non-local diffusion. Indeed, social influence can be exercised at long range and it is natural to consider descriptions that take long range diffusion into account. Such a non-local system arising in [@bn] reads: $$\label{crime_non-local} \begin{cases} &s_t(x,t) ={\mathcal L} s(x,t) -s(x,t)+s_b+\alpha(x) u(x,t)\\ &u_t(x,t) = \Lambda(s) - u(x,t). \end{cases}$$ The case when ${\mathcal L}=\Delta $ is a local diffusion operator provides the framework of the study in [@brr]. Here, ${\mathcal L}$ can also be a non-local operator such as the fractional Laplace operator or a general non-local interaction term: $${\mathcal L} s (x,t)= \int J(x,y) (s(y,t) - s(x,t) ) dy.$$ Observe that the steady states reduce to a single non-local equation: $$\label{eqs-stat} - {\mathcal L} s = s_b (x) -s +\alpha(x) \Lambda (s).$$ We note that the interaction between non-local diffusion and the mechanism for the formation of spikes is completely open. In particular, the description of the formation of spikes in (\[crime\_non-local\]) and (\[eqs-stat\]) are open problems. We expect that the decay of the kernel may come into play for the formation of spikes. [**Acknowledgment.**]{} The research of Henri Berestycki leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi -Reaction-Diffusion Equations, Propagation and Modelling. Part of this work was done while Henri Berestycki was visiting the Department of Mathematics at University of Chicago. He was also supported by an NSF FRG grant DMS - 1065979, “Emerging issues in the Sciences Involving Non-standard Diffusion”. Juncheng Wei was supported by a GRF grant from RGC of Hong Kong and a NSERC Grant from Canada. Matthias Winter thanks the Department of Mathematics of The Chinese University of Hong Kong for its kind hospitality. Lastly, the authors are thankful to the referees for careful reading of the manuscript and many constructive suggestions. [9]{} H. Berestycki and J. Nadal, Self-organised critical hotspots of criminal activity, [*European J.Appl. Math.*]{} 21 (2010), pp. 371-399. H. Berestycki , N. Rodriguez and L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity, [*Multiscale Modeling and Simulation (SIAM)*]{}, (2013), DOI: 10.1137/12089884X. H. Berestycki, M. Winter and J. Wei, in preparation. R. Cantrell, C. Cosner and R. Manasevich, Global bifurcation of solutions for crime modeling equations, [*SIAM J. Math. Anal.*]{} 44 (2012), 1340-1358. E.N. Dancer, Degree theory on convex sets and applications to bifurcation, appeared in: L. Ambrosio and E.N. Dancer, Calculus of Variations and Partial Differential Equations, Topics on Geometrical Evolutions Problems and Degree Theory, edited by G. Buttazzo, A. Marino, M.K.V. Murphy, Springer, 1999. P. Brantingham and P. Brantingham, Theoretical model of crime hot spot generation, [*Studies om Crime and Crime Prevention*]{}, 8 (1999), 7Ð26. A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, [*Indiana Univ. Math. J.*]{} 49 (4) (2000), 213-245. A. Doelman, T.J. Kaper and H. van der Ploeg, Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation, [*Methods Appl. Anal.*]{} 8 (2001), 387-414. J. Eck, Crime Hot Spots: What They Are, Why We Have Them, and How to Map Them, Chapter 1 in [*Mapping crime: Understanding hotspots,*]{} J. Eck, S. Chainey, J. Cameron and R. Wilson ed. National Institute of Justice, Washington DC, (2005). M. Holzer, A. Doelman and T.J. Kaper, Existence and stability of traveling pulses in a reaction-diffusion-mechanics system, [*J. Nonlinear Sci.*]{} 23 (2013), 129-177. T. Hillen and K.J. Painter, A user’s guide to PDE models for chemotaxis, [*J. Math. Biol.*]{} 58 (2009), 183–217. D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model. [*J. Math. Biol.*]{} 49 (2004), 358–390. T. Kolokolnikov, M.J. Ward and J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime, [*DCDS-B*]{}, to appear. T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, [*SIAM J. Appl. Math.*]{} 71 (2011), 1428-1457. T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, under revision in [*SIAM J. Appl. Math.*]{}. Y. Lou and W.-M. Ni, Diffusion vs. cross-diffusion: an elliptic approach, [*J. Differential Equations*]{} 154 (1999), 157–190. N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, [*Math. Models Methods Appl. Sci.*]{} 20 (2010), suppl. 1, 1425–1457. M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime - hotspots, bifurcations, and suppression, [*SIAM J. Appl. Dyn. Systems*]{} 9 (2010), 462–483. M. B. Short, A. L. Bertozzi, P. J. Brantingham and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion model of crime, [*Proc. Nat. Acad. Sci. USA*]{} 107 (2010), 3961-3965. M. B. Short, M.R. D’Orsogna, V.B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of crime behavior, [*Math. Methods Appl. Sci.* ]{} 18 (2008), 1249-1267. N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, [*Journal of Theoretical Biology* ]{} 79 (1979), 83-99. M.J. Ward and J. Wei, Asymmetric spike patterns for the Schnakenberg model [*Stud. Appl. Math.*]{} 109 (2002), 229–264. J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness, spectrum estimates, [*Europ. J. Appl. Math.*]{} 10 (1999), 353–378. J. Wei, Existence and Stability of Spikes for the Gierer-Meinhardt System, in [*HANDBOOK OF DIFFERENTIAL EQUATIONS, Stationary partial differential equations,*]{} volume 5 (M. Chipot ed.), Elsevier, pp. 489-581. J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $\mathbf R^1$, [*Methods Appl. Anal.*]{} 14 (2007), 119–163. J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, [*Discr. Cont. Dyn. Syst. A, Special issue for Prof. Mimura’s 65th Birthday*]{} 25 (2009), no.1, 363-398. J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, [*J. Math. Biol.*]{} 57 (2008), 53–89. J. Wei and M. Winter, [*Mathematical Aspects of Pattern Formation in Biological Systems*]{}, Applied Mathematical Sciences 189, Springer, London, 2014. J. Wei and L. Zhang, On a nonlocal eigenvalue problem, [*Ann. Scuola Norm. Sup. Pisa Cl. Sci.*]{} 30 (2001), 41–61. Yaping Wu and Qian Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, [*Discr. Cont. Dyn. Syst. A*]{} 29 (2011), 367-385. [^1]: École des hautes études en sciences sociales, CAMS, 190-198, avenue de France, 75244 Paris cedex 13, France. ([[email protected]]{}). [^2]: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China, and Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2. ([[email protected]]{}). [^3]: Brunel University, Department of Mathematical Sciences, Uxbridge UB8 3PH, United Kingdom ([[email protected]]{})

ArXiv

--- abstract: | We have performed [[*HST*]{}]{} imaging of a sample of 23 high-redshift ($1.8<z<2.75$) Active Galactic Nuclei, drawn from the [[combo-17]{}]{}survey. The sample contains moderately luminous quasars ($M_B \sim -23$). The data are part of the [[gems]{}]{} imaging survey that provides high resolution optical images obtained with the Advanced Camera for Surveys in two bands ([F606W]{} and [F850LP]{}), sampling the rest-frame UV flux of the targets. To deblend the AGN images into nuclear and resolved (host galaxy) components we use a PSF subtraction technique that is strictly conservative with respect to the flux of the host galaxy. We resolve the host galaxies in both filter bands in 9 of the 23 AGN, whereas the remaining 14 objects are considered non-detections, with upper limits of less than 5 % of the nuclear flux. However, when we coadd the unresolved AGN images into a single high signal-to-noise composite image we find again an unambiguously resolved host galaxy. The recovered host galaxies have apparent magnitudes of $23.0<\mathrm{{F606W}}<26.0$ and $22.5<\mathrm{{F850LP}}<24.5$ with rest-frame UV colours in the range $-0.2<(\mathrm{{F606W}}-\mathrm{{F850LP}})_\mathrm{obs}<2.3$. The rest-frame absolute magnitudes at 200 nm are $-20.0<M_{200~\mathrm{nm}}<-22.2$. The photometric properties of the composite host are consistent with the individual resolved host galaxies. We find that the UV colors of all host galaxies are substantially bluer than expected from an old population of stars with formation redshift $z\le5$, independent of the assumed metallicities. These UV colours and luminosities range up to the values found for Lyman-break galaxies (LBGs) at $z=3$. Our results suggest either a recent starburst, of e.g. a few per cent of the total stellar mass and 100 Myrs before observation, with mass-fraction and age strongly degenerate, or the possibility that the detected UV emission may be due to young stars forming continuously. For the latter case we estimate star formation rates of typically $\sim$$6\,\mathrm{M}_\odot\;\mathrm{yr}^{-1}$ (uncorrected for internal dust attenuation), which again lies in the range of rates implied from the UV flux of LBGs. Our results agree with the recent discovery of enhanced blue stellar light in AGN hosts at lower redshifts. author: - 'K. Jahnke, S. F. Sánchez, L. Wisotzki, M. Barden, S. V. W. Beckwith, E. F. Bell, A. Borch, J. A. R. Caldwell, B. Häu[ß]{}ler, S. Jogee, D. H. McIntosh, K. Meisenheimer, C. Y. Peng, H.-W. Rix, R. S. Somerville and C. Wolf' title: 'UV light from young stars in GEMS quasar host galaxies at $1.8<z<2.75$' --- Introduction {#sec:intro} ============ Around redshifts of $z\sim 2$–3, luminous quasars were orders of magnitude more numerous than today. Although the physics of how active galactic nuclei evolve is still not understood, several links between galaxy and quasar evolution have emerged over recent years. The observational confirmation of supermassive black holes in the nuclei of all galaxies with a substantial bulge component [e.g. @gebh00] makes every such galaxy a potential AGN host. The strong evolution of the AGN space density could therefore be related to the availability of accretion fuel in the host galaxies, or to the frequency of AGN triggering events. Gravitational interaction and major or minor merging of galaxies have long been suggested as important factors in driving nuclear activity in galaxies. Confirming any of these as the dominant process has proved difficult, mainly because the morphological characteristics found for relatively nearby AGN host galaxies are so diverse. Furthermore, the properties of the hosts in the ‘heyday’ of quasars ($z \ga 2$) are still elusive, a consequence of the substantial observational difficulties. The contrast between the bright nuclear point sources and the surrounding galaxy increases dramatically beyond $z\sim 1$ as a result from both surface brightness dimming and waveband shifts towards the rest frame UV. The last years have seen numerous attempts to resolve the host galaxies of high-redshift quasars. Owing to the observational challenges of detecting distant host galaxies the observational effort for each object is large, and the observed samples have consequently been very small, of the order of $\la 5$ per target group. While radio-loud quasars appear to be very extended and have been resolved out to $z \sim 4$ [e.g., @lehn92; @carb98; @hutc99; @kuku01; @hutc03; @sanc03], this is not the case for the large majority of radio-quiet quasars. At high redshifts two constraints dominate observational studies of host galaxies: On one side, very good seeing conditions are required to maximize the spatial contrast of the compact nuclear source compared to the extended host galaxy. On the other, large telescope apertures are preferrential to trace faint quasar hosts to as far away from the nucleus as possible. Thus significant progress had to wait for 8m-class telescopes at very good sites with active optics systems – with a very high light collecting power but atmospheric seeing limitations – and for the [[*HST*]{}]{} and its high space-based sensitivity, combined with unprecedented spatial resolution, but limited size that might miss light from faint outer structures of the hosts. Some host galaxies of radio-quiet quasars at $z\simeq 2$ have now been resolved both in the near infrared [@aret98; @kuku01; @ridg01; @falo04] and in the optical domains [@hutc02], showing these objects to be moderately luminous, corresponding to present-day $L^\star$ or slightly brighter. However, host galaxy colours have been unavailable, precluding estimates of the mass-to-light ratio ($M/L$). Thus, without colours the observed luminosities, and their evolution with redshift, cannot be mapped to the mass evolution if young stars contribute a major fraction of AGN host’s light. This is important as several high-luminosity quasars at $z \ga 2$ appear to be located in very UV-luminous host galaxies [@lehn92; @aret98; @hutc02]. Also, at low redshifts there is a link between nuclear activity and enhanced global star formation in the host galaxies. @kauf03 reported that SDSS spectra of local Seyfert 2 galaxies show a significant contribution from young stellar populations, and that this trend is strongly correlated with nuclear luminosity. In a multicolour study of low-$z$ QSO hosts [@jahn04a] as well as at intermediate redshifts [@sanc04a see below] we found that hosts of elliptical morphology can be significantly bluer than the bulk of inactive ellipticals. These results indicate that in the recent past the star formation activity in galaxies hosting an AGN may be different from normal galaxies. The details are far from understood. Clearly more information is required to investigate the relation of starformation and AGN activity, their common cause or causal order and the evolution of these properties with redshift. The new generation of wide-field imaging mosaics obtained with the Hubble Space Telescope ([[*HST*]{}]{}), especially in the conjunction with deep AGN surveys, has opened a new observational avenue towards AGN host galaxy studies. Here we present first results on AGN within the [[gems]{}]{} project [@rix04], the largest [[*HST*]{}]{} colour mosaic to date. In the present paper we investigate the presence of rest-frame ultraviolet light in a substantial sample of $z>1.8$ AGN, all with nuclear luminosities near $M_B = -23$. In a companion paper [@sanc04a] we study rest-frame colours and morphological properties of a sample of intermediate-redshift ($z \la 1$) AGN. The paper is organised as follows. We first describe the sample selection and properties together with a summary of the observational data (Sect. \[sec:data\]). We then comment on the decomposition of the nuclear and galaxy contribution, including a brief summary of the extensive simulations that we use to estimate measurement errors (Sect. \[sec:analysis\]). In Sect. \[sec:results\] we present the measured host galaxy magnitudes and describe our treatment of non-detections. We move on to discuss the results in Sect. \[sec:discussion\], followed by our conclusions in Sect. \[sec:conclusions\]. We use $H_0=70$kms$^{-1}$Mpc$^{-1}$, $\Omega_m=0.3$ and $\Omega_\Lambda = 0.7$ throughout this paper. All quoted magnitudes are zeropointed to the AB system with ZP$_\mathrm{F606W}=26.493$ and ZP$_\mathrm{F850LP}=24.843$. AGN in the GEMS survey {#sec:data} ====================== Overall survey properties ------------------------- [[gems]{}]{}, Galaxy Evolution from Morphologies and SEDs [@rix04] is a large imaging survey in two bands (F606W and F850LP) with the Advanced Camera for Surveys (ACS) aboard [[*HST*]{}]{}. Centered on the Chandra Deep Field South (CDFS), it covers an area of $\sim 28'\times28'$ (78 ACS fields). Each ACS field was integrated for $3\times 12$–13min exposures per filter (one orbit), dithered by $3''$ between exposures. The individual images were then combined, corrected for the ACS geometric distortion, and at the same time rebinned to a finer pixel grid of $0\farcs03$, achieving approximate Nyquist-sampling of the PSF. The image combination also removed artefacts such as cosmic ray hits and satellite trails. Resulting point source limiting magnitudes are $m_\mathrm{AB}(\mathrm{F606W})=28.3$ ($5\sigma$) and $m_\mathrm{AB}(\mathrm{F850LP})=27.1$ ($5\sigma$). In its central $\sim$1/5, [[gems]{}]{} incorporates the epoch 1 data from the [[goods]{}]{}[@giav03] project, that are similarly deep as the other [[gems]{}]{}fields. Further details of the data reduction procedure will be given in a forthcoming paper (Caldwell et al. 2004, in prep.). The area covered by [[gems]{}]{} coincides with one of four fields covered by the [[combo-17]{}]{} survey [@wolf04], which produced a low-resolution spectrophotometric data base (based on photometry in 17 filters) for about 10000 galaxies and 60 type 1 AGN brighter than $R\la 24$ (Vega zeropoint) in the CDFS area [@wolf03b; @wolf04]. The large number of filters permitted simultaneous assignment of accurate SEDs and redshifts for both galaxies and type 1 AGN. Galaxies and AGN are classified by matching an SED template library to the set of 17 photometric points. The AGN SED is composed of a range of continuum spectra with added broad emission lines [all details are given in @wolf04]. Type 2 AGN as well as very low luminosity AGN are invariably classified as galaxies. [[combo-17]{}]{} photometric redshifts are very reliable, with an rms scatter of $\sigma_z/(1+z) \simeq 0.02$ for galaxies (at $z < 1.2$) and $\sigma_z/(1+z) \simeq 0.03$ for AGN at all redshifts. In this paper we address specific [[combo-17]{}]{} sources just by their running identifiers; the full [[combo-17]{}]{} list of classifications in the CDFS will be made available in the future (Wolf et al. 2004, in prep.). ![\[z\_R\] [[combo-17]{}]{} redshifts and $R$ magnitudes of the AGN in the sample. The dashed line marks the expected $R$ band magnitude for an AGN of $M_B=-23$ at the corresponding redshifts, assuming a typical quasar spectrum [@wolf03], the dotted lines correspond to $M_B=-21.5$ and $M_B=-24.5$ for reference. ](f1.eps){width="8.5cm"} The AGN sample {#sec:gems} -------------- The [[combo-17]{}]{}-selected AGN in the [[gems]{}]{} field range over redshifts from $z\simeq 0.5$ up to $z\simeq 4$. In this study we investigate the high-$z$ part of the AGN distribution. Our sample contains all AGN brighter than $R = 24$ in the redshift range $1.8 < z < 2.75$ that show a meaningful counterpart in the [[gems]{}]{} images. This excludes three objects with a primary classification as AGN by [[combo-17]{}]{} that were apparently low redshift ($z\sim0.1$) emission line galaxies. At these redshifts, all light detected in the [[gems]{}]{} bands will originate in the rest frame ultraviolet. In fact, the lower redshift limit has been imposed to ensure that the long wavelength cutoff of the [F850LP]{} filter is still located below the Balmer jump for all objects. The upper redshift boundary, on the other hand, was set to avoid contamination from possible extended Ly$\alpha$ emission in the [F606W]{} filter. The resulting sample contains 23 AGNs, three of which are positioned in the overlap region of two tiles so two separate images exist for these. One object ([[combo-17]{}]{} 05696) shows inconsistent photometry between the two tiles due to variability over the 111 days between the two integrations. Another object ([[combo-17]{}]{} 19965) was classified in [[combo-17]{}]{} with a redshift of $z=0.634$ that had to be revised following spectroscopy observations (G. Worseck, private communication) and now entered this sample with $z=1.90$. Table \[tab:sample\] gives an overview over the sample properties, and Figure \[z\_R\] shows the distribution in $R$ and $z$. The average absolute magnitudes for these objects – which we call intermediately luminous quasars – place them close to the canonical division of $M_B \simeq -23$ between Seyfert galaxies (low-luminosity) and QSOs (high-luminosity). Several X-ray sources in the CDFS have already been studied by [[*HST*]{}]{} with the WFPC2 camera [@schr01; @koek02; @grog03], most of them faint AGN, but their sample is completely disjoint from the [[combo-17]{}]{} AGN selection for this redshift range. One object falls into our redshift range but has $R>24$, outside our selection limits. [cccccccc]{} ID& Tile& RA (2000)& DEC (2000)& $z$& $R$ (Vega)& [F606W]{} & [F850LP]{}\ 12325 &11& 033301.7& –275819& 1.843& 20.38& 20.13& 19.65\ 19965 &23& 033145.2& –275436& 1.90 & 19.96& 20.59& 20.29\ 30792 &82& 033243.3& –274914& 1.929& 22.36& 21.60& 22.06\ 02006 &04& 033232.0& –280310& 1.966& 19.76& 19.59& 18.98\ 04809 &08& 033136.3& –280150& 1.988& 22.31& 21.18& 20.91\ 06817 &09& 033127.8& –280051& 1.988& 21.59& 21.61& 20.86\ 18324 &19& 033300.9& –275522& 1.990& 22.58& 22.00& 21.33\ 05498 &01& 033316.1& –280131& 2.075& 22.29& 22.91& 22.28\ 11941 &10& 033326.3& –275830& 2.172& 20.51& 20.80& 20.40\ 62127 &62& 033136.7& –273446& 2.175& 23.57& 24.91& 24.37\ 51835 &55& 033140.1& –273917& 2.179& 22.92& 23.01& 22.41\ 00784 &05& 033227.1& –280336& 2.282& 23.29& 23.28& 22.80\ 36120 &39& 033149.4& –274634& 2.306& 22.37& 22.70& 22.23\ 05696 &02& 033321.8& –280121& 2.386& 23.06& 22.73& 22.32\ 05696 &03& $''$ & $''$ & $''$ & $''$ & 23.14& 22.68\ 07671 &07& 033151.8& –280026& 2.436& 22.35& 22.35& 22.24\ 07671 &15& $''$ & $''$ & $''$ & $''$ & 22.36& 22.22\ 06735 &02& 033306.3& –280056& 2.444& 21.98& 22.14& 21.88\ 01387 &08& 033144.0& –280320& 2.503& 23.24& 24.05& 23.17\ 33644 &31& 033259.9& –274748& 2.538& 21.87& 21.28& 21.17\ 11922 &11& 033309.1& –275827& 2.539& 22.25& 22.65& 21.93\ 16621 &19& 033309.7& –275614& 2.540& 19.98& 20.41& 20.06\ 15396 &21& 033216.2& –275644& 2.682& 22.64& 22.69& 22.41\ 33630 &33& 033140.1& –274746& 2.719& 21.74& 22.21& 21.98\ 42882 &45& 033201.6& –274328& 2.719& 23.18& 23.89& 23.48\ 42882 &95& $''$ & $''$ & $''$ & $''$ & 24.04& 23.54\ Data analysis {#sec:analysis} ============= Background and variances {#sec:bg} ------------------------ Even though space based, ACS shows a non-negligible background from stray light. In the reduction process already a global, outlier clipped median background was subtracted (Caldwell et al. 2004, in prep.). As the deblending of nuclear and host galaxy component with two-dimensional modelling is sensitive to background sources, we applied an extra procedure to remove net residuals in the local background. This included an iterative masking of all objects in the field and the determination of the local background from the object-free regions. For each square of $200\times200$ pixels an average from the unmasked pixels was computed, with a subsequent bilinear interpolation between these values to yield a background estimate for the whole field. After background adjustment, small subimages of $128\times128$ pixels were extracted around all AGNs, corresponding to a field of view of $3\farcs84\times3\farcs84$. This field size contains $\ge 99$ % of the ACS PSF flux. Since the AGN are not strongly resolved this fraction also applies to the AGN flux. The data reduction procedure kept record of the individual pixel weights throughout the process of reduction and combination. This information was then used in combination with the shot noise derived from pixel count rates to construct variance images which were later used in the error budget calculations. PSF estimation {#sec:psf} -------------- Compared to ground based telescopes, the point spread function (PSF) of ACS is very stable in time. However, coma, astigmatism and defocus from surface height variations of the two CCDs lead to variations over the field of view that need to be taken into account. The variations are much weaker than for the WFPC2 camera but remain non-negligible. Also the ‘breathing’ of [[*HST*]{}]{} changes the focal length which leads also to a small time variability in the PSF. In Figure \[fig:psf\] we show the mean ACS PSF compiled from $\sim$500 stars in the extensive [[gems]{}]{} area and the variation of the PSF over the FOV. \ The variations were also investigated by @kris03b from the crowded field of 47 Tucanae. Such an analysis was not possible for individual [[gems]{}]{} fields due to the small number of stars ($\sim$10) per field. Thus a simultaneous characterisation of the spatial and temporal variations was not possible. However, since the spatial variations dominate (further details of our investigations of the ACS PSF variations will be given in a dedicated technical paper, Jahnke et al. 2004, in prep.) we used the large number of unsaturated stars in the [[gems]{}]{} area to construct an empirical PSF individually for each object. At a given position we combined the nearest $\sim$35 undisturbed stars to create a position-specific PSF estimate. In this way we average over time, but only stars from a radius $\la40\arcsec$ were used, and PSF shape errors due to [*spatial*]{} variation were minimized, while a very high S/N was achieved for each of these PSF estimates. Finally, the subimage of each AGN and its connected PSF were registered to a common centroid. From our PSF analysis we found that while coherent large-scale variations were essentially absent within each stack of $\sim$35 PSF stars, there was still considerable mismatch between the individual stars, in particular in the central pixel regions. As such mismatched pixels could be spuriously assigned to a host galaxy, we took the variations within each PSF stack to derive rms frames describing an inherent PSF uncertainty; these were then also included in the variance images, artificially reducing the weight in the inner pixel regions. PSF rms errors per pixel range from up to 30% in individual pixels inside 1 FWHM to 5–15 % inside $0\farcs2$, and to generally below 5% outside. Peak scaled PSF subtraction {#sec:psfsub} --------------------------- For luminous AGN at high redshifts, separating the galaxy image from the nuclear point source is a daunting task. Even with a very good knowledge of the PSF, the problem is still that the relative scalings of galaxy and AGN are not known [*a priori*]{}. In fact, one cannot even be certain that the host galaxy is detectable at all and not swamped by the central point source. We therefore started out with the well-established technique of simple PSF subtraction. In each case, the PSF was scaled to the central flux of the AGN, integrated inside a circular aperture of 4 pixel (012) diameter centered on the nucleus. This radius encircles approximately 34% of the total energy of a point source. A smaller radius (e.g. 1 pixel) would become too sensitive to shot noise and PSF mismatch, while larger radii contain a significantly higher fraction of the total flux (50%/60%/70% at 3/4/5 pixel radius) and thus would make a detection of any host galaxy component successively harder. This procedure somewhat oversubtracts the nuclear component by an amount corresponding to the underlying host galaxy contribution inside the encircled region. However, this method yields strictly conservative estimates the host galaxy flux, i.e. always unserestimating it. We used extensive simulations to determine correction factors for this oversubtraction (see Section \[sec:errors\]). The peak-scaled PSF subtraction method has the advantage of being independent of any assumption about the host galaxy morphology. Two-dimensional deblending {#sec:galfit} -------------------------- As a second method we employed the modelling package [[galfit]{}]{} in Version 1.7a [@peng02] that allows the simultaneous fitting of several two-dimensional components to an image, convolved with a given PSF. We describe our application of [[galfit]{}]{} to quasar images in @sanc04a in detail. In total we ran [[galfit]{}]{} in three configurations, always fitting two components, which were the point-source nucleus and either an exponential disk [@free70], a $r^{1/4}$ de Vaucouleurs spheroid [@deva48], or a @serc68 model with the Sérsic index as a free parameter. However, simulations (Sect. \[sec:errors\]) showed that with the present data, the [[galfit]{}]{} version used[^1] was operating near its limit, due to the very high contrast between nuclei and host galaxies. In the context of this paper we thus used [[galfit]{}]{} only as a cross-check on the peak subtraction method. Detection sensitivity {#sec:detection} --------------------- To determine the limits for detecting host galaxies we constructed a sample of 200 randomly selected unsaturated stars, 100 in each observed band, to mimic unresolved quasars. This way we could investigate how our nucleus-removal techniques responded to a undetectable host galaxy, and we could set limits on the size and shape of expected residuals, thus lower flux limits for detectable host galaxies. For these stars the PSFs were created in exactly the same way as for the AGNs. The object itself was always excluded in the PSF production thus each test star and its PSF were fully independent. This set of simulated ‘naked quasars’ showed that in 88% (97%) of all cases, any residuals – which could be taken as spurious (g)host galaxy detections – had fluxes of less than 5% (10%) of the total object flux. From this we adopted the condition that a real detection should show a residual flux after peak subtraction of at least 5% of the total flux, corresponding to a maximum nucleus-to-host ratio of 20. Because of the systematic oversubtraction inherent in the procedure corrections for the flux have to be applied (see next section). The final decision if a host galaxy is resolved is based on this criterion. In addition we visually inspected whether the detected flux indeed came from a host galaxy or whether other, unmasked structures were present, using the peak subtracted images and radial profile. If this could be ruled out we classified a host galaxy as detected. {width="8cm"} {width="8cm"} Systematic offsets and errors {#sec:errors} ----------------------------- While the mere detection of an AGN host galaxy can be achieved with comparably little effort, the determination of flux error bars and systematic offsets is much more complicated. We performed extensive simulations of artificial quasar images, composed from empirical PSFs and host galaxy models plus artificial noise matching the actual flux and noise distribution in real images. These simulations are described in detail by @sanc04a. We applied the peak subtraction nucleus removal as well as [[galfit]{}]{} to a set of $\sim$2000 quasar images created in this way. Comparing input and output parameter values yielded mean magnitude offsets as well as statistical errors for the individual host galaxy magnitudes (Fig. \[fig:corrs\_and\_errors\_z\]). The simulations give reliability regions and error bars. The left panel in Figure \[fig:corrs\_and\_errors\_z\] shows which magnitudes are recovered for a given synthetic host galaxy. Since the input set covers a large range of different morphological configurations, scale lengths, nucleus-to-host ratios, etc., the recovered values will scatter. Close to the detection limit, the scatter and the corrections grow rapidly as a function of magnitude; additionally, the ability to differentiate between different morphological types will generally be lost. The combination of these effects is reflected in the spread of the output of the simulations. This measured spread is a direct estimate for the uncertainties of the total flux (right panel in Figure \[fig:corrs\_and\_errors\_z\]). From these simulations we adopt approximate regions in brightness where host galaxy magnitudes can be reliably determined, with correction of 0.25 to a maximum of 0.6 mag. These regions go down to $\mathrm{{F606W}}=26.2$, $\mathrm{{F850LP}}=24.6$ for the peak subtraction method and the present data. Outside the corrections and errors increase. In three cases, marked in Table \[tab:results\_vz\] with ‘?’ in column $Z_\mathrm{hg}$, the observed magnitudes extend to outside these regions; here we continue using the derived corrections for these three objects, but the so derived host galaxies are more uncertain and their magnitudes should be taken with care. Notice that the [F850LP]{} band data are substantially shallower than the [F606W]{} band, mainly a consequence of the ACS detector sensitivity. {width="\colwidth"} {width="\colwidth"}\ {width="\colwidth"} {width="\colwidth"} Results {#sec:results} ======= Detected host galaxies {#sec:resolved} ---------------------- Using the above criteria for detecting a residual host galaxy, we find nine of the 23 host galaxies to be resolved in both bands, although some lie close to the sensitivity limit. One object formally fell above the 5% level in one band but not the other; [[combo-17]{}]{} 33630 at $z=2.719$ might be marginally resolved in [F850LP]{} and shows a structure at 1distance that might be a tidal arm or a foreground object. With this object lying at the highest redshift of the sample we do not consider this a clear detection. As mentioned, for three further cases in the [F850LP]{} band the host galaxies are very faint (marked with a ‘?’ in the $Z_\mathrm{hg}$ column in Tab. \[tab:results\_vz\]). While their flux is above 5% of the total, their raw magnitudes fell 0.8–1.0 mag outside the reliability region where corrections and associated errors are still small. This low S/N is also reflected in the radial profiles (see Appendix \[sec:appendix\]). As described above, tests with field stars show that 12% of all objects ($\sim$3 objects) might show spurious ‘host galaxies’ at the 5% flux level, and 3% (0 or 1 objects) at the 10% flux level. In [F606W]{} five of our objects fall with their host fluxes between these two values. In the [F850LP]{}-band these are four which include the three uncertain ones from above. According to statistics 1–3 of these might be spurious detections. However, including or excluding these more uncertain data points in the following analysis does not have an influence on the conclusions drawn. For each object the host galaxy flux was determined by simple aperture photometry after subtraction of the scaled PSF, excluding resolved companion objects. The radius of the aperture was matched the used image size. All extracted magnitudes are collected in Table \[tab:results\_vz\], and shown together with the upper limit for the unresolved objects in the top panels of Figure \[fig:errorsimpeak\]. The extracted host galaxy images and radial surface brightness profiles are shown in Fig. \[fig:allimages\] in the appendix. To illustrate the behaviour of true point sources, we included a selection of 24 field stars, 12 each in the [F606W]{}- and [F850LP]{}-band (Figures \[fig:stars\_v\] and \[fig:stars\_z\]), that were subjected to the same PSF determination and peak subtraction as described in Section \[sec:detection\]. We plot the same profiles as for the AGN. This selection is random apart from the fact that the 24 stars were observed on 24 different tiles. For most of the stars there is no systematic positive residual flux visible, as expected. The few that do show positive fluxes form the spurious detection statistic described above. While with deeper images or at lower redshift [@sanc04a] the morphological appearance of the host galaxies can be determined, this was generally not possible for the present data. Apart from the extentions in [[combo-17]{}]{} 00784, apparent residual structure visible in the colour images shown in the Appendix is dominated by the PSF subtraction procedure. This includes elongations and apparent off-centering that is due to limited centering precision of the order of 0.05–0.1 pixel (see e.g. [[combo-17]{}]{} 05696 from tile 2). Thus we adopted the host galaxy morphological class in all cases as ‘undecided’, and then applied the systematic corrections for oversubtraction provided by the simulations (Fig. \[fig:corrs\_and\_errors\_z\]). The corrections were typically of the order of $\sim 0.4$ mag, as documented by the columns with ‘[*cor*]{}’ subscripts in Table \[tab:results\_vz\]. Here we also list the estimated uncertainties resulting from our extensive simulations. The distribution of corrected magnitudes is displayed in the lower panels of Figure \[fig:errorsimpeak\]. ![\[fig:unresplot\] The stacked image of individually unresolved host galaxies. Top: grey scale original image and PSF subtracted host galaxy image (both show $2\arcsec\times2\arcsec$) in the [F606W]{}-band. On top of the linear grey scale plot logarithmic contours with 0.5 dex spacing are overplotted. Bottom: radial profiles of [F606W]{}-band. The curves show the upper data points with error bars from the original image, the PSF (dashed red line) and peak subtracted host profile (lower points with error bars from bootstrapping). The host galaxy is shown without correction for any oversubtraction of the nucleus. ](f5.eps){width="8.5cm"} Detecting the undetected hosts {#sec:unresolved} ------------------------------ For 14 AGNs we find that the individual PSF-subtracted residuals are consistent with non-detections, i.e. the magnitudes of individual host galaxies lie below the 5% limit. These objects are marked by arrow symbols in Figures \[fig:errorsimpeak\] and \[fig:nuchost\]. It is interesting to note that these flux limits on the hosts are by no means all outstandingly faint. For many, the reason why they were not detected is the high contrast with the AGN which for these objects has above average brightness. Only three objects show non-detections that indicate exceptionally faint host galaxies. We will further discuss the implications of the detection limits in Section \[sec:colours\]. In an attempt to assess at least the mean host galaxy properties of the unresolved AGN, we simply coadded the images of all 14 objects, one of which was observed in two frames. This yielded a very deep image with effectively 15 orbits of integration time. We also combined their PSFs, weighted by the relative flux of the AGNs, and created combined variance and PSF rms frames. The higher S/N in combination with the lower PSF noise (PSF and AGN position are sampled at 14 different subpixel points) yields a nominal increase in surface brightness sensitivity of $\sim$1.5 mag. The resulting ‘object’ has a mean redshift of $z=2.3$ (weighted by AGN flux), and total magnitudes of $m_{\mathrm{{F606W}},\mathrm{tot}}=21.5$ and $m_{\mathrm{{F850LP}},\mathrm{tot}}=21.6$, respectively. This image is shown in Figure \[fig:unresplot\]. With the higher sensitivity we now indeed find a host galaxy component in the [F606W]{}-band image after PSF subtraction of 4.4% of the total flux. The radial surface brightness profile also shows a small excess over an unresolved point source. In both cases this flux is highly significant as we confirmed using a bootstrap simulation for the composition of the coadded frame from the 15 frames. In the bootstrap simulation we constructed 100 new sets of 15 frames each, drawn with repetition from the original 15 frames, coadded the images in each set and did the flux analysis as above. The uncertainty in the total flux estimated from these 100 realisations is $\sigma=1.05$% of the total flux, or 25% in host galaxy flux. All realisations yielded substantial positive fluxes. The error is resulting from a combination of PSF uncertainty and the noise inside the scaling aperture of 4 pixel diameter. We show the uncertainties in the radial surface brightness determined from bootstrapping as error bars for the derived host galaxy in Figure \[fig:unresplot\]. The so extracted magnitudes for host galaxy and nucleus in the [F606W]{}-band are listed in the last row of Table \[tab:results\_vz\] (the ‘stack’ object). The [F850LP]{}-band stack, however, with its lower sensitivity showed a much weaker signal than the [F606W]{}-band, too faint to reliably be classified as resolved. [cccccccccccc]{} ID&Tile& $V_\mathrm{tot}$& $V_\mathrm{hg}$& $V_\mathrm{hg,cor}$& $V_\mathrm{nuc,cor}$& N/H$_{V,\mathrm{cor}}$& $Z_\mathrm{tot}$& $Z_\mathrm{hg}$& $Z_\mathrm{hg,cor}$& $Z_\mathrm{nuc,cor}$& N/H$_{Z,\mathrm{cor}}$\ 19965&23& 20.59& 23.9& 23.3$\pm0.05$& 20.7& 11.3& 20.29& 23.8 & 23.5 $\pm0.2$& 20.4&17.5\ 30792&82& 21.60& 24.6& 24.4$\pm0.2$ & 21.7& 12.0& 22.06& 24.2 & 23.9 $\pm0.2$& 22.3& 4.2\ 18324&19& 22.00& 23.6& 23.0$\pm0.05$& 22.5& 1.6& 21.33& 22.7 & 22.4 $\pm0.2$& 21.9& 1.6\ 05498&01& 22.91& 25.5& 25.2$\pm0.2$ & 23.1& 7.1& 22.28& 25.0?& 24.4 $\pm1.2$& 22.5& 6.0\ 51835&55& 23.01& 26.0& 25.7$\pm0.3$ & 23.1& 11.2& 22.41& 23.8 & 23.4 $\pm0.2$& 23.0& 1.6\ 00784&05& 23.28& 25.0& 24.7$\pm0.2$ & 23.6& 2.8& 22.80& 24.0 & 23.7 $\pm0.2$& 23.4& 1.2\ 05696&02& 22.73& 24.7& 24.4$\pm0.2$ & 23.0& 3.6& 22.32& 23.8 & 23.5 $\pm0.2$& 22.8& 1.9\ 05696&03& 23.14& 24.2& 23.9$\pm0.15$& 23.9& 0.9& 22.68& 24.0 & 23.7 $\pm0.2$& 23.2& 1.6\ 07671&07& 22.35& 25.6& 25.2$\pm0.2$ & 22.4& 13.2& 22.24& 25.0?& 24.4 $\pm1.2$& 22.4& 6.3\ 07671&15& 22.36& 25.5& 25.1$\pm0.2$ & 22.5& 11.9& 22.22& 24.6 & 24.3 $\pm0.3$& 22.4& 5.5\ 11922&11& 22.65& 24.9& 24.6$\pm0.2$ & 22.9& 4.8& 21.93& 25.2?& 24.5 $\pm1.5$& 22.0& 9.3\ stack& & 21.47& 24.9& 24.3$\pm0.05$& 21.6& 12.3& 21.60& &&\ [ccccc]{} ID&Tile&$z$&$(\mathrm{{F606W}} - \mathrm{{F850LP}})$&SFR [F606W]{}\ & & & &\[$\mathrm{M}_\odot/\mathrm{year}$\]\ 19965& 23 &1.90&–0.2$\pm0.2$& 11\ 30792& 82 &1.929& 0.5$\pm0.3$& 4\ 18324& 19 &1.990& 0.6$\pm0.2$& 15\ 05498& 01 &2.075& 0.8$\pm1.2$& 2\ 51835& 55 &2.179& 2.3$\pm0.4$& 1.5\ 00784& 05 &2.282& 1.0$\pm0.3$& 4\ 05696& 02 &2.386& 0.9$\pm0.3$& 6\ 05696& 03 &2.386& 0.1$\pm0.3$& 9\ 07671& 07 &2.436& 0.8$\pm1.2$& 3\ 07671& 15 &2.436& 0.9$\pm0.4$& 3\ 11922& 11 &2.539& 0.1$\pm1.5$& 5\ stack& &2.3 & $<0.0$ & 6\ ![\[fig:SSP\_colours\_AB\] Observed colours ($\mathrm{{F606W}} - \mathrm{{F850LP}}$) of the sample from PSF peak subtraction (circles), the open symbol marks the upper limit for the ‘stacked’ AGN. Overplotted are two single burst models from [solar metallicity @bruz03] (solid lines). The upper curve is for a passively evolving burst at $z=5$, the lower for burst of 100 Myr age, relative to each redshift. The dot-dashed lines are mixtures between the two, with a (from top) 0.1%, 1% and 10% fraction of mass of the 100 Myr population on top of 99.9%, 99% and 90% of the $z=5$ population. ](f6.eps){width="8.5cm"} ![\[fig:nuc\_host\_col\] Nuclear vs. host galaxy colours for the objects in the sample with available colours. The two objects imaged twice are connected with a dotted line (in one case not visible to near identical colours). The dashed line is the 1:1 relation to guide the eye. ](f7.eps){width="8.5cm"} Discussion {#sec:discussion} ========== UV colours {#sec:colours} ---------- From the corrected magnitudes we have derived $(\mathrm{{F606W}}-\mathrm{{F850LP}})$ colours for the detected host galaxies. These are listed in Table \[tab:colours\], now including also correction for Galactic dust extinction. However, with $E(B-V)=0.008$ [@schl98] these values of $A(\mathrm{{F606W}})=0.024$ and $A(\mathrm{{F850LP}})=0.014$ negligibly affect the colours. At these redshifts, the observed photometric bands correspond to the rest frame ultraviolet, ranging from 2160Å at $z=1.8$ to 1616Å at $z=2.75$ in [F606W]{} and 3150Å to 2350Å in [F850LP]{}, respectively, so for this redshift range pure rest-frame UV colours are observed. Figure \[fig:SSP\_colours\_AB\] shows the measured values plotted against $z$. There is no discernible colour trend with redshift. All points fall within a relatively narrow range of colours; apart from [[combo-17]{}]{} 51835, the objects occupy a band of $-0.2<(\mathrm{{F606W}}-\mathrm{{F850LP}})_\mathrm{observed}<1.0$. The open symbol represents our stacked ‘average’ AGN constructed from the 14 unresolved objects. Although it was not resolved in the [F850LP]{} band, the upper limit on its colour is actually consistent with the values derived for several of the detected objects. One critical issue in measuring UV luminosities of barely resolved AGN hosts is the lingering possibility that flux may have spilled over from the nuclei. This could have happened as a purely observational artefact due to imperfect PSF removal, or physically by scattering of UV photons off dust in the the host galaxy. The latter phenomenon is known to be relevant in high redshift radio galaxies [@vern01]. Independently of the underlying mechanism, any such cross-contamination should be visible in a correlation of host galaxy with nuclear colours. Figure \[fig:nuc\_host\_col\] shows these colours plotted against each other. No correlation is visible and we conclude that a substantial contamination of the host galaxy light from the AGN is very unlikely. Notice also that when considering physical scattering in the hosts, powerful radio galaxies are huge massive entities of generally vastly different appearance compared to the relatively modest AGN hosts featuring in our sample. UV colours can be converted to UV spectral slope $\beta$ independent of redshift, when assuming that the SED can be described in the form $F_\lambda \propto \lambda^\beta$. With $\beta$ known, the absolute magnitude at 200 nm, $M_\mathrm{200nm}$, can be computed directly from the [F606W]{}- or [F850LP]{}-band apparent magnitudes. We do this for both the host and the nucleus and these values are shown for the full sample including upper limits in Figure \[fig:nuchost\], which illustrates the two main constraints for resolving a host galaxy, apart from compactness (see Sect. \[sec:unresolved\]). The reliability of host galaxy photometry is constrained by S/N, thus dependent on the filter which have different depths. This is marked by the horizontal dashed lines which show the $M_\mathrm{200nm}$ magnitude of host galaxies at the $\mathrm{{F606W}}=26.2$ mag and $\mathrm{{F850LP}}=24.6$ mag edges of the adopted regions of reliability (see Section \[sec:errors\]), at mean redshift and mean $\beta$. As the second effect the maximum nucleus–host contrast appears as the scatter of the unresolved objects around a line shifted by 3.2 mag from unity, corresponding to 5% of the total flux. Here the scatter is only induced by the assumption that all unresolved host galaxies have a the $\beta$ value the upper limit determined for the stacked image. The diagonal dotted lines mark lines of 10%, 20%, 33% and 50% of the total flux associated with the host galaxy. Thus in total the region right of the solid diagonal line is inaccessible to host galaxy detection with the current data and method of analysis. We would like to emphasise that the similarity of nuclear properties for resolved and unresolved host galaxies suggests that also the host galaxy properties are similar – thus supporting that the data point for the coadded stacked object is not far off the individually resolved objects in all plots. ![\[fig:nuchost\] Nuclear vs. host galaxy absolute magnitude at 200 nm. Shows are objects with resolved host galaxies (solid symbols), the coadded stacked object (open circle with upper limit arrow) and upper limits for objects with unresolved host (arrows). The diagonal lines mark positions of constant fraction of host galaxy light of 5%, 10%, 20%, 33% and 50% of the total light, assuming a constant spectral slope $\beta$. The horizontal dashed lines show the magnitude of host galaxies at the $\mathrm{{F606W}}=26.2$ mag and $\mathrm{{F850LP}}=24.6$ mag detection limit, at mean redshift and mean $\beta$. ](f8.eps){width="8.5cm"} In Figure \[fig:beta\] spectral slope is plotted against $M_\mathrm{200nm}$ for the host galaxies, compared to the mean value for a sample of 794 Lyman break galaxies at redshift $z\sim3$ [@shap03], all values uncorrected for the influence of dust in the galaxies. The slight anticorrelation that seems to be visible between $\beta$ and UV luminosity suggest that the more luminous host galaxies are having the steeper spectral slopes or bluer colours. This would mean that more luminous host galaxies had bluer colours and thus more UV light from young stars. To test this we computed the Spearman rank-order coefficient for this data set. The test gave a probability for the zero hypothesis – uncorrelated data – of 9%. Thus the zero hypothesis can not even be rejected on a 2$\sigma$ level and thus the anticorrelation is not significant. A few of the most UV-luminous host galaxies fall into the same magnitude–colour regime as the LBGs, while a number of objects are substantially redder. Thus we do not find a positive correlation of the amount of host UV light and luminosity. ![\[fig:beta\] Spectral slope $\beta$ in the UV ($F_\lambda \propto \lambda^\beta$) vs. 200 nm absolute magnitude, uncorrected for dust. The open symbol marks the ‘stacked’ object, the crossed square is the mean value obtained by @shap03 for 794 Lyman break galaxies at $z\sim3$. ](f9.eps){width="8.5cm"} Origin of the host galaxy UV flux --------------------------------- The top solid line in Figure \[fig:SSP\_colours\_AB\] shows the theoretical colour of a galaxy that formed all its stars at $z=5$ and evolved passively afterwards. This colour was computed from single stellar population (SSP) models taken from @bruz03. The assignment of solar metallicity in this context is arbitrary and subject to discussion (see below). However, qualitatively there is no strong dependence on metallicity, the exact formation redshift or the particular choice of IMF for a given model family (@salp55 or @chab03) in this wavelength range. This comparison shows that the measured UV colours of our detected AGN hosts as well as of the stacked ‘mean’ host galaxy are markedly bluer than expected for an ‘old’ population at that epoch, i.e.$t_{age}\sim t_\mathrm{Hubble}$. Any correction for dust in the host galaxies will strengthen this result. Thus the blue light detected in these galaxies must come from relatively young stars. These stars could be forming continuously, in which case the UV luminosities can be interpreted as indicators of the star formation rate in these galaxies. We follow up on this option in the next subsection. Alternatively, the UV flux could be the afterglow of a past starburst. This is the option we consider first. ### Recent starburst {#sec:starburst} Given the unknown dust absorption and metallicities in the host galaxies, a single UV colour is insufficient for performing a detailed age dating of starburst or a decomposition of stellar populations. However, we want to compare to the available theoretical colour range spanned by galaxy formation at $z=5$ and a very recent (100 Myr) starburst to illustrate how mixing of an old underlying population with most of the mass and a recent starburst influences the colour. In Figure \[fig:SSP\_colours\_AB\] the solid lines mark these extremes. In between these the dot-dashed lines show how contribution of a 100 Myr component would influence the colour of an otherwise old population. From top to bottom 0.1%, 1% and 10% in mass are added to the old population. There is a strong degeneracy between the choice of burst age and the mixing ratio. For 10 Myr less than one tenth in mass is required to produce the same UV colour compared to 100 Myr. If we choose 100 Myr as a timescale similar to the dynamical timescale in galaxies, the masses involved in that starburst would be of the order of a few percent of the total stellar mass. For the assumption of only one single-aged population we can rule out a very high formation redshift – $z=5$ corresponds to ages of 3.5–2.5 Gyr at $z=1.8$–$2.5$ in the chosen cosmology. For the adopted set of models, the resulting age estimates would range mostly between $\sim$0.1 and $\sim$0.7 Gyr. We note that the colour tracks in Figure \[fig:SSP\_colours\_AB\] are largely flat over the redshift range of interest, and that the two pure population and the different mixing ratios correspond to different colours almost independently of $z$. Since our measured colours are all quite similar, we conclude that the luminosity weighted ages of the the UV-dominating stellar population must be rather similar, unless younger ages and more reddening in some objects conspire. Clearly, a single UV colour is insufficient to perform a reliable age dating, with all broad band colours being affected by various degeneracies with respect to dust and metallicities. However, we have reason to believe that at least the central lines of sight towards the AGN are reasonably free of dust extinction (because the AGN sample is selected by optical/UV flux), and we therefore do not expect dust to play a major role. At any rate, significant dust extinction would make the host galaxies intrinsically bluer than what we observe. On the other hand, assuming a metallicity lower than solar would shift all curves in Figure \[fig:SSP\_colours\_AB\] downward, resulting in older age estimates. Reducing the metallicity to $Z = 0.004$ (1/5 solar) gives an single burst age increase by a factor of two (for Bruzual & Charlot models). We conclude that if the UV light in our host galaxies is emitted by a passively evolving population of young stars, this population is typically much younger than a Gyr. The diagram in Figure \[fig:nuchost\] shows that no correlation exists between the amount of stellar UV light from the host galaxies and the amounts of UV radiation produced by the nuclei. If the latter is taken as a measure of the amount of matter accreted by the nuclei then in the context of a recent starburst the size of the starburst and the amount of accreted matter must be governed by different mechanisms. If the accretion rate is primarily defined by the nuclear mass, and a correlation between galaxy and black hole mass is assumed, then the size of the starburst is independent of the bulk stellar mass of the host galaxy. Other factors must be dominating the amount of (gas) mass involved in the starburst. This can be either the total amount of gas available in the galaxy, the size of the region involved in the starburst, the strength of an interaction responsible for the starburst, etc. In any case the amounts of gas involved are variable for a given nuclear luminosity. Including the upper limits in Figure \[fig:nuchost\] the host luminosities span e.g. $\sim$4 mag at $M_{200~\mathrm{nm}}(\mathrm{nuc})= -22$, i.e. the amount of gas involved can vary by a factor of $\sim40$ or more. ### Estimating a host star formation rate {#sec:sfr} We now interpret the detected UV emission in the alternative framework of being due to young stars forming continuously in the AGN host galaxies. Under this assumption it is possible to estimate the star formation rate (SFR) of the host galaxies from the measured rest frame UV luminosities. Following @kenn98 and using the conversion of AB magnitudes into monochromatic fluxes we obtain $$\mathrm{SFR}\left(\frac{\mathrm{M}_\odot}{\mathrm{year}}\right) = 1.8 \times10^{-27} \left(\frac{d_l^2 10^{-0.4\left(m_\mathrm{AB} + 48.6\right)}}{1+z}\right)$$ where $d_l$ is the luminosity distance to the AGN in cm, and $m_\mathrm{AB}$ is the observed UV magnitude at an arbitrary wavelength between 1500–2800Å. With this formula we can now convert our [F606W]{} band luminosities (which are much deeper than the [F850LP]{}band data) into star formation rates. As long as intrinsic dust attenuation is neglected, these values are of course mere lower limits. The resulting SFR values are listed in Table \[tab:colours\] and generally amount to a few solar masses per year, with remarkably little variation. The maximum value found is $\sim$15 $\mathrm{M}_\odot \:\mathrm{yr}^{-1}$, and the mean is $\sim$6 $\mathrm{M}_\odot \:\mathrm{yr}^{-1}$, including the results from the stacked image of individually unresolved host galaxies. This value is thus representative for the full sample of 23 AGN. We show the distribution of star formation rates vs. redshifts in Figure \[fig:sfr\]. No trend emerges, consistent with the previously established observations that neither colours nor luminosities show any significant trend with $z$. This is again compared to the star formation rates for LBGs at $z=2\ldots2.6$ from @erb03, as determined from the UV flux and uncorrected for dust. There are a few host galaxies with higher UV flux while for the majority it is smaller by a factor of 2–3. However, there is a principal caveat in our guiding assumption of this subsection: The UV light from galaxies with strong ongoing star formation is expected to be totally dominated by the youngest stars. @kenn98 pointed out that the spectral shape of galaxies with a constant SFR over at least $\sim 100$ Myr is basically flat (in $f_\nu$) between 1500–2800Å, assuming a @salp55 IMF. This would lead to an expected UV colour for our objects of $(m_\mathrm{{F606W}} - m_\mathrm{{F850LP}})=0$ or spectral index $\beta=-2$, more or less independently of redshift (within the $z$ range of our sample), inconsistent with our observations for most objects (see previous subsection, Figure \[fig:SSP\_colours\_AB\] and Table \[tab:colours\]). In other words, most detected host galaxies of our sample have UV colours that are too ‘red’ for a simple continuous star formation scenario, while the stacked host galaxy is roughly consistent. If for the resolved hosts the [F850LP]{} band fluxes were used instead of the [F606W]{} band, SFR values would be higher by roughly a factor of 2. This apparent inconsistency could be resolved in several ways. The initial assumption could be wrong, and the UV flux originates not from freshly formed stars, but from a passively evolved starburst as outlined in Sect. \[sec:starburst\]. The star formation rate might not have been constant over the past, so that varying amounts of stars of different masses and ages would have been formed; such a configuration is always possible, and our only argument against it would invoke Occam’s razor. Finally, as discussed in the previous section, there could be an underlying older stellar population contributing more to [F850LP]{} and less to [F606W]{} (but note that by sample design, also [F850LP]{} is completely below the Balmer jump at all relevant redshifts). This is the scenario that is favoured by our intermediate redshift data from [[gems]{}]{} [@sanc04a]. Clearly, in this study our current data set of just two UV bands is insufficient to settle this ambiguity. However, in all three cases there would be no continuous star formation as inferred for LBGs. The blue light would be a result not of a continuous process but of one or more events in the past of the host galaxy that triggered star formation. Whether galaxy interaction or merger incidents were responsible or the formation of bars or spiral arms is involved can not be investigated with the present set of data. Comparison with other AGN host galaxy studies {#sec:other} --------------------------------------------- Even at low redshifts, colour data of AGN host galaxies are relatively scant, except for rather low-luminosity AGN where the host galaxy can be separated with relative ease. In those cases, colours were generally found to be consistent with morphological types, in particular for the prevailing disk-type host galaxies [@koti94; @scha00]. However, when higher nuclear luminosities are observed (which generally correlate with a larger bulge component), then a tendency towards abnormally blue colours and younger stellar populations emerges [@kauf03; @jahn04a]. This tendency appears to hold also at intermediate redshifts, as demonstrated in our companion [[*HST*]{}]{} paper [@sanc04a], where we find more than half of the investigated host galaxies to have bluer colours than what would be expected from their morphological types. However, we can not confirm increasing amounts of blue light from young stars with increasing host luminosity as found by [@kauf03]. Our data are consistent with constant UV flux for all hosts. At high redshifts ($z \ga 2$), colour information is available only for a handful of objects. In a ground-based study of six bright radio-loud quasars, @lehn92 found indications that the hosts were very blue, actively star-forming galaxies. For three high luminosity quasars, one radio-loud and two radio-quiet, @aret98 claimed very luminous envelopes and star formation rates of several hundreds solar masses per year. More recently, @hutc02 presented optical [[*HST*]{}]{} observations of three radio-loud and two radio-quiet quasars and found less extreme, but qualitatively similar results. Because of their high radio power and optical luminosities, many of the quasars observed in previous studies are probably quite different from the moderate-luminosity hosts of moderate-luminosity radio-quiet AGN that we focus on. It is nevertheless interesting to see that the presence of a significantly enhanced UV continuum and young stars seems to be at least qualitatively similar between QSOs of intermediate and high luminosities. Conclusions {#sec:conclusions} =========== We performed the hitherto largest study of host galaxies properties of a complete sample of high-redshift AGN. We detected the hosts and extract colour information in 9 of the 23 AGN, and we also achieved a statistical detection of the host in the remaining 14 from a stack analysis. The UV luminosities can be interpreted in three ways: either as contribution from a passively evolving population of relatively young stars, forming typically 0.5 Gyrs ago, as a mix between a population of old (e.g. $t_{age}\sim t_\mathrm{Hubble}$) stars and a small contribution of a recently formed young population (e.g.0.1%–10% in mass at an age of 100 Myrs or 1/10th of this for age 10 Myrs), or as an indicator of ongoing star formation at a level of $\sim$2–15$\mathrm{M}_\odot\:\mathrm{yr}^{-1}$ (uncorrected for internal dust attenuation). While the first possibility is very simplistic and appears unphysical, the UV colours actually favour the two burst interpretations; but the possibility of on-going star formation cannot be completely ruled out from our data. In the framework of combined old and young populations, it is remarkable how similar the host galaxy colours are within the sample, and, unless different mass–age combinations conspire, hence the estimated stellar mass fractions and ages. The derived young population mass fractions and ages are also very similar to the values estimated in our companion [[gems]{}]{} study of AGN at $z\la 1$ [@sanc04a], where we find abnormally blue rest-frame $U-V$ colours for a substantial fraction of host galaxies, particularly the most luminous AGN in the sample. Even more, these colours and ages are in turn very close to the mean values obtained from our ground-based low-$z$ multicolour sample [@jahn04a]. While the stellar population diagnostics of @kauf03 are not immediately convertible into our simple colour indicators, their impressive and highly significant results point in exactly the same direction. While the results from all these redshift regimes are similar and point to a connection of nuclear activity and the presence of young stars, and that mass fractions of young stars are similar, we always find that a larger range of absolute masses is involved, showing as a range in UV luminosity. Here we find a variation of a factor of $\ga$40 for a given nuclear luminosity. Our host galaxy colours span a range that reaches the colours of Lyman Break Galaxies for a few very luminous hosts, while, as mentioned, the colours of the majority are somewhat redder than these. A comparison of optical/UV properties to the general population of high redshift galaxies would be very illuminating, but large statistical samples only become available in the near future, e.g. from the [[goods]{}]{}project. This persistent trend to find AGN to be associated with blue stellar colours is intriguing and suggests a close connection between enhanced star formation and nuclear activity. Additional support for such a connection comes from the detection of submm CO emission in a number of extremely luminous high-redshift QSOs and radio galaxies [@omon03], although the current sensitivity of submm telescopes is insufficient to perform this test for less luminous AGN at high $z$. While the fact that there is a relation can hardly be denied, its physical origin remains obscure. Is the enhancement of star formation a prerequisite for nuclear activity? Is it a simultaneously occurring phenomenon, caused by the same trigger? Or is it a consequence of the AGN? Galaxy merging and interaction are clearly two possible candidates to connect these two phenomena, but neither the only ones nor are the involved physics understood. Much additional data will be required, in particular those helping reliably to reconstruct the star formation history in high-redshift galaxies, before any firm conclusions can be drawn. Based on observations taken with the NASA/ESA [*Hubble Space Telescope*]{}, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under NASA contract NAS5-26555. Support for the GEMS project was provided by NASA through grant number GO-9500 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS5-26555. EFB and SFS ackowledge financial support provided through the European Community’s Human Potential Program under contract HPRN-CT-2002-00316, SISCO (EFB) and HPRN-CT-2002-00305, Euro3D RTN (SFS). CW was supported by a PPARC Advanced Fellowship. SJ acknowledges support from the National Aeronautics and Space Administration (NASA) under LTSA Grant NAG5-13063 issued through the Office of Space Science. DHM acknowledges support from the National Aeronautics and Space Administration (NASA) under LTSA Grant NAG5-13102 issued through the Office of Space Science. [38]{} natexlab\#1[\#1]{} Aretxaga, I., Terlevich, R. J., & Boyle, B. J. 1998, MNRAS, 296, 643 , G., & [Charlot]{}, S. 2003, MNRAS, 344, 1000 , R., [S[' a]{}nchez]{}, S. F., [Gonz[' a]{}lez-Serrano]{}, J. I., [Benn]{}, C. R., & [Vigotti]{}, M. 1998, AJ, 115, 1234 , G. 2003, PASP, 115, 763 de Vaucouleurs, G. 1948, Ann. Astrophys., 11, 247 , D. K., [Shapley]{}, A. E., [Steidel]{}, C. C., [Pettini]{}, M., [Adelberger]{}, K. L., [Hunt]{}, M. P., [Moorwood]{}, A. F. M., & [Cuby]{}, J. 2003, ApJ, 591, 101 , R., [Kotilainen]{}, J. K., [Pagani]{}, C., [Scarpa]{}, R., & [Treves]{}, A. 2004, ApJ, 604, 495 Freeman, K. C. 1970, ApJ, 160, 812 Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber, S. M., Filippenko, A. V., Green, R., Grillmair, C., Ho, L. C., Kormendy, J., Lauer, T. R., Magorrian, J., Pinkney, J., Richstone, D., & Tremaine, S. 2000, ApJ, 539, L13 Giavalisco, M., [et al.]{} 2004, ApJ, 600, L93 , N. A., [Koekemoer]{}, A. M., [Schreier]{}, E. J., [Bergeron]{}, J., [Giacconi]{}, R., [Hasinger]{}, G., [Kewley]{}, L., [Norman]{}, C., [Rosati]{}, P., [Tozzi]{}, P., & [Zirm]{}, A. 2003, ApJ, 595, 685 , J. B. 2003, AJ, 125, 1053 Hutchings, J. B., Crampton, D., Morris, S. L., Durand, D., & Steinbring, E. 1999, AJ, 117, 1109 Hutchings, J. B., Frenette, D., Hanisch, R., Mo, J., Dumont, P. J., Redding, D. C., & Neff, S. G. 2002, AJ, 123, 2936 Jahnke, K., Kuhlbrodt, B., & Wisotzki, L. 2004, MNRAS (in press), astro-ph/0311123 Kauffmann, G., Heckman, T. M., Tremonti, C., Brinchmann, J., Charlot, S., White, S. D. M., Ridgway, S., Brinkmann, J., Fukugita, M., Hall, P., Ivezic, Z., Richards, G., & Schneider, D. 2003, MNRAS, 346, 1055 Kennicutt, C. K. 1998, ARAA, 36, 189 , A. M., [Grogin]{}, N. A., [Schreier]{}, E. J., [Giacconi]{}, R., [Gilli]{}, R., [Kewley]{}, L., [Norman]{}, C., [Zirm]{}, A., [Bergeron]{}, J., [Rosati]{}, P., [Hasinger]{}, G., [Tozzi]{}, P., & [Marconi]{}, A. 2002, ApJ, 567, 657 Kotilainen, J. K., & Ward, M. J. 1994, MNRAS, 266, 953 Krist, J., & Hook, R. 2003, STSCI Intrument Science Report ACS 2003-06 Kukula, M. J., Dunlop, J. S., McLure, R. J., Miller, L., Percival, W., Baum, S. A., & O’Dea, C. P. 2001, MNRAS, 326, 1533 Lehnert, M. D., Heckman, T. M., Chambers, K. C., & Miley, G. K. 1992, ApJ, 393, 68 , A., [Beelen]{}, A., [Bertoldi]{}, F., [Cox]{}, P., [Carilli]{}, C. L., [Priddey]{}, R. S., [McMahon]{}, R. G., & [Isaak]{}, K. G. 2003, A&A, 398, 857 Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266 Ridgway, S. E., Heckman, T. M., Calzetti, D., & Lehnert, M. 2001, ApJ, 550, 122 Rix, H.-W., Barden, M., Beckwith, S. V. W., Bell, E. F., Borch, A., Caldwell, J. A. R., Häu[ß]{}ler, B., Jahnke, K., Jogee, S., McIntosh, D. H., Meisenheimer, K., Peng, C. Y., Sánchez, S. F., Somerville, R. S., Wisotzki, L., & Wolf, C. 2004, ApJS, 152, 163 , E. E. 1955, ApJ, 121, 161 Sánchez, S. F., & González-Serrano, J. I. 2003, A&A, 406, 435 Sánchez, S. F., Jahnke, K., Wisotzki, L., Barden, M., Beckwith, S. V. W., Bell, E. F., Borch, A., Caldwell, J. A. R., Häu[ß]{}ler, B., Jogee, S., McIntosh, D. H., Meisenheimer, K., Peng, C. Y., Rix, H.-W., Somerville, R. S., & Wolf, C. 2004, submitted to ApJ, astro-ph/0403645 Schade, D., Boyle, B. J., & Letawsky, M. 2000, MNRAS, 315, 498 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 , E. J., [Koekemoer]{}, A. M., [Grogin]{}, N. A., [Giacconi]{}, R., [Gilli]{}, R., [Kewley]{}, L., [Norman]{}, C., [Hasinger]{}, G., [Rosati]{}, P., [Marconi]{}, A., [Salvati]{}, M., & [Tozzi]{}, P. 2001, ApJ, 560, 127 Sérsic, J. 1968, Atlas de Galaxes Australes, Observatorio Astronomico de Cordoba Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, ApJ, 588, 65 , J., [Fosbury]{}, R. A. E., [Villar-Mart[í]{}n]{}, M., [Cohen]{}, M. H., [Cimatti]{}, A., [di Serego Alighieri]{}, S., & [Goodrich]{}, R. W. 2001, A&A, 366, 7 Wolf, C., Meisenheimer, K., Kleinheinrich, M., Borch, A., Dye, S., Gray, M., Wisotzki, L., Bell, E. F., Rix, H.-W., Cimatti, A., Hasinger, G., & Szokoly, G. 2004, submitted to A&A, astro-ph/0403666 Wolf, C., Meisenheimer, K., Rix, H.-W., Borch, A., Dye, S., & Kleinheinrich, M. 2003, A&A, 401, 73 , C., [Wisotzki]{}, L., [Borch]{}, A., [Dye]{}, S., [Kleinheinrich]{}, M., & [Meisenheimer]{}, K. 2003, A&A, 408, 499 AGN images and surface brightness plots of AGN and stars {#sec:appendix} ======================================================== Figure \[fig:allimages\] shows plots for each of the nine resolved object plus the composite ‘stacked’ object. Two objects appear twice as they appear in overlapping areas of [[gems]{}]{}tiles. Figures \[fig:stars\_v\] and \[fig:stars\_z\] show a random selection of isolated stars used to show the zero case of a point sources without any host galaxy contribution, for comparison purposes. {width="\fullwidth"} {width="\fullwidth"} [^1]: The [[galfit]{}]{} version 1.7a was recentering the PSF to given coordinates using a convolution with a narrow gaussian, not by shifting the PSF by means of rebinning to a new position. The latter is better for the application to AGN decomposition and is now incorporated in later versions of [[galfit]{}]{}.

ArXiv

--- abstract: | The distribution of higher order level spacings, i.e. the distribution of $% \{s_{i}^{(n)}=E_{i+n}-E_{i}\}$ with $n\geq 1$ is derived analytically using a Wigner-like surmise for three Gaussian ensembles of random matrix as well as Poisson ensemble. It is found $s^{(n)}$ in Gaussian ensembles follows a generalized Wigner-Dyson distribution with rescaled parameter $\alpha=\nu C_{n+1}^2+n-1$, while that in Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Notably, the distribution of $% s^{(2n)}$ in GOE coincide with that for $s^{(n)}$ in GSE. Numerical evidence are provided through simulations of random spin systems as well as non-trivial zeros of Riemann zeta function. The higher order generalizations of gap ratios are also discussed. author: - 'Wen-Jia Rao$^1$' title: Wigner Surmise for Higher Order Level Spacings in Random Matrix Theory --- Introduction {#intro} ============ Random matrix theory (RMT) was introduced half a century ago when dealing with complex nuclei[@Porter], and since then has found various applications in fields ranging from quantum chaos to isolated many-body systems[@RMP; @PR]. This roots in the fact that RMT describes universal properties of random matrix that depend only on its symmetry while independent of microscopic details. Specifically, the system with time reversal invariance is represented by matrix that belongs to the Gaussian orthogonal ensemble (GOE); the system with spin rotational invariance while breaks time reversal symmetry belongs to the Gaussian unitary ensemble (GUE); while Gaussian symplectic ensemble (GSE) represents systems with time reversal symmetry but breaks spin rotational symmetry. Among various statistical quantities, the most widely used one is the distribution of nearest level spacings $s$, i.e. the gaps between adjacent energy levels, which measures the strength of level repulsion. The exact expression for the $P\left( s\right) $ can be derived analytically for random matrix with large dimension, which is cumbersome[@Mehta; @Haake2001]. Instead, for most practical purposes it’s sufficient to employ the so-called Wigner surmise[@Wigner] that deals with $2\times 2$ matrix (this will be reviewed in Sec. \[nearest\]), the out-coming result for $% P(s) $ has a neat expression that contains a polynomial part accounting for level repulsion and an Gaussian decaying part (see Eq. (\[equ:nearest\])). Different models may and usually do have different density of states (DOS), hence to compare the universal behavior of level spacings, an unfolding procedure is required to erase the model dependent information of DOS. This unfolding procedure is, however, not unique and suffers from subtle ambiguity raised by concrete unfolding strategy[@Gomez2002]. To overcome this obstacle, Oganesyan and Huse[@Oganesyan] proposed a new quantity to study the level statistics, i.e. the ratio between adjacent gaps $r_{n}=\frac{s_{n}}{s_{n-1}}$, whose distribution $P\left( r\right) $ is later analytically derived by Atas *et al.*[@Atas]. The gap ratio is independent of local DOS and requires no unfolding procedure, hence has found various applications, especially in the context of many-body localization (MBL)[@Huse1; @Huse2; @Huse3; @Sarma; @Luitz]. The gap ratio has been later generalized to higher order case to describe level correlations on longer ranges[@Tekur1; @Tekur; @Atas2; @Chavda], although the general analytical result is still lacking. In contrast, the higher order level spacing itself is much less studied. Motivated by a recent work that encountered the next-nearest level spacings[@Rubah], we proceed to pursuit the general distribution of higher order level spacings in this work. By using a Wigner-like surmise, we succeeded in obtaining an analytical expression for the distribution of higher order spacing $s_{n}=E_{i+n}-E_{i}$ in all the three Gaussian ensembles of RMT, as well as the Poisson ensemble. The results show the distribution of $s_{n}$ in the former class follows a generalized Wigner-Dyson distribution with rescaled parameter; while $s_{n}$ in Poisson ensemble follows generalized semi-Poisson distribution with index $n$. This paper is organized as follows. In Sec. \[nearest\] we review the Wigner surmise for deriving the distribution of nearest level spacings, and present numerical data to validate this surmise. In Sec. \[analytical\] we present the analytical derivation for higher order level spacings using a Wigner-like surmise, and numerical fittings are given in Sec. \[numerics\]. In Sec. \[ratio\] we discuss the generalization of gap ratios to higher order. Conclusion and discussion come in Sec. \[conclusion\]. Nearest Level Spacings {#nearest} ====================== We begin with the discussion about nearest level spacings, our starting point probability distribution of energy levels $P\left( \left\{ E_{i}\right\} \right) $ in three Gaussian ensembles, whose expression can be found in any textbook on RMT (e.g. Ref. \[\]),$$P\left( \left\{ E_{i}\right\} \right) \propto \prod_{i<j}\left\vert E_{i}-E_{j}\right\vert ^{\nu }e^{-A\sum_{i}E_{i}^{2}} \label{equ:Dist}$$where $\nu =1,2,4$ for GOE,GUE,GSE respectively. The distribution of nearest level spacing can then be written as $$P\left( s\right) =\int \prod_{i=1}^{N}dE_{i}P\left( \left\{ E_{i}\right\} \right) \delta \left( s-\left\vert E_{1}-E_{2}\right\vert \right) \text{,}$$whose result is quite complicated. Instead, Wigner proposes a surmise that we can focus on the $N=2$ case, the distribution then reduces to[$$P\left( s\right) \propto \int_{-\infty }^{\infty }\left\vert E_{1}-E_{2}\right\vert ^{\nu }\delta \left( s-\left\vert E_{1}-E_{2}\right\vert \right) e^{-A\sum_{i}E_{i}^{2}}dE_{1}dE_{2}\text{.}$$]{} By introducing $x_{1}=E_{1}-E_{2}$, $x_{2}=E_{1}+E_{2}$, we have$$\begin{aligned} P\left( s\right) &\propto &2\int_{-\infty }^{\infty }\left\vert x_{1}\right\vert ^{\nu }\delta \left( s-\left\vert x_{1}\right\vert \right) e^{-\frac{A}{2}\sum_{i}x_{i}^{2}}dx_{1}dx_{2} \notag \\ &=&Cs^{\nu }e^{-As^{2}/2}\text{.}\end{aligned}$$The constants $A,C$ can be determined by working out the integral about $% x_{2}$, but it is more convenient to obtain by imposing the normalization condition$$\int_{0}^{\infty }P\left( s\right) ds=1\text{, }\int_{0}^{\infty }sP\left( s\right) ds=1\text{.} \label{equ:normalization}$$From which we can reach to the famous Wigner-Dyson distribution $$P(s)=\left\{ \begin{array}{ll} \frac{\pi }{2}s\exp \big(-\frac{\pi }{4}s^{2}\big) & \nu =1\quad \text{GOE} \\[1mm] \frac{32}{\pi ^{2}}s^{2}\exp \big(-\frac{4}{\pi }s^{2}\big) & \nu =2\quad \text{GUE} \\[1mm] \frac{2^{18}}{3^{6}\pi ^{3}}s^{4}\exp \big(-\frac{64}{9\pi }s^{2}\big) & \nu =4\quad \text{GSE}% \end{array}% \right. \label{equ:nearest}$$ On the other hand, the levels are independent in Poisson ensemble, which means the occurrence of next level is independent of previous level, the nearest level spacings then follows a Poisson distribution $P\left( s\right) =\exp \left( -s\right) $. Although the Wigner surmise is for $2\times 2$ matrix, it works fairly good when the matrix dimension is large. To demonstrate this, we present numerical evidence from a quantum many-body system – the spin-$1/2$ Heisenberg model with random external field, which is the canonical model in the study of many-body localization (MBL),$$H=\sum_{i=1}^{L}\mathbf{S}_{i}\cdot \mathbf{S}_{i+1}+\sum_{i=1}^{L}\sum_{% \alpha =x,y,z}h^{\alpha }\varepsilon _{i}^{\alpha }S_{i}^{\alpha }, \label{equ:H}$$where we set coupling strength to be $1$ and assume periodic boundary condition in Heisenberg term. This $\varepsilon _{i}^{\alpha }$s are random numbers within range $\left[ -1,1\right] $, and $h^{\alpha }$ is referred as the randomness strength. We focus on two choices of $h^{\alpha }$: (i) $% h^{x}=h^{z}=h\neq 0$ and $h^{y}=0$, the Hamiltonian matrix is orthogonal; (ii) $h^{x}=h^{y}=h^{z}=h\neq 0$, the model being unitary. This model undergos a thermal-MBL transition at roughly $h_{c}\simeq3$ ($2.5$) in the orthogonal (unitary) model, where the level spacing distribution evolves from GOE (GUE) to Poisson[@Regnault16]. We choose a $L=12$ system to present a numerical simulation, and prepare $% 500 $ samples at $h=1$ and $h=5$ for both the orthogonal and unitary model. In Fig. \[fig:NN\_spacing\](Left) we plot the density of states (DOS) for the $h=1 $ case in orthogonal model. We can see DOS is much more uniform in the middle part of the spectrum, which is also the case for $h=5$ and unitary model. Therefore we choose the middle half of energy levels to do the spacing counting, and the results are shown in Fig. \[fig:NN\_spacing\](Right). We observe a clear GOE/GUE distribution for $h=1$ in orthogonal/unitary model and a Poisson distribution for $h=5$ in orthogonal model as expected, the fitting result for $h=5$ in unitary model is not shown since it almost coincides with that in orthogonal model. It is noted the fitting for Poisson distribution has minor deviations around the region $% s\sim 0$, this is due to finite size effect since there will always remain exponentially-decaying but finite correlation between levels in a finite system. As we will demonstrate in subsequent section, the fitting for higher order level spacings will be better since the overlap between levels decays exponentially with their distance in MBL phase. A technique issue is, when counting the level spacings, we choose to take the middle half levels of the spectrum, while we can also employ a unfolding procedure using a spline interpolation that incorporates all energy levels[@Avishai2002], and the fitting results are almost the same[Regnault162,Rao182]{}. ![(Left) The density of states (DOS) $\protect\rho (E)$ of random field Heisenberg model at $L=10$ and $h=1$ in orthogonal case, the DOS is more uniform in the middle part, we therefore choose the middle half levels to do level statistics. (Right) Distribution of nearest level spacings $P(E_{i+1}-E_{i})$, we see a GOE/GUE distribution for $h=1$ in the orthogonal/unitary model, while a Poisson distribution is found for $h=5$ in orthogonal model, the result for $h=5$ in unitary model is not displayed since it coincides with that in the orthogonal model.[]{data-label="fig:NN_spacing"}](Nearest_Spacing.pdf){width="8.7cm"} Higher Order Level Spacings =========================== Now we proceed to consider the distribution of higher order level spacings $% \left\{ s_{i}^{\left( n\right) }=E_{i+n}-E_{i}\right\} $, using a Wigner-like surmise. We first give the analytical derivation, then provide numerical evidence from simulation of spin model in Eq. (\[equ:H\]) as well as the non-trivial zeros of Riemann zeta function. Analytical Derivation {#analytical} --------------------- Introduce $P_{n}\left( s\right) =P\left( \left\vert E_{i+n}-E_{i}\right\vert =s\right)$, to apply the Wigner surmise, we are now considering $\left( n+1\right) \times \left( n+1\right) $ matrices, the distribution $% P_{n}\left( s\right) $ then goes to$$\begin{aligned} P_{n}\left( s\right) &\propto &\int_{-\infty }^{\infty }\prod_{i<j}\left\vert E_{i}-E_{j}\right\vert ^{\nu }\delta \left( s-\left\vert E_{1}-E_{n+1}\right\vert \right) \notag \\ &&\times e^{-A\sum_{i=1}^{n+1}E_{i}^{2}}\prod_{i=1}^{n+1}dE_{i}\end{aligned}$$We first change the variables to$$x_{i}=E_{i}-E_{i+1}\text{, }i=1,2,...,n\text{; }\quad x_{n+1}=\sum_{i=1}^{n+1}E_{i}\text{,}$$the $P_{n}\left( s\right) $ then evolves into $$P_{n}\left( s\right) \propto \int_{-\infty }^{\infty }\frac{\partial \left( E_{1},E_{2},...,E_{n+1}\right) }{\partial \left( x_{1},x_{2},...,x_{n+1}\right) }\left( \prod_{i=1}^{n}\prod_{j=i}^{n}\left\vert \sum_{k=i}^{j}x_{k}\right\vert ^{\nu }\right) \delta \left( s-\left\vert \sum_{i=1}^{n}x_{i}\right\vert \right) e^{-\frac{A}{n}\left[ \sum_{i=1}^{n}% \sum_{j=i}^{n}\left( \sum_{k=i}^{j}x_{k}\right) ^{2}+x_{n+1}^{2}\right] }\prod_{i=1}^{n+1}dx_{i}.$$ In this expression, the Jacobian $\frac{\partial \left( E_{1},E_{2},...,E_{n+1}\right) }{\partial \left( x_{1},x_{2},...,x_{n+1}\right) }$ and integral for $x_{n+1}$ are all constants that can be absorbed into the normalization factor, hence we can simplify $P_{n}\left( s\right) $ to$$\begin{aligned} P_{n}\left( s\right) &\propto &\int_{-\infty }^{\infty }\left( \prod_{i=1}^{n}\prod_{j=i}^{n}\left\vert \sum_{k=i}^{j}x_{i}\right\vert ^{\nu }\right) \delta \left( s-\left\vert \sum_{i=1}^{n}x_{i}\right\vert \right) \notag \\ &&\times e^{-\frac{A}{n}\sum_{i=1}^{n}\sum_{j=i}^{n}\left( \sum_{k=i}^{j}x_{k}\right) ^{2}}\prod_{i=1}^{n}dx_{i}.\end{aligned}$$Next, we introduce the $n$-dimensional spherical coordinate $$\begin{aligned} x_{1} &=&r\cos \theta _{1}\text{; }\quad x_{n}=r\prod_{i=1}^{n-1}\sin \theta _{i}\text{;} \notag \\ x_{i} &=&r\left( \prod_{j=1}^{i-1}\sin \theta _{j}\right) \cos \theta _{i-1}% \text{, \thinspace }i=2,3,...,n-1\text{;} \\ 0 &\leq &\theta _{i}\leq \pi \text{, }i=1,2,...,n-2\text{;}\quad 0\leq \theta _{n-1}\leq 2\pi \text{,} \notag\end{aligned}$$whose Jacobian is$$\frac{\partial \left( x_{1},x_{2},...,x_{n}\right) }{\partial \left( r,\theta _{1},\theta _{2},...,\theta _{n-1}\right) }=r^{n-1}% \prod_{i=1}^{n-2}\sin ^{n-1-i}\theta _{i} \label{equ:Jac}$$which reduces to the normal spherical coordinate when $n=3$. The resulting expression of $P_{n}\left( s\right) $ is complicated, while we are mostly interested in the scaling behavior about $s$, hence we can write the formula as$$\begin{aligned} P_{n}\left( s\right) &\propto &\int_{0}^{\infty }r^{n-1}\int r^{\nu C_{n+1}^{2}}\delta \left( s-r\left\vert G\left( \boldsymbol{\theta }\right) \right\vert \right) \notag \\ &&\times H\left( \boldsymbol{\theta }\right) e^{-\frac{A}{n}r^{2}J\left( \boldsymbol{\theta }\right) }drd\boldsymbol{\theta }\end{aligned}$$where $C_{n+1}^{2}=n\left( n+1\right) /2$, and $d\boldsymbol{\theta }$ $% =\prod_{i=1}^{n-1}d\theta _{i}$, the explanation goes as follows: (i) the first term $r^{n-1}$ comes from the radial part of the Jacobian in Eq. ([equ:Jac]{}); (ii) the second $r^{\nu C_{n+1}^{2}}$ comes number of terms in $% \prod_{i=1}^{n}\prod_{j=i}^{n}\left\vert \sum_{k=i}^{j}x_{i}\right\vert ^{\nu }$, where each term contributes a factor $r^{\nu }$; (iii) the auxiliary function $G\left( \boldsymbol{\theta }\right) =\sum_{i=1}^{n}x_{i}/r$; (iv) the second auxiliary function $H\left( \boldsymbol{\theta }\right) $ is comprised of the angular part of the Jacobian and the angular part of $\prod_{i=1}^{n}\prod_{j=i}^{n}\left\vert \sum_{k=i}^{j}x_{i}\right\vert ^{\nu }$; (v) $J\left( \boldsymbol{\theta }% \right) $ is the angular part of $\sum_{i=1}^{n}\sum_{j=i}^{n}\left( \sum_{k=i}^{j}x_{k}\right) ^{2}$. The key observation is that $G\left( \boldsymbol{\theta }\right) ,H\left( \boldsymbol{\theta }\right) ,J\left( \boldsymbol{\theta }\right) $ all depend only on $\boldsymbol{\theta }$ while independent of $r$. Since we are only interested in the scaling behavior about $s$, we can work out the delta function, and get$$P_{n}\left( s\right) \propto s^{\nu C_{n+1}^{2}+n-1}\int H\left( \boldsymbol{% \theta }\right) e^{-\frac{AJ\left( \boldsymbol{\theta }\right) }{n\left\vert G\left( \boldsymbol{\theta }\right) \right\vert ^{2}}s^{2}}d\boldsymbol{% \theta }$$Although the integral for $\boldsymbol{\theta }$ is tedious and difficult to handle, it will only make correction to the Gaussian factor while not influence the scaling behavior about $s$. Therefore we can write $% P_{n}\left( s\right) $ into a generalized Wigner-Dyson distribution$$\begin{aligned} P_{n}\left( s\right) &=&C\left( \alpha \right) s^{\alpha }e^{-A\left( \alpha \right) s^{2}}\text{, } \label{equ:GWD} \\ \alpha &=&\frac{n\left( n+1\right) }{2}\nu +n-1\text{.} \label{equ:rescale}\end{aligned}$$The normalization factors $C\left( \alpha \right) $ and $A\left( \alpha \right) $ can be determined by the normalization condition in Eq. ([equ:normalization]{}), for which we obtain$$A\left( \alpha \right) =\left( \frac{\Gamma \left( \alpha /2+1\right) }{% \Gamma \left( \alpha /2+1/2\right) }\right) ^{2}\text{, }C\left( \alpha \right) =\frac{2\Gamma ^{n+1}\left( \alpha /2+1\right) }{\Gamma ^{n+2}\left( \alpha /2+1/2\right) }\text{,}$$where $\Gamma \left( z\right) =\int_{0}^{\infty }t^{z-1}e^{-t}dt$ is the Gamma function. When $n=1$, $P_{n}\left( s\right) $ reduces to the conventional Wigner-Dyson distribution in Eq. (\[equ:nearest\]). Interestingly, there exists coincidence between distributions in different ensembles. For example, as can be easily checked, $P_{k}\left( s\right) $ in the GSE coincides with $P_{2k}\left( s\right) $ in GOE for arbitrary integer $k$, where the special case with $k=1$ has been well-known for circular ensembles[@GSE]; $P_{7}\left( s\right) $ in GOE coincides with $% P_{5}\left( s\right) $ in GUE, and so on. We also note similar results have been proposed for $n\geq 2$ using a phenomenological argument based on several assumptions[@Magd], while our derivation is rigorous without assumption. For the uncorrelated energy levels in the Poisson class, the distribution for higher order spacing can also be obtained. Let’s start with $n=2$, we can write $s^{\prime}=E_{i+2}-E_{i}=\left( E_{i+2}-E_{i+1}\right) +\left( E_{i+1}-E_{i}\right) =s_{i+1}+s_{i}$, where $s_{i+1}$ and $s_{i}$ can be treated as independent variables that both follows Poisson distribution, therefore the distribution $P_{2}\left( s^{\prime }\right) $ for unnormalized $s^{\prime }$ is$$P_{2}\left( s^{\prime }\right) \propto \int_{0}^{s^{\prime }}P_{1}\left( s^{\prime }-s_{1}\right) P_{1}\left( s_{1}\right) ds_{1}=s^{\prime }e^{-s^{\prime }}\text{.} \label{equ:recur}$$Then by requiring the normalization condition we arrive at $P_{2}\left( s\right) =4se^{-2s}$, which is nothing but the semi-Poisson distribution. Repeating this procedure $n-1\ $times, we reach to$$P_{n}\left( s\right) =\frac{n^{n}}{\Gamma \left( n\right) }s^{n-1}e^{-ns}% \text{.} \label{equ:Pn}$$which is a generalized semi-Poisson distribution with index $n$. Compared to the Poisson distribution for nearest level spacings, it’s crucial to note that $P_{n}\left( 0\right) =0$ for $n\geq 2$, this is not a result of level repulsion as in the Gaussian ensembles, rather, it simply states that $% n+1\left( n\geq 2\right) $ consecutive levels do not coincide. We note every $P_{n}\left( s\right) $ in the Gaussian and Poisson ensembles tends to be the Dirac delta function $\delta \left( s-1\right) $ in the limit $n\rightarrow \infty $, which is easily understood since in that limit only one spacing remains in the spectrum. Finally, we want to emphasize the the levels are well-correlated in the three Gaussian ensembles, hence the derivation of $P_{n}\left( s\right) $ for Poisson ensemble in Eq. ([equ:recur]{}) do not hold, otherwise the result will deviate dramatically[Rubah]{}. For convenience we list the order of the polynomial part in $P_{n}\left( s\right) $ for the three Gaussian ensembles as well as Poisson ensemble up to $n=8$ in Table \[tab:1\], note that the exponential parts in the former class are Gaussian type and that for Poisson ensemble is a exponential decay. $n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ --------- ----- ------ ------ ------ ------ ------ ------- ------- GOE $1$ $4$ $8$ $13$ $19$ $26$ $34$ $43$ GUE $2$ $7$ $14$ $23$ $34$ $47$ $62$ $79$ GSE $4$ $13$ $26$ $43$ $64$ $89$ $118$ $151$ Poisson $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ : The order of the polynomial term in $P_{n}(s)$ for the three Gaussian ensembles as well as Poisson ensemble, the decaying term is Gaussian type for the former class and exponential decay for the latter.[]{data-label="tab:1"} Numerical Simulation {#numerics} -------------------- To show how well the generalized Wigner-Dyson distribution in Eq. ([equ:GWD]{}) works for matrix with large dimension, we now perform numerical simulations for the random spin model in Eq. (\[equ:H\]), where we also pick the middle half levels to do statistics. We have tested the formula up to $n=5$, and in Fig. \[fig:higher\_spacing\] we display the fitting results for $n=2$ and $n=3$. ![Distribution of next-nearest level spacings $P(E_{i+2}-E_{i})$ (Left) and next-next-nearest level spacings $P(E_{i+3}-E_{i})$ (Right), where $\alpha$ and $n$ are the index in Eq. (\[equ:GWD\]) and Eq. (\[equ:Pn\]) respectively.[]{data-label="fig:higher_spacing"}](higher_spacing.pdf){width="8.7cm"} As expected, the fittings are quite accurate for both GOE and GUE as well as Poisson ensemble. In fact, the fittings for higher order spacings in the Poisson ensemble are better than that for nearest spacing in Fig. [fig:NN\_spacing]{}(Right). This is because in MBL phase the overlap between levels decays exponentially with their distance, hence the fitting for higher order level spacings is less affected by finite size effect. For another example we consider the non-trivial zeros of the Riemann zeta function$$\zeta \left( z\right) =\sum_{n=1}^{\infty }\frac{1}{n^{z}}\text{.}$$It was established that statistical properties of non-trivial Riemann zeros $% \left\{ \gamma _{i}\right\} $ are well described by the GUE distribution[Zeta]{}. Therefore, we expect the gaps $\left\{s^{(n)}_i=\gamma _{i+n}-\gamma _{i}\right\} $ follows the same distribution as those in GUE. The numerical results for $n=1,2,3$ are presented in Fig. \[fig:zeta\], as can be seen, the fittings are perfect. ![The distribution of $n$-th order spacings of the non-trivial zeros $\{\gamma_i\}$ of Riemann zeta function, where $\alpha$ is the index in generalized Wigner-Dyson distribution in Eq. (\[equ:GWD\]). The data comes from $10^4$ levels starting from the $10^{22}$th zero, taken from Ref. \[\].](ZetaFit.pdf){width="8cm"} [fig:zeta]{} Higher Order Spacing Ratios {#ratio} =========================== As mentioned in Sec. \[intro\], besides the level spacings, another quantity is also widely used in the study of random matrices, namely the ratio between adjacent gaps $\left\{ r_{i}=\frac{s_{i}}{s_{i-1}}\right\} $, which is independent of local DOS. The distribution of nearest gap ratios $P\left( \nu ,r\right) $ is given in Ref. \[\], whose result is$$P\left( \nu ,r\right) =\frac{1}{Z_{\nu }}\frac{\left( r+r^{2}\right) ^{\nu }% }{\left( 1+r+r^{2}\right) ^{1+3\nu /2}}$$where $\nu =1,2,4$ for GOE,GUE,GSE, and $Z_{\nu }$ is the normalization factor determined by requiring $\int_{0}^{\infty }P\left( \nu ,r\right) dr=1$. This gap ratio can also be generalized to higher order, but in different ways, i.e. the overlapping [Atas,Atas2]{} and non-overlapping [Tekur,Chavda]{} way. In the former case we are dealing with$$\widetilde{r}_{i}^{\left( n\right) }=\frac{E_{i+n}-E_{i}}{E_{i+n-1}-E_{i-1}}=% \frac{s_{i+n}+s_{i+n-1}+...+s_{i+1}}{s_{i+n-1}+s_{i+n-2}+...+s_{i}}\text{,}$$which is named overlaping ratio since there is shared spacings between the numerator and denominator. While the non-overlapping ratio is defined as$$r_{i}^{\left( n\right) }=\frac{E_{i+2n}-E_{i+n}}{E_{i+n}-E_{i}}=\frac{% s_{i+2n}+s_{i+2n-1}+...+s_{i+n+1}}{s_{i+n}+s_{i+n-1}+...+s_{i}}\text{.}$$These two generalizations are quite different when we are to study their distributions using Wigner surmise: for overlapping ratio $\widetilde{r}% _{i}^{\left( n\right) }$, the smallest matrix dimension is $\left( n+2\right) \times \left( n+2\right) $; while it is $\left( 1+2n\right) \times \left( 1+2n\right) $ for non-overlapping ratio; only for $n=1$ do these two coincide. Naively, we can expect the distribution for $\widetilde{r% }^{\left( n\right) }$ is more involved due to the overlapping spacings. Indeed, the $n=2$ case for $P\left( \widetilde{r}^{\left( n\right) }\right) $ has been worked out in Ref. \[\] and the result is very complicated. Instead, the non-overlapping ratio is less studied. Ref. \[\] provides compelling numerical evidence for the distribution of non-overlapping ratio$$\begin{aligned} P\left( \nu ,r^{\left( n\right) }\right) &=&P\left( \nu ^{\prime },r\right) \text{, } \label{equ:rn} \\ \nu ^{\prime } &=&\frac{n\left( n+1\right) }{2}\nu +n-1\text{.} \label{equ:rescale2}\end{aligned}$$Surprisingly, the rescaling relation Eq. (\[equ:rescale2\]) coincides with that for higher order level spacing in Eq. (\[equ:rescale\]). We have also confirmed this formula by numerical simulations in our spin model Eq. ([equ:H]{}), and the results for $n=2$ in GOE ($\nu =1$) case is presented in Fig. \[fig:ratiocom\], where we also draw the distribution of overlapping ratio $\widetilde{r}^{\left( 2\right) }$ for comparison. As can be seen, they differ dramatically, and the fitting for non-overlapping ratio is quite accurate. This result strongly suggest the non-overlapping ratio is more universal than the overlapping ratio, and its distribution $P\left( r^{\left( n\right) }\right) $ is homogeneously related with that for $n-$th order level spacing, for which we provide a heuristic explanation as follows. For a given energy spectrum $\left\{ E_{i}\right\} $ from a Gaussian ensemble with index $\nu $, we can make up a new spectrum $\left\{ E_{i}^{^{\prime }}\right\} $ by picking one level from every $n$ levels in $% \left\{ E_{i}\right\} $, then the $n$-th order level spacing $s^{\left( n\right) }$ in $\left\{ E_{i}\right\} $ becomes the nearest level spacing in $\left\{ E_{i}^{^{\prime }}\right\} $, and the $n$-th order non-overlapping ratio in $\left\{ E_{i}\right\} $ becomes the nearest gap ratio in $\left\{ E_{i}^{^{\prime }}\right\} $. Since we have analytically proven the rescaling relation in Eq. (\[equ:rescale\]), we conjecture the probability density for $\left\{ E_{i}^{^{\prime }}\right\} $ (to leading order) bear the same form as $% \left\{ E_{i}\right\} $ in Eq. (\[equ:Dist\]) with the rescaled parameter $% \alpha $ in Eq. (\[equ:rescale\]). Therefore, the higher order non-overlapping gap ratios also follow the same rescaling as expressed in Eq. (\[equ:rn\]) and Eq. (\[equ:rescale2\]). ![The distribution of second-order gap ratio in the orthogonal model, where red and blue dots correspond to overlapping and non-overlapping ratios respectively, the latter fits perfectly with the formula in Eq. (\[equ:rn\]) with $\nu^{\prime}=4$. Note the data is taken from the whole energy spectrum without unfolding.[]{data-label="fig:ratiocom"}](ratioComparison.pdf){width="8cm"} Conclusion and Discussion {#conclusion} ========================= We have analytically studied the distribution of higher order level spacings $\left\{ s_{i}^{\left( n\right) }=E_{i+n}-E_{i}\right\} $ which describes the level correlations on long range. It is shown $s^{\left( n\right) }$ in the Gaussian ensemble with index $\nu $ follows a generalized Wigner-Dyson distribution with index $\alpha =\nu C_{n+1}^{2}+n-1$, where $\nu =1,2,4$ for GOE,GUE,GSE respectively. This results in the coincidence of distribution for $s^{\left( 2k\right) }$ in GOE with that for $s^{\left( k\right) }$ in GSE. While $s^{\left( n\right) }$ in Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Our derivation is rigorous based on a Wigner-like surmise, and the results have been confirmed by numerical simulations from random spin system and non-trivial zeros of Riemann zeta function. We also discussed the higher order generalization of gap ratios, which come in two different ways – the overlapping and non-overlapping way – and point out their difference in studying their distributions using Wigner-like surmise. Notably, the distribution for the non-overlapping gap ratio has been studied numerically in Ref. \[\], in which the authors find a scaling relation Eq. (\[equ:rescale2\]) that is identical to the one we find analytically for higher order level spacings. This strongly indicates the distribution of higher order spacing and non-overlapping gap ratio is correlated in a homogeneous way, for which we provided a heuristic explanation. Our derivations are rigorous that based only on universal property of random matrix while independent of concrete physical Hamiltonian, hence can be applied to a variety of models in related areas. It is interesting to note the distribution of next-nearest level spacing in Poisson class is semi-Poisson $P_{2}\left( s\right) \propto s\exp \left( -2s\right) $, which is suggested to be the distribution for nearest level spacing at the thermal-MBL transition point in orthogonal model [@Serbyn]. This either is a mathematical coincidence or indicates the universality property of this transition point is more affected by the MBL phase than the thermal phase. Besides, in this paper the distribution of higher order level spacing is derived only in $\left( n+1\right) \times \left( n+1\right) $ matrix, its exact value in large matrix as well as the difference between them can in principle be estimated using the method in Ref. \[\], this is left for a future study. Acknowledgements {#acknowledgements .unnumbered} ================ The author acknowledges the helpful discussions with Xin Wan and Rubah Kausar. This work is supported by the National Natural Science Foundation of China through Grant No.11904069 and No.11847005. [99]{} C. E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York), 1965. T. A. Brody et al., Rev. Mod. **53**, 385 (1981). T. Guhr, A. Muller-Groeling, H. A. Weidenmuller, Phys. Rep. **299**, 189 (1998). M. L. Mehta, Random Matrix Theory, Springer, New York (1990). F. Haake, Quantum Signatures of Chaos (Springer 2001). E. P. Wigner, in Conference on Neutron Physics by Timeof-Flight (Oak Ridge National Laboratory Report No. 2309, 1957) p. 59. J. M. G. Gomez, R. A. Molina, A. Relano, and J. Retamosa, Phys. Rev. E **66**, 036209 (2002). V. Oganesyan and D. A. Huse, Phys. Rev. B **75**, 155111 (2007). Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev. Lett. **110**, 084101 (2013). V. Oganesyan, A. Pal, D. A. Huse, Phys. Rev. B **80**, 115104 (2009). A. Pal, D. A. Huse, Phys. Rev. B **82**, 174411 (2010). S. Iyer, V. Oganesyan, G. Refael, D. A. Huse, Phys. Rev. B **87**, 134202 (2013). X. Li, S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phy. Rev. Lett. **115**, 186601 (2015). David J. Luitz, Nicolas Laflorencie, and Fabien Alet, Phys. Rev. B **91**, 081103(R) (2015). Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, and E. Vivo, J. Phys. A: Math. Theor. **46**, 355204 (2013). S. H. Tekur, S. Kumar and M. S. Santhanam, Phys. Rev. E, **97**, 062212 (2018). S. H. Tekur, U. T. Bhosale, and M. S. Santhanam, Phys. Rev. B **98**, 104305 (2018). P. Rao, M. Vyas, and N. D. Chavda, arXiv:1912.05664v1. R. Kausar, W.-J. Rao, and X. Wan, arXiv:2005.00721. N. Regnault and R. Nandkishore, Phys. Rev. B **93**, 104203 (2016). Y. Avishai, J. Richert, and R. Berkovits, Phys. Rev. B **66**, 052416 (2002). S. D. Geraedts, R. Nandkishore, and N. Regnault, Phys. Rev. B **93**, 174202 (2016). W.-J. Rao, J. Phys.:Condens. Matter **30**, 395902 (2018). M. L. Mehta and F. J. Dyson, Journal of Mathematical Physics, **4** (1963). A. Y. Abul-Magd and M. H. Simbel, Phys. Rev. E **60**, 5371 (1999). H. L. Montgomery, Proc. Symp. Pure Math. **24**, 181 (1973); E. B. Bogomolny and J. P. Keating, Nonlinearity **8**, 1115 (1995); ibid Nonlinearity , 911 (1995); Z. Rudnick and P. Sarnak, Duke Math. J. **81**, 269 (1996); J. P. Keating and N. C. Snaith, Comm. Math. Phys. **214**, 57 (2000). A. Odlyzko, www.dtc.umn.edu/$\sim$odlyzko/zeta\_tables/index.html. M. Serbyn and J. E. Moore, Phys. Rev. B **93**, 041424(R) (2016).

ArXiv

--- abstract: 'In parametric sequence alignment, optimal alignments of two sequences are computed as a function of the penalties for mismatches and spaces, producing many different optimal alignments. Here we give a $3/(2^{7/3}\pi^{2/3})n^{2/3} +O(n^{1/3} \log n)$ lower bound on the maximum number of distinct optimal alignment summaries of length $n$ binary sequences. This shows that the upper bound given by Gusfield et. al. is tight over all alphabets, thereby disproving the “$\sqrt{n}$ conjecture". Thus the maximum number of distinct optimal alignment summaries (i.e. vertices of the alignment polytope) over all pairs of length $n$ sequences is $\Theta(n^{2/3})$.' address: 'Department of Mathematics, University of California, Berkeley 94720' author: - Cynthia Vinzant title: Lower Bounds for Optimal Alignments of Binary Sequences --- sequence alignment ,parametric analysis ,computational biology Introduction and Notation ========================= Finding optimal alignments of DNA or amino acid sequences is often used in biology to measure sequence similarity (homology) and determine evolutionary history. For a review of many problems relating to sequence alignment, see [@G; @book] and [@ASCB]. Here we deal with the question of how many different alignment summaries can be considered optimal for a given pair of sequences (though many different alignments may correspond to the same alignment summary). Given sequences $S$, $T$, an *alignment* $\Gamma$ is a pair $(S', T')$ formed by inserting spaces, “$-$", into $S$ and $T$. In each position, there is a *match*, in which $S'$ and $T'$ have the same characters, a *mismatch*, in which they have different characters, or a space in one of the sequences. Then for any alignment, we have an *alignment summary* $(w, x,y)$, where $w$ is the number of matches, $x$ is the number of mismatches, and $y$ is the number of spaces in one of the sequences. Notice that $n = w+x+y$, where $n$ is the length of both sequences. Given a pair of sequences, the convex hull of all such points $(w,x,y)$ is called their *alignment polytope*. We can score alignments by weighting each component. Since we have $w+x+y=n$, we can normalize so that the weight of $w$ is 1, the weight of $x$ is $-\alpha$ and the weight of $y$ is $-\beta$. Then $$score_{(\alpha, \beta)}(w,x,y) = w - \alpha x - \beta y.$$ A sequence is *optimal* if it maximizes this score. For biological relevance, we will only consider non-negative $\alpha$ and $\beta$, which penalizes mismatches and spaces. It is also possible to weight other parameters, such as *gaps* (consecutive spaces) or mismatches between certain subsets of characters. Here we will consider only the two parameter model described above. \[ex1\] For the sequences 111000 and 010110, we have an alignment $$\begin{matrix} - & 1 & - & 1&1& 0 &0 &0\\ 0 & 1 & 0 & 1&1&1&-&- \end{matrix} \qquad$$ which has 3 matches, 1 mismatch, and 2 spaces. So for a given $\alpha$ and $\beta$ the score of the this alignment would be $3 - \alpha - 2\beta$. Any value of $\alpha$ and $\beta$ will give an optimal alignment. Given $\alpha$ and $\beta$, we can use the Needleman-Wunsch algorithm to effectively compute optimal alignments [@NW] (for a review, see [@ASCB Ch. 2, 7]). Unfortunately, different choices for $\alpha, \beta$ give different optimal alignments, leaving the problem of which weights to use. To resolve this, Waterman, Eggert, and Lander proposed *parametric alignment*, in which the weights $\alpha$, $\beta$ are viewed as parameters rather than constants [@beginnings]. Since alignments are discrete, this creates a partition of the $(\alpha, \beta)$ plane into *optimality regions*, so that for each region $R$, there is an alignment that is optimal for all the points on its interior and $R$ is maximal with this property [@G]. Each optimality region is a convex cone in the plane [@G], [@ASCB Ch. 8]. Notice that because our scoring function is linear, the vertices of the alignment polytope are our optimal alignment summaries. Also, if we let $P_{xy}$ be the convex hull of all $(x,y)$ occurring in alignment summaries, then $$score_{(\alpha, \beta)} = w - \alpha x - \beta y = n - (\alpha +1)x - (\beta +1)y,$$ since $n= w+x+y$. Thus the vertices of $P_{xy}$ will be those that minimize $(x,y)\cdot (\alpha+1, \beta+1)$ for some $(\alpha, \beta$), thus maximizing $score_{(\alpha, \beta)}$ and corresponding to optimal alignments [@ASCB]. From this we can see that the the decomposition of the $(\alpha, \beta)$ plane into optimality regions can be obtained by shifting the normal fan of $P_{xy}$ by $(-1, -1)$ [@ASCB Ch. 8]. The goal of parametric alignment is to find all these optimality regions with their corresponding optimal alignments. The Needleman-Wunsch algorithm is also an effective method of computing the alignment polytope of sequences (and thus optimal alignments and the decomposition of the $(\alpha, \beta)$ plane) [@ASCB]. Gusfield et. al. showed that for two sequences of length $n$, the number of optimality regions of the $(\alpha, \beta)$ plane (equivalently the number of vertices in their alignment polytope) is $O(n^{2/3})$[@G]. Indeed for larger dimensional models (say with $d$ free parameters), this bound was extended to $O(n^{d-(1/3)})$ by Fernández-Baca et. al. [@Baca2] and improved to $O(n^{d(d-1)/(d+1)})$ by Pachter and Sturmfels [@alg; @bounds]. For $d=2$, Fernández-Baca et. al. refined this bound to $3(n/2\pi)^{2/3} + O(n^{1/3}\log (n))$ and showed it to be tight over an infinite alphabet [@Baca]. They also provide a lower bound of $\Omega(\sqrt{n})$ over a binary alphabet. Using randomly-generated sequences, Fernández-Baca et. al. observed that the average number of optimality regions closely approximates $\sqrt{n}$. This led them to conjecture that, over a finite alphabet, the expected number of optimality regions is $\Theta(\sqrt{n})$[@Baca]. The question remained of whether or not the upper bound of Gusfield et. al. was tight over a finite alphabet. For a discussion, see [@ASCB Ch. 8], which conjectures that the maximum number of optimality regions induced by any pair of length-$n$ binary strings is $\Theta(\sqrt{n})$ [@ASCB]. Here we construct a counterexample to this conjecture, which together with the above upper bounds shows it instead to be $\Theta(n^{2/3})$. Our main theorem is that Gusfield’s bound is tight for binary strings. The maximum number of optimality regions induced by binary strings of length $n$ is $\Theta(n^{2/3})$. Ideally, sequences would have few optimal alignments, making the “best" one more apparent. While this result may not tell us about the expected number of optimal alignments (or be biologically relevant), it does provide a worst case scenario for sequence alignment and show that the bound from [@G] cannot be improved. Luckily, the bound is still sublinear. Indeed parametric sequence alignment can be practical and has been achieved for whole genomes [@fly]. This paper is mainly motivated by [@Baca], [@G], and [@ASCB]. We largely follow their notation and presentation. Decomposing the $(\alpha, \beta)$ plane ======================================= Alignment Graphs ---------------- We can represent every alignment of two length-$n$ sequences as a path through their *alignment graph*. The graph can be thought of as an $(n+1) \times (n+1)$ grid, with rows and columns numbered consecutively from top to bottom (left to right), from 0 to $n$ [@Baca]. An *alignment path* is a path on these vertices, starting at $(0,0)$, ending at $(n,n)$, and only moving down, right or diagonally down and to the right. Each path corresponds to a unique alignment. In this path, a move down (or left) corresponds to a space in the first (or second) sequence, and a diagonal move corresponds to a match or mismatch (depending on the characters). See Figure \[fig: example\] for the alignment graph of our above example alignment. (45, 28) (4,4)(2,0)[7]{}[(0,1)[12]{}]{} (4,4)(0,2)[7]{}[(1,0)[12]{}]{} (2.5,4.5)(0,2)[3]{}[1]{} (2.5, 10.5)(0,4)[2]{}[0]{} (2.5, 12.5)[1]{} (4.5, 16.5)(2,0)[3]{}[1]{} (10.5, 16.5)(2,0)[3]{}[0]{} (4,14)[(1, -1)[2]{}]{} (6,10)[(1,-1)[6]{}]{} (4,16)[(0, -1)[2]{}]{} (6,12)[(0,-1)[2]{}]{} (12,4)[(1,0)[4]{}]{} (4,10)(.4,0)[15]{}[(0,-1)[6]{}]{} (4, 14)(.4,0)[15]{}[(0,-1)[2]{}]{} (10, 16)(.4,0)[15]{}[(0,-1)[2]{}]{} (10, 12)(.4,0)[15]{}[(0,-1)[2]{}]{} (31,0)[(0,1)[22]{}]{} (25,6)[(1,0)[18]{}]{} (30.5,2)(0,4)[5]{}[(1,0)[1]{}]{} (27,5.5)(4,0)[4]{}[(0,1)[1]{}]{} (27,2) (25,0)[(1,1) [18]{}]{} (26, 0)[(1,2)[11]{}]{} (32,19)[$\Gamma_1$]{} (35,14)[$\Gamma_2$]{} (38, 9)[$\Gamma_3$]{} (29,20)[$\beta$]{} (42,4)[$\alpha$]{} \[fig: example\] Optimality regions ------------------ Gusfield et. al. observed that the boundaries between optimality regions in the $(\alpha, \beta)$ plane must be lines passing through the point $(-1,-1)$. \[Gusfield et. al., [@G]\] All optimality regions on the $(\alpha, \beta)$ plane are semi-infinite cones, and are delimited by lines of the form $\beta = c + (c+1)\alpha$ for some constant $c$. In general, a boundary between two optimality regions consists of the $(\alpha, \beta)$ for which the optimal sequences from each region have equal, optimal scores. Since $$score_{(-1, -1)}(w,x,y) = w +x +y \equiv n,$$ for every $w, x, y$, each such line (specifically these boundary lines) must pass through the point $(-1,-1)$. They also note that all of these boundary lines must intersect the non-negative $\beta$-axis because none of them cross the positive $\alpha$-axis [@G]. This comes from observing that in any alignment, we can change a mismatch to a space (in each sequence) without affecting the number of matches. Thus all along the line $\beta =0$, the optimal alignment will have the maximum number of matches possible, without regard to spaces (since those are not penalized). So no boundary line can separate the nonnegative $\alpha$-axis into distinct optimality regions. Since all boundary lines must pass through the point $(-1,-1)$ and cannot intersect the positive $\alpha$-axis, we indeed have that \[Gusfield et. al., [@G]\] Each of the optimality regions must have nontrivial intersection with the non-negative $\beta$-axis. That is, for any path $\Gamma$ that is optimized by some $(\alpha, \beta)$, there must be some $\beta '$ so that $\Gamma $ is optimized by $(0, \beta')$. This allows us to restrict our attention to optimality regions on the $\beta$-axis. Then boundary regions are just points, $(0,\beta)$, for which consecutive optimal alignments have optimal $score_{(0,\beta)}$. Note that alignments with summaries $(w_1, x_1, y_1)$ and $(w_2, x_2, y_2)$ will have equal $score_{(0,\beta)}$ when $$w_1 - \beta y_1 = w_2 - \beta y_2,$$ meaning that $$\beta = \frac{\Delta w}{\Delta y} := \frac{w_2-w_1}{y_2-y_1}.$$ In order to find different optimality regions, we will find distinct $\frac{\Delta w}{\Delta y}$ forming boundary points on the $\beta$-axis. The Lower Bound =============== For each $2 \leq r$, define $F_r$ as $F_r:= \{\frac{a}{b}\leq 1 \; : \; \frac{a}{b}$ is reduced and $a+b = r\}$. Since $a/b$ is reduced and $a+b=r$, $a$ and $b$ must be relatively prime to $r$. Then each number relatively prime to $r$ will show up exactly once (in either the numerator or the denominator), so $|F_r| = \phi(r)/2$ for $r >2$ where $\phi$ is the Euler totient function, and $|F_2| = |\{1/1\}| =1$. Let $$\mathcal{F}_q = \bigcup_{r=2}^q F_r,$$ giving us $|\mathcal{F}_q| = \frac{1}{2}\sum_{r=3}^{q}\phi(r) +1$.\ \ Fixing $q$, let $a_1/b_1 < a_2/b_2 < \ldots < a_m/b_m=1$ be the elements of $\mathcal{F}_q$. We’re going to construct two sequences of length $n = 4 \sum_kb_k $, $S = s_1s_2 \hdots s_n$ and $T=t_1t_2\hdots t_n$. Since $b_k < a_k+ b_k$, this gives us $$n = 4\sum_{k=1}^m b_k < 4 \sum_{k=1}^m (a_k + b_k) = 4\sum_{r=2}^s r |F_r| = 2 \sum_{r=2}^s r \phi(r).$$ The Sequences ------------- Let’s construct the first sequence, $S$. To start, let the first $b_1+a_1$ elements of $S$ be 0, followed by $b_1-a_1$ 1’s. Then repeat for $k>1$ (i.e. next place $b_2+a_2$ 0’s followed by $b_2-a_2$ 1’s). Notice that for each $a_k/b_k \in \mathcal{F}_q$, we use $(b_k+a_k)+ (b_k-a_k)=2b_k$ places. To get the second half of the sequence, take the reverse complement of the first half (reflecting it and switching all the 1’s and 0’s). So $$S = 0^{b_1+a_1}1^{b_1 - a_1} 0^{b_2+a_2}\hdots 0^{b_m+a_m}1^{b_m - a_m} \;\; 0^{b_m-a_m}1^{b_m + a_m}\hdots 0^{b_1-a_1}1^{b_1 + a_1}.$$ More formally, define $$i(r) = \sum_{k=1}^r 2b_k \;\;\;\;\; \text{ and } \;\;\;\;\; j(r) = \sum_{k=r}^m 2b_k.$$ (So $n = 2i(m) = 2j(1)$). Then $$s_{i(r-1) +k} = \left\{ \begin{array}{rl} 0 & \text{for } 1 \leq k \leq b_r +a_r\\ 1 & \text{for } b_r+a_r +1\leq k \leq 2b_r \end{array} \right.$$ and $$s_{\frac{n}{2}+j(r+1)+k} = \left\{ \begin{array}{rl} 0 & \text{for } 1 \leq k\leq b_r -a_r\\ 1 & \text{for } b_r-a_r +1\leq k \leq 2b_r. \end{array} \right.$$ The second sequence, $T$, will just be $n/2$ 1’s followed by $n/2$ 0’s, that is, $$t_k = \left\{ \begin{array}{rl} 1 & \text{for } 1 \leq k\leq n/2\\ 0 & \text{for } n/2 +1 \leq k \leq n. \end{array} \right.$$ \[ex: main\]For $q = 4$, $\mathcal{F}_4 = \{1/3, 1/2, 1/1 \}$. Then $n = 4(3+2+1)= 24$. Our sequences are $$S = 000011000100\; 110111001111$$ $$T = 111111111111\; 000000000000$$ The Alignment Paths ------------------- We are going to construct $m+1$ alignment paths, $\Gamma_{m+1}, \Gamma_m, \hdots, \Gamma_1$. Let $\Gamma_{m+1} $ be the path along the main diagonal (corresponding to the alignment with no spaces). To get $\Gamma_r$, align the first $j(r) = \sum_{k=r}^m 2b_k$ 0’s of $S$ with spaces and align its remaining elements without spaces, ending by aligning the last $j(r)$ 0’s of $T$ with spaces. Note that because there are $n/2$ 1’s in both $S$ and $T$, we’ll have enough room to do this. In fact, in the last alignment, $\Gamma_1$, all the 1’s of $S$ will be matched with all the 1’s of $T$. See Figure \[fig: gammas\] for the graphs of the optimal alignments of our example. (56,51) (5,0)(2,0)[25]{}[(0,1)[48]{}]{} (5,0)(0,2)[25]{}[(1,0)[48]{}]{} (3.5,.5)(0,2)[4]{}[1]{} (3.5,8.5)(0,2)[2]{}[0]{} (3.5,12.5)(0,2)[3]{}[1]{} (3.5, 18.5)[0]{} (3.5,20.5)(0,2)[2]{}[1]{} (3.5, 24.5)(0,2)[2]{}[0]{} (3.5,28.5)[1]{} (3.5,30.5)(0,2)[3]{}[0]{} (3.5,36.5)(0,2)[2]{}[1]{} (3.5,40.5)(0,2)[4]{}[0]{} (5.7,48.5)(2,0)[12]{}[1]{} (29.7,48.5)(2,0)[12]{}[0]{} (5,48)[(1, -1)[48]{}]{} (5,44)[(1,-1)[44]{}]{} (5,40)[(1,-1)[4]{}]{} (9,32)[(1,-1)[32]{}]{} (9,30)[(1,-1)[2]{}]{} (11, 24)[(1,-1)[4]{}]{} (15, 18)[(1,-1)[6]{}]{} (21, 8)[(1,-1)[8]{}]{} (5,48)[(0, -1)[8]{}]{} (9,36)[(0,-1)[6]{}]{} (11,28)[(0,-1)[4]{}]{} (15, 20)[(0,-1)[2]{}]{} (21, 12)[(0,-1)[4]{}]{} (29,0)[(1,0)[24]{}]{} (29,48)(.4,0)[60]{}[(0,-1)[8]{}]{} (29,36)(.4,0)[60]{}[(0,-1)[6]{}]{} (29,28)(.4,0)[60]{}[(0,-1)[4]{}]{} (29,20)(.4,0)[60]{}[(0,-1)[2]{}]{} (29,12)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,40)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,30)(.4,0)[60]{}[(0,-1)[2]{}]{} (5,24)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,18)(.4,0)[60]{}[(0,-1)[6]{}]{} (5,8)(.4,0)[60]{}[(0,-1)[8]{}]{} (29,0) (29,12) (29,20) (29,24) \[fig: gammas\] (20, 32) (4,0)[(0,1)[30]{}]{} (0,4)[(1,0)[18]{}]{} (3.5,8)[(1,0)[1]{}]{} (3.5,10)[(1,0)[1]{}]{} (3.5, 16)[(1,0)[1]{}]{} (3.5, 28)[(1,0)[1]{}]{} (16, 3.5)[(0,1)[1]{}]{} (0,7.5)[1/3]{} (0,9.5)[1/2]{} (2,15.5)[1]{} (2,29)[$\beta$]{} (17, 2)[$\alpha$]{} (4,8)[(3,4)[14]{}]{} (4,10)[(2,3)[12]{}]{} (4,16)[(1,2)[6]{}]{} (5,24)[$\Gamma_4$]{} (8,20)[$\Gamma_3$]{} (13.5,16)[$\Gamma_2$]{} ( 13,10)[$\Gamma_1$]{} (13,17)[(-2,1)[2.5]{}]{} \[fig: ab decomp\] Alignment Scores ---------------- Let $w_r^1$ denote the number of matching 1’s in $\Gamma_r$ and similarly $w_r^0$ denote the number of matching 0’s in $\Gamma_r$, with $w_r$ being the total number of matches. Note that $$w_r^1 - w_{r+1}^1 = b_r +a_r \;\; \text{ and }\;\;w_r^0 - w_{r+1}^0 = - (b_r-a_r).$$ Since $w_r = w_r^1+w_r^0$, we have that $$w_r - w_{r+1} = (b_r+a_r) - (b_r-a_r) = 2a_r.$$ Let $y_r$ denote the number of spaces in $\Gamma_r$ (which equals $j(r)$). Then $$y_r - y_{r+1} = j(r) - j(r+1) = 2b_r.$$ Putting these together, we get that for every $r$, $$\frac{\Delta w_r}{\Delta y_r} := \frac{w_r - w_{r+1}}{y_r - y_{r+1}} = \frac{a_r}{b_r}.\label{a/b achieved}$$ Optimality ---------- We need to show that each of these paths is optimal for distinct optimality regions, which will be accomplished by the next two lemmata. \[optimal\] Let $\Gamma$ be any alignment of $S$ and $T$. Then for any $\beta \geq 0$, there is some $\Gamma_r$ so that $score_{(0,\beta)}(\Gamma_r) \geq score_{(0,\beta)}(\Gamma)$. Say that $\Gamma$ has alignment path $\sigma$ and alignment summary $(w,x,y)$. Let the coordinates of the alignment graph be $(t,s)$, with $(0,0)$ starting in the upper left corner. Say that $(n/2, n/2+k)$ is the first time $\sigma$ meets the vertical line $t=n/2$. Because of the symmetry of our sequences, we can take $k$ to be nonnegative (meaning that $\sigma$ hits the line $t=n/2$ below or at $s=n/2$). If $\sigma$ has $k<0$, we can rotate our picture $180^o$ to get another alignment path with the same summary and $k \geq 0$. So suppose $k \geq 0$ and take $r$ so that $j(r+1) < k \leq j(r)$.\ \ *(Case 1:* $k-j(r+1) \leq b_r - a_r$*)*. Since there are only $w_{r+1}^1$ 1’s above $s=n/2+k$, we have $w^1\leq w^1_{r+1}$. Similarly, there are at most $w_{r+1}^0$ 0’s below $s=n/2+k$, so $w^0 \leq w_{r+1}^0$. Furthermore, by going through the point $(n/2, n/2+k)$, $\sigma$ must have at least $k$ spaces, so $y \geq k \geq j(r+1) = y_{r+1}$. Putting these together gives that for any $\beta \geq 0$, $$score_{(0,\beta)}(\Gamma_{r+1}) - score_{(0,\beta)}(\Gamma) = (w_{r+1} - w) - \beta (y_{r+1} - y) \geq 0 .$$ Intuitively, $\Gamma$ can have at most as many matches and must have at least as many spaces as $\Gamma_{r+1}$, and thus cannot have a higher score.\ \ *(Case 2: $k-j(r+1) > b_r - a_r$ and $\beta \leq 1$)* There are $w_r^0$ 0’s in $S$ below $s= n/2+k$, so we have $w^0 \leq w^0_r$. In addition to the $w_{r+1}^1$ 1’s in $S$ above $s=n/2 + j(r+1)$, there are another $k-j(r)+(b_r+a_r)$ 1’s in $S$ between $s=n/2+j(r+1)$ and $s=n/2+k$. So $$w^1 \leq w_{r+1}^1 +k-j(r)+(b_r +a_r) = w_r^1 + k -j(r),$$ since $w_{r+1}^1 + (b_r +a_r)= w_r^1$. Thus $$w = w^0 +w^1 \leq w^0_r + w^1_r +k - j(r) = w_r +k -j(r).\label{w_r}$$ As is case 1, we have that $y\geq k$, so $$\begin{aligned} score_{(0,\beta)}(\Gamma_r) - score_{(0,\beta)}(\Gamma) &= (w_r - w) - \beta(y_r - y) \nonumber\\ &\geq (j(r) - k) - \beta(j(r) - k) \tag{by \eqref{w_r}} \nonumber \\ & \geq 0 \tag{as $\beta \leq 1$} \nonumber \end{aligned}$$\ *(Case 3: $k-j(r+1) > b_r - a_r$ and $\beta > 1$)* We’ll show that $score_{(0,\beta)}(\Gamma_{m+1}) \geq score_{(0,\beta)}(\Gamma)$. Remember that $\Gamma_{m+1}$ is the alignment with no spaces ($y_{m+1} = 0$), corresponding to the main diagonal of the alignment graph. Note for any $r$, $$\label{w_m+1} w_r = w_{m+1}+\sum_{k=r}^m 2a_k,$$ so using equation from case 2, we get $$w_{m+1} - w \geq j(r)-k - \sum_{k=r}^m 2a_k.$$ As in previous cases, $y \geq k$. Then, $$\begin{aligned} score_{(0,\beta)}(\Gamma_{m+1}) - score_{(0,\beta)}(\Gamma) & = (w_{m+1} - w) - \beta(y_{m+1} -y) \nonumber\\ & \geq j(r)-k - \sum_{k=r}^m 2a_k + \beta k \tag{by \eqref{w_m+1}} \nonumber\\ &\geq j(r) - \sum_{k=r}^m 2a_k \tag{as $\beta >1$} \nonumber \\ & = \sum_{k=r}^m 2b_k - \sum_{k=r}^m 2a_k \nonumber \\ & \geq 0. \nonumber\end{aligned}$$ Lemma \[optimal\] tells us that any optimality region has one of the $\Gamma_r$ as an optimal alignment. Now we need to check that each $score_{(0,\beta)}(\Gamma_r)$ is optimized by a different region. To see this, we use equation and following lemma. \[Fernández-Baca, et. al., [@Baca]\] Let $\Gamma_1, \Gamma_2, \hdots, \Gamma_q$ be paths in the alignment graph. Assume $score(\Gamma_i)=w_i - \beta y_i$, where $y_1 > y_2 > \hdots > y_q$. Let $\beta_0 = 0, \beta_q = \infty$, and for $r = 1, \hdots, q-1$, $\beta_r = (w_r - w_{r+1})/(y_r - y_{r+1})$. Suppose $\beta_0 < \beta_1 < \hdots < \beta_q$. Then for $\beta \in (\beta_{r-1}, \beta_r)$ and $p \neq r$, $score_{(0,\beta)}(\Gamma_r) > score_{(0,\beta)}(\Gamma_p)$. So each of the $\Gamma_r$ do indeed represent each of the different optimality regions on the $\beta$-axis, and thus in the $(\alpha, \beta)$ plane. The Actual Lower Bound ---------------------- The maximum number of optimality regions induced by any pair of length-$n$ sequences is $\Omega(n^{2/3})$. Above we have constructed sequences of length $n \leq 2 \sum_{r=2}^q r\phi(r)$ that gave $m = \frac{1}{2}\sum_{r=2}^q \phi(r)$ optimality regions. From analytic number theory, as calculated in [@Baca], $$m= \frac{1}{2}\sum_{r=3}^q \phi(r) +1 = \frac{3}{2 \pi^2}q^2 +O(q \log q),$$ and $$n \leq 2\sum_{r=2}^q r\phi(r) = \frac{4}{\pi^2}q^3 + O(q^2\log q).$$ Then $q \geq (\frac{\pi^2 n}{4})^{1/3} +O(\log n)$, meaning $$\begin{aligned} m= \frac{1}{2} \sum_{r=3}^q \phi(r) +1 & \geq \frac{3}{2\pi^2}\left((\frac{\pi^2 n}{4})^{1/3}\right)^2 +O(n^{1/3} \log n) \nonumber\\ & = \frac{3}{2^{7/3}\pi^{2/3}}n^{2/3} +O(n^{1/3} \log n). \nonumber\end{aligned}$$ With the upper bounds from [@G] and [@Baca], this gives The maximum number of optimality regions over all pairs of length-$n$ sequences is $\Theta(n^{2/3})$, and more specifically is between $\frac{3}{2^{7/3}\pi^{2/3}}n^{2/3} +O(n^{1/3} \log n)$ and $\frac{3}{(2\pi)^{2/3}}n^{2/3} +O(n^{1/3} \log n)$. It’s unclear whether the current bounds on optimality regions for scoring with $d>2$ parameters, $O(n^{d(d-1)/(d+1)})$, are also tight or whether better upper bounds exist. Another interesting open question (perhaps with more practical relevance) is the order of the expected number of optimality regions, rather than the maximum. Acknowledgements ================ Thanks to Lior Pachter for his advice and suggestion of this problem. This paper came out of his class at U.C. Berkeley, “Discrete Mathematics for the Life Sciences", in the spring of 2008. Thanks also to Bernd Sturmfels and Peter Huggins for their useful suggestions. [00]{} C. Dewey, P. Huggins, K. Woods, B. Sturmfels, L. Pachter, Parametric alignment of Drosophila genomes. *PLoS Computational Biology*, 2(6):e73 (2006). D. Fernández-Baca, T. Seppäläinen, G. Slutzki, Bounds for parametric sequence alignment. *Discrete Applied Math* 118 (2002), 181-198. D. Fernández-Baca, T. Seppäläinen, G. Slutzki, Parametric multiple sequence alignment and phylogeny construction. *Journal of Discrete Algorithms.* 2(2)(2004), 271-287. D. Gusfield, K. Balasubramanian, D. Naor, Parametric Optimization of Sequence Alignment. *Algorithmica* 12 (1994), 312-326. D. Gusfield, Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology, Cambridge University Press, Cambridge, New York, Melbourne, 1997. S. Needleman, C. Wunsch, A general method applicable to the search for similarities in the amino acid sequence of two proteins. *Journal of Molecular Biology*, 48 (1970) 443-445, 1970. L. Pachter, B. Sturmfels, Parametric inference for biological sequence alignment. *Proc. of the National Academy of Sciences, USA*, 101(46) (2004) 16138-43, 2004. L. Pachter, B. Sturmfels, editors. *Algebraic Statistics for Computational Biology.* Cambridge University Press, 2005. M.S. Waterman, M. Eggert, E.S. Lander, Parametric sequence comparisons. *Proc. of the National Academy of Sciences, USA*, 89 (1992) 6090-6093.

ArXiv

--- abstract: 'We consider a fluid-structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in $L^p$-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.' address: - 'Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany' - | Mathematisches Institut, Angewandte Analysis\ Heinrich-Heine-Universität Düsseldorf\ 40204 Düsseldorf, Germany author: - Robert Denk - Jürgen Saal bibliography: - 'fl\_str\_int.bib' date: 'September 20, 2019' title: '$L^p$-theory for a fluid-structure interaction model' --- \[section\] \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Assumption]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][Remarks]{} \[theorem\][Example]{} \[theorem\][Examples]{} Ø[[O]{}]{} §[[S]{}]{} Ł[[L]{}]{} 0.5ex plus 0.5ex minus 0.5ex Introduction and main result {#secintro} ============================ We consider the system $$\label{fsi} \left. \begin{array}{rcll} \rho (\partial_t u + (u \cdot \nabla) u)) - \mbox{div}\,T(u,\,q) & = & 0, & \quad t > 0,\ x \in \Omega(t), \\[0.5em] \mbox{div}\,u & = & 0, & \quad t > 0,\ x \in \Omega(t), \\[0.5em] u & = & V_\Gamma, & \quad t \geq 0,\ x \in \Gamma(t), \\[0.5em] \frac1{\nu\cdot e_n}e_n^\tau T(u,\,q)\nu& = & \phi_\Gamma, & \quad t \geq 0,\ x \in \Gamma(t), \\[0.5em] \Gamma(0) = \Gamma_0, \quad V_\Gamma(0)=V_0, \quad u(0) & = & u_0, & \quad x \in \Omega(0), \end{array} \right\}$$ which represents a (one-phase) fluid-structure interaction model. The fluid with density $\rho>0$ and viscosity $\mu>0$ occupies at a time $t \geq 0$ the region $\Omega(t) \subseteq \bR^n$ with boundary $\Gamma(t)=\partial\Omega(t)$. Furthermore, we assume the fluid to be incompressible, and we assume the stress to be given as $$T(u,\,q) = 2 \mu D(u) - q, \qquad D(u) = {\textstyle \frac{1}{2}} (\nabla u + (\nabla u)^{\tau}).$$ The unknowns in the model are the velocity $u$, the pressure $q$ and the interface $\Gamma$. We denote by $\nu$ the exterior unit normal field at $\Gamma$, by $V_\Gamma$ the velocity of the boundary $\Gamma$, and by $e_j$ the $j$-th standard basis vector in $\R^n$, i.e. $e_n=(0,\cdots,0,1)$. The function $\phi_\Gamma$ describes the elastic response at $\Gamma$ which is given by a damped Kirchhoff-type plate model. Throughout the paper we assume that $\Gamma$ is given as a graph of a function $\eta:\R_+\times\R^{n-1}\to \R$, that is $$\label{gammagraph} \Gamma(t) = \Big\{(x',\eta(t,x'));\ x'\in \R^{n-1}\Big\}, \quad t \geq 0,$$ and that $\Gamma(t)$ is sufficiently flat. Thus $\Omega(t)$ is a perturbed upper half-plane.In these coordinates, the elastic response is given as $$\label{phigamma} \phi_\Gamma=m(\partial_t,\partial')\eta :=\partial_{t}^2\eta+\alpha(\Delta')^2\eta-\beta\Delta'\eta -\gamma\partial_t\Delta'\eta$$ for $\alpha,\gamma>0$, $\beta\in\R$, where $\Delta'$ stands for the Laplacian in $\R^{n-1}$. Finally, the initial configuration and velocity of the interface resp.the initial fluid velocity are given by $\Gamma_0$ and $V_0$ resp.$u_0= (u_0',u_0^n)$. We remark that in the formulation of the boundary conditions in lines 3 and 4 of , one has to take into account that the Kirchhoff plate model is formulated in a Lagrangian setting, whereas for the fluid an Eulerian setting is used. This is discussed in more detail in the beginning of Section 2. The symbol of $m(\partial_t,\partial')$ is given as $$m(\lambda,\xi')= \lambda^2+\alpha|\xi'|^4+\beta|\xi'|^2 +\gamma\lambda|\xi'|^2,\quad \lambda\in\C,\ \xi'\in\R^{n-1},$$ which vanishes if $$\lambda=-\frac{\gamma|\xi'|^2}{2}\pm\sqrt{\frac{\gamma^2|\xi'|^4}{4}-\alpha |\xi'|^4-\beta|\xi'|^2}.$$ For $\gamma>0$ , the roots of $m(\cdot,\xi')$ lie in some sector which is a subset of $\{\lambda\in\C: \Re\lambda <0\}$. This indicates that the term $-\gamma\partial_t\Delta'\eta$ in $\phi_\Gamma$ parabolizes the problem. Physically, one also speaks of structural damping of the plate. We notice that basically the same results as proved in this note can be expected by considering layer like domains or rectangular type domains with periodic lateral boundary conditions. For simplicity, however, we restrict the approach given here to the just introduced geometry. Model (\[fsi\]) was introduced in [@Quarteroni2000] in connection to applications to cardiovascular systems. In the 2D case, this system was investigated in [@batak2017] in the $L^2$-setting. In fact, in [@batak2017 Proposition 3.12] it is proved that the linear operator associated to (\[fsi\]) generates an analytic $C_0$-semigroup in a suitable Hilbert space setting. This exhibits the parabolic character of the problem. Therefore, it is reasonable to consider $L^p$-theory for the system (\[fsi\]) which is the main purpose of this note. Alternative approaches to system (\[fsi\]) in the $L^2$-setting also for the hyperbolic-parabolic case, i.e. $\gamma=0$, are given, e.g., in [@cdeg2005; @grandmont2008; @Lengeler-Ruzicka14; @Lengeler; @Muha-Canic15], concerning weak solutions and, e.g., in [@bdav2004; @Coutand-Shkoller06; @lequeurre2011; @lequeurre2013] concerning (local) strong solutions. A more recent approach in an two-dimensional $L^2$-framework concerning global strong solutions is presented in [@grahil2016]. In the present paper, we develop an $L^p$-approach in general dimension for system (\[fsi\]). We show the existence of strong solutions for small data and give a precise description of the maximal regularity spaces for the unknowns. More precisely, we prove the following main result for (\[fsi\]). \[main\] Let $n\ge 2$, $p\ge (n+2)/3$, $T>0$, and $J=(0,T)$. Assume that $$\|u_0\|_{W^{2-2/p}_p(\Omega(0))} +\|\eta_0\|_{W^{5-3/p}_p(\R^{n-1})} +\|\eta_1\|_{W^{3-3/p}_p(\R^{n-1})} <\kappa,$$ where $\Gamma_0=\graph(\eta_0)$ and $V_0=\graph(\eta_1)$, for some $\kappa>0$. Then, there exists a unique solution $(u,q,\Gamma)$ of system (\[fsi\]) such that $\Gamma=\graph(\eta)$ and such that $$\begin{aligned} u&\in H^1_p(J;L^p(\Omega(t)))\cap L^p(J;H^2_p(\Omega(t))),\\ q&\in L^p(J;\dot{H}^1_p(\Omega(t))),\\ \eta&\in \mE_\eta :=W_{p}^{9/4 - 1/(4p)}(J; L^p(\R^{n-1})) \cap H_{p}^2(J; W_p^{1-1/p}(\R^{n-1}))\\ & \quad \cap L^p(J;W_p^{5-1/p}(\R^{n-1})),\end{aligned}$$ provided that $\kappa=\kappa(T)$ is small enough and that the following compatibility conditions are satisfied: 1. $ \div u_0 = 0$, 2. if $p>\tfrac32$, then $u_0'|_{\Gamma_0} = 0$ and $u_0^n|_{\Gamma_0} - \eta_1=0$ almost everywhere, 3. there exists an $\eta_*\in\mE_\eta$ with $\eta_*|_{t=0} = \eta_0$, $\partial_t\eta_*|_{t=0}=\eta_1$ and $$\partial_t \eta_*\in H_{p}^1(\R_+; {}_0\dot H_p^{-1}(\R^n_+)),$$ where $$\partial_t \eta_*(\phi) := - \int_{\R^{n-1}} \partial_t \eta_*\phi dx',\quad \phi\in\dot H_{p'}^1(\R^n_+).$$ The solution depends continuously on the data. a\) The compatibility conditions (1)–(3) are natural in the sense that they are also necessary for the existence of a strong solution. Condition (3) appears in a similar way for the two-phase Stokes problem, see, e.g., [@Pruess-Simonett16], Section 8.1. b\) We remark that the maximal regularity space $\mE_\eta$ for $\eta$ describing the boundary is not a standard space. It is given as an intersection of three Sobolev spaces. This is due to the fact that the symbol of the complete system has an inherent inhomogeneous structure, and therefore the Newton polygon method is the correct tool to show maximal regularity. For the details, see Section 3 below. c\) We note that in the physically relevant situations $n=2$ and $n=3$, the case $p=2$ is included. This might be of importance when considering the singular limit $\gamma\to0$ for vanishing damping of the plate. d\) We formulated the result in the form of existence for fixed time and small data. By the same methods, one can also show short time existence for arbitrarily large data. The proof of Theorem \[main\] is based on several ingredients: First, we transform the system to a fixed domain and consider the linearization of the transformed system. By an application of the Newton polygon approach (see, e.g., [@Denk-Kaip13] and [@Denk-Saal-Seiler08]), we obtain maximal regularity for the linearized system. To deal with the nonlinearities, we employ embedding results on anisotropic Sobolev spaces given in [@koehnesaal]. The transformed system {#sectrans} ====================== We start with a short discussion of the boundary conditions, where the Eulerian approach for the fluid has to be coupled with the Lagrangian description for the plate (see also [@Lengeler-Ruzicka14] and [@grandmont2008]). Let $\Gamma$ be given as in (\[gammagraph\]) and assume that $\eta$ is sufficiently smooth. Following the Kirchhoff plate model, in-plate motions are ignored, and the velocity of the plate at the point $(x',\eta(t,x'))^\tau$ is parallel to the vertical direction and given by $(0,\partial_t \eta(t,x'))^\tau = \partial_t \eta(t,x') e_n$. As the fluid is assumed to adhere to the plate, we have no-slip boundary conditions for the fluid, and the equality of the velocities yields the first boundary condition $$\label{eq1-1} u(t,x',\eta(t,x')) = \partial_t \eta(t,x') e_n \quad (t>0, \, x'\in \R^n).$$ The exterior normal at the point $(x',\eta(t,x'))$ of the boundary $\Gamma(t)$ is given by $$\nu=\nu(t,x')=\frac1{\sqrt{1+|\nabla'\eta(t,x')|^2}} \left( \begin{array}{c} \nabla'\eta(t,x')\\ -1 \end{array} \right).$$ We define the transform of variables $$\theta : J \times \R^{n}_+ \to \bigcup_{t \in J} \{t\} \times \Omega(t), \ (t,x',x_n)\mapsto\theta(t,x',x_n):= (t,x',x_n+\eta(t,x')).$$ Obviously we have $\theta^{-1}(t,x',y)=(t,x',y-\eta(t,x'))$. As it was discussed in [@Lengeler-Ruzicka14], Section 1.2, the force $F$ exerted by the fluid on the boundary is given by the evaluation of the stress tensor at the deformed boundary in the direction of the inner normal $-\nu(t,x')$. More precisely, we obtain ([@Lengeler-Ruzicka14], Eq. (1.4)) $$F = - \sqrt{1+|\nabla'\eta(t,x')}\; e_n^\tau (T(u,q)\circ \theta(t,x)) \nu(t,x').$$ As $\sqrt{1+|\nabla' \eta|^2} = - \nu(t,x')\cdot e_n$, the equality of the forces gives the second boundary condition $$\label{eq1-2} \begin{aligned} \frac{1}{\nu(t,x')\cdot e_n} \; e_n^\tau [ T(u,q) ] (t,x',\eta(t,x')) \, \nu(t,x') & = [m(\partial_t,\partial')\eta] (t,x')\\ & \qquad (t>0,x'\in\R^{n-1}). \end{aligned}$$ Conditions and are the precise formulation of the boundary conditions in . To solve the problem , we first note that by a re-scaling argument we may assume that $\rho=\mu=1$ for the density $\rho$ and viscosity $\mu$ from now on. Next, we transform the problem (\[fsi\]) to a problem on the fixed half-space $\R^n_+$, using the above transformation $\theta$. To this end, we set $J:=(0,T)$ and write $x=(x',x_n)\in\R^n_+$ with $x'\in\R^{n-1}$. With the corresponding meaning we write $v'$, $\nabla'$, etc. The pull-back is then defined as $$v:=\Theta^\ast u:= u\circ\theta, \qquad p:= \Theta^\ast q := q\circ\theta,$$ and correspondingly the push-forward as $$u:=\Theta_\ast v:= v\circ\theta^{-1}, \qquad q:= \Theta_\ast p := p\circ\theta^{-1}.$$ We also set $\Gamma_0=\Gamma(0)=\{(x',\eta_0(x'));\ x'\in\R^{n-1}\}$ and $V_0=V_\Gamma(0)=(0,\eta_1(\cdot))^\tau$. Applying the transform of variables to (\[fsi\]) leads to the following quasilinear system for $(v,p,\eta)$: $$\label{tfsi} \begin{array}{rclll} \partial_t v - \Delta v + \nabla p& = & F_v(v,p,\eta) & \mbox{in} & J \times \R^n_+, \\ \div v & = & G(v,\eta)& \mbox{in} & J \times \R^n_+, \\ v'&=&0 & \mbox{on} & J \times \R^{n-1}, \\ \partial_t\eta - v^n & = & 0 & \mbox{on} & J \times \R^{n-1}, \\ -2 \partial_n v^n + p -m(\partial_t,\partial')\eta& = & H_\eta(v,\eta) & \mbox{on} & J \times \R^{n-1}, \\ v|_{t=0} &=& v_0 &\mbox{in} & \R^n_+,\\ \eta|_{t=0}&= & \eta_0 & \mbox{in} & \R^{n-1},\\ \partial_t\eta|_{t=0} & = & \eta_1 & \mbox{in} & \R^{n-1}. \end{array}$$ The non-linear right-hand sides are given as $$\begin{array}{rcl} F_v(v,p,\eta) & = & ( \partial_t \eta - \Delta' \eta) \partial_n v - 2 (\nabla' \eta \cdot \nabla') \partial_n v + |\nabla' \eta|^2 \partial^2_n v \\[0.5em] & & \quad - (v \cdot \nabla) v + (v' \cdot \nabla'\eta) \partial_n v + (\nabla'\eta,0)^\tau\partial_n p, \\[0.5em] G(v,\eta) & = & \nabla'\eta \cdot \partial_n v', \\[0.5em] H_\eta(v,\eta) & = & - \nabla'\eta \cdot \partial_n v' - \nabla'\eta \cdot \nabla' v^n. \end{array}$$ The linearized system ===================== The aim of this section is to derive maximal regularity for the linearized system $$\label{linfsi} \begin{array}{rclll} \partial_t v - \Delta v + \nabla p& = & f_v & \mbox{in} & \R_+ \times \R^n_+, \\ \div v & = & g& \mbox{in} & \R_+ \times \R^n_+, \\ v'&=&0 & \mbox{on} & \R_+ \times \R^{n-1}, \\ \partial_t\eta - v^n & = & 0 & \mbox{on} & \R_+ \times \R^{n-1}, \\ -2 \partial_n v^n + p -m(\partial_t,\partial')\eta& = & f_\eta & \mbox{on} & \R_+ \times \R^{n-1}, \\ v|_{t=0} &=& v_0 &\mbox{in} & \R^n_+,\\ \eta|_{t=0}&= & \eta_0 & \mbox{in} & \R^{n-1},\\ \partial_t\eta|_{t=0} & = & \eta_1 & \mbox{in} & \R^{n-1}. \end{array}$$ We will consider this system in Sobolev spaces with exponential weight with respect to the time variable. Let $\rho\in\R$ and $X$ be a Banach space. For $u\in L^p(\R_+,X)$, we define $\Psi_\rho$ as the multiplication operator with $e^{-\rho t}$, i.e. $\Psi_\rho u(t) := e^{-\rho t}u(t),\; t\in\R_+$. The spaces with exponential weights are defined by $$\begin{aligned} H_{p,\rho}^s(\R_+,X) & := \Psi_{-\rho}(H_p^s(\R_+,X)),\\ W_{p,\rho}^s(\R_+,X) & := \Psi_{-\rho}(W_p^s(\R_+,X))\end{aligned}$$ with canonical norms $\|u\|_{H_{p,\rho}^s(\R_+,X) } := \| \Psi_\rho u\|_{H_p^s(\R_+,X)}$ and $\|u\|_{W_{p,\rho}^s(\R_+,X) } := \| \Psi_\rho u\|_{W_p^s(\R_+,X)}$. For $\rho\ge 0$ and $s>0$, we define ${}_0 H_{p,\rho}^s(\R_+,X)$ and ${}_0 W_{p,\rho}^s(\R_+,X)$ analogously, replacing $H_p^s$ and $W_p^s$ by ${}_0H_p^s$ and ${}_0W_p^s$, respectively. For mapping properties and interpolation results under the condition that $X$ is a UMD space, we refer, e.g., to [@Denk-Saal-Seiler08], Lemma 2.2. We also make use of homogeneous spaces, e.g., for $\Omega\subset\R^n$ we set $$\dot H_p^1(\Omega):=\{v\in L^1_{\loc}(\Omega):\ \nabla v\in L^p(\Omega)\}$$ and $\dot H_{p,0}^1(\Omega):=\overline{C^\infty_c(\Omega)}^{\|\nabla\cdot\|_p}$. The corresponding dual spaces are defined as $$\dot H^{-1}_p(\Omega):=\bigl(\dot H^1_{p,0}(\Omega)\bigl)' \quad\text{and}\quad \dot H^{-1}_{p,0}(\Omega):=\bigl(\dot H^1_{p}(\Omega)\bigl)',$$ see [@Pruess-Simonett16] Section 7.2. The homogeneous Sobolev-Slobodeckii spaces over $\R^n$ are defined as usual [@Triebel15] and we have $$\dot W^{s}_p(\R^n)=\dot B^s_{pp}(\R^n)$$ for $1<p<\infty$, $n\in\N$, and $s\in\R\setminus\Z$, where the latter one denotes the homogeneous Besov space. In the following, we denote the time trace $u\mapsto \partial_t^k u|_{t=0}$ by $\gamma_k^t$ and the trace to the boundary $u\mapsto \partial_n^k u|_{\R^{n-1}}$ by $\gamma_k$. The solution $(v,p,\eta)$ of will belong to the spaces $$\begin{aligned} v\in \mE_v & := H_{p,\rho}^1(\R_+;L^p(\R^n_+))\cap L^p_\rho(\R_+;H_p^2(\R^n_+)),\\ p\in\mE_p & := L^p_\rho(\R_+; \dot H_p^1(\R^n_+)),\\ \eta\in\mE_\eta & := W_{p,\rho}^{9/4 - 1/(4p)}(\R_+; L^p(\R^{n-1}))\cap H_{p,\rho}^2(\R_+; W_p^{1-1/p}(\R^{n-1}))\\ & \quad \cap L^p_\rho(\R_+;W_p^{5-1/p}(\R^{n-1})).\end{aligned}$$ The function spaces for the right-hand side of are given by $$\begin{aligned} f_v \in \mF_v & := L^p_\rho(\R_+; L^p(\R^n_+)),\\ g\in \mF_g & := H_{p,\rho}^1(\R_+;\dot H_p^{-1}(\R^n_+))\cap L^p_\rho(\R_+;H_p^1(\R^n_+)),\\ f_\eta\in \gamma_0\mE_p & := L^p_\rho(\R_+; \dot W_p^{1-1/p}(\R^{n-1})).\end{aligned}$$ By trace results with respect to the time trace, the spaces for the initial values are given by $$\begin{aligned} v_0 \in \gamma_0^t\mE_v &:= W_p^{2-2/p}(\R^n_+),\\ \eta_0 \in \gamma_0^t\mE_\eta & := W_p^{5-3/p}(\R^{n-1}),\\ \eta_1 \in \gamma_1^t\mE_\eta & := W_p^{3-3/p}(\R^{n-1}).\end{aligned}$$ We will also need the following compatibility conditions: 1. $ \div v_0 = g|_{t=0}$ in $\dot H_p^{-1}(\R^{n}_+)$. 2. If $p>\tfrac32$, then $v_0'|_{\R^{n-1}} = 0$ almost everywhere in $\R^{n-1}$. 3. If $p>\tfrac32$, then $v_0^n|_{\R^{n-1}} - \eta_1=0$ almost everywhere in $\R^{n-1}$. 4. There exists an $\eta_*\in\mE_\eta$ with $\eta_*|_{t=0} = \eta_0$, $\partial_t\eta_*|_{t=0}=\eta_1$ and $$\label{eq2-6} (g,\partial_t \eta_*)\in H_{p,\rho}^1(\R_+; \dot H_{p,0}^{-1}(\R^n_+)).$$ Here we define $$(g,\partial_t \eta_*)(\phi) := \int_{\R^n_+} g\phi dx - \int_{\R^{n-1}} \partial_t \eta_*\phi dx'$$ for $\phi\in\dot H_{p'}^1(\R^n_+)$. Additionally, we have $(g|_{t=0},\eta_1)=(g|_{t=0}, v_0^n|_{\R^{n-1}})$ in $\dot H_{p,0}^{-1}(\R^n_+)$. We remark that only is an additional condition, as it was shown in [@Denk-Saal-Seiler08], Theorem 4.5, that for every $\eta_0\in\gamma_0^t\mE_\eta$ and $\eta_1\in\gamma_1^t\mE_\eta$ there exists an $\eta_*\in \mE_\eta$ with $\eta_*|_{t=0} = \eta_0$ and $\partial_t\eta_*|_{t=0} = \eta_1$. The main result of this section is the following maximal regularity result. \[2.2\] Let $p>1$, $p\neq3/2$. Then there exists a $\rho_0>0$ such that for every $\rho\ge \rho_0$, system has a unique solution $(v,p,\eta)\in \mE_v\times\mE_p\times\mE_\eta$ if and only if the data $f_v,g,f_\eta,v_0,\eta_0,\eta_1$ belong to the spaces above and satisfy the compatibility conditions . The solution depends continuously on the data. The proof of this theorem will be done in several steps and follows from Subsections \[sec\_necessity\]–\[sec\_uniqueness\]. Necessity {#sec_necessity} --------- Let $(v,p,\eta)\in\mE_v\times \mE_p\times \mE_\eta$ be a solution of . By standard continuity and trace results, the right-hand sides $f_v,$ and $g$ as well as the time trace $v_0$ belong to the spaces above. Noting that $\div\colon L^p(\R^n_+)\to \dot H_p^{-1}(\R^n_+)$ is continuous, we have $g = \div u \in H_p^1(\R_+;\dot H_p^{-1}(\R^n_+)) \subset C([0,\infty);\dot H_p^{-1}(\R^n_+))$, and as for all $p>1$ we also have $v_0\in W_p^{2-2/p}(\R^n_+)\subset L^p(\R^n_+)$, we obtain the compatibility condition (C1) for all $p>1$ (see also [@Pruess-Simonett16], Theorem 7.2.1). For $f_\eta$ we have $\mE_\eta\subset H_{p,\rho}^2(\R_+;W_p^{1-1/p}(\R^{n-1})$ by the mixed derivative theorem (see, e.g., [@Denk-Saal-Seiler08], Lemma 4.3), and therefore $$\partial_t^2\eta \in L^p_\rho(\R_+; W_p^{1-1/p}(\R^{n-1})\subset \gamma_0\mE_p.$$ It is easy to see that the other terms of $m(\partial_t,\partial')\eta$ belong to the same space. By standard trace results, we also obtain $\gamma_0 u\in \gamma_0\mE_p$. Concerning the pressure, we remark that $\gamma_0\colon \dot H_p^1(\R^n_+) \to \dot W_p^{1-1/p}(\R^{n-1})$ is a retraction, see, e.g., [@Jawerth78], Theorem 2.1, and therefore $\gamma_0 p\in \gamma_0\mE_p$. This yields $f_\eta\in \gamma_0\mE_p$. For the time traces of $\eta$, we can apply [@Denk-Saal-Seiler08], Theorem 4.5 which gives $\eta_0\in\gamma_0^t\mE_\eta$ and $\eta_1\in \gamma_1^t\mE_\eta$. If $p>\frac32$, then the boundary trace of $v_0$ exists in the space $W_p^{2-3/p}(\R^{n-1})$. This yields the compatibility conditions (C2) and (C3) as equality in the space $W_p^{2-3/p}(\R^{n-1})$, hence in particular as equality almost everywhere. To show (C4), we can set $\eta_* := \eta$. For $\phi\in \dot H_{p'}^1(\R^n_+)$, we obtain $$(g,\partial_t\eta)(\phi) = \int_{\R^n_+} \div u \,\phi dx - \int_{\R^{n-1}} u^n \phi dx' = -\int_{\R^n_+} u \cdot \nabla \phi dx$$ and therefore $(g, \partial_t\eta)\in H_{p,\rho}^1(\R_+; \dot H_{p,0}^{-1}(\R^n_+))$. Setting $t=0$, we obtain $(g|_{t=0}, \eta_1) = (g|_{t=0}, v_0^n)$ as equality in $\dot H_{p,0}^{-1}(\R^n_+)$. Reductions {#sec_reduction} ---------- We can reduce some part of the right-hand side of to zero by applying known results on the Stokes system. For this, let $(v^{(1)}, p^{(1)})\in \mE_v\times \mE_p$ be the unique solution of the Stokes problem in the half space $$\label{eq2-3} \begin{array}{rclll} \partial_t v^{(1)} - \Delta v + \nabla p^{(1)}& = & f_v & \mbox{in} & \R_+ \times \R^n_+, \\ \div v^{(1)} & = & g& \mbox{in} & \R_+ \times \R^n_+, \\ (v^{(1)})'&=&0 & \mbox{on} & \R_+ \times \R^{n-1}, \\ (v^{(1)})^n & = & \partial_t\eta_* & \mbox{on} & \R_+ \times \R^{n-1}, \\ v^{(1)}|_{t=0} &=& v_0 &\mbox{in} & \R^n_+. \end{array}$$ The unique solvability of follows from [@Pruess-Simonett16], Theorem 7.2.1. To show that this theorem can be applied, we remark in particular that the compatibility condition (e) in [@Pruess-Simonett16 p. 324] holds because of (C4). Let $\tilde v := v-v^{(1)} $, $\tilde p := p - p^{(1)}$, and $\tilde\eta := \eta - \eta_*$. Then $(v,p,\eta)$ is a solution of if and only if $(\tilde v,\tilde p,\tilde \eta)$ is a solution of $$\label{eq2-4} \begin{array}{rclll} \partial_t \tilde v - \Delta \tilde v + \nabla \tilde p& = & 0 & \mbox{in} & \R_+ \times \R^n_+, \\ \div \tilde v & = & 0& \mbox{in} & \R_+ \times \R^n_+, \\ \tilde v'&=&0 & \mbox{on} & \R_+ \times \R^{n-1}, \\ \tilde v^n - \partial_t\tilde \eta& = & 0 & \mbox{on} & \R_+ \times \R^{n-1}, \\ -2 \partial_n \tilde v^n + \tilde p -m(\partial_t,\partial')\tilde \eta& = & \tilde f_\eta & \mbox{on} & \R_+ \times \R^{n-1}, \\ \tilde v|_{t=0} &=& 0 &\mbox{in} & \R^n_+,\\ \tilde\eta|_{t=0}&= & 0 & \mbox{in} & \R^{n-1},\\ \partial_t\tilde\eta|_{t=0} & = & 0 & \mbox{in} & \R^{n-1}. \end{array}$$ Here, $$\begin{aligned} \tilde f_\eta & := f_\eta + 2\partial_n (v^{(1)})^n - p^{(1)} + m(\partial_t, \partial') \eta_*.\end{aligned}$$ By the trace results in Subsection \[sec\_necessity\], we have $\tilde f_\eta \in \gamma_0\mE_p$. Solution operators for the reduced linearized problem ----------------------------------------------------- In the following, we show solvability for the reduced problem , omitting the tilde again. An application of the Laplace transform formally leads to the resolvent problem $$\label{resfsi} \begin{array}{r@{\ =\ }ll} \lambda v-\Delta v+\nabla p & 0&\text{in} \ \R^n_+,\\ \mbox{div}\,v & 0&\text{in} \ \R^n_+,\\ v'&0&\text{on} \ \partial\R^n_+,\\ v^n- \lambda\eta&0&\text{on} \ \partial\R^n_+,\\ -2\partial_n v^n+p-m(\lambda,\partial')\eta&f_\eta&\text{on} \ \partial\R^n_+ \end{array}$$ with $$m(\lambda,\partial')\eta= \lambda^2\eta+\alpha(\Delta')^2\eta-\beta\Delta'\eta -\gamma\lambda\Delta'\eta.$$ We observe that the second and the third line of imply that $$\partial_n v^n(\cdot,0)=-\nabla'\cdot v'(\cdot,0)=0.$$ Hence the fifth line reduces to $$p-m(\lambda,\partial')\eta=f_\eta\quad\text{on}\ \partial\R^n_+.$$ Applying partial Fourier transform in $x'\in\R^{n-1}$, we obtain the following system of ordinary differential equations in $x_n$ for the transformed functions $\hat v$, $\hat p$ and $\hat \eta$: $$\begin{aligned} \omega^2 \hat v - \partial_n^2 \hat v + (i \xi',\partial_n)^\tau \hat p & = 0, \quad x_n>0, \nonumber \\i \xi \cdot \hat v' + \partial_n \hat v^n & = 0, \quad x_n>0, \\\hv'&=0,\quad x_n=0,\\ \lambda \heta -\hat v^n & = 0, \quad x_n=0,\\ \hp-m(\lambda,|\xi'|)\heta&=\hf_\eta,\quad x_n=0,\\\end{aligned}$$ Here we have set $\omega:= \omega(\lambda,\xi'):=\sqrt{\lambda + |\xi'|^2}$ and $$m(\lambda,\xi'):=\lambda^2+\alpha|\xi'|^4+\gamma\lambda|\xi'|^2 +\beta|\xi'|^2.$$ Multiplying the first equation with $(i\xi',\partial_n)$ and combing it with the second one yields $(-|\xi'|^2+\partial_n^2)\hp = 0$ for $x_n>0$. The only stable solution of this equation is given by $$\label{(2.7)} \hp(\xi',x_n) = \hp_0(\xi') e^{-|\xi'|x_n}, \quad \xi' \in {\mathbb R}^{n-1}, \ x_n > 0.$$ To solve the above system we employ the ansatz $$\begin{aligned} \hat v'(\xi',x_n) & = -\int_0^\infty k_+(\lambda,\xi',x_n,s) i\xi'\hp(\xi',s) ds + \hat \phi'(\xi') e^{-\omega x_n},\label{rd_eq1}\\ \hat v^n(\xi',x_n) & = -\int_0^\infty k_-(\lambda,\xi',x_n,s) \partial_n \hp(\xi',s)ds + \hat \phi^n(\xi') e^{- \omega x_n}\label{rd_eq1a}\end{aligned}$$ with $$k_{\pm} (\lambda,\xi,x_n,s) := \frac{1}{2 \omega} \left( e^{- \omega|x_n-s|} \pm e^{-\omega (x_n+s)} \right).$$ Here the traces $\hat p_0$ and $\hat\phi = (\hat \phi', \hat \phi^n)^\tau$ still have to be determined. The fact that $\div v=0$ enforces $$\label{divbed} i\xi'\cdot\hphi'(\xi')=\omega\hphi^n(\xi').$$ The kinematic boundary condition instantly gives us $$\label{kinbc} \lambda\heta-\hphi^n=0.$$ Next, by utilizing (\[(2.7)\]), from the tangential boundary condition we obtain $$0=\hv'(\xi',0)=-\int_0^\infty \frac{e^{-\omega s}}{\omega} i\xi'\hp(\xi',s) ds + \hat \phi'(\xi'),$$ which implies $$\label{eq2-9} \frac{i\xi'}{\omega+|\xi'|}\hp_0=\omega\hphi'.$$ Multiplying this with $i\xi'$ and employing the relations (\[divbed\]), (\[kinbc\]) yields $$\label{tangbc} -\frac{|\xi'|^2}{\omega+|\xi'|}\hp_0 =\omega^2\hphi^n=\lambda\omega^2 \heta.$$ Plugging this into the last line of the transformed system we obtain $$\label{normaltrace} \left(\frac{\lambda\omega^2(\omega+|\xi'|)}{|\xi'|^2} +m(\lambda,\xi')\right)\heta=-\hf_\eta.$$ This yields $$\label{eq2-8} \heta=-\frac{|\xi'|^2}{N_L(\lambda,|\xi'|)}\hf_\eta$$ with $$N_L(\lambda,|\xi'|)=|\xi'|^2m(\lambda,\xi') +\lambda\omega^2(\omega+|\xi'|).$$ Formula defines the solution operator for $\eta$ as a function of $f_\eta$ on the level of its Fourier-Laplace transform. The following result is based on the Newton polygon approach and shows that the solution operator is continuous on the related Sobolev spaces. In the following, we consider $(-\Delta')^{1/2}$ as an unbounded operator in $L^p_\rho(\R_+; L^p(\R^{n-1})$, and define $N_L(\partial_t,(-\Delta')^{1/2})$ by the joint $H^\infty$-calculus of $\partial_t$ and $(-\Delta')^{1/2}$ (for details, we refer to, e.g., [@Denk-Saal-Seiler08], Corollary 2.9). We will apply the Newton polygon approach on the Bessel potential scale $H_p^s$ with respect to time and on the Besov scale $B_{pp}^r$ with respect to space. \[2.3\] a) There exists a $\rho_0>0$ such that for all $\rho\ge \rho_0$, the operator $ N_L(\partial_t, (-\Delta')^{1/2})\colon H_N \to L^p_\rho(\R_+;B_{pp}^{-1-1/p}(\R^{n-1}) ) $ is an isomorphism, where $$\begin{aligned} H_N & := {}_0 H_{p,\rho}^{5/2}(\R_+;B_{pp}^{-1-1/p}(\R^{n-1}))\cap {}_0H_{p,\rho}^2(\R_+;B_{pp}^{1-1/p}(\R^{n-1})) \\ & \quad \cap L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1})). \end{aligned}$$ b\) Let $\rho\ge \rho_0$. Then for every $f_\eta\in \gamma_0\mE_p$, we have $$\begin{aligned} \eta & := \Delta' \big[ N_L(\partial_t, (-\Delta')^{1/2})\big]^{-1} f_\eta \in \mE_\eta,\\ \phi^n & := \partial_t \eta \in \gamma_0\mE_v,\\ p_0 & := f_\eta + m(\partial_t,\partial')\eta \in \gamma_0\mE_p. \end{aligned}$$ a\) We apply the Newton-polygon approach developed in [@Denk-Saal-Seiler08]. Replacing $z=|\xi'|$, the $r$-principle symbols, i.e., the leading terms of $N_L$ associated to the relation $\lambda\sim z^r$, are easily calculated as $$P_r(\lambda,z)= \left\{ \begin{array}{rl} \alpha z^6,& 0<r<2,\\ m_0(\lambda,z)z^2,& r=2,\\ \lambda^2 z^2,&2<r<4,\\ \lambda^2 z^2+\lambda^{5/2},&r=4,\\ \lambda^{5/2},& r>4, \end{array} \right.$$ where $m_0=m$ for $\beta=0$, that is $$m_0(\lambda,z):=\lambda^2+\alpha z^4+\gamma\lambda z^2.$$ In other words, the associated Newton-polygon has the three relevant vertices $(6,0)$, $(2,2)$, and $(0,\frac52)$ and two relevant edges which again reflects the quasi-homogeneity of $N_L$. Now, let $\vp\in(0,\pi/2)$ and $\theta\in(0,\vp/4)$. For $r\neq 2$ we then obviously have $$\label{rsymbcond} P_r(\lambda,z)\neq 0\quad \left((\lambda,z)\in\Sigma_{\pi-\vp}\times\Sigma_\theta\right).$$ For $r=2$ we deduce $$P_2(\lambda,z)=0 \quad\Leftrightarrow \quad \lambda=\frac{z^2}{2} \left(-\gamma\mp \sqrt{\gamma^2-4\alpha}\right).$$ By the fact that $\gamma>0$ we see that $$\vp_0:=\pi-\arg \left(-\gamma\mp \sqrt{\gamma^2-4\alpha}\right) <\frac{\pi}2.$$ Thus, assuming $\vp\in(\vp_0,\pi/2)$ and $\theta\in\left(0,(\vp-\vp_0)/4\right)$ we see that (\[rsymbcond\]) is satisfied for all $r>0$. This allows for the application of [@Denk-Saal-Seiler08 Theorem 3.3] (setting $s=0$ and $r=-1-1/p$ in the notation of [@Denk-Saal-Seiler08]) which yields a). b\) We write $$\eta = [ N_L(\partial_t, (-\Delta')^{1/2})\big]^{-1}\Delta' f_\eta.$$ As $\Delta'$ is an isomorphism from $\dot H_p^{2+t}(\R^{n-1})$ to $\dot H_p^{t}(\R^{n-1})$ for each $t\in\R$, by real interpolation of these spaces (see [@Jawerth78], Lemma 1.1) we see that it is also an isomorphism from $\dot B_{pp}^t(\R^{n-1})$ to $\dot B_{pp}^{t-2}(\R^{n-1})$ for each $t\in\R$. In particular, $\Delta' f_\eta \in L^p_\rho(\R_+; \dot B_{pp}^{-1-1/p}(\R^{n-1}))$. Using the fact that for $s<0$ the embedding $\dot B_{pp}^s (\R^{n-1}) \subset B_{pp}^s( \R^{n-1})$ holds (see [@Triebel14 p. 104, (3.339)], [@Triebel15 Section 3.1]), we obtain the embedding $$L^p_\rho(\R_+; \dot B_{pp}^{-1-1/p}(\R^{n-1})) \subset L^p_\rho(\R_+; B_{pp}^{-1-1/p}(\R^{n-1})).$$ An application of a) yields $$\begin{aligned} \eta & \in {}_0H_{p,\rho}^{5/2}(\R_+;B_{pp}^{-1-1/p}(\R^{n-1})) \cap {}_0H_{p,\rho}^{2}(\R_+;B_{pp}^{1-1/p}(\R^{n-1}))\\ & \quad \cap L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1})). \end{aligned}$$ Now, the mixed derivative theorem in mixed scales (see [@Denk-Kaip13], Proposition 2.7.6) implies $$\begin{aligned} {}_0H_{p,\rho}^{5/2}(\R_+;&B_{pp}^{-1-1/p}(\R^{n-1})) \cap L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1}))\\ & \subset B_{pp,\rho}^{9/4 - 1/(4p)}(\R_+; L^p(\R^{n-1}))\end{aligned}$$ and we obtain $\eta\in \mE_\eta$. For $u^n := \partial_t \eta$, we immediately get $$u^n \in W_{p,\rho}^{5/4-1/(4p)}(\R_+; L^p(\R^{n-1}))\cap L^p_\rho(\R_+;W_p^{3-1/p}(\R^{n-1})\subset\gamma_0\mE_v.$$ Finally, the fact that $m(\partial_t, \partial')\eta \in \gamma_0\mE_p$ for $\eta\in \mE_\eta$ was already remarked in Subsection \[sec\_necessity\]. Due to the last result, we obtain the existence of a solution $(v,p,\eta)$ of . For $\eta$, $\phi^n$, and $p_0$ defined as in Lemma \[2.3\] b), we can define $p$ and $v$ by and –, respectively. Here, $\phi'$ is given by . As we know that $\phi^n$ and $p_0$ belong to the canonical spaces by Lemma \[2.3\] b), we get $v\in \mE_v$ and $p\in\mE_p$ by standard results on the Stokes equation (see, e.g., [@Hieber-Saal18], Section 2.6, and [@Pruess-Simonett16], Section 7.2). By construction, $(v,p,\eta)$ is a solution of . Uniqueness of the solution {#sec_uniqueness} -------------------------- To show that the solution of is unique, let $(v,p,\eta)$ be a solution with zero right-hand side and zero initial data. Then the Laplace transform in $t$ and partial Fourier transform in $x'$ are well-defined, and the calculations above show, in particular, that $$\heta=-\frac{|\xi'|^2}{N_L(\lambda,|\xi'|)}\hf_\eta = 0$$ for almost all $\xi'\in\R^{n-1}$. Therefore, $\eta=0$ which implies that $(v,p)$ is the solution of the Dirichlet Stokes system with zero data. Therefore, $v=0$ and $p=0$. This finishes the proof of Theorem \[2.2\]. \[remark-finite-time\] Theorem \[2.2\] was formulated on the infinite time interval $(0,\infty)$ with exponentially weighted spaces with respect to $t$. As usual in the theory of maximal regularity, we obtain the same results on finite time intervals $t\in J = (0,T)$ with $T<\infty$ without weights, i.e., with $\rho=0$. This is due to the fact that on finite time intervals the weighted and unweighted norms are equivalent and that there exists an extension operator from $(0,T)$ to $(0,\infty)$ acting on all spaces above. Therefore, the results of Theorem \[2.2\] hold with $\rho=0$ on the finite interval $J=(0,T)$. As we consider the nonlinear equation on a finite time interval, we will replace the function spaces above by $\mE_v := H^1(J; L^p(\R^n_+))\cap L^p(J;H_p^2(\R^n_+))$ etc., keeping the same notation. The nonlinear system ==================== To prove mapping properties of the nonlinearities we employ sharp estimates for anisotropic function spaces provided in [@koehnesaal]. In fact, we can proceed very similar as in [@koehnesaal Section 5.2, Proposition 5.6]. For $\omega_j\in\N_0$, $j=1,\ldots,\nu$, we define a weight vector as $\omega:=(\omega_1,\ldots,\omega_\nu)$ and denote by $\dot\omega:=\mathrm{lcm}\{\omega_1,\ldots,\omega_\nu\}$ the lowest common multiple. Further, for $n=(n_1,\ldots,n_\nu)\in\N^\nu$ we write $$\R^n=\R^{n_1}\times\cdots\times\R^{n_\nu}.$$ The (generalized) Sobolev index of an $E$-valued anisotropic function space then reads as $$\frac{1}{\dot{\omega}} \left( s - \frac{\omega \cdot n}{p} \right) =: \left\{ \begin{array}{rc} \mbox{ind}(B^{s, \omega}_{p, q}(\bR^n,\,E)), & s \in\R,\ 1 < p < \infty,\ 1 \leq q \leq \infty, \\[0.5em] \mbox{ind}(H^{s, \omega}_p(\bR^n,\,E)), & -\infty < s < \infty,\ 1 < p < \infty, \\[0.5em] \mbox{ind}(W^{s, \omega}_p(\bR^n,\,E)), & 0 \leq s < \infty,\ 1 \leq p < \infty, \end{array} \right.$$ where $\omega\cdot n=\sum_{j=1}^\nu \omega_jn_j$. Note that we have the corresponding definition, if $\R^n$ is replaced by a cartesian product of Intervals. For an introduction to anisotropic spaces such as $H^{s, \omega}_{p}(\R^n,E)$ we refer to [@koehnesaal] and the references cited therein. In the situation considered here we always have $\omega=(2,1)$ and the anisotropic spaces below can be represented as an intersection such as $$\begin{aligned} &H^{1,(2,1)}_p(J\times \R^{n-1},L^p(\R_+))\\ &=H^{1/2}_p(J,L^p(\R^{n-1},L^p(\R_+))) \cap L^p(J,H^1(\R^{n-1},L^p(\R_+))),\end{aligned}$$ for instance. In this case we have $$\mbox{ind}\bigl(H^{1,(2,1)}_p(J\times \R^{n-1},L^p(\R_+))\bigr) =\frac12\left(1-\frac{2+n-1}{p}\right) =\frac12-\frac{n+1}{2p}.$$ Now, let $J=(0,T)$. By the [*mixed derivative theorem*]{}, see e.g.[@Denk-Saal-Seiler08 Lemma 4.3], we have $$H^{2}_p(J,W^{1-1/p}_p(\R^{n-1}))\cap L^p(J,W^{5-1/p}_p(\R^{n-1}))\hook H^1(J,W^{3-1/p}(\R^{n-1})).$$ This yields $$\label{est-ht} \begin{split} \partial_t \eta &\in W^{5/4 - 1/4p}_p(J,L_p(\R^{n-1})) \cap L_p(J,W^{3 - 1/p}_p(\R^{n-1})) \\ &\hook W^{1-1/2p}_p(J,L^p(\R^{n-1})) \cap L^p(J,W^{2-1/p}_p(\R^{n-1}))\\ &=W^{2 - 1/p, (2,1)}_p(J\times \R^{n-1}) \end{split}$$ for $\eta \in \mE_3$. Again by the mixed derivative theorem we have $$H^{2}_p(J,W^{1-1/p}_p(\R^{n-1})) \cap L^p(J,W^{5 - 1/p}_p(\R^{n-1})) \hookrightarrow W^{2- 1/2p}_p(J,H^1_p(\R^{n-1})),$$ which gives us $$\label{est-hx} \begin{split} \partial_j \eta &\in W^{2 - 1/2p}_p(J,L_p(\R^{n-1})) \cap L_p(J,W^{4 - 1/p}_p(\R^{n-1}))\\ &= W^{4 - 1/p, (2,1)}_p(J \times \R^{n-1}) \end{split}$$ for $\eta \in \mE_3$ and $j = 1,\,\dots,\,n - 1$. Analogously we obtain that $$\label{est-hxy} \begin{split} \partial_j \partial_k \eta &\in W^{3/2 - 1/2p}_p(J,L_p(\R^{n-1})) \cap L_p(J,W^{3 - 1/p}_p(\R^{n-1})) \\ &= W^{3 - 1/p, (2,1)}_p(J \times \R^{n-1})\\ &\hook\ W^{2 - 1/p, (2,1)}_p(J \times \R^{n-1}) \end{split}$$ for $\eta \in \mE_3$ and $j,\,k = 1,\,\dots,\,n - 1$. For the velocity we have $$\label{est-u} v \in H^{2,(2,1)}_p(J \times \R^n_+) \hookrightarrow H^{2,(2,1)}_p(J \times \R^{n-1},L^p(\R_+)).$$ Another application of the mixed derivative theorem yields $$\begin{aligned} \partial_j v & \in H^{1, (2,1)}_p(J \times \R^n_+) \hookrightarrow H^{1,(2,1)}_p(J \times \R^{n-1},L_p(\R_+)),\label{est-ux} \\ \partial_j\partial_k v & \in L_p(J \times \R^n_+) = L_p(J \times \R^{n-1},L_p(\R_+)),\label{est-uxy}\end{aligned}$$ for $j,\,k = 1,\,\dots,\,n$. Taking trace this also implies $$\begin{aligned} v|_{\partial\R^n_+} & \in W^{1 - 1/2p}_p(J,L_p(\R^{n-1})) \cap L_p(J,W^{2 - 1/p}_p(\R^{n-1}))\nonumber\\ &= W^{2 - 1/p, (2,1)}_p(J \times \R^{n-1})\label{est-ux-tr1}\\ \partial_j v|_{\partial\R^n_+} & \in W^{1/2 - 1/2p}_p(J,L_p(\R^{n-1})) \cap L_p(J,W^{1 - 1/p}_p(\R^{n-1}))\nonumber\\ &= W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1})\label{est-ux-tr2}\end{aligned}$$ for $j = 1,\,\dots,\,n$. Now, we denote by $L$ the linear operator on the left hand side of system (\[tfsi\]) and by $N=(F_v,G,0,0,H_\eta,0,0,0)$ its nonlinear right-hand side. Then (\[tfsi\]) is reformulated as $$L(v,p,\eta)=N(v,p,\eta)+(0,0,0,0,0,v_0,\eta_0,\eta_1).$$ We also set $$\begin{aligned} \widetilde\mE&:=\mE_v\times\mE_p\times\mE_\eta,\\ \widetilde\mF&:=\mF_v\times\mF_g\times\{0\}\times\{0\} \times \gamma_0\mE_p \times\gamma_0^t\mE_v\times\gamma_0^t\mE_\eta \times\gamma_1^t\mE_\eta.\end{aligned}$$ The nonlinearity admits the following properties. \[mappropnl\] Let $p\ge(n+2)/3$. Then $N\in C^\omega(\widetilde\mE,\widetilde\mF)$, $N(0)=0$, and we have $DN(0)=0$ for the Fréchet derivative of $N$. [*Mapping properties of $F_v$.*]{} Gathering (\[est-ht\]), (\[est-hxy\]), and (\[est-ux\]) we can estimate the term $$(\partial_t \eta - \Delta' \eta)\,\partial_n v,$$ as desired, provided the vector-valued embedding $$\begin{split} \label{emb-fu1} &\underbrace{W^{2 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_1 = 1 - \frac{n + 2}{2p}} \cdot \underbrace{H^{1, (2,1)}_p(J \times \R^{n-1},L^p(\R_+))}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n + 1}{2p}}\\ &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1},L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}} \end{split}$$ does hold. Applying [@koehnesaal Theorem 1.7] this readily follows if at least one of the two indices $\mbox{ind}_1$, $\mbox{ind}_2$ is non-negative. The strictest condition to be fulfilled by [@koehnesaal Theorem 1.7], however, is $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$ in case that both of the indices on the left-hand-side are negative which can occur for small $p$. It is easily seen that this condition is equivalent to $$\label{reqp1} p \geq \frac{n + 2}{3}.$$ For the terms $$2 (\nabla' \eta \cdot \nabla')\,\partial_n v, \quad |\nabla' \eta|^2 \partial^2_n v, \quad (\nabla' \eta,0)^\tau\,\partial_n p$$ we employ (\[est-hx\]), (\[est-uxy\]) and the vector-valued embeddings $$\begin{split} \label{emb-fu2} &\big[ \underbrace{W^{4 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_1 = 2 - \frac{n + 2}{2p}} \big]^m \cdot \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1},L^p(\R_+))}_{\textrm{ind}_2 = - \frac{n + 1}{2p}}\\ &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1}, L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}} \end{split}$$ for $m = 1,\,2$. Due to [@koehnesaal Theorem 1.9] the above embeddings are valid, provided that $\mbox{ind}_1 > 0$ or, equivalently, $$\label{reqp1weak} p > \frac{n + 2}{4}.$$ Next, (\[est-u\]) and (\[est-ux\]) show that we obtain the desired estimate of the term $(v \cdot \nabla)v$, if $$\underbrace{H^{2,(2,1)}_p(J \times \R^n_+)}_{\textrm{ind}_1 = 1 - \frac{n + 2}{2p}} \cdot \underbrace{H^{1, (2,1)}_p(J \times \R^n_+)}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n + 2}{2p}} \hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times \R^n_+)}_{\textrm{ind} = - \frac{n + 2}{2p}}.$$ This is guaranteed by [@koehnesaal Theorem 1.7] if $\max\,\{\,\mbox{ind}_1,\,\mbox{ind}_2\,\} \geq 0$. Again, for small values of $p$ both of the indices on the left-hand-side can become negative. Then [@koehnesaal Theorem 1.7] implies the embedding above if $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$, which is equivalent to (\[reqp1\]). Thanks to (\[est-hx\]) and (\[est-ux\]) the term $(v' \cdot \nabla' \eta)\partial_nv$ can be estimated by utilizing the embedding $$\label{3factoremb} \begin{split} &\underbrace{H^{1, (2,1)}_p(J \times \R^{n-1},H^1_p(\R_+))}_{\textrm{ind}_1 = \frac{1}{2} - \frac{n + 1}{2p}}\cdot \underbrace{W^{4 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_2 = 2 - \frac{n + 2}{2p}}\\ &\cdot \underbrace{H^{1, (2,1)}_p(J \times \R^{n-1},L^p(\R_+))}_{\textrm{ind}_3 = \frac{1}{2} - \frac{n + 1}{2p}} \quad\hookrightarrow\quad \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1}, L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}}. \end{split}$$ Note that here we also employ $$H^{2, (2,1)}_p(J \times \R^{n}_+) \ \hook \ H^{1, (2,1)}_p(J \times \R^{n-1},H^1_p(\R_+))$$ and $H^1_p(\R_+)\cdot L^p(\R_+)\hook L^p(\R_+)$ which is valid due to the Sobolev embedding $H^1_p(\R_+)\hook L^\infty(\R_+)$ for $p>1$. Thanks to [@koehnesaal Theorem 1.7] (\[3factoremb\]) holds, if $\min\,\{\,\mbox{ind}_1,\,\mbox{ind}_2,\,\mbox{ind}_3\,\} \geq 0$. If at least one of the three indices on the left-hand side is negative, then the sum of the negative indices on the left hand side has to exceed the index on the right hand side. The most restrictive constraint hence results from $\mbox{ind}_1 + \mbox{ind}_2 + \mbox{ind}_3 \geq \mbox{ind}$, which is fullfilled if $$\label{anothconstr} p\ge\frac{2n+3}{6}.$$ Consequently, by our assumptions $F_v$ has the desired mapping properties, since (\[reqp1\]) also yields (\[reqp1weak\]) and (\[anothconstr\]). [*Mapping properties of $G$.*]{} First we show $G(v,\eta) \in H^1_p(J,\,\dot{H}^{-1}_p(\bR^n_+))$. Integration by parts yields $\partial_n \in \sL(L_p(J \times \bR^n_+),\,L_p(J,\,\dot{H}^{-1}_p(\bR^n_+)))$. Using this property and the fact that $\eta$ does not depend on $x_n$, it is sufficient to estimate the terms $$\partial_t \nabla'\eta \cdot v', \ \nabla' \eta \cdot \partial_t v'$$ in $L_p(J \times \bR^n_+)$. Thanks to (\[est-ht\]) and the mixed derivative theorem we know $$\partial_t \nabla'\eta \in W^{1 - 1/p, (2, 1)}_p(J \times \R^{n-1}).$$ The first term can thus be estimated by the vector-valued embedding $$\begin{aligned} &\underbrace{W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_1 = \frac12- \frac{n + 2}{2p}} \cdot \underbrace{H^{2, (2,1)}_p(J \times \R^{n-1},\,L^p(\R_+))}_{\textrm{ind}_2 = 1 - \frac{n + 1}{2p}}\\ &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1},\,L^p(\R_+))}_{\textrm{ind} = - \frac{n +1}{2p}}.\end{aligned}$$ According to [@koehnesaal Theorem 1.7] this embedding is again valid, if we have $\max\,\{\,\mbox{ind}_1,\,\mbox{ind}_2\,\} \geq 0$ or if $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$ in case that both indices on the left hand side are negative. The latter condition is again equivalent to (\[reqp1\]). The second term may be estimated by employing (\[est-hx\]), the vector-valued embedding (\[emb-fu2\]) for $m = 1$, and $\partial_t v\in L^p(J\times\R^{n-1},L^p(\R_+))$ under constraint (\[reqp1weak\]). To see that also $G(v,\eta) \in L^p(J,H^1_p(\R^n_+))$, we estimate the terms $$\partial_j \nabla' \eta \cdot \partial_n v', \ \nabla'\eta \cdot \partial_j \partial_n v', \ \nabla'\eta \cdot \partial^2_n v', \qquad j = 1,\,\dots,\,n - 1,$$ in $L^p(J \times \R^n_+)$. Similar as above this may be accomplished by utilizing (\[est-hx\]), (\[est-hxy\]), (\[est-ux\]), (\[est-uxy\]) in combination with the vector-valued embeddings (\[emb-fu1\]), and (\[emb-fu2\]). Once more this is feasible if (\[reqp1\]) holds. [*Mapping properties of $H_\eta$.*]{} Note that $W^{1 - 1/p, (2,1)}_p(J\times \R^{n-1}))\,\hook\, \gamma_0\mE_p$. Hence, according to (\[est-hx\]) and (\[est-ux\]) we can estimate the terms $$-\nabla'\eta \cdot \partial_n v',\ -\nabla'\eta \cdot \nabla'v^n$$ as desired provided that the embedding $$\begin{aligned} &\big[ \underbrace{W^{4 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_1 = 2 - \frac{n + 2}{2p}} \big] \cdot \underbrace{W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1}))}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n + 2}{2p}}\\ &\hookrightarrow \underbrace{W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1}))}_{\textrm{ind} = \frac{1}{2} - \frac{n + 2}{2p}}\end{aligned}$$ is at our disposal. By [@koehnesaal Theorem 1.9] this is the case if $\mbox{ind}_1 > 0$. Hence, the nonlinearity $H_\eta$ has the desired mapping properties, provided that $p>(n+2)/4$. This, in turn, is true since (\[reqp1\]) is satisfied. Altogether we have proved the asserted embeddings, i.p. that $N(\widetilde\mE)\subset \widetilde\mF$. The claimed smoothness of $N$ as well as $N(0)=0$ and $DN(0)=0$ follow obviously by the fact that $N$ consists of polynomial nonlinearities which are of quadratic or higher order. For a Banach space $E$ we denote by $B_E(x,r)$ the open ball in $E$ with radius $r>0$ centered in $x\in E$. Based on Theorems \[2.2\] and \[mappropnl\] we can derive well-posedness of (\[tfsi\]) for small data. For simplicity we also set $$\begin{aligned} \mE&:=\left\{(v,p,\eta)\in\mE_v\times\mE_p\times\mE_\eta; \ \partial_t\eta=v^n,\ v'=0\ \text{on } \partial\R^n_+\right\},\\ \mF&:= \biggl\{(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1) \in\widetilde\mF;\ f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1\text{ satisfy}\biggr.\\ &\biggl.\qquad \text{the compatibility conditions (C1)-(C4)}\biggr\}.\end{aligned}$$ \[mainnonlinsys\] Let $p\ge (n+2)/3$ and $T>0$. Then there is a $\kappa=\kappa(T)>0$ such that for $(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)\in B_{\widetilde\mF}(0,\kappa)$ satisfying the compatibility conditions (C2)-(C4) and $$\label{nonlincd} \div v_0=\nabla'\eta_0\cdot\partial_n v'_0+g|_{t=0} \quad \text{in } \dot{H}^{-1}_p(\R^n_+)$$ there is a unique solution $(v,p,\eta)\in\mE$ of system (\[tfsi\]). The solution depends continuously on the data. We pick $f:=(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)$ as assumed. System (\[tfsi\]) (including exterior forces) reads as $$\label{fullnlsys} L(v,p,\eta)=N(v,p,\eta)+f.$$ We first have to verify that the right hand side belongs to $\mF$. Observe that (\[nonlincd\]) gives (C1). Hence, by our assumptions the compatibility conditions (C1)-(C3) are satisfied. To see compatibility condition (C4) we have to verify that there exists an $\eta_*\in\mE_\eta$ satisfying $(\eta_*,\partial_t\eta_*)|_{t=0}=(\eta_0,\eta_1)$ and $$(g+G(v,\eta),\partial_t\eta_*)\in H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+))$$ for every triple $(v,p,\eta)\in\mE$ such that $(v,\eta,\partial_t\eta)|_{t=0}=(v_0,\eta_0,\eta_1)$. Note that by assumption there is an extension $\eta_*\in\mE_\eta$ with the prescribed traces such that $$(g,\partial_t\eta_*)\in H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+)).$$ Hence it suffices to prove that $$\label{nonlinginhm1} (\nabla'\eta\cdot\partial_n v',0)\in H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+))$$ For $\phi\in \dot{H}^1_{p}(\R^n_+)$ we observe that thanks to $v'(x',0)=0$ we obtain $$\int_{\R_+}\phi(x)\nabla'\eta(x')\cdot\partial_n v'(x)\,dx_n =-\int_{\R_+}\nabla'\eta(x')\cdot v'(x)\partial_n\phi(x)\,dx_n.$$ In order to deduce (\[nonlinginhm1\]) it hence suffices to prove that $$\nabla'\eta\cdot v' \in H^1_{p}(J;L^p(\R^n_+)).$$ Thanks to (\[est-hx\]) and (\[est-u\]) this follows from the embedding $$\begin{aligned} &W^{4-1/p,(2,1)}_p(J\times \R^{n-1}) \cdot H^{2,(2,1)}_p(J\times \R^{n-1};L^p(\R_+))\\ &\hook\, H^{2,(2,1)}_p(J\times \R^{n-1};L^p(\R_+)) \,\hook\, H^1_{p}(J;L^p(\R^n_+)).\end{aligned}$$ Applying once again [@koehnesaal Theorem 1.9] we see that this is fulfilled if $\mbox{ind}\bigl(W^{4-1/p,(2,1)}_p(J\times \R^{n-1})\bigr)>0$. This, in turn, holds if $p>(n+2)/4$ which is implied by our assumption $p\ge(n+2)/3$. Thus, (\[nonlinginhm1\]) follows. Altogether we have proved that $(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)\in B_{\widetilde\mF}(0,\kappa)$ satisfying the compatibility conditions (C2)-(C4) and (\[nonlincd\]) implies that $N(w)+f\in\mF$ for $w\in\overline{B_\mE(0,r)}$. Hence, (\[fullnlsys\]) is well-defined. Now, we set $$K(w)=L^{-1}(N(w)+f), \quad w\in\overline{B_\mE(0,r)}$$ and prove that it is a contraction on $\overline{B_\mE(0,r)}$ for $r>0$ small enough. Theorem \[2.2\] yields that $L\in\sLis(\mE,\mF)$. This and the mean value theorem imply $$\begin{aligned} \|K(w)-K(z)\|_\mE&\le C\|N(w)-N(z)\|_\mE\\ &\le C\sup_{v\in B_\mE(0,r)}\|DN(v)\|_{\sL(\mE,\widetilde{\mF})} \|w-z\|_\mE\quad (w,z\in \overline{B_\mE(0,r)}).\end{aligned}$$ Fixing $r>0$ such that $\sup_{v\in B_\mE(0,r)}\|DN(v)\|_{\sL(\mE,\widetilde{\mF})}\le 1/2C$, which is possible thanks to Theorem \[mappropnl\], we see that $K$ is contractive. The estimate above and Theorem \[mappropnl\] also imply $$\begin{aligned} \|K(w)\|_\mE &\le \|K(w)-K(0)\|_\mE+ C\|f\|_\mF\\ &\le \frac{r}2+ C\kappa \quad (w\in \overline{B_\mE(0,r)}).\end{aligned}$$ Choosing $\kappa\le r/2C$ we see that $K$ is indeed a contraction on $\overline{B_\mE(0,r)}$. The contraction mapping principle gives the result. By the equivalence of the systems (\[fsi\]) and (\[tfsi\]) given through the diffeomorphic transform introduced in Section \[sectrans\], it is clear that Theorem \[mainnonlinsys\] implies our main result Theorem \[main\].

ArXiv

--- abstract: | The flavour degree of freedom in non-charged $q\bar q$ mesons is discussed in a generalisation of quantum electrodynamics including scalar coupling of gauge bosons, which yields to an understanding of the confinement potential in mesons. The known “flavour states” $\sigma$, $\omega$, $\Phi$, $J/\Psi$ and $\Upsilon$ can be described as fundamental states of the $q\bar q$ meson system, if a potential sum rule is applied, which is related to the structure of vacuum. This indicates a quantisation in fundamental two-boson fields, connected directly to the flavour degree of freedom.\ In comparison with potential models additional states are predicted, which explain the large continuum of scalar mesons in the low mass spectrum and new states recently detected in the charm region. PACS/ keywords: 11.15.-q, 12.40.-y, 14.40.Cs, 14.40.Gx/ Generalisation of quantum electrodynamics with massless elementary fermions (quantons, $q$) and scalar two-boson coupling. Confinement potential. Flavour degree of freedom of mesons described by fundamental $q^+q^-$ states. Masses of $\sigma$, $\omega$, $\Phi$, $J/\Psi$ and $\Upsilon$. --- version 30.3.2011 [Two-boson field quantisation and flavour in $q^+q^-$ mesons]{} H.P. Morsch[^1]\ Institute for Nuclear Studies, Pl-00681 Warsaw, Poland The flavour degree of freedom has been observed in hadrons, but also in charged and neutral leptons, see e.g. ref. [@PDG]. It is described in the Standard Model of particle physics by elementary fermions of different flavour quantum number. The fact that flavour is found in both strong and electroweak interactions could point to a supersymmetry between these fundamental forces, which should give rise to a variety of supersymmetric particles, which in spite of extensive searches have not been observed. A very different interpretation of the flavour degree of freedom is obtained in an extension of quantum electrodynamics, in which the property of confinement of mesons as well as their masses are well described. This is based on a Lagrangian [@Moinc], which includes a scalar coupling of two vector bosons $$\label{eq:Lagra} {\cal L}=\frac{1}{\tilde m^{2}} \bar \Psi\ i\gamma_{\mu}D^{\mu}( D_{\nu}D^{\nu})\Psi\ -\ \frac{1}{4} F_{\mu\nu}F^{\mu\nu}~,$$ where $\Psi$ is a massless elementary fermion (quanton, q) field, $D_{\mu}=\partial_{\mu}-i{g_e} A_{\mu}$ the covariant derivative with vector boson field $A_{\mu}$ and coupling $g_e$, and $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ the field strength tensor. Since our Lagrangian is an extension of quantum electrodynamics, the coupling $g_e$ corresponds to a generalized charge coupling $g_e\geq e$ between charged quantons $q^+$ and $q^-$. By inserting the explicite form of $D^{\mu}$ and $D_{\nu}D^{\nu}$ in eq. (\[eq:Lagra\]), this leads to the following two contributions with 2- and 3-boson ($2g$ and $3g$) coupling, if Lorentz gauge $\partial_\mu A^\mu =0$ and current conservation is applied $$\label{eq:L2} {\cal L}_{2g} =\frac{-ig_e^{2}}{\tilde m^{2}} \ \bar \Psi \gamma_{\mu} \partial^{\mu} (A_\nu A^\nu) \Psi $$ and $$\label{eq:L3} {\cal L}_{3g} =\frac{ -g_e^{3}\ }{\tilde m^{2}} \ \bar \Psi \gamma_{\mu} A^\mu (A_\nu A^\nu)\Psi \ .$$ Requiring that $A_\nu A^\nu$ corresponds to a background field, ${\cal L}_{2g}$ and ${\cal L}_{3g}$ give rise to two first-order $q^+q^-$ matrix elements $$\label{eq:P2} {\cal M}_{2g} =\frac{-\alpha_e^{2}}{\tilde m^{3}} \bar \psi(\tilde p') \gamma_\mu~\partial^\mu \partial^\rho w(q)g_{\mu\rho}~\gamma_\rho \psi(\tilde p)$$ and $$\label{eq:P3} {\cal M}_{3g} = \frac{-\alpha_e^{3}}{\tilde m} \bar \psi(\tilde p')\gamma_{\mu} ~w(q)\frac{g_{\mu\rho} f(p_i)}{p_i^{~2}} w(q)~ \gamma_{\rho} \psi(\tilde p)~,$$ in which $\alpha_e={g_e^2}/{4\pi}$ and $\psi(\tilde p)$ is a two-fermion wave function $\psi(\tilde p)=\frac{1}{\tilde m^3} \Psi(p)\Psi(k)$. The momenta have to respect the condition $\tilde p'-\tilde p=q+p_i=P$. Further, $w(q)$ is the two-boson momentum distribution and $f(p_i)$ the probability to combine $q$ and $P$ to $-p_i$. Since $f(p_i)\to 0$ for $\Delta p\to 0$ and $\infty$, there are no divergencies in ${\cal M}_3$. By contracting the $\gamma$ matrices by $\gamma_\mu\gamma_\rho+ \gamma_\rho\gamma_\mu=2g_{\mu\rho}$, reducing eqs. (\[eq:P2\]) and (\[eq:P3\]) to three dimensions, and making a transformation to r-space (details are given in ref. [@Moinc]), the following two potentials are obtained, which are given in spherical coordinates by $$V_{2g}(r)= \frac{\alpha_e^2\hbar^2 \tilde E^2}{\tilde m^3}\ \Big (\frac{d^2 w(r)}{dr^2} + \frac{2}{r}\frac{d w(r)}{dr}\Big )\frac{1}{\ w(r)}\ , \label{eq:vb}$$ where $\tilde E=<E^2>^{1/2}$ is the mean energy of scalar states of the system, and $$\label{eq:vqq} V^{(1^-)}_{3g}(r)= \frac{\hbar}{\tilde m} \int dr'\rho(r')\ V_{g}(r-r')~,$$ in which $w(r)$ and $\rho(r)$ are two-boson wave function and density (with dimension $fm^{-2}$), respectively, related by $\rho(r)=w^2(r)$. Further, $V_{g}(r-r')$ is an effective boson-exchange interaction $V_{g}(r)=-\alpha_e^3\hbar \frac{f(r)}{r}$. Since the quanton-antiquanton parity is negative, the potential (\[eq:vqq\]) corresponds to a binding potential for vector states (with $J^\pi =1^-$). For scalar states angular momentum L=1 is needed, requiring a p-wave density, which is related to $\rho(r)$ by $$\label{eq:spur} \rho^{ p}(\vec r)=\rho^{ p}(r)\ Y_{1,m}(\theta,\Phi) = (1+\beta R\ d/dr) \rho(r)\ Y_{1,m}(\theta,\Phi)\ .$$ $\beta R$ is determined from the condition $<r_{\rho^p}>\ =\int d\tau\ r \rho^p(r)=0$ (elimination of spurious motion). This yields a boson-exchange potential given by $$\label{eq:vqq0} V^{(0^+)}_{3g}(r)= \frac{\hbar}{\tilde m} \int d\vec r\ '\rho^{ p}(\vec r\ ')\ Y_{1,m}(\theta',\Phi')\ V_{g}(\vec r-\vec r') = 4\pi \frac{\hbar}{\tilde m} \int dr'\rho^{ p}(r')\ V_{g}(r-r')~.$$ We require a matching of $V^{(0^+)}_{3g}(r)$ and $\rho(r)$ $$\label{eq:con1} V^{(0^+)}_{3g}(r)=c_{pot} \ \rho(r)\ ,$$ where $c_{pot}$ is an arbitrary proportionality factor. Eq. (\[eq:con1\]) is a consequence of the fact that $V_{g}(r)$ should be finite for all values of r. This can be achieved by using a form $$\label{eq:veff} V_{g}(r)=f_{as}(r) (-\alpha_e^3 \hbar /r)\ e^{-cr}$$ with $f_{as}(r)=(e^{(ar)^{\sigma}}-1)/(e^{(ar)^{\sigma}}+1)$, where the parameters $c$, $a$ and $\sigma$ are determined from the condition (\[eq:con1\]). Self-consistent two-boson densities are obtained assuming a form $$\label{eq:wf} \rho(r)=\rho_o\ [exp\{-(r/b)^{\kappa}\} ]^2\ \ with\ \ \kappa \simeq 1.5\ .$$ The matching condition (\[eq:con1\]) is rather strict (see fig. 1) and determines quite well the parameter $\kappa$ of $\rho(r)$: using a pure exponential form ($\kappa$=1) a very steep rise of $\rho(r)$ is obtained for $r\to 0$ , but an almost negligible and flat boson-exchange potential, which cannot satisfy eq. (\[eq:con1\]). Also for a Gaussian form ($\kappa$=2) no consistency is obtained, the deduced potential falls off more rapidly towards larger radii than the density $\rho(r)$. The agreement between $<r^2_{\rho}>$ and $<r^2_{V_{3g}}(r)>$ cannot be enforced by using a different parametrisation for $f_{as}(r)$. Only by a density with $\kappa\simeq 1.5$ a satisfactory solution is obtained. For our solution (\[eq:wf\]) it is important to verify that $V_{g}(r)$ is quite similar in shape to $\rho^{p}(r)$ required from the modification of the boson-exchange propagator. This is indeed the case, as shown in the upper part of fig. 2, which displays solution 4 in the tables. Further, the low radius cut-off function $f_{as}(r)$ is shown by the dashed line, which falls off to zero for $r\to 0$. A transformation to momentum ($Q$) space leads to $f_{as}(Q)\to 0$ for $Q\to \infty$. Interestingly, this decrease of $f_{as}(Q)$ for large momenta is quite similar to the behaviour of quantum chromodynamics, a slowly falling coupling strength $\alpha(Q)$ related to the property of asymptotic freedom [@GWP]. In the two lower parts of fig. 2 the resulting two-boson density and the boson-exchange potential (\[eq:vqq0\]) are shown in r- and Q-space[^2] for solution 4 in the tables, both in very good agreement. In the Fourier transformation to Q-space the process $gg\rightarrow q\bar q$ is elastic and consequently the created $q\bar q$-pair has no mass. However, if we take a finite mass of the created fermions of 1.4 GeV (such a mass has been assumed for a comparable system in potential models [@qq]), a boson-exchange potential is obtained (given by the dashed line in the lower part of fig. 2), which cannot be consistent with the density $\rho(r)$. Thus, our solutions require [**massless**]{} fermions. This allows to relate the generated system to the absolute vacuum of fluctuating boson fields with energy $E_{vac}=0$. The mass of the system is given by $$\label{eq:mass} M^m_{n}=-E_{3g}^{~m}+E_{2g}^{~n} \ ,$$ where $E_{3g}^{~m}$ and $E_{2g}^{~n}$ are binding energies in $V_{3g}(r)$ and $V_{2g}(r)$, respectively, calculated by using a mass parameter $\tilde m=1/4~\tilde M =1/4 <Q^2_\rho>^{1/2}$, where $\tilde M$ is the average mass generated, and $\tilde E$ given in table 2. The coupling constant $\alpha_e$ is determined by the matching of the binding energies to the mass, see eq. (\[eq:mass\]). The boson-exchange potential is attractive and has negative binding energies, with the strongest bound state having the largest mass and excited states having smaller masses. These energies do not increase the mean energy $E_{vac}$ of the vacuum: writing the energy-momentum relation $E_{vac}=0=\sqrt{<Q^2_\rho>}+E_{3g}$, this relation is conserved, if $E_{3g}$ is compensated by the root mean square momentum of the deduced density $<Q^2_\rho>^{1/2}$. Differently, the binding energy in the self-induced two-boson potential (\[eq:vb\]), which does not appear in normal gauge theory applications (see ref. [@PDG]), is positive and corresponds to a real mass generation by increasing the total energy by $E_{2g}$. Therefore, this potential allows a creation of stationary $(q\bar q)^n$ states out of the absolute vacuum of fluctuating boson fields, if two rapidly fluctuating boson fields overlap and cause a quantum fluctuation with energy $E_{2g}$. The two-boson potential $V_{2g}(r)$ (with density parameters from solution 4 in the tables) is compared to the confinement potential from lattice gauge calculations [@Bali] in the upper part of fig. 3, which shows good agreement. The corresponding potentials obtained from the other solutions are very similar, if a small decrease of $\kappa$ is assumed for solutions of stronger binding (as given in table 2). Solution (meson) $M^1_1$ $M^1_{2}$ $M^1_{3}$ $M_1^{exp}$ $M_{2}^{exp}$ $M_{3}^{exp}$ ------------ ------------ --------- ----------- ----------- -------------- --------------- --------------- 1   scalar $\sigma$ 0.55 1.28 1.88 0.60$\pm$0.2 1.35$\pm$0.2 2   scalar $ f_o $ 1.38 2.25 2.9 1.35$\pm$0.2     vector $\omega$ 0.78 1.65 2.3 0.78 1.65$\pm$0.02 3   scalar $ f_o $ 2.68 3.34 3.9 —     vector $\Phi$ 1.02 1.68 2.23 1.02 1.68$\pm$0.02 4   scalar not seen 11.7 12.3 12.8 —     vector $J/\Psi$ 3.10 3.69 4.16 3.097 3.686 (4.160) 5   scalar not seen 40.5 41.0 41.4 —     vector $\Upsilon$ 9.46 9.98 10.38 9.46 10.023 10.355 : Deduced masses (in GeV) of scalar and vector $q^+q^-$ states in comparison with known $0^{++}$ and $1^{--}$ mesons [@PDG] (for $V_{3g}(r)$ only the lowest bound state is given). We have seen in fig. 1 that the functional shape of the two-boson density (\[eq:wf\]) (given by the parameter $\kappa$) is quite well determined. In contrary, we find that the slope parameter $b$ (which governs the radial extent $<r^2_\rho>$) is not constrained by the different conditions applied. This allows a continuum of solutions with different radius. However, on the fundamental level of overlapping boson fields quantum effects are inherent and should give rise to discrete solutions. Such a (new) quantisation can only arise from an additional constraint orginating from the structure of the vacuum. This may be formulated in the form of a vacuum potential sum rule. Sol. $\kappa$ $b$ $\alpha_e$ $c$ $a$ $\sigma$ $<Q^2_{\rho}>^{1/2}$ $\tilde E$ $<r^2_{\rho}>$ $<r^2>_{exp}$ ------ ---------- ------- ------------ ------ ------- ---------- ---------------------- ------------ ---------------- --------------- 1  1.50 0.77 0.26 2.4  6.4 0.86 0.59 0.9 0.65  –  2  1.46 0.534 0.385 3.3 12.0 0.86 0.81 1.0 0.33  0.33  3  1.44 0.327 0.44 5.35 16.4 0.85 1.44 1.3 0.13  0.21  4  1.40 0.125 0.58 13.6 50.7 0.83 3.50 1.6 0.02  0.04  5  1.37 0.042 0.635 46.0 132.6 0.82 10.46 2.3 0.002   –  : Parameters and deduced values of $<Q^2_{\rho}>^{1/2}$, $\tilde E$ in GeV and $<r^2>$ in $fm^2$ for the different solutions. $b$ is given in $fm$, $c$ and $a$ in $fm^{-1}$. The values of $<r^2>_{exp}$ are taken from ref. [@Mo]. We assume the existence of a global boson-exchange interaction in the vacuum $V_{vac}(r)$, which has a radial dependence similar to the boson-exchange interaction (\[eq:veff\]) discussed above, but with an additional $1/r$ fall-off, which leads to $V_{vac}(r)\sim 1/r^2$. Further, we require that the different potentials $V^i_{3g}(r)$ (where $i$ are discrete solutions) sum up to $V_{vac}(r)$ $$\label{eq:sum} \sum_i V^i_{3g}(r)=V_{vac}(r)= \tilde f_{as}(r) (-\tilde \alpha_e^3 \hbar\ r_o/{r^2})\ e^{-\tilde cr} \ ,$$ where $\tilde f_{as}(r)$ and $e^{-\tilde cr}$ are cut-off functions as in eq. (\[eq:veff\]). Actually, we expect that the cut-off functions should be close to those for the state with the lowest mass. Interestingly, the radial forms of $V_g(r)$ and $V_{vac}(r)$ are the only two forms, which lead to equally simple forms in Q-space: $1/r\to1/Q^2$ and $1/r^2\to1/Q$. This supports our assumption. If we assume that the new quantisation is related to the flavour degree of freedom, the different “flavour states” of mesons $\omega$, $\Phi$, $J/\Psi$, and $\Upsilon$ should correspond to eigenstates of the sum rule (\[eq:sum\]). Indeed, we find that the sum of the boson-exchange potentials with g.s. masses of 0.78, 1.02, 3.1 and 9.4 add up to a potential, which is in reasonable agreement with the sum (\[eq:sum\]). However, the needed cut-off parameters $a$, $\sigma$, and $c$ correspond to those for the $\sigma(600)$ solution (see ref. [@Moinc]). This can be regarded as strong evidence for the $\sigma(600)$ being to the lowest flavour state. By inclusion of this solution also, a good agreement with the sum rule (\[eq:sum\]) is obtained. This is shown in the lower part of fig 3, where the different potentials are given by dashed and dot-dashed lines with their sum given by the solid line. The resulting masses of scalar and vector states together with their excited states in $V_{2g}(r)$ are given in table 1, which are in good agreement with experiment for the known states. The corresponding density parameters are given in table 2 with mean square radii in reasonable agreement with the meson radii extacted from experimental data (see ref. [@Mo]). It is evident that in this multi-parameter fit there are ambiguities, which can be reduced only by detailed studies of the contributions of the different states to the average mass $\tilde E$ and its relation to $<Q^2_{\rho}>^{1/2}$. However, the reasonable account of the experimental masses and the quantitative fit of the sum rule (\[eq:sum\]) in fig. 3 indicates that our results are quite correct. As compared to potential models using finite fermion (quark) masses (see e.g. ref. [@qq]), we obtain significantly more states, bound states in $V_{2g}(r)$ and in $V_{3g}(r)$. The solutions in table 1 correspond only to the 1s level in $V_{3g}(r)$, in addition there are Ns levels with N=2, 3, ... Most of these states, however, have a relatively small mass far below 3 GeV. As the boson-exchange potential is Coulomb like, it creates a continuum of Ns levels with masses, which range down to the threshold region. This is consistent with the average energy $\tilde E$ of scalar excitations in table 2, which increases much less for heavier systems as compared to the energy of the 1s-state. These low energy states give rise to large phase shifts at low energies, in particular large scalar phase shifts. Concerning masses above 3 GeV, solution 5 yields additional scalar 2s and 3s states at masses of about 12 and 8.8 GeV, respectively, whereas an extra vector 2s state is obtained (between the most likely $\Psi$(3s) and $\Psi$(4s) states at 4.160 GeV and 4.415 GeV) at a mass of about 4.2 GeV. This state may be identified with the recently discovered X(4260), see ref. [@PDG]. Corresponding excited states in the confinement potential (\[eq:vb\]) should be found at masses of 4.9, 5.3 and 5.5 GeV with uncertainties of 0.2-0.3 GeV. In summary, the present model based on an extension of electrodynamics leads to a good understanding of the confinement and the masses of fundamental $q\bar q$ mesons. The flavour degree of freedom is described by stationary states of different radial extent, whose potentials exhaust a vacuum potential sum rule. In a forthcoming paper a similar description will be discussed for neutrinos, which supports our conclusion that the flavour degree of freedom is related to the structure of overlapping boson fields in the vacuum. Fruitful discussions and valuable comments from P. Decowski, M. Dillig (deceased), B. Loiseau and P. Zupranski among many other colleagues are appreciated. [99]{} Review of particle properties, C. Amsler et al., Phys. Lett B 667, 1 (2008);\ http://pdg.lbl.gov/ and refs. therein H.P. Morsch, “Inclusion of scalar boson coupling in fundamental gauge theory Lagrangians”, to be published D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973) (2005) and refs. therein R. Barbieri, R. Kögerler, Z. Kunszt, and R. Gatto, Nucl. Phys. B 105, 125 (1976); E. Eichten, K.Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, Phys. Rev. D 17, 3090 (1978); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985); D. Ebert, R.N. Faustov, and V.O. Galkin, Phys. Rev. D 67, 014027 (2003); and refs. therein G.S. Bali, K. Schilling, and A. Wachter, Phys. Rev. D 56, 2566 (1997);\ G.S. Bali, B. Bolder, N. Eicker, T. Lippert, B. Orth, K. Schilling, and T. Struckmann, Phys. Rev. D 62, 054503 (2000) H.P. Morsch, Z. Phys. A 350, 61 (1994) M. Ablikim, et al., hep-ex/0406038 (2004); see also D.V. Bugg, hep-ex/0510014 (2005) and refs. therein 16, 229 (2003) ![Comparison of two-boson density for $<r^2_{\rho}>$=0.2 fm$^2$ (dot-dashed lines) and boson-exchange potential $|V_{3g}(r)/c_{pot}|$ (solid lines) for $\kappa$=1.5, 1 and 2, respectively.[]{data-label="fig:f1"}](g1ex2sneu.eps){height="16cm"} \[ht\] ![Self-consistent solution with $<r^2_{\rho}>$=0.013 fm$^2$. Low radius cut-off function $f_{as}(r)$, shape of interaction (\[eq:veff\]) and $\rho^{p}(r)$ given by dashed, solid and dot-dashed lines, respectively. Two-boson density and boson-exchange potential $|V_{3g}/c_{pot}|$ in r- and Q-space, for the latter multiplied by $Q^2$, given by the overlapping dot-dashed and solid lines, respectively. The dashed line in the lower part corresponds to a calculation assuming fermion masses of 1.4 GeV.[]{data-label="fig:simdens"}](g1ex2c.eps "fig:"){height="18cm"} ![ Deduced confinement potential $V_{2g}(r)$ (\[eq:vb\]) taken from solution 4 (solid line) in comparison with lattice gauge calculations [@Bali] (solid points) . Boson-exchange potentials for the different solutions in the tables (given by dot-dashed and dashed lines) and sum given by solid line. This is compared to the vacuum sum rule (\[eq:sum\]) given by the dot-dashed line overlapping with the solid line. A pure potential $V=-const/r^2$ is shown also by the lower dot-dashed line.[]{data-label="fig:confine"}](confinesum.eps){height="18cm"} [^1]: postal address: Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany\ E-mail: [email protected] [^2]: In Q-space multiplied by $Q^2$.

ArXiv

=0.5cm =1em #### Introduction. {#introduction. .unnumbered} Two new large colliders with relativistic heavy nuclei, the RHIC and the LHC, are scheduled to be in operation in the nearest future. The charge numbers $Z_1=Z_2=Z$ of the nuclei with masses $M_1=M_2=M$ and their Lorentz factors $\gamma_1=\gamma_2=\gamma=E/M$ are the following $$\begin{aligned} Z=79\,, \ \gamma &=&\,\;108 \ {\mathrm {for \ RHIC \ (Au--Au \ collisions)}}\, \nonumber \\ Z=82\,, \ \gamma &=&3000 \ {\mathrm {for \ LHC \ \ (Pb--Pb \ collisions)}}\,. \label{1}\end{aligned}$$ Here $E$ is the heavy ion energy in the c.m.s. One of the important processes at these colliders is $$Z_1Z _2\to Z_1Z_2 \, e^+e^- \,. \label{2}$$ Its cross section is huge. In the Born approximation (see Fig. \[f1\] with $n=n'=1$) the total cross section according to the Racah formula [@R] is equal to $\sigma_{\mathrm{Born}} = 36 $ kbarn for the RHIC and 227 kbarn for the LHC. Therefore it will contribute as a serious background to a number of experiments, besides, this process is the leading beam loss mechanism (for details see review [@BB]). The cross sections of the process (\[2\]) in the Born approximation are known with accuracy $\sim 1/ \gamma^2$ (see, for example, Refs. [@R; @KLBGMS] and more recent calculations reviewed in Refs. [@BB; @BHT]). However, besides of the Born amplitude $M_{\mathrm {Born}} =M_{11}$, also other amplitudes $M_{nn'}$ (see Fig. \[f1\]) have to be taken into account for heavy nuclei since in this case the parameter of the perturbation series $Z\alpha$ is of the order of unity. Therefore, the whole series in $Z\alpha$ has to be summed to obtain the cross section with sufficient accuracy. Following Ref. [@BM], we call the Coulomb correction (CC) the difference $d\sigma_{\mathrm{Coul}}$ between the whole sum $d \sigma$ and the Born approximation $$d\sigma = d \sigma_{\mathrm{Born}} + d \sigma_{\mathrm{Coul}}\,. \label{4}$$ Such kind of CC is well known in the photoproduction of $e^+e^-$ pairs on atoms (see Ref. [@BM] and §98 of [@BLP]). The Coulomb correction to the total cross section of that process decreases the Born contribution by about 10 % for a Pb target. For the pair production of reaction (\[2\]) with $Z_1\alpha \ll 1$ and $ Z_2 \alpha \sim 1$ CC has been obtained in Refs. [@NP; @BB]. Recently this correction has been calculated for the pair production in the collisions of muons with heavy nuclei [@IKSS]. The results of Refs. [@NP; @BB; @IKSS] agree with each other in the corresponding kinematic regions and noticeably change the Born cross sections. Formulae for CC for two heavy ions were suggested ad hoc in Sect. 7.3 of  [@BB]. However, our calculations presented here do show that this suggestion is incorrect. In the present paper we calculate the Coulomb correction for process (\[2\]) omitting terms of the order of $1$ % compared with the main term given by the Born cross section. We find that these corrections are negative and quite important: $$\begin{aligned} \sigma_{\mathrm{Coul}}/ \sigma_{\mathrm{Born}} &=& -25\, \% \;\; {\mathrm for \ \ RHIC}\,, \nonumber \\ \sigma_{\mathrm{Coul}}/ \sigma_{\mathrm{Born}} &=& -14\, \% \;\; {\mathrm for \ \ LHC}\,. \label{5}\end{aligned}$$ This means that at the RHIC the background process with the largest cross section will have a production rate 25 % smaller than expected. Our main notations are given in Eq. (\[1\]) and Fig. \[f1\], besides, $(P_1+P_2)^2 = 4E^2 = 4 \gamma^2 M^2$, $q_i= (\omega_i,\, {\bf q}_i)= P_i-P_i'$, $\varepsilon= \varepsilon_++\varepsilon_-$ and $$\sigma_0=\frac{\alpha^4 Z_1^2 Z_2^2}{\pi m^2} \,, \;\; L= \ln{P_1P_2 \over 2M_1 M_2}= \ln{\gamma^2} \label{3}$$ where $m$ is the electron mass. The quantities ${\mathbf q}_{i\perp}$ and ${\mathbf p}_{\pm\perp}$ denote the transverse part of the corresponding three–momenta. Throughout the paper we use the well known function[@BM] $$f(Z) = Z^2\alpha^2 \sum_{n=1}^{\infty} {1\over n(n^2+Z^2\alpha^2)}\,,$$ its particular values for the colliders under discussion are $f(79)=0.313$ and $f(82)=0.332$. #### Selection of the leading diagrams and the structure of the amplitude. {#selection-of-the-leading-diagrams-and-the-structure-of-the-amplitude. .unnumbered} Let ${\cal M}$ be the sum of the amplitudes $M_{nn'}$ of Fig. \[f1\]. It can be presented in the form $$\begin{aligned} \label{7a} {\cal M}&=& \sum_{nn'\geq 1 } M_{nn'}= M_{\mathrm{Born}} +M_1+{\tilde M}_1+ M_2\,,\\ M_1 &=& \sum_{n'\geq 2} M_{1n'}\,, \ \ \tilde M_1 = \sum_{n\geq 2} M_{n1}\,, \ \ M_2= \sum_{nn'\geq 2} M_{nn'} \,. \nonumber\end{aligned}$$ The Born amplitude $M_{\mathrm{Born}}$ contains the one–photon exchange both with the first and the second nucleus, whereas the amplitude $M_1$ ($\tilde M_1$) contains the one–photon exchange only with the upper (lower) nucleus. In the last amplitude $M_2$ we have no one–photon exchange. According to this classification we write the total cross section as $$\sigma = \sigma_{\mathrm{Born}} +\sigma_1 +\tilde\sigma_1 + \sigma_2 \label{7}$$ where $$\begin{aligned} &&d\sigma_{\mathrm{Born}} \propto |M_{\mathrm{Born}}|^2\,, \nonumber \\ &&d\sigma_1 \propto 2 {\mathrm Re}(M_{\mathrm{Born}} M_1^*) +|M_1|^2 \,, \nonumber \\ &&d\tilde\sigma_1 \propto 2 {\mathrm Re}(M_{\mathrm{Born}} \tilde M_1^*) +| \tilde M_1|^2 \,, \nonumber \\ &&d\sigma_2 \propto 2 {\mathrm Re}\left( M_{\mathrm{Born}} M_2^* + M_1\tilde M_1^* +M_1M_2^* \right. \nonumber \\ && \left. \hspace{1cm}+\tilde M_1M_2^* \right) + |M_2|^2 \,. \nonumber\end{aligned}$$ It is not difficult to show that the ratio $\sigma_i / \sigma_{\mathrm{Born}}$ is a function of $(Z\alpha)^2$ only but not of $Z \alpha$ itself. Additionally we estimate the leading logarithms appearing in the cross sections $\sigma_i$. The integration over the transfered momentum squared $q_1^2 $ and $q_2^2$ results in two large Weizsäcker–Williams (WW) logarithms $\sim L^2$ for the $\sigma_{\mathrm{Born}}$, in one large WW logarithm $\sim L$ for $\sigma_1$ and $\tilde\sigma_1$. The cross section $\sigma_2$ contains no large WW logarithm. Therefore, the relative contribution of the cross sections $\sigma_i$ is $\sigma_1 / \sigma_{\mathrm{Born }} =\tilde\sigma_1 / \sigma_{\mathrm{Born}} \sim (Z\alpha)^2 /L$ and ${\sigma_2 / \sigma_{\mathrm{Born}}} \sim (Z\alpha)^2 /L^2 \, < 0.4$ % for the colliders (\[1\]). As a result, with an accuracy of the order of $1 \%$ we can neglect $\sigma_2$ in the total cross section and use the equation $$\sigma = \sigma_{\mathrm{Born}} +\sigma_1 +\tilde\sigma_1\,. \label{9}$$ With that accuracy it is sufficient to calculate $\sigma_1$ and $\tilde\sigma_1$ in the leading logarithmic approximation (LLA) only since the next to leading log terms are of the order of $(Z\alpha /L)^2$. This fact greatly simplifies the calculations. The calculation in the LLA can be performed using the equivalent photon or WW approximation. The main contribution to $\sigma_1$ and $\tilde\sigma_1$ is given by the region $(\omega_1/\gamma)^2 \ll -q_1^2 \ll m^2$ and $(\omega_2/\gamma)^2 \ll -q_2^2 \ll m^2$, respectively. In the first region the main contribution arises from the amplitudes $M_{\mathrm{Born}}+M_1$ (in the second region $M_{\mathrm{Born}}+\tilde M_1$). The virtual photon with four–momentum $q_1$ is almost real and the amplitude can be expressed via the amplitude $M_\gamma$ for the real photoproduction $\gamma Z_2\to Z_2 e^+e^-$ (see, for example, §99 of Ref. [@BLP]) $$M_{\mathrm{Born}}+M_1\approx \sqrt{4 \pi \alpha} Z_1 \frac{ |{\mathbf{q}}_{1\perp}|} {(-q_1^2)} \, \frac{2 E}{\omega_1} \, M_\gamma \,. \label{amp}$$ The amplitude $M_\gamma$ has been calculated in Ref. [@BM]. We use the convenient form of that amplitude derived in the works [@OM] and [@IM]: $$M_\gamma= ( f_1 \, M_\gamma^{\mathrm{Born}} + {\mathrm i} f_2 \, \Delta M_\gamma) \, {\mathrm e}^{{\mathrm i} \Phi} \label{Mgamma}$$ where $M_\gamma^{\mathrm{Born}}$ is the Born amplitude for the $\gamma Z_2 \to Z_2 e^+ e^-$ process. This Born amplitude depends on the transverse momenta ${\mathbf p}_{\pm\perp}$ only via the two combinations $A=\xi_+-\xi_-$ and ${\mathbf B}=\xi_+ {\mathbf p}_{+\perp} + \xi_- {\mathbf p}_{-\perp}$ where $\xi_\pm= m^2/( m^2+{\mathbf p}_{\pm\perp}^2)$. The quantity $\Delta M_\gamma$ is obtained from $M_\gamma^{\mathrm{Born}}$ replacing $A\to \xi_++\xi_--1$ and ${\mathbf B}\to \xi_+ {\mathbf p}_{+\perp} - \xi_- {\mathbf p}_{-\perp}$. All the nontrivial dependence on the parameter $Z_2 \alpha \equiv \nu$ are accumulated in the Bethe-Maximon phase $$\Phi=\nu \, \ln\frac{(p_+P_2) \xi_+}{(p_-P_2) \xi_-} \label{phase}$$ and in the two functions (with $z=1 - (-q_2^2/m^2) \xi_+\xi_-$) $$f_1=\frac{ F({\mathrm i} \nu,-{\mathrm i} \nu; 1 ; z)} { F({\mathrm i} \nu,-{\mathrm i} \nu; 1 ; 1 )} \,,\ \ f_2=\frac{1-z}{\nu} f_1'(z)\,. \label{f1f2}$$ The function $f_1(z)$ and its derivative $f_1'(z)$ are given with the help of the Gauss hypergeometric function $F(a,b;c;z)$. It can be clearly seen that in the region ${\mathbf p}_{\pm\perp}^2 \sim m^2$ the amplitude $M_\gamma$ differs considerably from the $M_\gamma^{\mathrm{Born}}$ amplitude and, therefore, the whole amplitude ${\cal M}$ differs from its Born limit $M_{\mathrm{Born}}$. Let us stress that just this transverse momentum region ${\mathbf p}_{\pm\perp}^2 \sim m^2$ gives the main contribution into the total Born cross section $\sigma_{\mathrm{Born}}$ and into $\sigma_1$. Outside this region the CC vanishes. Indeed, for ${\mathbf p}_{\pm\perp}^2 \ll m^2$ or ${\mathbf p}_{\pm\perp}^2 \gg m^2$ the variable $ z \approx 1$, therefore, $f_1\approx 1$, $f_2 \approx 0$ and $$M_{\mathrm{Born}}+M_1= M_{\mathrm{Born}} {\mathrm e}^{{\mathrm i} \Phi} \,. \label{limit}$$ Note that the region ${\mathbf p}_{\pm\perp}^2 \gg m^2$ gives a negligible contribution to the total cross section $\sigma$, however, this region might be of interest for some experiments. The results of Ref. [@BM] which are used here in the form of Eqs. (\[amp\])-(\[limit\]) are the basis for our consideration. These results were confirmed in a number of papers (see, for example, Refs. [@Qclas; @IM]) using various approaches. Recently in Refs. [@DIRACEQ] the Coulomb effects were studied within the frame–work of a light–cone or an eikonal approach. However, the approximations used in Refs. [@DIRACEQ] fail to reproduce the classical results of Bethe and Maximon [@BM]. To show this explicitly, we consider the simple case $Z_1 \alpha \ll 1, \; Z_2 \alpha \equiv \nu \sim 1$ in which the principal result of Refs. [@DIRACEQ] for the amplitude takes the form ${\cal M}=M_{\mathrm{Born}}+ M_1= M_{\mathrm{Born}} \exp ({\mathrm i} \Psi)$ with $\Psi = \nu \ln {\mathbf q}_{2\perp}^2$ in obvious contradiction to Eqs. (\[amp\])-(\[Mgamma\]). Since in the works [@DIRACEQ] different statements on the applicability range of their results can be found, we take as an example the common region $(\omega_i/\gamma_i)^2 \ll {\mathbf q}_{i\perp}^2 \ll m^2$. But even in that region their expression for the matrix element does not reproduce Eq. (\[limit\]) since their phase $\Psi$ does not coincide with the Bethe–Maximon phase $\Phi$, i.e. $\Psi\neq \Phi$. #### CC to the energy distribution and to the total cross section. {#cc-to-the-energy-distribution-and-to-the-total-cross-section. .unnumbered} As it was explained in the previous section, the basic expression for the cross section $d\sigma_1$ in the LLA can be directly obtained using the WW approximation. To show clearly the terms omitted in the LLA, we start with a more exact expression for $d \sigma_1$ derived for the case of $\mu Z $ collisions considered in Ref. [@IKSS]. The reason is that for the most interesting region (when the energy of relativistic $e^{\pm}$ pairs is much smaller than the nucleus energy) the muon in the $\mu Z$ scattering as well as the upper nucleus of the ion–ion collision can be equally well treated as spinless and pointlike particles. Using Eqs. (14) and (17) from Ref. [@IKSS] (given in the lab frame of the muon projectile on a nucleus target) and the invariant variables $x_{\pm}= (p_{\pm} P_2)/(q_1 P_2)$, $y= (q_1 P_2) / (P_1P_2)$ we obtain $d \sigma_1$ for the pair production in $Z_1Z_2$ collisions in the invariant form (and at $y\ll 1$) $$\begin{aligned} d\sigma_1& =& - \frac{4}{3} \sigma_0 f(Z_2) \left\{ \left[ (1+\xi) a-1\right] \ln \frac{1+\xi}{\xi} - \right. \nonumber \\ &-& \left. a + \frac{4-a}{1+\xi} \, \right\} {dy\over y} dx_+dx_- \delta(x_++x_- - 1) \label{10}\end{aligned}$$ with $a= 2 (1+x_+^2+x_-^2) \,, \ \ \xi= \left( M_1 y/m\right)^2 x_+x_- $. The main contribution to $\sigma_1$ is given by the region $$\frac{M_1^2 M_2^2 }{(P_1 P_2)^2} \ll \xi \ll 1\,. \label{11}$$ The corresponding expression for $d\tilde\sigma_1$ can be obtained by making the replacements $$d\tilde\sigma_1 = d\sigma_1(q_1 \to q_2, P_1 \leftrightarrow P_2, Z_1\leftrightarrow Z_2)\,. \label{12}$$ Below we consider only the experimentally most interesting case when in the collider system ($\gamma_1=E_1/M_1 \sim \gamma_2=E_2/M_2$) both $e^+$ and $e^-$ are ultrarelativistic ($\varepsilon_\pm \gg m$). We assume that the $z$-axis is directed along the initial three-momentum of the first nucleus ${\mathbf P}_1$. To obtain the energy distribution of $e^+$ and $e^-$ in the LLA we have to take into account two regions $p_{\pm z} \gg m$ and $(-p_{\pm z}) \gg m$ where the lepton pair is produced either in forward or backward direction. In the first region we have $ x_\pm= \varepsilon_{\pm}/\varepsilon$, $y=\varepsilon/E_1$, and from Eq. (\[10\])-(\[11\]) we obtain in the LLA $$\begin{aligned} &d\sigma_1^{(1)}&= \nonumber \\ &-&4 \,\sigma_0 f(Z_2) \left(1 - \frac{4\varepsilon_+ \varepsilon_-}{3 \varepsilon^2} \right) \, \ln \frac{(m \gamma_1)^2}{ \varepsilon_+ \varepsilon_-} \, {d\varepsilon_+ d \varepsilon_- \over \varepsilon^2} \,, \label{13new} \\ && m \ll \varepsilon_\pm \ll m \gamma_1 \,. \nonumber\end{aligned}$$ In the second region we have $x_\pm \approx \varepsilon_{\mp}/ \varepsilon$, $y\approx m^2 \varepsilon /( 4 E_1 \varepsilon_+\varepsilon_-)$) and $$\begin{aligned} &d\sigma_1^{(2)}&= \nonumber \\ &-&4 \,\sigma_0 f(Z_2) \left(1 - \frac{4\varepsilon_+ \varepsilon_-}{3 \varepsilon^2} \right) \, \ln \frac{\gamma_1^2 \varepsilon_+ \varepsilon_-}{m^2} \, {d\varepsilon_+ d \varepsilon_- \over \varepsilon^2} \,, \label{14new} \\ && m \ll \varepsilon_\pm \ll m \gamma_2 \,. \nonumber\end{aligned}$$ Summing up these two contributions, we find $$d\sigma_1= - 8 \, \sigma_0 f(Z_2) \left(1 - \frac{4\varepsilon_+ \varepsilon_-}{3 \varepsilon^2} \right) \, \ln \gamma_1^2 \, {d\varepsilon_+ d \varepsilon_- \over \varepsilon^2} \,. \label{13}$$ To obtain $\sigma_1$ we have to integrate the expressions (\[13new\]) and (\[14new\]) over $\varepsilon_-$ (with logarithmic accuracy) $$\begin{aligned} d\sigma_1^{(1)} &=& - \frac{28}{9} \sigma_0 f(Z_2) \, \ln \frac{(m \gamma_1)^2}{\varepsilon_+^2} \, \frac{d\varepsilon_+} {\varepsilon_+} \,, \label{15new} \\ &&m\ll \varepsilon_+\ll m \gamma_1 \,, \nonumber\end{aligned}$$ $$\begin{aligned} d\sigma_1^{(2)} &=& - \frac{28}{9} \sigma_0 f(Z_2) \, \ln \frac{(\gamma_1 \varepsilon_+)^2}{m^2} \, \frac{d\varepsilon_+} {\varepsilon_+} \,, \label{16new} \\ &&m\ll \varepsilon_+\ll m \gamma_2 \nonumber\end{aligned}$$ from which it follows that $$d\sigma_1= - \frac{28}{9} \sigma_0 f(Z_2)\, \ln \gamma_1^2 \, \frac{d\varepsilon_+} {\varepsilon_+} \,. \label{17new}$$ The further integration of Eqs. (\[15new\]), (\[16new\]) over $\varepsilon_+$ results in $$\sigma_1=- \frac{28}{9} \sigma_0 f(Z_2)\, \left[ \ln \frac{P_1 P_2}{ 2 M_1 M_2} \right]^2 \,. \label{18new}$$ This expression is in agreement with the similar result for the $\mu Z$ scattering (see Eq. (31) from [@IKSS] for $Z_1 =1, Z_2=Z$). The corresponding formulae for $\tilde\sigma_1$ can be obtained from Eqs. (\[13\]), (\[17new\]) and (\[18new\]) by replacing $\gamma_1\leftrightarrow \gamma_2$, $Z_1\leftrightarrow Z_2$. The whole CC contribution $d \sigma_{\mathrm{Coul}}= d( \sigma_1+ \tilde\sigma_1)$ for the symmetric case $Z_1=Z_2=Z$ and $\gamma_1=\gamma_2=\gamma$ takes the following form $$d\sigma_{\mathrm Coul}= - 16 \,\sigma_0 f(Z) \left(1 - \frac{4\varepsilon_+ \varepsilon_-}{3 \varepsilon^2} \right) \, L \, \frac{d\varepsilon_+ d \varepsilon_-} {\varepsilon^2} \label{133}$$ at $ m\ll \varepsilon_{\pm} \ll m \gamma \,$, $$d\sigma_{\mathrm Coul}= - \frac{112}{9} \sigma_0 f(Z) \, L\, \frac {d\varepsilon_+ } {\varepsilon_+} \label{1333} \\$$ at $ m\ll \varepsilon_+ \ll m \gamma \,$, and $$\sigma_{\mathrm Coul}=- \frac{56}{9} \sigma_0 f(Z)\, L^2 \,. \label{188new}$$ The size of this correction for the two colliders was given before in Eq. (\[5\]). The total cross section with and without Coulomb correction as function of the Lorentz factor $\gamma$ is illustrated in Fig. \[f3\] for Pb nuclei. #### Conclusion. {#conclusion. .unnumbered} We have calculated the Coulomb corrections to $e^+e^-$ pair production in relativistic heavy ion collisions for the case of colliding beams. Our main results are given in Eqs. (\[133\])-(\[188new\]). We have restricted ourselves to the Coulomb corrections for the energy distribution of electrons and positrons and for the total cross section. In our analysis we neglected contributions which are of the relative order of $\sim (Z\alpha)^2/L^2$. The CC to the angular distribution of $e^+e^-$ can be easily obtained in a similar way, however only with an accuracy $Z\alpha/L^2$. Since our basic formulae (\[10\]), (\[12\]) are given in the invariant form, a similar calculation can be easily repeated for fixed–target experiments. This interesting question will be considered in a future work. [*Acknowledgments.*]{} — We are very grateful to G. Baur, Yu. Dokshitzer, U. Eichmann, V. Fadin, I. Ginzburg and V. Telnov for useful discussions. V.G.S. acknowledges support from Volkswagen Stiftung (Az. No. I/72 302). D.Yu.I. and V.G.S. are partially supported by the Russian Foundation for Basic Research (code 96-02-19114). [99]{} Email address: [email protected] Email address: [email protected] Email address: [email protected] G. Racah, Nuovo Cim. [**14**]{}, 93 (1937). C. A. Bertulani, G. Baur, Phys. Rep. [**163**]{}, 299 (1988). V. N. Baier, V. S. Fadin, ZhETF [**61**]{}, 476 (1971); E. A. Kuraev, V. G. Lasurik-Elzufin, Pis’ma ZhETF [**13**]{}, 391 (1971); V. M. Budnev, I. F. Ginzburg, G. V. Meledin, V. G. Serbo, Nucl. Phys. B [**63**]{}, 519 (1973). G. Baur, K. Henken, D. Trautman, J. Phys. G [**24**]{}, 1657 (1998). H. Bethe, L. C. Maximon, Phys. Rev. [**93**]{}, 768 (1954); H. Davies, H. Bethe, L. C. Maximon, Phys. Rev. [**93**]{}, 788 (1954). V. B. Berestetskii, E. M. Lifshitz, L. B. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 1989). A. I. Nikishov, N. V. Pichkurov, Sov. J. Nucl. Phys. [**35**]{}, 561 (1982). D. Yu. Ivanov, E. A. Kuraev, A. Schiller, V. G. Serbo, Phys. Lett. B [**442**]{}, 453 (1998). H. Olsen, L. C. Maximon, Phys. Rev. [**114**]{}, 887 (1959). D. Ivanov, K. Melnikov, Phys. Rev. D [**57**]{}, 4025 (1998). B. Segev and J. C. Wells, Phys. Rev. A [**57**]{}, 1849 (1998); A. J. Baltz, L. McLerran, Phys. Rev. C [**58**]{}, 1679 (1998); U. Eichmann, J. Reinhardt, S. Schramm, W. Greiner, nucl-th/9804064; U. Eichmann, J. Reinhardt, W. Greiner, nucl-th/9806031. H. Olsen, L. C. Maximon, and H. Wergeland, Phys. Rev. [**106**]{}, 27 (1957); V. N. Baier, V. M. Katkov, ZhETF [**55**]{}, 1542 (1965).

ArXiv

--- abstract: 'Deep learning based natural language processing model is proven powerful, but need large-scale dataset. Due to the significant gap between the real-world tasks and existing Chinese corpus, in this paper, we introduce a large-scale corpus of informal Chinese. This corpus contains around 37 million book reviews and 50 thousand netizen’s comments to the news. We explore the informal words frequencies of the corpus and show the difference between our corpus and the existing ones. The corpus can be further used to train deep learning based natural language processing tasks such as Chinese word segmentation, sentiment analysis.' author: - | Jianyu Zhao\ School of Data Science\ Fudan University, China\ `[email protected]`\ `[email protected]`\ Zhuoran Ji\ GSQ Tec.\ Shen Zhen, China\ `[email protected]`\ bibliography: - 'references.bib' title: 'LSICC: A Large Scale Informal Chinese Corpus' --- Introduction ============ Deep learning has been the mainstay for natural language processing, ranging from text summarization [@paulus2017deep] to sentiment analysis [@zhang2018deep] to text generation [@sutskever2014sequence] and automated question-answering system [@yu2014deep]. Unlike traditional rule-based methods, the scale and quality of the corpus significantly influence the performance of the deep learning models. In Chinese NLP field, there are many famous large-scale corpora with high quality, such as Baidu Encyclopedia, People’s Daily News and Sina Weibo News. Various powerful Chinese deep learning models are trained on these corpora [@li2018analogical], [@min2015bosonnlp], [@cui2016consensus], [@nallapati2016abstractive], [@gu2016incorporating]. However, most Chinese corpora are in written Chinese, while most real-world deep learning based NLP systems deal with informal Chinese, such as products reviews, netizens’ opinions, and microblogs. There are great gaps between informal Chinese and written Chinese, especially in words usages and sentences structures. The pre-trained deep learning model trained from written Chinese corpus, such as words embedding and Chinese words segmentation tools, may perform badly on tasks with informal Chinese. To address this issue, we introduce LSICC, a large-scale corpus of informal Chinese. Containing around 37 million book reviews and 50 thousand netizens’ opinions to news, LSICC is a typical informal Chinese corpus. Most sentences of LSICC are in spoken Chinese and even Internet slang. As far as we know, LSICC is the first large-scale, well-formatted, cleansed corpus focusing on informal Chinese. This paper makes the following contributions: 1. collect a large scale corpus of informal Chinese 2. filter out the informationless data items 3. compare the proportions of informal words in several corpus Informal Chinese ================ Informal Chinese, including spoken Chinese and Chinese Internet Slang, has a substantial difference with the formal one, in both grammar and words usage. In this section, we discuss the difference between formal Chinese and informal Chinese. Spoken Chinese -------------- For most language, there are differences between the spoken one and the written one. In Chinese, the gap is even more significant due to the long history of written Chinese. Similar to another language, spoken Chinese sometimes does not follow the rules as strictly as written Chinese, especially for the elliptical sentences. For example, in spoken Chinese, the subjects sometimes are omitted. In addition to the grammar, the usage of the words influences the neural network based Chinese natural language processing model most. There are various interchangeable words pairs between written Chinese and spoken Chinese, such as [UTF8]{}[gbsn]{}“脑袋” and [UTF8]{}[gbsn]{}“头部” , which both mean “head” in Chinese. The two words in each interchangeable words pair usually have almost the same meanings, but the one in written Chinese is more formal, while the one in spoken Chinese is informal. Internet Slang -------------- Born in the 1990s, Chinese Internet slang refers to various kinds of slang created by netizens and used in chat rooms, social networking services, and online community. Nowadays, Chinese Internet slang is not little memes within internet ingroup, but becoming popular language style of all Chinese speakers. From 2012, Xinhuanet selects “Top 10 Chinese Internet Slang” [@topten] every year, and Chinese Internet slang is used even by Chinese official institutions. The first kind of Internet slang is the phonetic substitution, whose pronunciation is same or similar to the formal phrase. For example, in Internet slang, people may use [UTF8]{}[gbsn]{}“神马” to replace [UTF8]{}[gbsn]{}“什么” . Both [UTF8]{}[gbsn]{}“神马” and [UTF8]{}[gbsn]{}“什么” are pronounced as “cien ma” and has the meaning of “what”. However, in written Chinese, [UTF8]{}[gbsn]{}“神马” means “horse-god”, while [UTF8]{}[gbsn]{}“什么” means “what”. Transliteration is also a primary way to form Internet slang. As the words are transliterated from another language, both the meaning and pronunciation of the transliterated words are similar to the source language. For example, [UTF8]{}[gbsn]{}“伐木累” is transliterated from English word “family” and only used as Chinese Internet slang [@Li:2008:MMR:1613715.1613849]. Meanwhile, Internet slang is also created by giving new meanings to the old words. For example, in written Chinese, [UTF8]{}[gbsn]{}“酱油” means “soy s sauce”. However, in the Chinese Internet slang, it refers to “passing by". Data Collection =============== LSICC collects book reviews from DouBan Dushu and netizen’s opinions from Chiphell. This section describes these two datasets and pre-processing methods briefly. DouBan DuShu ------------ DouBan DuShu[^1] is a Chinese website where users can share their reviews about various kinds of books. Most of the users on this website are unprofessional book reviewers. Therefore, the comments are usually spoken Chinese or even Internet slang. In addition to the comments, users can mark the books from one star to 5 stars according to the quality of the books. We have collected more than 37 million short comments from about 18 thousand books with 1 million users. The great number of users provide diversities of the language styles, from moderate formal to informal. An example of the data item is shown in table \[douban\]. [l|l|l]{} Key & Description & Value Example\ Book Name & The name of the book & [UTF8]{}[gbsn]{}理想国 \ User Name & Who gives the comment (anonymized) & 399\ Tag & The tag the book belongs to & [UTF8]{}[gbsn]{}思想 \ Comment & Content of the comment & [UTF8]{}[gbsn]{}我是国师的脑残粉 \ Star & Stars given to the book (from 1 star to 5 stars) & 5 stars\ Date & When the comment posted & 2018-08-21\ Like & Count of “like” on the comment & 0\ Chiphell -------- Chiphell [^2] is a web portal where netizens share their views to news and discuss within groupuscule. We have collected discussion forums from several subjects, such as computer hardware, motors and clothes. There are more than 50 thousand discussions in the corpus. Similar to the DouBan DuShu corpus, most of the sentences collected from Chiphell are informal Chinese and some of them are in particular domains. An example from each subject is shown in table \[chh\]. [l|p[6cm]{}|p[6cm]{}]{} Subject & Topic & Example\ News & [UTF8]{}[gbsn]{}美机场航空业希望修改客机降落的Emoji表情：机头朝下不吉利 & [UTF8]{}[gbsn]{}那我还说改完的意思是无限复飞呢,飞到没油不又gg了 \ Computer Hardware & [UTF8]{}[gbsn]{}请问现在大船货除开3610还有其他性价比的大船大容量吗 & [UTF8]{}[gbsn]{}我1T的PM1633。。卖1300都木有人接 \ Mobile Phones & [UTF8]{}[gbsn]{}努比亚X 综合讨论帖 & [UTF8]{}[gbsn]{}MIX3辣鸡被友商各种吊打 \ Clothes & [UTF8]{}[gbsn]{}程序媛的皮艺生活 & [UTF8]{}[gbsn]{}花点时间在复杂又感兴趣的事情上是一件快乐又有成就感的体验 \ Data Pre-processing ------------------- In addition to the raw dataset, we extracted the comments and preprocessed them to provide a clean, formal formatted and comprehensive Chinese corpus. After carefully investigate the raw text, mainly three preprocessing methods are applied: 1. convert Traditional Chinese to Simplified Chinese 2. remove over-short comments (less than 4 characters) 3. add identifier to special characters, such as special signs, English words and emoticons Experiments =========== To further explore the informal Chinese corpus, we calculate the proportion of informal words in the corpus. The experiment is conducted on Weibo News [@hu2015lcsts], Sougou News, People’s Daily [@yu2001guideline] and the LSICC. We manually collected 70 informal words as the benchmark, which covers both spoken Chinese words and Chinese network slang words. We counted the frequencies of informal words and the number of total words to calculate the proportion of the informal words in the whole corpus. As shown in table \[proportion\], the LSICC has the highest proportion of the informal words, which is more than two times the second highest one, the Weibi News. Noted that the more formal the media is, the lower the proportion of the informal words in it. Corpus Informal Words Total Words Proportion ---------------- ---------------- ------------- ------------ LSICC 621807 705231306 8.82 Weibo News 46831 125082112 3.74 Sougou News 1238 14160148 0.87 People’s Daily 25 3482887 0.07 : Proportion of the informal words in each corpus \[proportion\] The result indicated that the gap between the language that the real-world natural language models deal with the existing corpora is significant. Using the vector representations extracted from the corpus of formal Chinese as the word embedding may attribute to poor performance. Conclusions and Future Work =========================== We constructed a large-scale Informal Chinese dataset and conducted a basic words frequency statistic experiment on it. Compared to the existing Chinese corpus, LSICC is more typical dataset for real-world natural language processing tasks, especially for sentiment analysis. As a next step, we should conduct embedding extraction Chinese words segmentation and sentiment analysis on LSICC. Meanwhile, as the raw information, such as the usernames and book names is kept, LSICC can also be used to build recommendation systems and explore social network. [^1]: available on: https://github.com/JaniceZhao/Douban-Dushu-Dataset.git [^2]: available on: https://github.com/JaniceZhao/Chinese-Forum-Corpus.git

ArXiv

--- abstract: 'X-rays trace accretion onto compact objects in binaries with low mass companions at rates ranging up to near Eddington. Accretion at high rates onto neutron stars goes through cycles with time-scales of days to months. At lower average rates the sources are recurrent transients; after months to years of quiescence, during a few weeks some part of a disk dumps onto the neutron star. Quasiperiodic oscillations near 1 kHz in the persistent X-ray flux attest to circular motion close to the surface of the neutron star. The neutron stars are probably inside their innermost stable circular orbits and the x-ray oscillations reflect the structure of that region. The long term variations show us the phenomena for a range of accretion rates. For black hole compact objects in the binary, the disk flow tends to be in the transient regime. Again, at high rates of flow from the disk to the black hole there are quasiperiodic oscillations in the frequency range expected for the innermost part of an accretion disk. There are differences between the neutron star and black hole systems, such as two oscillation frequencies versus one. For both types of compact object there are strong oscillations below 100 Hz. Interpretations differ on the role of the nature of the compact object.' address: | Laboratory for High Energy Astrophysics\ NASA/GSFC Greenbelt, MD 20771 author: - 'Jean H. Swank' title: 'X-Ray Observations of Low-Mass X-Ray Binaries: Accretion Instabilities on Long and Short Time-Scales' --- \#1[[A&A,]{} [\#1]{}]{} \#1[[Acta Astr.,]{} [\#1]{}]{} \#1[[A&AS,]{} [\#1]{}]{} \#1[[ARA&A,]{} [\#1]{}]{} \#1[[AJ,]{} [\#1]{}]{} \#1[[ApJ,]{} [\#1]{}]{} \#1[[ApJS,]{} [\#1]{}]{} \#1[[MNRAS,]{} [\#1]{}]{} \#1[[Nature,]{} [\#1]{}]{} \#1[[PASJ,]{} [\#1]{}]{} Introduction {#introduction .unnumbered} ============ Low-mass X-ray binaries (LMXB) are the binaries of a low-mass “normal” star and a compact star. The compact star could be a white dwarf, a neutron star, or a black hole. The Rossi X-Ray Timing Explorer ([*RXTE*]{}) has been observing since the beginning of 1996 and has obtained qualitatively new information about the neutron star and black hole systems. In this paper I review the new results briefly in the context of what we know about these sources. The brightest, Sco X–1, was one of the first non-solar X-ray sources detected, but only with [*RXTE*]{} have sensitive measurements with high time resolution been made that could detect dynamical time-scales in the region of strong gravity. [*RXTE*]{} also has a sky monitor with a time-scale of hours that keeps track of the long term instabilities and enables in depth observations targeted to particular states of the sources. The LMXB have a galactic bulge or Galactic Population II distribution. The mass donor generally fills its Roche lobe, is less than a solar mass, and is optically faint, in contrast to the early type companions of pulsars like Cen X–3 or the black hole candidate Cyg X–1. In many cases the optical emission is dominated by emission from the accretion disk, and that is dominated by reprocessing of the X-ray flux from the compact object [@vPM95]. The known orbital periods of these binaries range from 16 days (Cir X–1) to 11 minutes (4U 1820–30). The very short period systems ($< 1$ hr) are expected to have degenerate dwarf mass donors and probably the mass transfer is being driven by gravitational radiation. The different properties of the sources indicate several populations. The longer period systems with more massive companions are probably slightly evolved from the main sequence. There are about 50 persistent neutron star LMXB [@vP95]. Distances can be estimated in a variety of ways. The hydrogen column density indicated by the X-ray spectrum should include a minimum amount due to the interstellar medium. Many of the sources emit X-ray bursts associated with thermonuclear flashes that reach the hydrogen or helium Eddington limits. In some cases the optical source provides clues. The resulting luminosity distribution appears to range from several times the Eddington limit for a neutron star down below the luminosity of about $10^{35}$ ergs s$^{-1}$, corresponding to $\approx 10^{-11}$ $\msun$ yr$^{-1}$ [@CS97]. The lower limit has come from instrument sensitivity, but it may also reflect the luminosity below which the accretion flow is not steady, so that the source must be a transient. “X-Ray Novae” that are among the brightest X-ray sources for a month to a year are sufficiently frequent that they were seen in rocket flights in the beginning of X-ray astronomy. The X-ray missions that monitored parts of the sky during the last three decades found that on average there are 1–2 very bright transient sources each year (e.g. [@CSL97]) with durations of a month to a year. In 5 years of [*RXTE*]{} operations, we know of 20 transient neutron star sources and and an equal number of transient black hole sources. If they have a 20 yr recurrence time we have seen only a quarter of them and if we have only been watching a third of the region in the sky, 20 observed sources implies more than 240 sources exist. In reality there is a distribution of the recurrence times, some as short as months, others longer than 50 years, if optical records are good. On the basis of such estimates, the number of potential black hole transients is estimated to be on the order of thousands [@TS96]. The separation of sources into persistent and transient sources is a very gross simplification. One of the discoveries of recent missions, and especially of the All Sky Monitor (ASM) [@Bradt00] has been that the persistent sources have cycles of variations with time-scales ranging from many months to days. If the transient outbursts originate in accretion instabilities, perhaps these variations are related. In the next section I show some of the kinds of behavior being observed. At radii close to the compact objects the dynamical time-scale gets shorter, till it is the milliseconds of the neutron star or black hole. RXTE’s large area detectors detect oscillations on these time-scales which must reflect the dynamics at the innermost stable circular orbit (ISCO) of these neutron stars and black holes. The neutron stars of this sample are expected to have magnetic dipole moments and surface fields about $10^8 - 10^9$ gauss. Of course the neutron stars have a surface such that matter falling from the accretion disk to the neutron star crashes into the surface and generates X-ray emission. In the case of the black holes matter could fall through the event horizon and disappear with no further emission of energy. Thus the X-rays produced and the dynamics that dominates in the two cases (neutron star versus black hole) could be different. However, a number of similarities appear in the signals we receive. Long Time-scale Variabilities {#long-time-scale-variabilities .unnumbered} ============================= High Accretion Rate - Persistent Sources {#high-accretion-rate---persistent-sources .unnumbered} ---------------------------------------- Among the persistent LMXB there are characteristic variations on time-scales of months in some sources and days in others[@Bradt00]. Quasiperiodic modulations were pointed out at 37 days for Sco X–1 (IAUC 6524), 24.7 days for GX 13+1 (IAUC 6508), 77.1 days for Cyg X–2 (IAUC 6452), 37 days for X 2127+119 in M15 (IAUC 6632). The obviously important, but not strictly periodic modulations in 4U 1820–30 and 4U 1705–44 at time-scales of 100–200 days are shown in Figure 1. For Sco X–1, the changes in activity level occur in a day and the activity time-scale is hours. The hardness is often correlated with the rate, although this measure does not bring out more subtle spectral changes. These time-scales are less regular than the 34 day cycle time of Her X-1, and similar modulations in LMC X-4 and SMC X-1, which are thought to be due to the precession of a tilted accretion disk. The latter sources are high magnetic field pulsars in which the disk is larger than in the LMXB, and is truncated by the magnetosphere at a radius as large as $10^8$ cm. The LMXB spectral changes are also different than those of the pulsars. In the LMXB case the changes are thought to be real changes in the accretion onto the neutron star, at least the production of X-rays, rather than a change in an obscuration of the X-rays that we see. The spectral changes are captured in the color-color diagrams that give rise to the names “Z” and “Atoll” for subsets of the LMXB. These were identified with EXOSAT observations by Hasinger and van der Klis [@HvdK89]. Characteristics of the bursts from 4U 1636–53 depended on the place of the persistent flux in the atoll color-color diagram [@vdK90]. This implied that the real mass accretion rate was correlated with the position on the diagram (although other possibilities such as the distribution of accreted material on the surface of the neutron star may play a role). That the position in the diagram in not uniquely correlated to the flux is as yet not understood. Transients atoll sources like Aql X–1 and 4U 1608–52 go around the atoll diagram during the progress of the outburst. Low Average Accretion Rate - Transients {#low-average-accretion-rate---transients .unnumbered} --------------------------------------- There are only a few persistent LMXB in which the compact object is a black hole. Black hole binaries are for some reason more likely to be transients. Perhaps the binaries harboring them are not being driven to have as much mass exchange, so that it happens that these systems are in the range of mass flow through the disk that makes them transient. There are also neutron star transients with low average mass exchange rates. Figure 2 shows on the left two neutron star transients, a well known atoll burster Aql X–1 and the pulsar GRO J1744–28, which had two outbursts a year apart, but has otherwise not been seen. On the right are two black hole candidates, 4U 1630–47, which recurs approximately every two years, and XTE J1550–564, which like GRO J1744–28, had a dramatic outburst, with a weaker recurrence after a year’s hiatus. Black hole candidates can get brighter than the transient bursters, consistent with the Eddington limit for more massive compact objects and they probably go through more different spectral and timing “states”, but there are also similarities in the kinds of behavior that are exhibited. From both BeppoSAX and RXTE results it is clear that there is a population of systems which have transient episodes, but which are an order of magnitude less luminous at peak. BeppoSax has seen bursts from a number of sources for which the persistent flux is below their sensitivity limit. RXTE has seen a dozen sources which may not rise above $10^{36}$ ergs s$^{-1}$ during transient episodes. Several of these are believed to be neutron stars because Type I (cooling) bursts were observed. They include the source SAX J1808.4-3658, unique to date, that both pulses (2.5 msec) and has Type I bursts. Some sources have spectral and timing properties consistent with black hole candidates which go into the black hole “low hard” state, with strong white noise variability below 10 Hz and hard spectra. One of these was V4641 Sgr, which went into much brighter outburst, with a radio jet, before disappearing. Instabilities Close to the Compact Object {#instabilities-close-to-the-compact-object .unnumbered} ========================================= Kilohertz Oscillations for Neutron Stars - near the ISCO {#kilohertz-oscillations-for-neutron-stars---near-the-isco .unnumbered} -------------------------------------------------------- More than 22 LMXB have now exhibited a signal at kilohertz frequencies in the power spectra of the x-ray flux (See [@vdK00]). Figure 3 (thanks to T. Strohmayer) shows results for samples of data from an atoll and a Z source. Usually this signal is two peaks at 1–15 % power. They indicate quasi-periodic oscillations with coherence (mean frequency/frequency width) as much as 100. The centroid frequencies are not constant for a source, but vary. Over a few hours the frequency is correlated with the X-ray flux, increasing with the flux. The flux variations of a factor of two are correlated with changes of frequency between 500 Hz and 1000 Hz, approximately [@SSZ98]. The highest reported is 1330 Hz, from 4U 0614+09. Considering that for a circular orbit at the Kepler radius $r_K$, the observed frequency is $(2183/M_1) (r_{ISCO}/r_K)^{3/2}$, where $r_{ISCO} = 6GM/c^2$ is the innermost stable circular orbit for a spherical mass $M = M_1 \msun$ of smaller radius, neutron stars of masses $M_1$ = 1.6–2.0 would have Kepler frequencies at the ISCO of just such maximum frequencies as are observed. While the luminosities of the sources exhibiting these QPO range from $10^{36}$ ergs s$^{-1}$ to above $10^{38}$ ergs s$^{-1}$, the maximum values of the upper frequency range only between 820 Hz and 1330 Hz. This suggests [@Zhang98; @Kaaret99] that it represents a characteristic of the neutron stars fairly independently of the accretion rate. The ISCO and the neutron star radius are candidates. For lower fluxes, the frequencies, at least locally in the light curve, decline, as if the Keplex orbit were further out. Which is more likely, that the inner radius is then at the ISCO or at the radius of the neutron star? In the latter case the neutron star is outside the innermost stable circular orbit. Understanding the boundary requires consideration of the radiation pressure, the magnetic fields, and the optical depth of the inner disk. For sources with flux near the Eddington limit, the optical depth of the material near the surface should be much larger than the optical depth of the material accreting at rates 100 times less. For the inner disk being at the ISCO, and fairly compact neutron stars, this plausibly does not matter. For the inner disk at the surface or a large neutron star, it seems hard to explain the similarity of appearance between luminous Z sources and fainter atoll sources. There are in fact differences in the appearance of the QPOs; one is that the amplitude of the QPOs is larger for the atoll sources than for the Z sources. So the situation is not completely clear. If a disk is truncated at an inner radius which moves in toward the neutron star as the mass flow through the disk increases and a QPO is generated at near this inner edge, the frequency would be likely to increase with the luminosity. The frequency would not be able to increase beyond the value corresponding to the minimum orbit in which the disk could persist. Miller, Psaltis, and Lamb [@MLP98] argued that if radiation drag was responsible for the termination of the disk, optical depth effects would lead to the sonic point radius moving in as the accretion rate increases. There would be a highest frequency corresponding to the minimum possible sonic point radius. In the cycles of 4U 1820-30 the frequency approached a maximum which it maintained as the flux increased further before the feature became too broad to detect. This kind of behavior would arise from a sonic point explanation. From Figure 4, it can be seen that if the equation of state (EOS) of the nuclear matter at the center of a neutron star is very stiff, near the L equation of state, for $1.4-2 \msun$ neutron stars the radius of the star is close to its own ISCO; whether it is inside or outside it is depends sensitively on the mass. If the equation of state is softer, closer to the FPS EOS, interpretation of the maximum frequencies observed as a Kepler frequency [*at the surface*]{} would imply a mass significantly less than the $1.4 \msun$ with which many neutron stars are probably formed. In either case, moderately stiff EOS and maximum frequency at the ISCO, or stiff equation of state and maximum frequency either at the ISCO or the surface, the frequency would be from near the ISCO, if not just outside it. Accurate considerations require the rotation rate of the neutron star to be taken into account. A characteristic of the twin kilohertz peaks is that when the frequency changes, the two frequencies approximately move together, with the difference approximately constant, at least until near the maximum frequencies (and luminosities) for which they are observed in a given source. This suggests a beat frequency and the relation between the difference frequency and the frequencies seen during bursts (See Strohmayer, this volume) suggest the neutron star spin as the origin of the beats. Miller, Lamb and Psaltis [@MLP98] explored how the two frequencies could be generated and Lamb and Miller refined the model in agreement with the 5 % changes in the frequency separation, that are observed [@LMiller00]. However, this varying separation between the two QPO also suggested identification as the radial epicyclic frequency of a particle moving in an eccentric orbit in the field of the neutron star. The lower of the two frequencies is then identified, not with a beat frequency, but with the precession of the periastron [@Stella99], although efforts to fit the predictions of this model in terms of particle dynamics produce implausibly large eccentricities, neutron star masses and spins [@MarkovicL00]. Psaltis and Norman proposed that similar frequencies could be resonant in a hydrodynamic disk [@PN99]. In these models, at least in their current forms, the difference between the two QPO peaks is not related to the spin, but to something like the radial epicycle frequency. A quite different class of models are those in which the disk has a boundary layer with the neutron star and the plasma is excited by the magnetic field of the neutron star [@TOK99]. The magnetic pole makes a small angle with the neutron star rotation axis. In this case the lower kilohertz QPO frequency is the Kepler frequency, while both the upper frequency and the low frequency oscillation (corresponding to the Horizontal Branch Oscillations in Z sourses) are related to oscillations of plasma interacting with the rotating magnetic field. Hectohertz oscillations for Black Holes {#hectohertz-oscillations-for-black-holes .unnumbered} --------------------------------------- Although accreting neutron stars and black holes should have important differences, they both presumably have an accretion disk with an inner radius, when the mass flow is high enough. Possible signals from the ISCO of black holes were discussed when accretion onto black holes was first considered [@Suny73] and anticipation of [*RXTE*]{} inspired detailed calculations [@NW93]. The [*RXTE*]{} PCA has detected QPO in 5 black hole candidates at frequencies that are suitable to be signals from the ISCO of black holes in the range of $5 - 30 \msun$. They have been observed only in selected observations and are generally of lower amplitude (a few %) than the neutron star kilohertz QPO. For GRS 1915+105, the frequency has always been 67 Hz [@Morgan97] . For GRO 1655–40, Remillard identified 300 Hz [@1655R99]. For XTE J1550–564, at different times it has been between 185 and 205 Hz [@1550R99]. For XTE J1859–262, a broad signal at 200 Hz is observed in the bright phases near the peak of the outburst [@Cui00]. For 4U 1630–47 as well, which has had 3 outbursts during the [*RXTE*]{} era, Remillard has reported 185 Hz. The black hole candidates have appeared to differ from the neutron stars in having one QPO rather than two. An obvious question is whether the second QPO is associated with the presence of a neutron star with a surface and a rotating magnetic dipole. Recent work by Strohmayer [@Stroh01] casts doubt on it. There were other black hole candidates observed with [*RXTE*]{}, which did not exhibit high frequency oscillations and the properties of the high frequency signal are not very well defined. Interpretation in terms of Kepler frequency at the ISCO, non-radial g-mode oscillations in the relativistic region of the accretion disk, and Lense-Thirring precession have been discussed. GRO J1655-40 is very interesting because the radial velocities of absorption lines of the secondary have given rather precise measurement of the mass. (The best estimates are so far $5.5-7.9 \msun$ [@Shahbaz99].) In this case the mass well known and the black hole’s angular momentum can be the goal. The 300 Hz frequency is high enough that for a g-mode the black hole would have near maximal angular momentum, but if it represents a Kepler velocity, a Schwarzschild black hole would still be possible[@Wagoner98]. The question has been asked whether the microquasars GRS 1915+105 and GRO J1655-40 have powerful radio jets associated with outbursts because they have fast rotation [@Mirabel99]. Decahertz Oscillations for Neutron Stars and Black Holes {#decahertz-oscillations-for-neutron-stars-and-black-holes .unnumbered} -------------------------------------------------------- In the Z source LMXBs the first QPOs discovered were the Horizontal Branch Oscillations (HBO), first seen by EXOSAT, but then by Ginga. They occur in the range 15–50 Hz, have amplitudes as high as 30 %, increase in frequency with the luminosity, and have strong harmonic structure. With [*RXTE*]{} observations the atoll LMXB have also been seen to have these signals, although often the coherence is less and there are other signals (See [@Wijnands00]). These QPO tend to be near in frequency to the break frequency of band-limited white noise at low frequencies. The black hole transients had already exhibited very similar features in Nova Muscae and GX 339–4 in the range 1-15 Hz. They have very similar properties to the HBO. [*RXTE*]{} PCA observations have found these QPO in the power spectra of most black hole candidates [@Sw01]. Different origins have been discussed for the neutron star and black hole QPOs, but their similarity is noted. Figure 5 shows examples from a Z source and a black hole candidate (See [@Focke96; @Dieters00]). The HBO were originally ascribed to a magnetic beat frequency model, assuming the Kepler frequency and the spin were both not seen. Stella and Vietri identified them with the Lense-Thirring precession (See [@Stella99]). They appear to have the correct quadratic relation to the high frequency kilohertz QPO. But the magnitude was too high, by even a factor of about four. Assigning them to twice the nodal frequency, a reasonable possibility for the x-ray modulation, relieves the problem in some cases, but still leaves a factor of two in many. Psaltis argues that a magnitude discrepancy of a factor of two can be accommodated in situations where there is actually complex hydrodynamic flow rather than single particle orbits [@Psaltis01]. In the case of the black holes, the energy spectra seem to distinguish contributions of an optically thick disk and non-thermal, that is “power-law” emission, attributed to scattering of low energy photons off more energetic electrons. This division of components is not observationally so clear in the neutron star LMXB (There are many plausible reasons for this: lower central mass and smaller inner disk, X-rays generated on infall to the surface, possible spinning magnetic dipole.) For the black hole transients, this low frequency QPO is clearly a modulation of the power-law photons. However, there appear to be a variety of correlations with the disk behavior, so that the two components are clearly coupled. In the case of the neutron stars Psaltis, Belloni, and van der Klis [@PBvdK99] have noted that the HBO and the lower kilohertz oscillation are correlated over a broad range of frequency (1–1000 Hz). Wijnands and van der Klis [@WvdK99] showed that [*both*]{} the noise break and the low frequency QPO are correlated in the same way for certain neutron stars and black hole candidates. Psaltis [*et al.*]{} went on to point out that if some broad peaks in the power spectra of some black holes were taken to correspond to the lower kilohertz frequency in the neutron star sources, these points also fell approximately on the same relation. While the degree to which this relation was meaningful, given the scatter in the points, selection effects, and distinctions of more than one branch of behavior, recent work is suggestive that in some way three characteristic frequencies of the disk in a strong gravitational field are significant, where these correspond to Kepler motion, precession of the perihelion and nodal precession. There remain difficulties however with specific assignments. It has often been noted that different interpretations implied weaker features in the spectrum, for example modulation of frequencies by the Lense-Thirring precession [@MarkovicL00] or excitation of higher modes in the case of g-modes [@Wagoner98]. In the case of the neutron stars, adding together large amounts of data to build up the statistical signal, while Sco X-1 did not show sidebands [@Mendez00], Jonker et al. [@Jonker00] found evidence of sidebands at about 60 Hz to the lower kilohertz frequency in three sources. The frequency separation is not the same as the low frequency QPO in those sources although it is in the same range and Psaltis argues is close enough that second order effects can be responsible for the difference. It is not clear yet whether the sidebands imply a modulation of the amplitude or whether they represent a beat phenomenon and are one-sided. Conclusions {#conclusions .unnumbered} =========== While it has not yet been possible to fit all the properties of LMXB neatly into a model, it is hard to imagine alternatives for some important results. One of these is that in accordance with the theory of General Relativity, there is an innermost stable orbit, such that quasistatic disk flow does not persist inside it. Nuclear matter at high densities does not meet such a stiff equation of state that the neutron star extends beyond the ISCO. Instead the results suggest the neutron star lies inside the ISCO for its mass. The accretion flows for both neutron stars and black holes have resonances which, from the observations, are apparently successfully coupled to X-ray flux. QPO are observed with high coherence. They can already be compared to assignments of various frequencies, but they do not match exactly with the identifications that have been made. However before it is possible to use it as diagnostic of gravity, it is necessary to sort out further the physics of the situations. Extending the measurements to signals an order of magnitude fainter taxes even the abilities of [*RXTE*]{}. Continued observations are pushing the limits lower by reducing statistical errors, but must deal with intrinsic source variability on longer time-scales. Observations are also being sought of especially diagnostic combinations of flux and other properties. van Paradijs, J., and McClintock, J. E., “Optical and Ultraviolet Observations of X-Ray Binaries”, in [*X-Ray Binaries*]{}, edited by W. H. G. Lewin, J. van Paradijs, and E. P. J. van den Heuvel, Cambridge Univ. Press, New York, 1995, pp. 58-125. van Paradijs, J., “A Catalogue of X-Ray Binaries”, in [*X-Ray Binaries*]{}, edited by W. H. G. Lewin, J. van Paradijs, and E. P. J. van den Heuvel, Cambridge Univ. Press, New York, 1995, pp. 536-577. Christian, D. J., and Swank, J. H., [*ApJS*]{}, [**109**]{}, 177-224 (1997). Chen, W., Shrader, C. R., and Livio, M., [*ApJ*]{}, [**491**]{}, 312-338 (1997). Tanaka, Y., and Shibazaki, N., [*Annu. Rev. A&A*]{}, [**34**]{}, 607-644 (1996). Bradt, H., Levine, A. M., Remillard, R. E., Smith, D. A., “Transient X–Ray Sources Observed with the Rxte All-Sky Monitor after 3.5 Years”, in [*Multifrequency Behaviour of High Energy Cosmic Sources: III*]{}, edited by F. Giovannelli and Lola Sabau-Graziati, [*Mem SAIt*]{}, [**71**]{}, (2000), in press. Hasinger, G. and van der Klis, M., [*A&A*]{}, [**225**]{}, 79-96 (1989). van der Klis, M., Hasinger, G., Damen, E., Penninx, W., van Paradijs, J., and Lewin, W. H. G., [*ApJ*]{}, [**360**]{}, L19-L22 (1990). van der Klis, M., [*Annu. Rev. A&A*]{}, [**38**]{}, 717-760 (2000). Strohmayer, T. E., Swank, J. H., and Zhang, W., “The periods discovered by [*RXTE*]{} in thermonuclear flash bursts”, in [*The Active X-Ray Sky*]{}, edited by L. Scarsi, H. Bradt, P. Giommi, and F. Fiore, Elsevier, New York, 1998, pp. 129-134. Zhang, W., Smale, A. P., Strohmayer, T. E., and Swank, J. H., [*ApJ*]{}, [**500**]{}, L171-L174 (1998). Kaaret, P., Piraino, Bloser, P. F., Ford, E. C., Grindlay, J. E., Santangelo, A., Smale, A. P., and Zhang, W., [*ApJ*]{}, [**520**]{}, L37-L40 (1999). Miller, M. C., Lamb, F. K., and Psaltis, D., [*ApJ*]{}, [**508**]{}, 791-830 (1998). Lamb, F. K., Miller, M. C., [*ApJ*]{}, submitted (2000) (astro-ph/0007460). Stella, L., Vietri, M., and Morsink, S. M., [*ApJ*]{}, [**524**]{}, L63–L66 (1999). Markovic, D., and Lamb, F. K., [*MNRAS*]{}, submitted (2000) (astro-ph/0009169). Psaltis, D., and Norman, C., [*ApJ*]{}, submitted (1999) (astro-ph/0001391). Titarchuk, L., Osherovich, V., and Kuznetsov, S., [*ApJ*]{}, [**525**]{}, L129-L132 (1999). Sunyaev, R., [*Sov. Astronom. AJ*]{} [**16**]{}, 941– 946 (1973). Nowak, M. A., and Wagoner, R. V., [*ApJ*]{}, [**418**]{}, 187-201 (1993). Morgan, E. H., Remillard,T.E., Greiner, J., [*ApJ*]{}, [bf, 482]{}, 993-1009 (1990). Remillard, R. E., Morgan, E. H., McClintock, J. E., Bailyn, C. D., and Orosz, J. A., [*ApJ*]{}, [**522**]{}, 397-412 (1999). Remillard, R. E., MClintock, J. E., Sobczak, G. J., Bailyn, C. D., Orosz, J. A., Morgan, E. H., and Levine, A. M., [*ApJ*]{}, [**517**]{}, L127-L130 (1999). Cui, W. E., Shrader, C. R., GHaswell, C. A., and Hynes, R. I., [*ApJ*]{}, [**535**]{}, L123-L127 (2000). Strohmayer, T. E., [*ApJ*]{}, submitted (2001). Shahbaz, T., van der Hooft, F., Casares, J., Charles, P. A., and van Paradijs, J., [*MNRAS*]{}, [**306**]{}, 89-94 (1999). Wagoner, R., [*Phys. Rep.*]{}, [**311**]{}, 259-269 (1998) (astro-ph/9805028). Mirabel, I. F., and Rodriguez, L. F., [*Ann. Rev. A&A*]{}, [**37**]{}, 409-443 (1999). Wijnands, R., [*Adv. Space Res.*]{}, submitted (2000) (astro-ph/0002074). Swank, J. H., “Disk Corona Oscillations”, in [*The Third Microquasar Workshop*]{}, editors A. Castrado and J. Greiner, in press (2000) (astro-ph/0011494). Focke, W., [*ApJ*]{}, [**470**]{}, [L127-L130]{} (1996). Dieters, S. [*et al.*]{}, [*ApJ*]{}, [**538**]{}, 307-314 (2000). Psaltis, D., [*ApJ*]{}, submitted (2000) (astro-ph/0101118). Psaltis, D., Belloni, T., and van der Klis, M., [*ApJ*]{}, [**520**]{}, 262-270 (1999). Wijnands, R., and van der Klis, M., [*ApJ*]{}, [**514**]{}, 939-944 (1999). Mendez, M. , and van der Klis, M., [*MNRAS*]{}, [**318**]{}, 938-942 (2000). Jonker, P. G., Mendez, M., and van der Klis, M., [*ApJ*]{}, [**540**]{}, L29-L32 (2000).

ArXiv

--- abstract: 'We study the phase behavior of bowl-shaped particles using computer simulations. These particles were found experimentally to form a meta-stable worm-like fluid phase in which the bowl-shaped particles have a strong tendency to stack on top of each other \[M.Marechal *et al*, Nano Letters **10**, 1907 (2010)\]. In this work, we show that the transition from the low-density fluid to the worm-like phase has an interesting effect on the equation of state. The simulation results also show that the worm-like fluid phase transforms spontaneously into a columnar phase for bowls that are sufficiently deep. Furthermore, we describe the phase behavior as obtained from free energy calculations employing Monte Carlo simulations. The columnar phase is stable for bowl shapes ranging from infinitely thin bowls to surprisingly shallow bowls. Aside from a large region of stability for the columnar phase, the phase diagram features four novel crystal phases and a region where the stable fluid contains worm-like stacks.' author: - Matthieu Marechal - Marjolein Dijkstra bibliography: - 'bowls.bib' title: 'Phase behavior and structure of colloidal bowl-shaped particles: simulations' --- Introduction ============ The concept of a mesogenic particle in the form of a bowl is relatively old in the molecular liquid crystal community. Such molecules are expected to form a columnar phase, which can be ferroelectric, i.e., a phase with a net electric dipole moment, when the particles possess a permanent dipole moment. Ferroelectric phases have potential applications for optical and electronic devices. In fact, crystalline (as opposed to liquid crystalline) ferroelectrics are already applied in sensors, electromechanical devices and non-volatile memory [@FerroApp]. A columnar ferroelectric phase may have the advantage over a crystal, that grain boundaries and other defects anneal out faster due to the partially fluid nature of the columnar phase. In reality, columnar phases of conventional disc-like particles often exhibit many defects, as flat thin discs can diffuse out of a column and columns can split up. The presence of these defects limits their potential use for industrial applications [@simulation-bowls]. Less defects are expected in a columnar phase of bowl-shaped mesogens, where particles are supposed to be more confined in the lateral directions. A whole variety of bowl-like molecules have already been synthesized and investigated experimentally [@Sawamura2002; @simpson2004; @xu1993rbl; @malthete1987icc]. In addition, buckybowlic molecules, *i.e.* fragments of $C_{60}$ whose dangling bonds have been saturated with hydrogen atoms, have been shown to crystallize in a columnar fashion [@Rabideau1996; @Forkey1997; @Matsuo2004; @Sakurai2005; @Kawase2006]. However, the number of theoretical studies is very limited as it is difficult to model the complicated particle shape in theory and simulations. In a recent simulation study, the attractive-repulsive Gay-Berne potential generalized to bowl-shaped particles has been used to investigate the stacking of bowl-like mesogens as a function of temperature [@simulation-bowls]. The authors reported a nematic phase and a columnar phase. This columnar phase did not exhibit overall ferroelectric order, although polar regions were found. In another very recent simulation study [@Cinacchi2010] of hard contact lenses (infinitely thin, shallow bowls), a new type of fluid phase was found in which the particles cluster on a spherical surface for bowls which are not too shallow. No columnar phase was found since the focus was on rather shallow bowls at a relatively low densities. Recently, a procedure has been developed to synthesize bowl-shaped colloidal particles [@Carmen]. This method starts with the preparation of highly uniform oil-in-water emulsion droplets. Subsequently, the droplets were used as templates around which a solid shell with tunable thickness is grown. In the next step of the synthesis, the oil in the droplets is dissolved and finally, during drying, the shells collapse into hemispherical double-walled bowls. In addition to these larger, more easily imaged colloids, a whole variety of bowl-shaped nanoparticles and smaller colloids have been synthesized and characterized [@Charnay2003; @Wang2004; @Liu2005; @Jagadeesan2008; @Love2002; @Hosein2007], and possible applications of these systems have been put forward. We also note that recently hemispherical particles were synthesized at an air-solution interface [@higuchi] and on a substrate [@Xia]. These hemispherical particles are intended to be used as microlense arrays, but they can also serve as a new type of shape-anisotropic colloidal particle. In our simulations, we model the particles as the solid of revolution of a crescent (see Fig. \[fig:particles\]a). The diameter $\sigma$ of the particle and the thickness $D$ are defined as indicated in Fig. \[fig:particles\]a. We define the shape parameter of the bowls by a reduced thickness $D/\sigma$, such that the model reduces to infinitely thin hemispherical surfaces for $D/\sigma=0$ and to solid hemispheres for $D/\sigma=0.5$. The advantages of this simple model is that it interpolates continuously between an infinitely thin bowl and a hemispherical solid particle (the two colloidal model systems, which we discussed above), and that we can derive an algorithm that tests for overlaps between pairs of bowls, which is a prerequisite for Monte Carlo simulations of hard-core systems. In a recent combined experimental and simulation study (for which we performed the simulations), the phase behavior of repulsive bowl-shaped colloids was investigated [@Marechal2010bowls]. The colloids were shown to form a worm-like fluid phase, in which the particles form long curved stacks running in random directions. By comparing the distribution of stack lengths, the simulation model was shown to describe the colloidal particles well. No evidence of columnar ordering was found in the experiments and in simulations of bowls with corresponding deepness, which was explained by the glassy behavior of the particles preventing rearrangements. The phase behavior of the model particles is expected to also describe other repulsive bowl shaped particles well, provided that the dimensions of the simulation particle are chosen such that the diameter of a stack and the inter-particle distance in the stack are the same as for the particles to be modeled. In this work, we expand the simulation results on the hard bowl-shaped particles. First, we elaborate on the model for the collapsed shells; the overlap algorithm is described in the appendix. Also, the (free energy) methods are explained in more detail than in Ref. [@Marechal2010bowls]. In the results section, we study the properties of the isotropic phase. We investigate the nature and the location of the transition between the homogeneous fluid phase and the fluid phase that contains the worm-like stacks. Furthermore, we show the packing diagram and the phase diagram with a tentative homogeneous–to–worm-like fluid transition line. In the last section we summarize and discuss the results. ![ (a) The theoretical model of the colloidal bowl is the solid of revolution of a crescent around the axis indicated by the dashed line. The thickness of the double-walled bowl is denoted by $D$ and the diameter of the bowl by $\sigma$. (b) The bowls are defined using two spheres of radii $R_1$ and $R_2$, that are a distance of $L$ apart. The direction vector, $\mathbf{u}_i$ and the reference point of the particle, $\mathbf{r}_i$, (the dot in the center of the smaller sphere) are indicated. \[fig:particles\]](particles){width="45.00000%"} Methods ======= Model ----- We describe the model that we use to represent the bowls in more detail. Consider a sphere with a radius $R_1$ at the origin and a second sphere with radius $R_2>R_1$ at position $-L\mathbf{u}_i$, where $\mathbf{u}_i$ is the unit vector denoting the orientation of the bowl and $L>0$. The bowl is represented by that part of the sphere with radius $R_1$ that has no overlap with the larger sphere, see Fig. \[fig:particles\]b. We have chosen the values for $L$ and $R_2$ such that the bowls are hemispherical (see appendix for explicit expressions for $L$ and $R_2$). We define the thickness of the bowls by $D=L-(R_2-R_1)$, such that the model reduces to the surface of a hemisphere for $D=0$ and to a solid hemisphere for $D=R_1$. The volume of the particle is $\frac{\pi}{4}\, D\, (\sigma^2 - D\sigma + \frac{2}{3} D^2)$, where $\sigma\equiv 2R_1$ is our unit of length. The algorithm to determine overlap between our bowls is described in the appendix. Fluid phase ----------- We employ standard $NPT$ MC simulations to obtain the equation of state (EOS) for the fluid phase. In addition, we obtain the compressibility by measuring the fluctuations in the volume: $$\frac{\langle V^2\rangle - \langle V\rangle^2}{\langle V\rangle}=\frac{k_B T}{\rho} \, \left(\frac{\partial\rho}{\partial P}\right)_T,$$ where $\rho=N/V$ is the number density and the derivative of the pressure is taken at constant temperature is denoted by the subscript $T$. We determine the free energy at density $\rho_1$ by integrating the EOS from reference density $\rho_0$ to $\rho_1$: $$\frac{F(\rho_1)}{N}=\mu(\rho_0)-\frac{P(\rho_0)}{\rho_0} + \int_{\rho_0}^{\rho_1} \frac{P(\rho)}{\rho^2} d \rho\label{eqn:widom}$$ where the chemical potential $\mu(\rho_0)$ is determined using the Widom particle insertion method [@widom], and $P(\rho_0)$ is determined by a local fit to the EOS. To investigate the structure of the fluid phase, we measure the positional correlation function [@Veerman_Frenkel], $$g_c(z)=\frac{1}{N \rho A_\text{col}} \langle \sum_{i=1}^N \sum_{j=1}^{N_\text{col}(i)} \delta(\mathbf{r}_{ij} \cdot \mathbf{u}_i-z) \rangle\label{eqn:gcz},$$ where the sum over $j$ runs over $N_\text{col}(i)$ particles in a column of radius $\sigma/2$ with orientation $u_i$ centered around particle $i$, and where the area of the column is denoted by $A_\text{col}=\pi \sigma^2/4$. At sufficiently high pressure the particles stack on top of each other to form disordered worm-like piles which resemble the stacks observed in the experiments [@Marechal2010bowls]. As the stacks have a strong tendency to buckle, we cannot use $g_c(z)$ to determine the length of the stacks. We therefore determine the stack size distribution using a cluster criterion. Particle $i$ and $j$ belong to the same cluster if $$\begin{aligned} |\mathbf{r}_{ij} + (\zeta D/2 + \sigma/4) (\mathbf{u}_j-\mathbf{u}_i)| & < & \sigma/2 \quad\text{and} \nonumber\\ \mathbf{u}_i \cdot \mathbf{u}_j & > & 0, \label{eqn:cluster_crit}\end{aligned}$$ and where the first condition has to be satisfied for $\zeta=-1$, $0$ or $1$ and $\mathbf{r}_{ij}=\mathbf{r}_j-\mathbf{r}_i$, with $\mathbf{r}_i$ denoting the center of the sphere with radius $R_1$ of particle $i$, see Fig. \[fig:particles\]b. If both conditions are satisfied, particle $j$ is just above ($\zeta=1$) or below ($\zeta=-1$) particle $i$ in the stack, or, when the stack is curved, particle $j$ can be next to particle $i$ ($\zeta=0$). We now define the cluster distribution as the fraction of particles that belongs to a cluster of size $n$: $\mathcal{P}_\text{stack}(n)\equiv n N_n/N$, where $N_n$ is the number of clusters of size $n$. We checked that the cluster size distribution does not depend sensitively to the choice of parameters in Eq. (\[eqn:cluster\_crit\]). Columnar phases --------------- We also perform $NPT$ Monte Carlo simulations of the columnar phase using a rectangular simulation box with varying box lengths in order to relax the inter-particle distance in the $z$ direction, along the columns, independently from the lattice constant in the horizontal direction. The difference between the free energy of the columnar phase at a certain density and the free energy of the fluid phase at a lower density is determined using a thermodynamic integration technique [@Bates_Frenkel]. We apply a potential which couples a particle to its column: $$\Phi_\mathrm{hex}(\mathbf{r}^N,\lambda)=\lambda\sum_{i=1}^N\cos(2 \pi N_x x_i/L_x)\sin(\pi N_y y_i/L_y), \label{eqn:col_pot}$$ where $x_i$ and $y_i$ are the $x$ and $y$ components respectively of $\mathbf{r}_i$, $N_\alpha$ is the number of columns in the $\alpha$ direction and $L_\alpha$ is the size of the box in the $\alpha$ direction. In our simulations, we calculate Eq. (\[eqn:col\_pot\]) while fixing the center of mass. To do so efficiently, we first calculate all four combinations $$\lambda\sum_{i=1}^N \mathrm{trig1}(2 \pi N_x x_i/L_x) \mathrm{trig2}(\pi N_y y_i/L_y)\label{eqn:col_gen_trigs}$$ for $\mathrm{trig1}=\cos,\sin$ and $\mathrm{trig2}=\cos,\sin$. The change in these four expressions upon displacement of a single particle while keeping the center of mass fixed can be expressed in terms of single particle properties and the previous values of the expressions by using some basic trigonometry. In this way, $\Phi_\mathrm{hex}(\mathbf{r}^N,\lambda)$, which is Eq. \[eqn:col\_gen\_trigs\] for $\mathrm{trig1}=\cos$ and $\mathrm{trig2}=\sin$, can be calculated without performing the full summation over all particles in Eq. (\[eqn:col\_gen\_trigs\]) every time we displace a particle. Unfortunately, this calculation requires the evaluation of many more trigonometric functions than the simple expression (\[eqn:col\_pot\]), but the extra computation time is negligible compared to the overlap check. In addition to this positional potential, we also constrain the direction of the particle, using the potential $$\Phi_\text{ang}(\mathbf{u}^N,\lambda)=\lambda' \sum_{i=1}^N u_{i,z},\label{eqn:col_pot_ang}$$ where we used $\lambda'=0.1\lambda$ and where $u_{i,z}$ is the $z$ component of $\mathbf{u}_i$. The thermodynamic integration path from the columnar phase to the fluid is as follows: We start from the columnar phase at a certain density $\rho_2$. Subsequently, we slowly turn on the two potentials, *i.e.* we increase $\lambda$ from 0 to $\lambda_\mathrm{max}$. Next, we integrate the equation of state to go from $\rho_2$ to $\rho_1$, while keeping $\lambda=\lambda_\text{max}$ fixed. During this step the columnar phase will only be stable below the coexistence density, if $\lambda_\text{max}$ is sufficiently high. We find that $\lambda_\text{max}=20k_BT$ suffices to guarantee stability of the columnar phase. Finally, fixing the density $\rho_1$, we gradually turn off the potentials, while integrating over $\lambda$ from $\lambda_\text{max}$ to 0. During this last step, the columnar phase melts continuously, provided that the density $\rho_1$ is low enough and that $\lambda$ is high enough to prevent melting during the density integration step. The resulting free energy difference between the columnar phase and fluid phase is given by $$\begin{gathered} F_\text{col}(\rho_2)-F_\text{fluid}(\rho_1)=\\\int_0^{\lambda_\text{max}} \big\langle \Phi_\text{hex}(\mathbf{r}^N,\lambda)/\lambda+\Phi_\text{ang}(\mathbf{u}^N,\lambda)/\lambda\big\rangle\big|_{\rho=\rho_1}+\\ \int_{\rho_1}^{\rho_2} d\rho\left.\frac{NP(\rho)}{\rho^2}\right|_{\lambda=\lambda_\text{max}} \\ -\int_0^{\lambda_\text{max}} \big\langle \Phi_\text{hex}(\mathbf{r}^N,\lambda)/\lambda+\Phi_\text{ang}(\mathbf{u}^N,\lambda)/\lambda\big\rangle\big|_{\rho=\rho_2}\end{gathered}$$ The positional potential (\[eqn:col\_pot\]) is designed to stabilize a hexagonal array of columns, but, strictly speaking, it does not have the hexagonal symmetry of the columnar phase, since it is not invariant under a 60 degrees rotation of the whole system around a lattice position. However, we find that replacing Eq. (\[eqn:col\_pot\]) by a positional potential that does have this symmetry, does not have a significant effect on the free energy difference. A second type of columnar phase can be constructed by flipping half of the bowls. In this way we obtain alternating vertical sheets (*i.e.* rows of columns) of bowls that point upwards and sheets of bowls that point downwards, we will refer to this phase as the inverted columnar phase. We calculate the free energy of this phase using the method described above, with the modification that the angular potential now reads, $$\Phi_\text{ang}(\mathbf{u}^N,\lambda)=\lambda' \sum_i u_{i,z}^2. \label{eqn:col_pot_ang2}$$ This potential could also have been used for the non-inverted columnar phase, and we have found that the result of the free energy integration for the columnar phase is the same whether we use Eq. (\[eqn:col\_pot\_ang2\]) or Eq. (\[eqn:col\_pot\_ang\]). Crystals {#sec:cryst} -------- ### Packing As the crystal phases of the bowls are not known a priori, we developed a novel pressure annealing method to obtain the possible crystal phases [@PhysRevLettSSS], which we named after the thermal annealing technique commonly used to find energy minima. Fully variable box shape $NPT$ simulations were performed on system of only 2-6 particles. By construction, the final configuration of such a simulation is a crystal, where the unit cell is the simulation box. One cycle of such a simulation consists of the following steps: We start at a pressure of $10 k_B T/\sigma^3$. Subsequently, we run a series of simulations, where the pressure increases by a factor of ten each run: $P\sigma^3/k_B T=10,100,\ldots, 10^6$. At the highest pressure ($10^6 k_B T/\sigma^3$) we measure the density and angular order parameters, $S_1\equiv\|\langle \mathbf{u}_i\rangle\|$ and $S_2\equiv \lambda_2$, where $\lambda_2$ is the highest eigenvalue of the matrix whose components are $Q_{\alpha_\beta}=\frac{3}{2} \langle u_{i\alpha} u_{i\beta} \rangle - \frac{1}{2}\delta_{\alpha\beta}$, where $\alpha,\beta=x,y,z$. We store the density if it is the highest density found so far for these values of $S_1$ and $S_2$. We ran 1000 of such cycles for each aspect ratio, which is enough to visit each crystal phase multiple times. After completing the simulations, we tried to determine the lattice parameters of the resulting crystal by hand. Although this last step is not necessary, it is convenient to have analytical expressions for the lattice vectors and the density. The pressure annealing runs were performed for $D/\sigma=0.1,0.15,\ldots,0.5$. For many of the crystals, we were not able to find analytical expressions for the lattice parameters. For these crystals, we obtain the densities of the close packed crystals for intermittent values of $D$ by averaging the density in single simulation runs at a pressure of $10^6 k_BT/\sigma^3$. The initial configurations for the value of $D$ of interest were obtained from the final configurations of the pressure annealing simulations for another value of $D$ by one of the following two methods, depending on whether we needed to decrease or increase $D$: When decreasing $L$ no overlaps are created so the final configuration of the simulation for the previous value of $L$ can be used as initial configuration. On the other hand, increasing $L$ results in an overlap, which is removed by scaling the system uniformly. Subsequently, the pressure is stepwise increased from 1000$k_BT/\sigma^3$ to $10^6 k_BT/\sigma^3$, by multiplying by 10 each step. ### Free energies We calculate the free energy of the various crystal phases by thermodynamic integration using the Einstein crystal as a reference state [@FrenkelSmit]. The Einstein integration scheme that we employ here is similar to the one that was used to calculate the free energies of crystals of dumbbells in Ref. [@dumbbell_article]. We briefly sketch the integration scheme here and discuss the modifications that we applied. We couple both the positions and the direction of the particles with a coupling strength $\lambda$, such that for $\lambda\rightarrow \infty$, the particles are in a perfect crystalline configuration. First, we integrate $\partial F/\partial\lambda$ over $\lambda$ from zero to a large but finite value for $\lambda$. Subsequently we replace the hard-core particle–particle interaction potential by a soft interaction, where we can tune the softness of the potential by the interaction strength $\gamma$. We integrate over $\partial F/\partial \gamma$ from a system with essentially hard-core interaction (high $\gamma=\gamma_\text{max}$), to an ideal Einstein crystal ($\gamma=0$). Some minor alterations to the scheme of Ref. [@dumbbell_article] were introduced, which were necessary, because of the different shape of the particle. For the coupling of the orientation of bowl $i$, *i.e.*, $\mathbf{u}_i$, to an aligning field, we have to take into account that the bowls have no up down symmetry, while the dumbbells are symmetric under $\mathbf{u}_i\rightarrow -\mathbf{u}_i$. The potential energy function that achieves the usual harmonic coupling of the particles to their lattice positions, as well as the new angular coupling, reads: $$\begin{gathered} \beta U({\bf r}^N,{\bf u}^N;\lambda) =\\ \lambda \sum_{i=1}^{N} ({\bf r}_i-{\bf r}_{0,i})^2/\sigma^2 + \sum_{i=1}^N \lambda (1-\cos(\theta_{i0})) , \label{eqn:einstein}\end{gathered}$$ where ${\bf r}_i$ and ${\bf u}_i$ denote, respectively, the center-of-mass position and orientation of bowl $i$ and ${\bf r}_{0,i}$ the lattice site of particle $i$, $\theta_{i0}$ is the angle between $\mathbf{u}_i$ and the ideal tilt vector of particle $i$, and $\beta=1/k_BT$. The Helmholtz free energy [@dumbbell_article] of the noninteracting Einstein crystal is modified accordingly, but the only modification is the integral over the angular coordinates: $$J(\lambda)=\int_{-1}^1 e^{\lambda (x-1)} dx=\frac{1-e^{-2\lambda}}{\lambda}.$$ Although the shape of the bowls is more complex than that of the dumbbell, we can still use a rather simple form for the pairwise soft potential interaction: $$\beta U_{\mathrm{soft}}({\bf r}^N,{\bf u}^N;\gamma)=\sum_{i<j} \beta \varphi({\bf r}_{i}-{\bf r}_{j},\mathbf{u}_i,\mathbf{u}_j,\gamma)$$ with $$\begin{gathered} \beta\varphi({\bf r}_{j}-{\bf r}_{i},\mathbf{u}_i,\mathbf{u}_j,\gamma) =\\ \left\{\begin{array}{cc} \gamma (1-A (r_{ij}'/\sigma_\text{max})^2) & \text{if $i$ and $j$ overlap}\\ 0 & \mathrm{otherwise}\end{array} \right. , \label{eqn:softpot_part}\end{gathered}$$ where $r_{ij}'\equiv | \mathbf{r}_j-\mathbf{r}_i+\frac{\sigma-D}{2}(\mathbf{u}_i-\mathbf{u}_j)|$ *i.e.* the distance between the “centers” of bowl $i$ and bowl $j$, $\sigma_\text{max}$ is the maximal $r_{ij}'$ for which the particles overlap: $\sigma_\text{max}^2=\sigma^2+(\sigma-D)^2$, $A$ is an adjustable parameter that is kept fixed during the simulation at a value $A=0.5$, and $\gamma$ is the integration parameter. It was shown in Ref. [@Fortini_soft_pot] that in order to minimize the error and maximize the efficiency of the free energy calculation, the potential must decrease as a function of $r$ and must exhibit a discontinuity at $r$ such that both the amount of overlap and the number of overlaps decrease upon increasing $\gamma$. Here, we have chosen this particular form of the potential because it can be evaluated very efficiently in a simulation, although it does not describe the amount of overlap between bowls $i$ and $j$ very accurately. We checked that adding a term that tries to describe the angular behavior of the amount of overlap does not significantly change our results of the free energy calculations. Also, we checked that by employing the usual Einstein integration method (*i.e.* only hard-core interactions) at a relatively low density we obtained the same result as by using the method of Fortini [*et al.*]{}[@Fortini_soft_pot]. Finally, we set the maximum interaction strength $\gamma_\text{max}$ to 200. We perform variable box shape NPT simulations [@Parrinello] to obtain the equation of state for varying $D$. In these simulations not only the edge length changes, but also the angles between the edges are allowed to change. We employ the averaged configurations in the Einstein crystal thermodynamic integration. We calculate the free energy as a function of density by integrating the EOS from a reference density to the density of interest: $$F(\rho_1^*)= F(\rho_0) +\int_{\rho_0}^{\rho_1} d\rho \left\langle \frac{N P(\rho)}{\rho^2}\right\rangle \label{eqn:int_eos}$$ Results ======= ![ The final configuration obtained from simulations at $P\sigma^3/k_B T=50$ and $D=0.3\sigma$ The colors denote different stacks.\[fig:snapshots\_worms\]](worms2.png){width="45.00000%"} Stacks ------ We perform standard Monte Carlo simulations in the isobaric-isothermal ensemble (NPT). Fig. \[fig:snapshots\_worms\] shows a typical configuration of bowl-shaped particles with $D = 0.3~\sigma$ at $P\sigma^3/k_BT=50$, displaying stacking behavior typical for the worm-like phase. ![The equation of state for bowl-shape particles with $D=0.1\sigma$, reduced pressure $P^*=\beta P\sigma^3$ (left axis), and the reduced compressibility $\frac{1}{\rho}\,\frac{\partial \rho}{\partial P^*}$ on a log scale (right axis) as a function of packing fraction $\phi$. The points are data obtained from $NPT$ simulations. The solid line is a fit to the pressure; the dashed line is the corresponding reduced compressibility, $\frac{1}{\rho}\big(\frac{\partial \mathsf{fit}(\rho)}{\partial \rho}\big)^{-1}$. \[fig:eos\]](compr_lab){width="49.00000%"} The equation of state (EOS) of the fluid is somewhat peculiar: the pressure as a function of density is not always convex for all densities, although the compressibility does decrease monotonically with packing fraction $\phi$ for $D=0.1\sigma$, see Fig. \[fig:eos\], where the packing fraction is defined as $\phi=\frac{\pi D}{4}(\sigma^2-D\sigma+\frac{2}{3}D^2)N/V$. This behavior persist for all $D\leq0.2\sigma$, but for $D\geq 0.25\sigma$ the pressure is always convex. We investigate the origin of these peculiarities using $g_c(z)$, the positional correlation function along the director of a particle, which includes only the particles in a column around a particle, as defined in Eq. (\[eqn:gcz\]). ![The pair correlation function, $g_c(z)$, of a fluid of bowl-shaped particles with $D=0.2\sigma$ as a function of the dimensionless inter-particle distance $z/\sigma$ along the axis of a reference bowl for various reduced pressures $P^*\equiv\beta P\sigma^3$. Only particles within a cylinder of diameter $\sigma$ around the bowl are considered, as indicated by the subscript ‘c’. We show typical two-particle configurations that contribute to $g_c(z)$ for $z/\sigma=-0.5,-0.2,0.2,0.4$ and $1$, where the filled bowls denote the reference particle, and the open bowls with thick outlines denote the other particle. \[fig:gcz\]](rdf_c_exp){width="49.00000%"} As can be seen from $g_c(z)$ in Fig. \[fig:gcz\], the structure of the fluid changes dramatically as the pressure is increased. At $P^*\equiv\beta P\sigma^3 =1$, the correlation function is typical for a low density isotropic fluid of hemispherical particles; no effect of the dent of the particles is found at low densities. The only peculiar feature of $g_c(z)$ for $P^*=1$ is that it is not symmetric around zero, but this is caused by our choice of reference point on the particle (see Fig. \[fig:particles\]b), which is located below the particle if the particle points upwards. In contrast, at $P^*=10$ $g_c(z)$ already shows strong structural correlations. Most noteworthy is the peak at $z=D$, that shows that the fluid is forming short stacks of aligned particles. Also, note that the value of $g_c(z)$ is nonzero around $z=0$. This is caused by pairs of bowls that align anti-parallel and form a sphere-like object, as depicted in Fig. \[fig:gcz\]. Finally, at $P^*=50$ and higher, long worm-like stacks are fully formed and $g_c(z)$ shows multiple peaks at $z=D n$ for both positive and negative integer values of $n$. Furthermore, at these pressures, there are no sphere-like pairs, as can be observed from the value of $g_c(0)$. The formation of stacks explains the peculiar behavior of the pressure: At low densities, the bowls rotate freely, which means that the pressure will be dominated by the rotationally averaged excluded volume. The excluded volume of two particles that are not aligned is nonzero, even for $D=0$, and gives rise to the convex pressure which is typical for repulsive particles. As the density increases and the bowls start to form stacks, the available volume increases, and the pressure increases less than expected, which can even cause the EOS to be concave. At even higher densities the worm-like stacks are fully formed, and the pressure is again a convex function of density for $D>0$, dominated by the excluded volume of locally aligned bowls. The excluded volume of completely aligned infinitely thin bowls is zero, and, therefore, the pressure increases almost linearly with density for $D=0$ when the stacks are fully formed. ![The probability, $\mathcal{P}_\text{stack}(n)$, to find a particle in a stack of size $n$ for $D/\sigma=0.2,0.3$ and $0.4$ and $P\sigma^3/k_BT=50$ \[fig:clust\]](clusters){width="49.00000%"} To quantify the length of the stacks we calculated the stack distribution as shown in Fig. \[fig:clust\]. As can be seen from the figure, the length of the stacks is strongly dependent on $D/\sigma$. However, we have found that above a certain threshold pressure the distribution of stacks is nearly independent of pressure. We investigated whether the worm-like stacks could spontaneously reorient to form a columnar phase. We increased the pressure in small steps of 1 $k_BT/\sigma^3$ from well below the fluid–columnar transition to very high pressures, where the system was essentially jammed. At each pressure, we ran the simulation for $4\cdot 10^6$ Monte Carlo cycles, where a cycle consists of $N$ particle and volume moves. These simulations show that the bowls with a thickness $D\geq 0.25 \sigma$ always remained arrested in the worm-like phase, which is similar to the experimental observations [@Marechal2010bowls]. However, for $D/\sigma=0.1$ and $0.2$, we find that the system eventually transforms into a columnar phase in the simulations (see Fig. \[fig:snapshot\_col\]). This might be explained by the fact that the isotropic-to-columnar transition occurs at lower packing fractions for deeper bowls (smaller $D$), which facilitates the rearrangements of the particles into stacks and the alignment of the stacks into the columnar phase. ![ The final configuration of a simulation of bowls with $L=0.1D$ at $P\sigma^3/k_B T=38$. The gray values denote different columns. \[fig:snapshot\_col\]](columnar){width="45.00000%"} Packing ------- ![The various crystal phases that were considered as possible stable structures. Five of these were found using the pressure annealing method: X, IX, B, IB and IX’. X, IX, B and IB are densely packed structures for $D \lesssim 0.5\,\sigma$ and fcc${}^2$ and IX’ are densely packed crystal structures for (nearly) hemispherical bowls ($D \simeq 0.5\,\sigma$). \[fig:crystals\]](crystals6){width="49.00000%"} We found six candidate crystal structures, denoted X,IX,IX’,B,IB and fcc${}^2$, using the pressure annealing method. Snapshots of a few unit cells of these crystal phases are shown in Fig. \[fig:crystals\]. We will describe these crystal structures using the order parameters $S_1$, that measures alignment of the particles, and the nematic order parameter ($S_2$), that is nonzero for both parallel and anti-parallel configurations. Crystal structure X has $S_1\simeq 1$ and $S_2\simeq 1$, and the particles are stacked head to toe in columns. The lattice vectors are $$\begin{array}{c} \displaystyle \mathbf{a_1}=\sigma \hat{x} \qquad \mathbf{a_2}=D \hat{z} \\[1.0em] \displaystyle \mathbf{a_3}=\frac{\sigma}{2}\hat{x} + \frac{1}{2} \sqrt{\sigma^2-D^2+2\sigma \sqrt{\sigma^2-D^2} }\;\hat{y}+\frac{D}{2} \hat{z}, \end{array}$$ and the density is $$\rho\sigma^3=\left[\frac{D \sigma}{2} \sqrt{\sigma^2-D^2+2\sigma \sqrt{\sigma^2-D^2} }\right]^{-1}.$$ The order parameters of the second crystal structure, are $S_1\simeq 0$ and $S_2\simeq 1$, which is caused by the fact that half of the particles point upwards, and the other half downwards. Further investigation shows that there are two phases with $S_2 \simeq 1$ and $S_1 \simeq0$: one at low $D$ (IX) and one at $D\simeq \sigma/2$ (IX’). The structure within the columns of the first (IX) of these two structures is the same as for the X structure, but one half of these columns are upside down, like in the inverted columnar phase (in fact, the IX crystal melts into the inverted columnar phase). The lattice vectors of crystal structure IX are $$\begin{array}{c} \displaystyle \mathbf{a_1}=\sigma \hat{x} \qquad \mathbf{a_2}=D \hat{z} \\[1.0em] \displaystyle \mathbf{a_3}=\frac{\sigma}{2}\hat{x} + \frac{1}{2} \sqrt{3\sigma^2-4 D^2}\;\hat{y}, \end{array}\label{eqn:IXa}$$ and the density is $$\rho\sigma^3=\left[\frac{D \sigma}{2} \sqrt{3\sigma^2-4 D^2}\right]^{-1}.\label{eqn:IXd}$$ The columns in the IX crystal are arranged in such a way that the rims of the bowls can interdigitate. The IX’ crystal can be obtained from the IX phase at $D=\sigma/2$ by shifting every other layer by some distance perpendicular to the columns, such that the particles in these layers fit into the gaps in the layers below or above. In this way a higher density than Eq. (\[eqn:IXd\]) is achieved. The columns of the third crystal phase (B) resemble braids with alternating tilt direction of the particles within each column. Because of this tilt $S_1$ and $S_2$ have values between 0 and 1, that depend on $D$. Furthermore, the inverted braids structure (IB), that has $0<S_2<1$ and $S_1=0$, can be obtained by flipping one half of the columns of the braid-like phase (B) upside down. These braid-like columns piece together in such a way that the particles are interdigitated. In other words, this phase is related to the B phase in exactly the same way as the IX phase is related to the X phase. ![Packing diagram: maximum packing fraction ($\phi$) of various crystal phases as a function of the thickness ($D$) of the bowls. The points are the results of the pressure annealing simulations. The thin dot-dashed lines are obtained from the pressure annealing results by slowly increasing or decreasing $D$ as described in Sec. \[sec:cryst\], except for the IX phase (thin dashed line with open squares) and the X phase (thin solid line with filled squares), for which the packing fraction can be expressed analytically. The thick lines denote the packing fractions of the perfect hexagonal columnar phase (col) and the paired fcc phase (fcc${}^2$). Any points that lie below these lines are expected to be thermodynamically unstable (see text). \[fig:packing\]](packing){width="49.00000%"} Finally, in the paired face-centered-cubic (fcc${}^2$) phase, pairs of hemispheres form sphere-like objects that can rotate freely and that are located at the lattice positions of an fcc crystal. The density at close packing is $2\sqrt{2}/\sigma^3$, *i.e.* twice the density of fcc. In Fig. \[fig:packing\] the results of the pressure annealing method are shown, along with the known packing fraction of the perfect hexagonal columnar phase (col). Since the columnar phase has positional degrees of freedom and the fcc${}^2$ phase has rotational degrees of freedom, we expect these phases to have a higher entropy (lower free energy) than any crystal phase with the same or lower maximum packing fraction whose degrees of freedom have all been frozen out. Therefore, any crystal structure with a packing fraction below the thick lines in Fig. \[fig:packing\] is most likely thermodynamically unstable. At first, we were unable to find the fcc${}^2$ using the pressure annealing method as described in Sec. \[sec:cryst\]. However, if we increase the pressure slowly to 100$k_BT/\sigma^3$ in simulations of 12 particles, we did observe the fcc${}^2$ phase for hemispherical particles ($D=\sigma/2$). In these simulations at finite pressure, it is important to constrain the length of all box vectors such that they remain larger than say $1.5\sigma$. Otherwise the box will become extremely elongated, such that the particles can interact primarily with their own images. When a particle interacts with it is neighbors, the Gibbs free energy $G=F+PV$ decreases, because the volume decreases without any decrease in entropy due to restricted translational motion (if a particle moves, its image moves as well, so a particle translation will never cause overlap of the particle with its image). The decrease in Gibbs free energy is of course an extreme finite size effect, which should be avoided if we wish to predict the equilibrium phase behavior. For the pressure annealing simulations at very high pressures, these effects are not important, because the entropy term in the Gibbs free energy is small compared to $PV$. We did not attempt to find the columnar phase using the modified pressure annealing method, as we were only interested in finding candidate crystal structures. Furthermore, the columnar phase was already found in more standard simulations with a larger number of particles. Free energies ------------- In order to determine the regions of the stability of the fluid, the columnar phase and the six crystal phases, we calculated the free energies of all phases as explained in the Methods section. The results of the reference free energy calculations are shown in Tbls. \[tbl:col\_fs\] and \[tbl:fs\]. We find that the columnar phase with all the particles pointing in the same direction is more stable than the inverted columnar phase, where half of the columns are upside down. However, the free energy difference between the two phases is only $0.013\pm 0.002 k_B T$ per particle at $\phi=0.5193$ and $D=0.3\sigma$. Based on this small free energy difference we do not expect polar ordering to occur spontaneously. Similar conclusions, based on direct simulations, were already drawn in Ref. [@simulation-bowls]. ![Dimensionless free energies $\beta F\, \sigma^3/V$ for hard bowls with $L=0.3\sigma$ and the fluid-columnar, columnar-IX and IX-IB coexistences, which were calculated using common tangent constructions. The columnar phases is denoted “col”. The irrelevant free energy offset is defined in such a way that the free energy of the ideal gas reads $\beta F/V=\rho (\log(\rho\sigma^3)-1)$. The free energies of the various phases are so close, that they are almost indistinguishable. \[fig:f\_of\_rho\]](f_cumb){width="49.00000%"} The densely-packed crystal structures in Fig. \[fig:crystals\] at $D \lesssim 0.3$, the worm-like fluid phase (Fig. \[fig:snapshots\_worms\]) and the columnar phase (Fig. \[fig:snapshot\_col\]) show striking similarity in the local structure: in all these phases the bowls are stacked on top of each other, such that (part of) one bowl fits into the dent of another bowl. As a result, the free energies and pressures of the various phases, are often almost indistinguishable near coexistence. For this reason it was sometimes difficult to determine the coexistence densities for $D<0.3\sigma$. Exemplary free energy curves for the various stable phases consisting of bowls with $D=0.3\sigma$ are shown in Fig. \[fig:f\_of\_rho\]. Phase diagram ------------- In Fig. \[fig:ph-dia\_shells\], we show the phase diagram in the packing fraction $\phi$ - thickness $D/\sigma$ representation. The packing fraction is defined as $\phi=\frac{\pi D}{4}(\sigma^2-D\sigma+\frac{2}{3}D^2)N/V$. For $D/\sigma\leq0.3$, we find an isotropic-to-columnar phase transition at intermediate densities, which resembles the phase diagram of thin hard discs [@Veerman_Frenkel]. However, the fluid-columnar-crystal triple point for discs is at a thickness-to-diameter ratio of about $L/\sigma \sim 0.2-0.3$, while in our case the triple point is at about $D/\sigma \sim 0.3-0.4$. The shape of the bowls stabilizes the columnar phase compared to the fluid and the crystal phase. We find four stable crystal phases IX, IB, IX’ and fcc${}^2$, while we had six candidate crystals. The two phases that were not stable are the X and B crystals, which are very similar to the stable IX and IB crystals respectively, except that X and B have considerable lower close packing densities. Therefore, one could have expected these phases to be unstable. On the other hand, we observe from the phase diagram, that IX is stable at intermediate densities for $0.25\sigma < D< 0.45\sigma$, while IB packs better than IX. In other words, stability can not be inferred from small differences in packing densities. ![Phase diagram in the packing fraction ($\phi$) versus thickness ($D$) representation. The light gray areas are coexistence areas, while the state points in the dark gray area are inaccessible since they lie above the close packing line. IX, IB, IX’ and fcc${}^2$ denote the crystals as shown in Fig. \[fig:crystals\], “F” is the fluid and “col” is the columnar phase. The lines are a guide to the eye. Worm-like stacks were found in the area marked “worms” bounded from below by the dashed line. This line denotes the probability to find a particle in a cluster that consists of more than two particles, $\mathcal{P}_\text{stack}(n>2)=1/2$. \[fig:ph-dia\_shells\]](ph-dia){width="49.00000%"} [0.49]{}[@\*[7]{}[c@]{}]{} $D/\sigma$ & phase 1 & phase 2 & $\rho_1\sigma^3$ & $\rho_2\sigma^3$ & $\beta P \sigma^3$ & $\mu^*$\ 0 & fluid & col & 4.083 & 4.824 & 26.11 & 15.22\ \ [0.49]{}[@\*[7]{}[c@]{}]{} $D/\sigma$ & phase 1 & phase 2 & $\phi_1$ & $\phi_2$ & $\beta P d^3$ & $\mu^*$\ 0.1 & fluid & col & 0.2778 & 0.3297 & 26.35 & 15.59\ 0.1 & col & IX & 0.8095 & 0.8104 & $2.7\!\cdot\!10^3$ & -\ 0.2 & fluid & col & 0.4096 & 0.4688 & 27.23 & 16.68\ 0.2 & col & IX & 0.7021 & 0.7108 & 325 & -\ 0.3 & fluid & col & 0.5286 & 0.5472 & 49.52 & 26.13\ 0.3 & col & IX & 0.6864 & 0.6944 & 281.4 & 91.03\ 0.3 & IX & IB & 0.6117 & 0.6226 & 110.9 & 44.92\ 0.4 & fluid & IB & 0.6098 & 0.6455 & 105.9 & 51.06\ 0.45 & fluid & IB & 0.6026 & 0.6545 & 87.92 & 46.90\ 0.5 & fluid & fcc${}^2$ & 0.4878 & 0.5383 & 28.34 & 22.10\ 0.5 & fcc${}^2$ & IX’ & 0.6870 & 0.7278 & 139.2 & 67.36\ Almost all coexistence densities were calculated by employing the common tangent construction to the free energy curves, except for the col–IX coexistence at $D=0.1\sigma$ and $0.2\sigma$. At these values of $D$ the transition occurs at very high pressures, while the free energy of the columnar phase is calculated at the fluid–col transition, which occurs at a low pressure. To get a value for the free energy of the columnar phase we would have to integrate the equation of state up to these high pressures, accumulating integration errors. Furthermore, we expect the coexistence to be rather thin, which would further complicate the calculation. So, instead we just ran long variable box shape $NPT$ simulations to see at which pressure the IX phase melts into the inverted columnar phase. As the free energy difference between the inverted columnar phase and the columnar phase is small, we assume that this is the coexistence pressure for the col–IX transition, although technically it is only a lower bound. The density of the columnar phase at this pressure is determined using a local fit of the equation of state. All coexistences are tabulated in Tbl. \[tbl:coexes\]. We draw a tentative line in the phase diagram to mark the transition from a structureless fluid to a worm-like fluid *i.e.* a fluid with many stacks. In a dense but structureless fluid, stacks of size 2 are quite probable, but larger stacks occur far less frequently. We calculate the probability to find a particle in a stack that contains more than 2 particles $\mathcal{P}_\text{stack}(n>2)=1-\mathcal{P}_\text{stack}(1)-\mathcal{P}_\text{stack}(2)$ and define the worm-like phase by the criterion $\mathcal{P}_\text{stack}(n>2)\geq 1/2$ in Fig. \[fig:ph-dia\_shells\]. We do not imply that the transition to the worm-like phase is a true thermodynamic phase transition; the transition is rather continuous. The type of stacks in the fluid changes from worm-like for $D=0.3\sigma$ to something resembling the columns in the braid-like crystals B and IB (see Fig. \[fig:crystals\]) for $D=0.4\sigma$. Therefore, the region of stability worm-like phase was ended at $D=0.35\sigma$, where there are similar amounts of braid-like and worm-like stacks. Summary and discussion ====================== We have studied the phase behavior of hard bowls in Monte Carlo simulations. We find that the bowls have a strong tendency to form stacks, but the stacks are bent and not aligned. We measured the equation of state and the compressibility in Monte Carlo $NPT$ simulations. The pressure we obtained from these simulations is concave for some range of densities for deep bowls. This is due to the increase in free volume when large stacks form. Using $g_c(z)$, the pair correlation function along the direction vector, we showed that the concavity of the pressure coincides with a dramatic change in structure from a homogeneous fluid to the worm-like fluid. We measured the three-dimensional stack length distribution in the simulations. When the pressure is increased slowly, the deep bowls spontaneously order into a columnar phase in our simulations. This poses severe restrictions on the thickness of future bowl-like mesogens (molecular or colloidal), which are designed to easily order into a globally aligned lyotropic columnar phase. We determined the phase diagram using free energy calculations for a particle shape ranging from an infinitely thin bowl to a solid hemisphere. We find that the columnar phase is stable for $D\leq0.3\sigma$ at intermediate packing fractions. In addition, we show using free energy calculations that the stable columnar phase possesses polar order. However, the free energy penalty for flipping columns upside down is very small, which makes it hard to achieve complete polar ordering in a spontaneously formed columnar phase of bowls. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Rob Kortschot, Ahmet Demirörs and Arnout Imhof for useful discussions. Financial support is acknowledged from an NWO-VICI grant and from the High Potential Programme of Utrecht University. Overlap algorithm ================= The overlap algorithm for our bowls checks whether the surfaces of two bowls intersect. Fig. \[fig:particles\] shows that the surface of the bowl consists of two parts. Part $p$ of the surface contains the part of the surface of the sphere of radius $R_p$, within an angle $\theta_p$ from the $z$-axis, where $p=1$ denotes the smaller sphere and the larger sphere is labeled $p=2$. We set $\theta_1=\pi/2$, to get a hemispherical outer surface. The edges of both surfaces have to coincide, such that our particles have a closed surface. Using this restriction $L$, $\theta_2$ and $R_2$ can all be expressed in terms of the radius of the smaller sphere, $R_1$, and the thickness of the bowl $D$, in the following way: $$\begin{aligned} R_2 & = & R_1+\frac{D^2}{2(R_1-D)}\\ \theta_2 & = & \arcsin(R_1/R_2)\\ L & = & R_2 \cos(\theta_2).\end{aligned}$$ Overlap occurs if either of the two parts of the surface of a bowl overlaps with either of the two parts of another bowl. So we have to check four pairs of infinitely thin (and not necessarily hemispherical) bowls, labeled $i$ and $j$, for overlap. The algorithm for two such surfaces that are equal in shape was already implemented by He and Siders [@He1990UFOs] as part of their overlap algorithm for their “UFO” particles, which are defined as the intersection between two spheres. An equivalent overlap algorithm was used by Cinacchi and Duijneveldt [@Cinacchi2010] to simulate infinitely thin contact lense-like particles, but the overlap algorithm was not described explicitly. We can not use one of these algorithms, since the two parts of the surface of our particle are unequal in shape. Therefore, we implemented a slightly different version of the overlap algorithm, which we describe in the remainder of this section. In our overlap algorithm, the existence of a overlap or intersection between two infinitely thin bowls is checked in three steps. - First, we check whether the full surfaces of the spheres intersect, i.e. $|R_i-R_j|<r_{ij}\equiv|\mathbf{r}_j-\mathbf{r}_i|<R_i+R_j$. If this intersection does not exist, there is no overlap, otherwise we proceed to the next step. - Secondly, we determine the intersection of the surface of each sphere with the other bowl. The intersection of bowl $i$ with the sphere of bowl $j$ exists if $$|\omega_{ij}+\zeta \phi_{ij}| < \theta_i \label{eqn:sphere_bowl}\\$$ for $\zeta=1$ or $-1$, where $$\begin{aligned} \cos(\phi_{ij})& =& \frac{R_i^2-R^2_j+r_{ij}^2}{2 r_{ij} R_i}\ \mathrm{and}\\ \cos(\omega_{ij})&=& \frac{\mathbf{u}_i \cdot \mathbf{r}_{ij}}{r_{ij}}.\end{aligned}$$ see Fig. \[fig:calc\_shells\]a. ![ The relevant lengths and angles which are used in the first and second steps (a) and in the third step (b) of the overlap algorithm. Shown are bowl $i$ and (part of) the sphere of bowl $j$ (a), the arcs of $i$ and $j$ and the circular intersection of the spheres of $i$ and $j$ (b). In (a) $\mathbf{r}_{ij}$ lies in the plane, while the plane of view in (b) is perpendicular to $\mathbf{r}_{ij}$. In this case, the sphere of particle $j$ overlaps with bowl $i$, but the arcs do not overlap, so particle $i$ and particle $j$ do not overlap. \[fig:calc\_shells\]](calcs){width="49.00000%"} This intersection is an arc, which is part of the circle that is the intersection between the two spheres. If in fact this arc is a full circle and the other particle has a nonzero intersection, the particles overlap. This is the case when Eq. (\[eqn:sphere\_bowl\]) holds for $\zeta=1$ *and* $\zeta=-1$. If, on the contrary, either of the two arcs does not exist, there is no overlap. Otherwise, if both arcs exist, but neither of them is a full circle, proceed to the next step. - Finally, if the two arcs overlap there is overlap, otherwise the particles do not overlap. The arcs overlap if $$|\alpha_{ij}|<|\gamma_i|+|\gamma_j|,\label{eqn:arc_overlap}$$ where $$\begin{aligned} \cos(\alpha_{ij})&=&\frac{ \mathbf{n}_i^\perp \cdot \mathbf{n}_j^\perp}{|\mathbf{n}_i^\perp| |\mathbf{n}_j^\perp|} \\ \cos(\gamma_i)&=& \frac{\cos(\theta_i)-\cos(\phi_{ij})\cos(\omega_{ij})}{\sin(\phi_{ij})\sin(\omega_{ij})},\end{aligned}$$ where $\mathbf{n}_i^\perp=\mathbf{n}_i-(\mathbf{r}_{ij}\cdot\mathbf{n}_i)\mathbf{r}_{ij}/r^2_{ij}$ and the expressions for $\gamma_j$ and $\mathbf{n}_j^\perp$ are equal to the expressions for $\gamma_i$ and $\mathbf{n}_i^\perp$ with $i$ and $j$ interchanged. The arcs together with the relevant angles are drawn in Fig. \[fig:calc\_shells\]b. The inequalities (\[eqn:sphere\_bowl\]) and (\[eqn:arc\_overlap\]) are expressed in cosines and sines using some simple trigonometry. In this way no inverse cosines need to be calculated during the overlap algorithm. For $D=0.5\sigma$ the bottom surface is a disk rather than an infinitely thin bowl. So the overlap check consists of bowl–bowl, bowl–disc and disc–disc overlap checks. For brevity, we will not write down the bowl–disk overlap algorithm, but it can be implemented in a similar way as the algorithm for bowl–bowl overlap described above. The disk–disk overlap algorithm was already implemented by Eppenga and Frenkel [@Eppenga].

ArXiv

--- author: - Chen Sun - Austin Myers - Carl Vondrick - Kevin Murphy - Cordelia Schmid bibliography: - 'egbib.bib' title: 'VideoBERT: A Joint Model for Video and Language Representation Learning' ---

ArXiv

--- abstract: 'This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra  [@FH]. For the topos of sets, we show that torsion-free functors on Loganathan’s category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an $S$-torsor, of an inverse semigroup $S$ in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$. We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on $L(S)$.' address: - 'Ganna Kudryavtseva, Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000, Ljubljana, SLOVENIA; Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, SLOVENIA ' - 'Primož Škraba, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, SLOVENIA ' author: - Ganna Kudryavtseva - Primož Škraba title: The principal bundles over an inverse semigroup --- Introduction ============ The classifying topos ${\mathcal{B}}(S)$ of an inverse semigroup $S$ has recently begun to be investigated [@F; @FH; @FLS; @FS; @KL; @St]. This topos is by definition the presheaf topos over Loganathan’s category $L(S)$ of $S$. There are several equivalent characterizations of this topos, cf. [@F; @FS; @KL]. An immediate question one can ask about ${\mathcal{B}}(S)$ is “What does ${\mathcal{B}}(S)$ classify?” A direct application of well-known results [@MM Theorems VII.7.2, VII.9.1] of topos theory, provides the answer: for an arbitrary Grothendieck topos ${\mathcal E}$, the presheaf topos ${\mathcal{B}}(S)$ classifies filtered functors $L(S)\to {\mathcal E}$. The category of geometric morphisms from ${\mathcal E}$ to ${\mathcal{B}}(S)$ is equivalent to the category of filtered functors $L(S)\to {\mathcal E}$. The construction of the functors establishing the correspondence in the above theorem can be found in [@MM]. In particular, if $\gamma^*\colon {\mathcal{B}}(S)\to {\mathcal E}$ is the inverse image functor of a geometeric morphism, its composition with the Yoneda embedding $L(S)\to {\mathcal{B}}(S)$ is a filtered functor, and any such a functor is obtained this way. This answer, however, is not quite satisfactory. We expect more structure given that for groups the category of filtered functors $G\to {\mathcal E}$ is equivalent to the category of $G$-[*torsors*]{}. The latter are just objects of ${\mathcal E}$ with a particular type of internal action of the group object obtained by applying the canonical constant sheaf functor $\Delta$ to $G$ [@MM VIII.2]. This naturally raises a question of how to define $S$-torsors, where $S$ is an inverse semigroup. The latter question was raised by Funk and Hofstra in [@FH], where in [@FH Theorem 3.9] they show that, for the topos of sets, $S$-torsors can be defined as (well-supported) transitive and free $S$-sets, where an inverse semigroup $S$-set is a homomorphism from $S$ to the symmetric inverse semigroup ${\mathcal{I}}(X)$. This description naturally generalizes the description of $G$-torsors in the topos of sets. It is mentioned (without providing details) by Lawson, Margolis and Steinberg [@LMS] that $S$-torsors in the topos of sets are precisely universal representations of $S$ defined and systematically studied in [@LMS]. Funk and Hofstra, in [@FH Definition 2.14], proposed a definition of an $S$-torsor, where $S$ is an inverse semigroup, in an arbitrary Grothendieck topos ${\mathcal E}$. They also state an equivalence of categories between $S$-torsors in ${\mathcal E}$ and filtered functors $L(S)\to {\mathcal E}$ ([@FH Theorem 3.10]). Their approach is based on internalizing $S$ in ${\mathcal E}$ as a semigroup (rather than as an inverse semigroup). Implicitly, [@FH Definition 2.14] considers actions (in an arbitrary Grothendieck topos) by partial bijections [@private_comm]. However, actions of by partial bijections in an arbitrary Grothendieck topos are not defined in [@FH], nor, to the best or our knowledge, anywhere else in the literature. Therefore, while the main theorem [@FH Theorem 3.10] is correct, the ommision of this definition along with other details, can make some of the definitions (e.g., [@FH Definition 2.14]) and proofs in [@FH] hard to follow or verify for the non-expert in topos theory. One of the main goals in the present paper is to try to make the constructions as detailed, simple and explicit as possible and particularly tailored to researchers in semigroup theory. Additionally, we provide a counter-example to a claim in [@FH Section 6] (see Section \[sec:fin\] for details). The paper is structured as follows. Section \[sec:prel\] provides some preliminaries needed to read this paper (as well as suggestions of the literature for further reading). In Section \[sec:sets\], we focus on the topos of sets and describe (possibly non-strict) $S$-sets, where $S$ is an inverse semigroup, attached to classes of functors from $L(S)$ to this topos. Some of these results were already given in [@FH], but we provide detailed proofs. We introduce a class of [*connected*]{} non-strict $S$-sets and prove that they are in a categorical equivalence with torsion-free functors on $L(S)$ (Theorem \[th:equiv\]). We then show that connected non-strict $S$-sets form a proper coreflective subcategory of the category of all non-strict $S$-sets which corrects [@FH Proposition 3.6] (see Example \[ex:ce\] and Proposition \[prop:ce\]). We also discuss the connection of transitive and universal $S$-sets with appropriate classes of functors on $L(S)$. This in particular leads to a new perspective on the classical result due to Schein [@Sch] on transitive and effective representations of an inverse semigroup. In Section \[sec:bundles\], we define $S$-torsors in the topos of sheaves ${\mathsf{Sh}}(X)$ over a topological space $X$ and prove that these are categorically equivalent to filtered functors $L(S)\to {\mathsf{Sh}}(X)$ (Theorem \[th:sheaves\]). It follows that in the topos ${\mathsf{Sh}}(X)$ the classifying topos ${\mathcal B}(S)$ classifies universal $S$-bundles.This can be seen as an instance of [@FH Theorem 3.10], with more details which hopefully provide a better insight into why this works. Finally, in Section \[sec:fin\], we outline an approach, which is substantially different from that used in [@FH], to the notion of an $S$-set in an arbitrary Grothendieck topos. We start from a functor on $L(S)$, construct an objects of action as a certain colimit (similarly as this is done for the topos of set) and then lift $S$ to the topos ${\mathcal H}$-class-wise, that is, we consider objects $\Delta H$, where $H$ is an ${\mathcal H}$-class of $S$ and $\Delta$ is the constant sheaf functor. It would be interesting to connect and compare this approach with the approach proposed in [@FH]. An important task which remains for future investigation is to further develop the general theory of actions of inverse semigroups by partial bijections in arbitrary Grothendieck toposes extending [@FH] and the present paper to the level of corresponding well-established theory of group actions [@MM V.6, VIII.2]. Preliminaries {#sec:prel} ============= For more complete exposition on inverse semigroups, we refer the reader to [@L], on categories to [@Awo; @MacL], and on toposes to [@MM; @M]. Inverse semigroups and their representations {#subs:inv} -------------------------------------------- Let $S$ be an inverse semigroup. By $E(S)$, we denote the semilattice of idempotents of $S$. For $s\in S$ we write ${\mathbf{d}}(s)=s^{-1}s$ and ${\mathbf{r}}(s)=ss^{-1}$. These idempotents are abstractions of the notions of the domain and the range idempotents, respectively, of a partial bijection. The natural partial order on $S$ is defined by $s\leq t$ if and only if $s=te$ for some $e\in E(S)$. For $X\subseteq S$, we write $$X^{\uparrow}=\{s\in S\colon s\geq x \text{ for some } x\in X\}.$$ The set $X^{\uparrow}$ is sometimes called the [*(upward) closure*]{} of $X$. The set $X$ is [*closed*]{} if $X^{\uparrow}=X$. For a set $X$, let ${\mathcal I}(X)$ denote the [*symmetric inverse semigroup*]{} on $X$ which consists of all bijections between subsets of $X$ (we refer to such maps as [*partial bijections*]{}). If $s\in {\mathcal I}(X)$ we set ${\mathrm{dom}}(s)$ and ${\mathrm{im}}(s)$ to be the domain and the image of $s$. A [*representation*]{} of an inverse semigroup $S$ on a set $X$, is an inverse semigroup homomorphism $\theta\colon S\to {\mathcal I}(X)$. Given a such a representation, we have a left action of $S$ on $X$ by partial bijections such that $s\cdot x$ is defined if and only if $x\in {\mathrm{dom}}(\theta(s))$ in which case $s\cdot x= \theta(s)(x)$. We say that $(X,\theta)$ is a [*left*]{} $S$-[*set*]{}. Where $\theta$ is clear, we will write $(X,\theta)$ as simply $X$. Unless otherwise stated, we assume that actions are left actions, and we refer to left $S$-sets as $S$-[*sets*]{}. Throughout the paper, we assume that the $S$-sets are [*effective*]{}, meaning that for every $x\in X$ there exists some $s\in S$ such that $s\cdot x$ is defined. An $S$-set $(X,\mu)$ is called [*transitive*]{} if for any $x,y\in X$ there is $s\in S$ such that $\mu(s)(x)=y$. It is called [*free*]{}, if the equality $\mu(s)(x)=\mu(t)(x)$ implies that there is $c\leq s,t$ such that $\mu(c)(x)=\mu(s)(x)$. Finally, we call a transitive and free $S$-set an $S$-[*torsor*]{}. Toposes in a nutshell --------------------- By a [*topos*]{}, we restrict ourselves to a [*Grothendieck topos*]{}, that is, a category ${\mathcal{E}}$ that satisfies the [*Giraud’s axioms*]{}. We refer the reader, for example, to [@M 1.1] for a detailed introduction to the notion of a topos. For our purposes, we do not need to recount the definition of a topos. It is important however to mention the following examples of toposes: 1. The category ${\mathsf{Sets}}$ of sets and maps between sets. 2. The category ${\mathrm{Et}}(X)$ of étale spaces over a topological space $X$. 3. The category ${\mathcal B}({\mathcal{C}})$ of presheaves of sets $F\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$ over a small category ${\mathcal{C}}$. Let us look at these examples at greater detail. An étale space over a topological space $X$ is a triple $(E,p,X)$ where $E$ is a topological space and $p\colon E\to X$ is a local homeomorphism. A [*morphism*]{} $ (E,p,X) \to (G,q,X)$ between étale spaces is a continuous map $\alpha\colon E\to G$ such that $q\alpha=p$. Given the well known equivalence between étale spaces and sheaves, the topos ${\mathrm{Et}}(X)$ is equivalent to the topos ${\mathsf{Sh}}(X)$ of sheaves over $X$. From the topos of sheaves ${\mathsf{Sh}}(X)$ one can recover the frame of opens of $X$, and thus, if $X$ is a sober space, $X$ itself can also be recovered [@M; @MM]. It follows that a topos can be thought of as a generalization of a (sober) topological space. Bearing this in mind, it is useful (for example, to interpret the definition of a point of a topos) to consider the topos ${\mathsf{Sets}}$ as an analogue of a one-point space. Turning to the third example, a [*presheaf of sets*]{} over a small category ${\mathcal{C}}$ is a contravariant functor $F$ from ${\mathcal{C}}$ to the category of sets ${\mathsf{Sets}}$, $F\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$. If $\alpha\colon c\to d$ is a morphism in ${\mathcal{C}}$, then the map $F(\alpha)\colon F(d)\to F(c)$ is called the [*translation map*]{} along $\alpha$. Let $F,G\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$ be presheaves of sets. By a morphism from $F$ to $G$, we mean a natural transformation $\pi$ from $F$ to $G$, that is, a collection of maps, $\pi_c\colon F(c)\to G(c)$, where $c$ runs through the objects of ${\mathcal{C}}$, which commute with the translation maps. The topos ${\mathcal B}({\mathcal{C}})$ is called [*the classifying topos*]{} of the small category ${\mathcal{C}}$. For a detailed verification that each of our examples satisfies the Giraud’s axioms, we refer the reader to [@M]. The category of elements of a functor ------------------------------------- Let ${\mathcal C}$ be a small category and $P\colon {\mathcal C}\to {\mathsf{Sets}}$ a covariant functor. The [*category of elements of*]{} $P$ is the category $\int_{\mathcal C}P$ whose objects are all pairs $(C,p)$ where $C$ is an object of ${\mathcal C}$ and $p\in P(C)$. Its morphisms $(C,p)\to (C',p')$ are those morphisms $u\colon C\to C'$ of ${\mathcal C}$ for which $P(u)(p)=p'$. The category of elements $\int_{\mathcal C} P$ of a contravariant functor $P\colon {\mathcal C}^{op}\to {\mathsf{Sets}}$ is defined similarly. Filtered and directed categories and functors {#subs:2.5} --------------------------------------------- A small category $I$ is called [*filtered*]{} if it satisfies the following axioms: 1. $I$ has at least one object. 2. For any two objects $i,j$ of $I$ there is a diagram $i\leftarrow k \to j$ in $I$, for some object $k$. 3. For any two parallel arrows $i\rightrightarrows j$ there exists a commutative diagram $k\to i\rightrightarrows j$ in $I$. Equivalently, a small category $I$ is filtered if for any finite diagram in $I$ there is a cone on that diagram. A small category $I$ is called [*directed*]{} if it satisfies axioms (F1) and (F2) above. A covariant functor $A\colon {\mathcal C}\to {\mathsf{Sets}}$ is called a [*filtered functor*]{} (resp. a [*directed functor*]{}) if its category of elements $\int_{\mathcal C} A$ is a filtered category (resp. a directed category). Geometric morphisms ------------------- Let ${\mathcal E}, {\mathcal F}$ be toposes. A [*geometric morphism*]{} $f\colon {\mathcal F} \to {\mathcal E}$ consists of a pair of functors $$f^*\colon {\mathcal E} \to {\mathcal F} \text{ and } f_*\colon {\mathcal F} \to {\mathcal E},$$ called the [*inverse image functor*]{} and the [*direct image functor*]{}, respectively, such that the following two axioms are satisfied: 1. $f^*$ is a left adjoint to $f_*$. 2. $f^*$ is left exact, that is, it commutes with finite limits. Since $f^*$ is a left adjoint, it commutes with colimits (by the dual to the well-known RAPL theorem [@Awo]). It follows from the uniqueness of adjoints that a geometric morphism $f\colon {\mathcal F} \to {\mathcal E}$ is determined by its inverse image functor $f^*\colon {\mathcal E}\to {\mathcal F}$ which is required to commute with any colimits and finite limits. Let $X,Y$ be topological spaces and $f\colon X\to Y$ a continuous map. This gives rise to a functor $f^*\colon {\mathrm{Et}}(Y)\to {\mathrm{Et}}(X)$, as follows. Let $(E,p,Y)$ be an étale space over $Y$ and put $$X\times_Y E =\{(x,e)\in X\times E\colon f(x)=p(e)\}.$$ Then the projection to the first coordinate $\pi_1\colon X\times_Y E \to X$ is a local homeomorphism. Indeed, assume that $(x,e)\in X\times_Y E$ and let $A$ be a neighborhood of $e$ such that $A$ is homeomorphic to $p(A)$. Then the set $$\{(x,t)\in X\times_Y E\colon t\in A\}$$ is homeomorphic to $f^{-1}(p(A))$ via $\pi_1$. The local homeomorphism $\pi_1$ is said to be obtained by [*pulling*]{} $p$ [*back along*]{} $f$. We set $$f^*(E,p,Y)=(X\times_Y E, \pi_1, X).$$ It is easy to see that $f^*$ preserves colimits and finite limits and thus gives rise to a geometric morphism $(f^*,f_*)$ from ${\mathrm{Et}}(X)$ to ${\mathrm{Et}}(Y)$. For sober spaces $X$ and $Y$ this construction gives rise to a bijective correspondence between continuous maps from $X$ to $Y$ and geometric morphisms from ${\mathrm{Et}}(X)$ to ${\mathrm{Et}}(Y)$. Thus, as toposes can be looked at as generalizations of topological spaces, geometric morphisms between toposes are generalizations of continuous maps. The constant sheaf functor and the global section functor --------------------------------------------------------- For any topos ${\mathcal E}$, there is a unique (up to isomorphism) geometric morphism $\gamma\colon {\mathcal E}\to {\mathsf{Sets}}$, given by $$\gamma^*(S)=\sum_{s\in S}1, \,\, \gamma_*(E)={\mathrm{Hom}}_{\mathcal E}(1,E),$$ where $1$ denotes the terminal object of ${\mathcal E}$. The inverse image part $\gamma^*$ of $\gamma$ is usually denoted by $\Delta$ and is called the [*constant sheaf functor*]{}, and the direct image part $\gamma_*$ is usually denoted by $\Gamma$ and is called the [*global section functor.*]{} This geometric morphism may be looked at as an analogue of the only continuous map from a topological space $X$ to a one-element topological space. Filtered functors and geometric morphisms ----------------------------------------- A [*point*]{} of a topos ${\mathcal E}$ is a geometric morphism $\gamma: {\mathsf{Sets}}\to {\mathcal E}$. This is parallel to looking at a point of a topological space $X$ as an inclusion of a one-element space into $X$. Note that such an inclusion $i\colon \{x\}\to X$ defines a filter $F$ in $X$ consisting of those $A\in X$ such that $i(x)\in A$. We now describe how this idea can be extended to a correspondence between points of the classifying topos of a category and filtered functors on this category. Let ${\mathcal C}$ be a small category, and let $A\colon {\mathcal C}\to {\mathsf{Sets}}$ be a functor. We describe a construction to be found in [@MM] of a pair of adjoint functors $f^*\colon {\mathcal{B}}({\mathcal C})\to {\mathsf{Sets}}$ and $f_*\colon {\mathsf{Sets}}\to {\mathcal{B}}({\mathcal C})$. The functor $f_*$ is easier to define and thus we start from its description. We have $f_*=\underline{\mathrm{Hom}}_{\mathcal C}(A,-)$, where the latter is the presheaf defined for each set $R$ and $C\in {\mathcal C}$ by $$\underline{\mathrm{Hom}}_{\mathcal C}(A,R)(C)={\mathrm{Hom}}_{\mathsf{Sets}}(A(C),R).$$ For a presheaf $P\in {\mathcal{B}}({\mathcal C})$, we define $f^*(P)$ to be the colimit $$f^*(P)=\lim_{\longrightarrow}\left(\int_{\mathcal C}P\stackrel{\pi_1}{\to} {\mathcal C} \stackrel{A}{\to} {\mathsf{Sets}}\right),$$ where $\pi_1(C,p)=C$. This colimit is the set which we denote by $P\otimes_{\mathcal{C}}A$. It is the quotient of the set $\bigcup_{C\in {\mathcal{C}}}(P(C)\times A(C))$ by the equivalence relation $\sim$ generated by $$(pu,a')\sim (p,ua'), \, p\in P(C), u\colon C\to C', a'\in A(C'),$$ where we denote $pu=P(p)(u)$ and $ua'=A(u)(a')$. We denote the elements of $P\otimes_{\mathcal{C}}A$ by $p\otimes a$ and treat them as tensors where ${\mathcal{C}}$ ‘acts’ on $P$ on the right and on $A$ on the left. The described adjoint pair $(f^*, f_*)$ is not in general a geometric morphism between toposes. By definition, it is a geometric morphism if and only if the tensor product functor $f^*$ is left exact. If this condition holds, the functor $A$ is called [*flat*]{}. Flat functors can be characterized precisely as filtered functors [@MM Theorem VII.6.3]. Let ${\mathsf{Filt}}({\mathcal C})$ denote the category of filtered functors ${\mathcal C}\to {\mathsf{Sets}}$, where morphisms are natural transformations, and ${\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C}))$ the category of geometric morphisms from ${\mathsf{Sets}}$ to the classifying topos ${\mathcal{B}}({\mathcal C})$ of ${\mathcal C}$ (or, equivalently the points of ${\mathcal{B}}({\mathcal C})$), where morphisms are natural transformations between the inverse image functors. \[th:filt\] There is an equivalence of categories $${\mathsf{Filt}}({\mathcal C})\, \,{\mathrel{ \settowidth{\@tempdima}{$\scriptstyle\tau$} \settowidth{\@tempdimb}{$\scriptstyle\rho$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}}\, \,{\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C}))$$ where the functors $\tau$ and $\rho$ are defined, for a filtered functor $A\colon {\mathcal C}\to {\mathsf{Sets}}$ and a point $f\in {\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C})) $, by $$\tau(A)^*=- \otimes_{\mathcal C} A, \,\, \tau(A)_*= \underline{\mathrm{Hom}}_{\mathcal C}(A,-),$$ $$\rho(f)=f^*\cdot {\mathbf{y}}\colon {\mathcal C}\to {\mathcal{B}}({\mathcal C})\to {\mathsf{Sets}},$$ where ${\mathbf{y}}$ denotes the Yoneda embedding of ${\mathcal C}$ into ${\mathcal{B}}({\mathcal C})$. We remark that Theorem \[th:filt\] remains valid in a wider setting where the topos ${\mathsf{Sets}}$ is replaced by an arbitrary topos ${\mathcal E}$. To formulate this result, known as Diaconescu’s theorem, one needs a suitable definition of a filtered functor from a small category to a topos, such that being filtered is equivalent to being flat, cf. [@MM VII.8]. For our purposes, we will need filtered functors to the topos of sheaves over a topological space, which we discuss in Section \[sec:bundles\]. Principal group bundles and group torsors ----------------------------------------- The connection between filtered functors on a small category and geometric morphisms to the classifying topos is a well known and fundamental result in topos theory. In the special case where the category is a group, denote it by $G$, it is known [@MM VIII.2] that the category of filtered functors $G\to {\mathcal E}$, where ${\mathcal E}$ is an arbitrary topos, is equivalent to the category of so-called $G$-torsors over ${\mathcal E}$. A $G$-[*torsor over*]{} ${\mathcal E}$ is an object $T$ of ${\mathcal E}$ equipped with an internal action (cf. [@MM V.6]) of a group object $\Delta G$ on it which satisfies some technical conditions. We remind the reader that $\Delta G$ is the value of the constant sheaf functor $\Delta\colon {\mathsf{Sets}}\to {\mathcal E}$ on $G$. Since $G$ has a structure of a group, $\Delta G$ inherits a structure of an internal group in ${\mathcal E}$. The aforementioned technical conditions arise as an abstraction of the well-known notion of a $G$-torsor in the topos of sheaves over a topological space $X$. In this setting, a $G$-torsor is just synonymous with a principal $G$-bundle. A [*principal*]{} $G$-[*bundle*]{} over a topological space $X$ can be characterized as an étale space $(E,p,X)$ with a continuous action $G\times E\to E$ over $X$ such that 1. For each point $x\in X$ the stalk $E_x=p^{-1}(x)$ is non-empty; 2. Stalks are invariant under the action; 3. The action map $G\times E_x \to E_x$ on each stalk $E_x$ is free meaning that $g\cdot x=x$ implies that $g$ is the identity element; 4. The action map $G\times E_x \to E_x$ on each stalk $E_x$ is free transitive, meaning that for every $a,b\in E_x$ there exists some $g\in G$ such that $g\cdot a=b$. A $G$-torsor in the topos of sets is a set $X$ equipped with a free and transitive action of $G$ on it. Such a set is in a bijection with $G$ and the action is equivalent to the action of $G$ on itself by left translations (this is a direct consequence of the elementary fact that a transitive group action is equivalent to the left action on the set of cosets over the stabilizer of any point). In particular, up to isomorphism, there is only one $G$-torsor in the topos of sets. The equivalence between filtered functors $G\to {\mathcal E}$ and $G$-torsors over ${\mathcal{E}}$, together with Theorem \[th:filt\], yields the result that the classifying topos ${\mathcal{B}}(G)$ classifies $G$-torsors in the sense that for an arbitrary topos ${\mathcal E}$, there is a categorical equivalence between geometric morphisms ${\mathcal E}\to {\mathcal{B}}(G)$ and $G$-torsors over ${\mathcal{E}}$. This parallels the topological result that the classifying space of $G$ classifies principal $G$-bundles. Covariant functors $L(S)\to {\mathsf{Sets}}$ vs representations of $S$ in ${\mathsf{Sets}}$ {#sec:sets} =========================================================================================== The relationship between various classes of (possibly non-strict) representations of an inverse semigroup $S$ in the topos of sets and covariant functors $L(S)\to {\mathsf{Sets}}$ was first observed and studied by Funk and Hofstra in [@FH]. In particular, they observe that filtered functors on $L(S)$ correspond to representations of $S$ which are transitive and free ([@FH Theorem 3.9], though these representations are wrongly referred to as another kind of representations). They also consider torsion-free and pullback preserving functors. In this section we prove that torsion-free functors on $L(S)$ correspond to a class of $S$-sets which we call [*connected*]{}. (This corrects an inaccuracy in [@FH Proposition 3.6].) We also put in correspondence directed functors on $L(S)$ and transitive effective representations of $S$, providing a different approach to the classical theory due to Schein [@Sch]. Finally, we explain that filtered functors on $L(S)$ correspond to a class of representations, called [*universal*]{}, which were introduced and studied by Lawson, Margolis and Steinberg in [@LMS]. Torsion-free functors and connected non-strict representations -------------------------------------------------------------- A map $\varphi\colon S\to T$ between inverse semigroups is called a [*prehomomorphism*]{} if $\varphi(ab)\leq \varphi(a)\varphi(b)$ for any $a,b\in S$. A prehomomorphism $S\to {\mathcal{I}}(X)$ will be called a [*non-strict*]{} representation of $S$. Similarly as representations correspond to $S$-sets, non-strict representations correspond to [*non-strict*]{} $S$-[*sets*]{}[^1], where the latter means a set $X$ together with a partial map $S\times X \to X$, $(s,x)\mapsto s\cdot x$, where defined, such that if $st\cdot x$ is defined then $t\cdot x$ and $s\cdot (t\cdot x)$ are defined and $st\cdot x= s\cdot (t\cdot x)$. Just as $S$-sets, the non-strict $S$-sets we consider are effective. The following constructions connecting non-strict $S$-sets and some covariant functors on $L(S)$ were introduced in [@FH]. We give here their slightly different but equivalent description. We also provide more details and notice the property of connectedness. Let $(X,\mu)$ be a non-strict $S$-set where $(s,x)\mapsto \mu(s,x)=s\cdot x$, where defined. For each $e\in E(S)$ let $\Phi(X,\mu)(e)$ be the domain of the action of $e$, that is to say, $$\Phi(X,\mu)(e)=\{x\in X\colon e\cdot x\text{ is defined}\}.$$ If $(f,s)$ is an arrow in $L(S)$, we define $\Phi(X,\mu)(f,s)$ to be the map from $\Phi(X,\mu)({\mathbf{d}}(s))$ to $\Phi(X,\mu)(f)$ given by $x\mapsto s\cdot x$. Since $e\cdot x$ is defined and $e={\mathbf{d}}(s)=s^{-1}s$, we have that $s\cdot x$ is defined. Observe that $s \cdot x=(fs)\cdot x$, so that $f\cdot (s\cdot x)$ is defined. Thus $s\cdot x\in \Phi(X,\mu)(f)$. We have constructed the covariant functor $\Phi(X,\mu)$ on $L(S)$. We need to record that the functor $\Phi(X,\mu)$ has one important property. We first define this property. Assume that $F\colon L(S)\to {\mathsf{Sets}}$ is a functor and put $\Psi(F)$ to be the colimit of the following composition of functors: $$E(S)\longrightarrow L(S)\stackrel{F}{\longrightarrow} {\mathsf{Sets}}.$$ This colimit is, by definition, equal to the quotient set $$\label{eq:colimit} \Psi(F)= \left(\bigcup_{e\in E(S)}\{e\}\times F(e)\right)/\sim,$$ where the equivalence $\sim$ on $\bigcup_{e\in E(S)}\{e\}\times F(e)$ is generated by $(e,x)\sim (e',F(e',e)(x))$. The functor $F$ is called [*torsion-free*]{} if $(e,x)\sim (e,y)$ implies that $x=y$. The constructed functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$ is torsion-free. This follows from the definition of $\sim$ since $$\Phi(X,\mu)(e',e)(x)=e'\cdot x=e\cdot x=x$$ for any $e'\geq e$ in $E$ and any $x\in X$ such that $e\cdot x$ is defined. We have therefore assigned to $(X,\mu)$ a torsion-free functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$. We now describe the reverse direction. Assume that $F$ is a torsion-free functor $L(S)\to {\mathsf{Sets}}$. By $[e,x]$ we will denote the $\sim$-class of $(e,x)$. For $s\in S$ and $\alpha\in \Psi(F)$ we define $$\label{eq:action} s\circ\alpha=\left\lbrace\begin{array}{ll}[{\mathbf{r}}(s), F({\mathbf{r}}(s),s)(x)],& \text{ if } \alpha=[{\mathbf{d}}(s),x];\\ \text{undefined,} & \text{otherwise.} \end{array}\right.$$ If $\alpha\in \Psi(F)$ we define $$\label{eq:connected}\pi_1(\alpha)=\{e\in E\colon \text{ there is some }(e,x)\in \alpha\}=\{e\in E\colon e\circ \alpha \text{ is defined}\}.$$ It follows that $s\circ \alpha$ is defined if and only if ${\mathbf{d}}(s)\in\pi_1(\alpha)$. \[lem:lem3\] 1. The map $\alpha\mapsto s\circ\alpha$, given by , is injective on its domain. 2. The assignment defines on $\Psi(F)$ the structure of a non-strict $S$-set $(\Psi(F), \nu)$. 3. For any $\alpha\in \Psi(F)$ and $e,f\in \pi_1(\alpha)$, there are $$e=e_1,e_2,\dots, e_k=f$$ in $\pi_1(\alpha)$ such that $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$. \(1) Follows from , since $F$ is torsion-free and thus all the translation maps are injective. \(2) We use the fact that a map $\varphi\colon S\to T$ between inverse semigroups is a prehomomorphism if and only if $\varphi(st)=\varphi(s)\varphi(t)$ for any $s,t$ such that ${\mathbf{r}}(t)={\mathbf{d}}(s)$ and $\varphi(ef)\leq \varphi(e)\varphi(f)$ for any $e,f\in E(S)$. It is immediate from that both of these conditions hold for $\Psi(F)$. \(3) Follows from the construction of $\Psi(F)$ and . Since the set $\pi_1(\alpha)$ is expressable in terms of the action, as is given in , we can define a non-strict $S$-set $X$, $(s,x)\mapsto s\cdot x$, where defined, to be [*connected*]{} if for any $x\in X$ and any $e,f\in E$ such that $e\cdot x$ and $f\cdot x$ are defined, there is a sequence of idempotents $e=e_1,e_2,\dots, e_k=f$, called a [*connecting sequence over*]{} $x$, such that $e_i\cdot x$ is defined and $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$. [*If $X$ is an $S$-set (that is, given by a homomomorphism), it is connected with $e,ef,f$ being a connecting sequence between $e$ and $f$ over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.* ]{} [*If $S$ is a monoid, any non-strict $S$-set is connected with $e,1,f$ being a connecting sequence between $e$ and $f$, again over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.*]{} It is not true that every non-strict $S$-set is connected, as the following example shows. \[ex:ce\] *Let $S=\{e,f,g\}$ be a three-element semilattice, given by the following Hasse diagram:* \(e) [$e$]{}; (aux) \[node distance=0.6cm, right of=e\] ; (f) \[node distance=1.2cm, right of=e\] [$f$]{}; (g) \[node distance=1cm, below of=aux\] [$g$]{}; (e) edge node\[above\] (g) (f) edge node\[above\] (g); Let $X=\{1,2\}$ and define the domains of action of $e$ and $f$ to be equal $\{1,2\}$, and the domain of action of $g$ to be equal $\{1\}$ (that is, $e$ and $f$ act by the identity map on $\{1,2\}$, and $g$ by the identity map on $\{1\}$). Thus $X$ becomes a non-strict $S$-set. It is however not connected, as both $e\cdot 2$ and $f\cdot 2$ are defined but there is no connecting sequence between $e$ and $f$ over $2$ as $g\cdot 2$ is undefined. In view of Lemma \[lem:lem3\], it follows that the non-strict $S$-set from Example \[ex:ce\] can not be equal $\Psi(F)$ for any torsion-free functor $F$ on $L(S)$. We now describe the correspondence between morphisms of non-strict $S$-sets and natural transformations of torsion-free functors on $L(S)$. Assume we are given non strict $S$-sets $(X,\mu)$, $(s,x)\mapsto s\cdot x$, where defined, and $(Y,\nu)$, $(s,x)\mapsto s\circ x$, where defined. A [*morphism*]{} from $(X,\mu)$ to $(Y,\nu)$ is a map $f\colon X\to Y$ such that if $s\cdot x$ is defined then $s\circ f(x)$ is also defined and $$f(s\cdot x) = s\circ (f(x)).$$ Let $f\colon (X,\mu)\to (Y,\nu)$ be a morphism, $e\in E$ and $x\in \Phi(X, \mu)(e)$. Then $f(x)\in \Phi(Y, \nu)(e)$ which defines a map $$\widetilde{f}_e\colon \Phi(X, \mu)(e)\to \Phi(Y, \nu)(e).$$ It is immediate that the maps $\widetilde{f}_e$ commute with the translation maps along any $(f,s)\in L(S)$ and thus define a natural transformation $\widetilde{f}\colon \Phi(X, \mu)\to \Phi(Y, \nu)$. We set $\Phi(f)=\widetilde{f}$. In the reverse direction, let $F$ and $F'$ be torsion-free functors $L(S)\to {\mathsf{Sets}}$ and let $\alpha\colon F\to F'$ be a natural transformation. Let $\alpha(e)$ denote the component of $\alpha$ at $e$. Further, let $\sim$ denote the congruence on the set $\bigcup_{e\in E(S)}\{e\}\times F(e)$ which defines the set $\Psi(F)$, and $\sim'$ denote a similar congruence which defines the set $\Psi(F')$. Let $x\in F(e)$, $y\in F(f)$ and $(e,x)\sim (f,y)$. Then $(e,\alpha(e)(x))\sim' (f,\alpha(f)(y))$. Without loss of generality, we may assume that $e\leq f$ and that $y=F(f,e)(x)$. Since compotents of $\alpha$ commute with the translation maps, we can write $$\alpha(f)(y)=\alpha(f)(F(f,e)(x))=F'(f,e)(\alpha(e)(x)),$$ which yields that $(e,\alpha(e)(x))\sim' (f,\alpha(f)(y))$. The proved lemma shows that the assignment $[e,x]\mapsto [e,\alpha(e)(x)]$ results in a well-defined map $$\widetilde{\alpha}\colon \Psi(F)\to \Psi(F').$$ The map $\widetilde{\alpha}$ is a morphism of non-strict $S$-sets. Let the structure of an $S$-set on $\Psi(F)$ (defined in ) be given by $(s,\alpha)\mapsto s\circ \alpha$, where defined, and that on $\Psi(F')$ be given by $(s,\alpha)\mapsto s*\alpha$, where defined. Let $s\in S$ and assume that $s\circ [e,x]$ is defined. We may then assume that $e={\mathbf{d}}(s)$. Then $\widetilde{\alpha}([{\mathbf{d}}(s),x])=[{\mathbf{d}}(s),\alpha({\mathbf{d}}(s))(x)]$ showing that $s*\widetilde{\alpha}([{\mathbf{d}}(s),x])$ is defined, too. The proof is completed by the following calculation using the fact that components of $\alpha$ commute with the translation maps: $$\widetilde{\alpha}(s\circ ([{\mathbf{d}}(s),x]))=\widetilde{\alpha}([{\mathbf{r}}(s),F({\mathbf{r}}(s),s)(x)])= [{\mathbf{r}}(s), \alpha({\mathbf{r}}(s))F({\mathbf{r}}(s),s)(x)];$$ $$s*\widetilde{\alpha}([{\mathbf{d}}(s),x])=s*[{\mathbf{d}}(s), \alpha({\mathbf{r}}(s))(x)]=[{\mathbf{r}}(s),F'({\mathbf{r}}(s),s)(\alpha({\mathbf{d}}(s),x))].$$ We set $\Psi(\alpha)=\widetilde{\alpha}$. Let ${\mathsf{Repr}}(S)$ denote the category of all non-strict $S$-sets, ${\mathsf{ConRepr}}(S)$ the category of all connected non-strict $S$-sets and ${\mathsf{TF}}(L(S))$ the category of torsion-free functors on $L(S)$. It is routine to verify that the assignments $\Phi\colon {\mathsf{Repr}}(S)\to {\mathsf{TF}}(L(S))$ and $\Psi\colon {\mathsf{TF}}(L(S))\to {\mathsf{ConRepr}}(S)$ are functorial. We denote the restriction of the functor $\Phi$ to the category ${\mathsf{ConRepr}}(S)$ by $\Phi'$. We obtain the following result. \[th:equiv\] There is an equivalence of categories $${\mathsf{ConRepr}}(S) \,\, {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle\Phi'$} \settowidth{\@tempdimb}{$\scriptstyle\Psi$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \leftarrowfill\cr}}}\limits^{\!\Phi'}_{\!\Psi}}} \,\, {\mathsf{TF}}(L(S)).$$ Let $F\colon L(S)\to {\mathsf{Sets}}$ be a torsion-free functor and show that $F$ is naturally isomorphic to the functor $\Phi\Psi(F)$. By construction, for $e\in E$ we have $$\Phi\Psi(F)(e)=\{[e,x]\colon e\cdot x \text{ is defined}\}.$$ Clearly, the maps $\tau_e\colon x\to [e,x]$, $x\in F(e)$, $e\in E$, are bijections. In addition, these maps commute with the translation maps because for any arrow $(f,s)$ in $L(S)$, we have $$[{\mathbf{d}}(s),x]\stackrel {\Phi\Psi(F)(f,s)}{\xrightarrow{\hspace*{1.2cm}}} [f, F(f,s)(x)].$$ It follows that we have constructed a natural isomorphism $\tau\colon F\to \Phi\Psi(F)$. In the reverse direction, let $(X,\mu)$ be a connected non-strict $S$-set, where $(s,x)\mapsto s\cdot x$, where defined. The elements of the set $\Psi\Phi(X,\mu)$ are equivalence classes $[e,x]$ where $e\in E$ and $e\cdot x$ is defined. We define the map $$\label{eq:beta} \beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$$ by $[e,x]\mapsto x$. Let $x\in X$. Since $\mu$ is effective, there exists an $s\in S$ such that $s\cdot x$ is defined. But then ${\mathbf d}(s)\cdot x$ is defined, as well, which implies that the map $\beta_{\mu}$ is surjective. To show injectivity of $\beta_{\mu}$, we note that the condition $[e,x] \neq [f,x]$ is equivalent to the claim that there is no connecting sequence between $e$ and $f$, which does not happen as $\mu$ is connected. We obtain that the non-strict $S$-set $\Psi\Phi(X,\mu)$, $[e,x]\mapsto s\circ [e,x]$, where defined, is equivalent to $(X,\mu)$. Indeed, $s\circ [e,x]$ is defined if and only if $s\cdot x$ is defined, and in the case where $s\cdot x$ is defined we have the equality $$s\circ [e,x]=[{\mathbf{r}}(s), s\cdot x].$$ Moreover, this equivalence is natural in $(X,\mu)$. \[cor:strict\] The category of $S$-sets is equivalent to the category of pull-back preserving functors on $L(S)$. The statement follows from Theorem \[th:equiv\] using the facts that any $S$-set is connected, and that a non-strict $S$-set $(X,\mu)$ is strict if and only if $\mu(ef)=\mu(e)\mu(f)$ for any $e,f\in E(S)$. We now establish a relationship between all non-strict $S$-sets and those of them which are connected. Recall that a subcategory ${\mathcal A}$ of a category ${\mathcal B}$ is called [*coreflective*]{} if the inclusion functor ${\mathrm{i}}\colon {\mathcal A}\to {\mathcal B}$ has a right adjoint. This adjoint is called a [*coreflector*]{}. \[prop:ce\] The category ${\mathsf{ConRepr}}(S)$ is a coreflective subcategory of the category ${\mathsf{Repr}}(S)$. The coreflector is given by the functor $\Psi\Phi$. Let $(X,\mu)$ be a non-strict $S$-set. Just as in the proof of Theorem \[th:equiv\], we have the map $\beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$ given by . This map is surjective, and is injective if and only if $\mu$ is connected. We show that the functor $\Psi\Phi$ is a right adjoint to the functor ${\mathrm{i}}\colon {\mathsf{ConRepr}}(S) \to {\mathsf{Repr}}(S)$ where the maps $\beta_{\mu}$ are the components of the counit $\beta\colon i\circ \Psi\Phi \to {\mathrm{id}}_{{\mathsf{Repr}}(S)}$. Let $(X,\mu)$ be any connected non-strict $S$-set, $(Y,\nu)$ be any non-strict $S$-set, and $$g\colon (X,\mu) \to (Y,\nu)$$ a morphism. To define the morphism $$f\colon (X,\mu) \to \Psi\Phi(Y,\nu),$$ let $x\in X$ and $e\in E(S)$ be such that $\mu(e)(x)$ is defined. Then it follows that $\nu(e)(f(x))$ is defined, as well. We set $$\label{eq:def_f} f(x)=[e,g(x)]\in \Psi\Phi(Y,\nu).$$ For brevity, in this proof, we write $s\cdot x$ for $\mu(s)(x)$, $s\circ x$ for $\nu(s)(x)$ and $s*x$ for $\Psi\Phi(\nu)(s)(x)$. Note that if $h\cdot x$ is defined where $h\in E(S)$ then $h\circ f(x)$ is defined, and an induction shows that $(e,g(x))\sim (h,g(x))$ follows from $(e,x)\sim (h,x)$, where the latter equivalence holds because $\mu$ is connected. Therefore, the map $f$ is well-defined. Let us show that $f$ is a morphism of non-strict $S$-sets. Assume that $s\cdot x$ is defined. This is equivalent to that ${\mathbf{d}}(s)\cdot x$ is defined. It follows that $s*[{\mathbf{d}}(s),g(x)]$ is defined as well, and applying we have $$s*[{\mathbf{d}}(s),g(x)]=[{\mathbf{r}}(s),s\circ g(x)]=[{\mathbf{r}}(s),g(s\cdot x)].$$ On the other hand, $f(s\cdot x)=[{\mathbf{r}}(s),g(s\cdot x)]$ holds by . All that remains is to note that the equality $g=\beta_{\nu} f$ is a direct consequence of the definitions of $f$ and $\beta_{\nu}$. Transitive representations of $S$ as directed functors on $L(S)$ {#sub:3.2} ---------------------------------------------------------------- \[prop:trans\] The equivalence in Corollary \[cor:strict\] restricts to an equivalence between the category of transitive $S$-sets and directed pullback preserving functors on $L(S)$. Let $(X,\mu)$, $(s,x)\mapsto s\cdot x$, if defined, be a transitive $S$-set. We show that the functor $\Phi(X,\mu)$ is directed. Let $(e,x)$ and $(f,y)$ be objects of the category of elements $\int_{L(S)}\Phi(X,\mu)$ of $\Phi(X,\mu)$ and $s\in S$ be such that $s\cdot x=y$. We put $t=fse$ and observe that $t\cdot x=y$ and also ${\mathbf{d}}(t)\leq e$, ${\mathbf{r}}(t)\leq f$. Observe that $({\mathbf{d}}(t),x)$ is an object of the category $\int_{L(S)}\Phi(X,\mu)$. Since $e\cdot x=x$, the arrow $(e,{\mathbf{d}}(t))$ of $L(S)$ is an arrow from $({\mathbf{d}}(t),x)$ to $(e,x)$ of the category $\int_{L(S)}\Phi(X,\mu)$. Since $t\cdot x=y$, the arrow $(f,t)$ is an arrow from $({\mathbf{d}}(t),x)$ to $(f,y)$ of the category $\int_{L(S)}\Phi(X,\mu)$. It follows that the functor $\Phi(X,\mu)$ is directed. In the reverse direction, assume that the functor $\Phi(X,\mu)$ is directed and let $x,y\in X$. Let $e,f\in E(S)$ be such that $e\cdot x$ and $f\cdot y$ are defined (such $e$ and $f$ exist since $X$ is effective: for some $s$ we have that $s\cdot x$ is defined, but then ${\mathbf d}(s)\cdot x$ is defined, as well). Since the category $\int_{L(S)}\Phi(X,\mu)$ is directed, there are $z\in X$ and $g\in E(S)$ such that $g\cdot z$ is defined, and in the category $\int_{L(S)}\Phi(X,\mu)$ there are arrows $(e,x)\leftarrow (g,z)\to (f,y)$. The arrow $(g,z)\to (f,y)$ is by definition an arrow $(f,s)$ in $L(S)$ from $g$ to $f$ such that $s\cdot z=y$. Likewise, the arrow $(g,z)\to (e,x)$ is an arrow $(e,t)$ in $L(S)$ from $g$ to $e$ such that $t\cdot z=x$. It follows that $st^{-1}\cdot x=y$ which implies that the action is transitive. We now briefly recall the classical result due to Boris Schein [@Sch] (see also [@H; @LMS]) of the structure of transitive $S$-sets[^2]. An inverse subemigroup $H$ of $S$ is called [*closed*]{} if it is upward closed as a subset of $S$, i.e. $H^{\uparrow}=H$. Let $H$ be a closed inverse subsemigroup of $S$. A [*coset*]{} with respect to $H$ is a set $(xH)^{\uparrow}$ where ${\mathbf{d}}(x)\in H$. Let $X_H$ be the set of cosets with respect to $H$. Define the structure of an $S$-set on $X_H$ by putting $s\cdot (xH)^{\uparrow}$ is defined if and only if $(sxH)^{\uparrow}$ is a coset in which case $$\label{eq:structure} s\cdot (xH)^{\uparrow}=(sxH)^{\uparrow}.$$ The obtained $S$-set $X_H$ is transitive and any transitive $S$-set is equivalent to one so constructed. [*It follows that Proposition \[prop:trans\] provides a link, which was not previously explicitly mentioned in the literature, between closed inverse subsemigroups of $S$ and directed and pullback preserving functors on $L(S)$.*]{} Universal representations and filtered functors on $L(S)$ {#sub:3.3} --------------------------------------------------------- Let $H$ be a closed inverse subsemigroup of $S$. Recall that a [*filter*]{} in a semilattice is an upward closed subset $F$ such that $a\wedge b\in F$ whenever $a,b\in F$. Since the meet in $E(S)$ coincides with the product of idempotents, it follows that $E(H)$ is a filter in $E(S)$. Since $H$ is closed, $H\supseteq E(H)^{\uparrow}$ always holds. On the other hand, for any filter $F$ in $E(H)$ we have that $F^{\uparrow}$ is a closed inverse subsemigroup of $S$. An $S$-set $(X,\mu)$ is called [*universal*]{} [@LMS], if it is equivalent to a representation of $S$ on cosets with respect to a closed inverse subsemigroup $F^{\uparrow}$, where $F$ is a filter in $E(S)$. The following result is mentioned without proof in [@LMS]. We provide a proof for completeness. \[prop:torsors\] An $S$-set $(X,\mu)$ is an $S$-torsor if and only if it is universal. Let $(X,\mu)$, $(s,x)\mapsto s\cdot x$, where defined, be an $S$-set. Let $x\in X$ and put $$H=\{s\in S\colon s\cdot x \text{ is defined and } s\cdot x=x\}.$$ Then $H$ is a closed inverse subsemigroup of $S$, and $(X,\mu)$ is equivalent to the structure of an $S$-set, $(X_H,\nu)$, given in , on the set $X_H$ of cosets with respect to $H$. We may thus assume that $(X,\mu)=(X_H,\nu)$. Assume that $(X_H,\nu)$ is an $S$-torsor. We show that $H=E(H)^{\uparrow}$. It is enough to verify that $H\subseteq E(H)^{\uparrow}$. Let $s\in H$. Since $(X_H,\nu)$ is free, the equalities $$s\cdot x = {\mathbf d}(s)\cdot x=x$$ imply that there is some $c\leq s, {\mathbf d}(s)$ such that $c\cdot x=x$. Therefore $c\in E(H)$ and $s\geq c$, so that we have the inclusion $H\subseteq E(H)^{\uparrow}$. Conversely, assume that $(X_H,\mu)$ is universal and let $s,t\in S$ and $x\in X_H$ be such that $s\cdot x=t\cdot x$. Then there are some $e,f\in E(H)$ such that $s\geq e$, $t\geq f$ such that $e\cdot x$ and $f\cdot x$ are defined, and then of course $e\cdot x=f\cdot x=x$. We put $h=ef$. Then $s,t\geq h$ and $h\cdot x=x$, so that $(X_H,\mu)$ is an $S$-torsor. The following result follows from Proposition \[prop:torsors\] and [@FH Proposition 3.9] stated there without proof. \[prop:ff\] The equivalence in Proposition \[prop:trans\] restricts to an equivalence between the category of universal $S$-sets and the category of filtered functors on $L(S)$. Consequently, the category of points of the topos ${\mathcal B}(S)$ is equivalent to the category of universal $S$-sets. Let $(X,\mu)$ be a universal $S$-set. Assume that we have two objects $(e,x)$ and $(f,y)$ and two arrows $$(e,x) {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle(f,s)$} \settowidth{\@tempdimb}{$\scriptstyle(f,t)$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \rightarrowfill\cr}}}\limits^{\!(f,s)}_{\!(f,t)}}} (f,y)$$ in the category of elements $\int_{L(S)}\Phi(X,\mu)$. This implies that ${\mathbf d}(s)={\mathbf d}(t)=e$ and $s\cdot x=t\cdot x=y$. Since $(X,\mu)$ is free, there is $c\leq c,s$ such that $c\cdot x=y$. Since $c\leq s$ we have that $c=sg$ for some $g\in E(S)$ where we may assume that $g\leq e$. Then ${\mathbf d}(c)=g$. This and $c\leq t$ yield that $c=tg$. It follows that there is an arrow $$(g,x)\stackrel{(e,g)}{\longrightarrow} (e,x)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. Since $(f,s)(e,g)=(f,c)=(f,t)(e,g)$ in $L(S)$, the diagram $$(g,x) \stackrel{(e,g)}{\longrightarrow}(e,x) {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle(f,s)$} \settowidth{\@tempdimb}{$\scriptstyle(f,t)$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \rightarrowfill\cr}}}\limits^{\!(f,s)}_{\!(f,t)}}} (f,y)$$ is commutative. Therefore, the functor $\Phi(X,\mu)$ satisfies axiom (F3) from the definition of a filtered functor (see Subsection \[subs:2.5\]). It satisfies (F1) and (F2) due to Proposition \[prop:trans\], since universal $S$-sets are transitive. Conversely, let $(X,\mu)$ be an $S$-set such that the functor $\Phi(X,\mu)$ is filtered. Assume that $s,t\in S$ and $x\in X$ are such that $s\cdot x=t\cdot x$. Let $e={\mathbf d}(s){\mathbf d}(t)$ and $h={\mathbf r}(s){\mathbf r}(t)$. Then $hse\cdot x=hte\cdot x$ and also ${\mathbf d}(hse)={\mathbf d}(hte)$, ${\mathbf r}(hse)={\mathbf r}(hte)$. We put $p={\mathbf d}(hse)$ and $q={\mathbf r}(hse)$. It follows that in the category $\int_{L(S)}\Phi(X,\mu)$ we have two parallel arrows $$(p,x) {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle(q,hse)$} \settowidth{\@tempdimb}{$\scriptstyle(q,hte)$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y).$$ By axiom (F3), there is a commutative diagram $$(r,z) \stackrel{(p,a)}\longrightarrow (p,x) {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle(q,hse)$} \settowidth{\@tempdimb}{$\scriptstyle(q,hte)$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. This means that $hsea=htea$. Then $hse{\mathbf r}(a)=hte{\mathbf r}(a)$ and also $$hse{\mathbf r}(a)\cdot y =hte{\mathbf r}(a)\cdot y = z,$$ which proves that $(X,\mu)$ is free. Applying Proposition \[prop:trans\] (noting that filtered functors preserve pullbacks) and Proposition \[prop:torsors\], we conclude that $(X,\mu)$ is an $S$-torsor. Principal bundles over inverse semigroups {#sec:bundles} ========================================= In this section, we obtain an equivalence between the category of universal representations of an inverse semigroup on étale spaces over a topological space $X$ and the category of principal $L(S)$-bundles over $X$. This extends the well known result for groups [@MM VIII.1, VIII.2], and also is an analogue of Proposition \[prop:ff\], if in the latter one replaces the topos of sets by the topos ${\mathsf{Sh}}(X)$. The following definition is taken from [@M]. Let $X$ be a topological space and ${\mathcal C}$ a small category. A functor $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is called a ${\mathcal C}$-[*bundle*]{}. If $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is a ${\mathcal C}$-[*bundle*]{}, $\alpha\colon c\to d$ is an arrow in ${\mathcal C}$ and $y\in E(c)$, we put $$\alpha\cdot y=E(\alpha)(y)\in E(d).$$ A ${\mathcal C}$-bundle $E$ is called [*principal*]{}, if for each point $x\in X$ the following axioms are satisfied by the stalks $E(C)_x$: 1. (non-empty) There is an object $c$ of $C$ such that $E(c)_x\neq\varnothing$; 2. (transitive) For any $y\in E(c)_x$ and $z\in E(d)_x$, there are arrows $\alpha\colon b\to c$ and $\beta\colon b\to d$ for some object $b$ of $C$, and a point $w\in E(b)_x$, so that $\alpha\cdot w=y$ and $\beta\cdot w=z$. 3. (free) For any two parallel arrows $\alpha,\beta\colon c\rightrightarrows d$ and any $y\in E(c)_x$, for which $\alpha\cdot y=\beta\cdot y$, there exists an arrow $\gamma\colon b\to c$ and a point $z\in E(b)_x$ so that $\alpha\gamma=\beta\gamma$ and $\gamma\cdot z=y$. Principal ${\mathcal C}$-bundles are known to coincide with filtered functors from ${\mathcal C}$ to the topos ${\mathsf{Sh}}(X)$. It is immediate that given a principal ${\mathcal C}$-bundle $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ and $x\in X$, the induced restriction to stalk functor $E_x\colon {\mathcal C}\to {\mathsf{Sets}}$, $c\mapsto E(c)_x$, is a filtered functor. For two principal ${\mathcal C}$-bundles $E$ and $E'$, a [*morphism*]{} from $E$ to $E'$ is simply a natural transformation $\varphi\colon E\to E'$, that is, a collection of sheaf maps $\varphi_c\colon E(c)\to E'(c)$, where $c$ runs through objects of ${\mathcal C}$, such that for each arrow $\alpha\colon c\to d$ in ${\mathcal C}$ and each $y\in E(c)$ we have that $\varphi_d(\alpha\cdot y)=\alpha\cdot\varphi_c(y)$. We have therefore defined the category ${\mathsf{Prin}}({\mathcal C}, X)$ of [*principal bundles*]{} over ${\mathcal C}$. In the case where ${\mathcal C}$ is Loganathan’s category $L(S)$ for an inverse semigroup $S$, we will call a principal bundle over $L(S)$ a [*principal bundle over*]{} $S$ and the category ${\mathsf{Prin}}(L(S), X)$ the category of [*principal bundles over*]{} $S$. We will write ${\mathsf{Prin}}(S, X)$ for ${\mathsf{Prin}}(L(S), X)$. We now define the notion of a [*universal*]{} $S$-[*set*]{} in the topos ${\mathsf{Sh}}(X)$. Let $\pi \colon E\to X$ be an étale space and assume that a structure of an $S$-set $(E,\mu)$, $(s,x)\mapsto s\cdot x$, if defined, is given on $E$ such that the following conditions are met: 1. (effective on each stalk) For any $x\in X$, there is at least one point $e\in E_x$ such that $s\cdot e$ is defined for some $s\in S$ (this in particular implies that all stalks are non-empty, that is, the map $\pi$ is surjective). 2. (domains are open) For any $s\in S$ the set $\{e\in E\colon s\cdot e \text{ is defined}\}$ is open. 3. (stalks are invariant) For any $s\in S$ and $e\in E$, if $s\cdot e$ is defined then $\pi(s\cdot e)=\pi(e)$. 4. (universal on stalks) For any $s\in S$ and $x\in X$, $(E_x,\mu|_{S\times E_x})$ (which is well-defined by (U3)) is a universal $S$-set. 5. (continuous) The partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous ($S$ is considered as a discrete space and $S\times E$ as a product space). It is easy to see that (U4) implies (U1), so that (U1) may be omitted from the above list. Let $\pi \colon E\to X$, $\pi'\colon E'\to X$ be étale spaces and $(E,\mu)$, $(s,e)\mapsto s\cdot x$, if defined, $(E',\nu)$, $(s,e)\mapsto s\circ x$, if defined, be universal $S$-sets in the topos ${\mathsf{Sh}}(X)$. A morphism $$f\colon (E,\mu) \to (E',\nu)$$ is defined as a morphism $f\colon E\to E'$ of étale spaces (that is, a continuous map such that $\pi=\pi'f$, cf. [@MM]) which is simultaneously a morphism of $S$-sets (that is, if $s\cdot e$ is defined then $s\circ f(e)$ is defined and $f(s\cdot x)=s\circ f(x)$). We denote the category of universal $S$-sets in the topos ${\mathsf{Sh}}(X)$ by ${\mathsf{Univ}}(S,X)$. \[th:sheaves\] There is an equivalence of categories $${\mathsf{Prin}}(S, X) \,\, {\mathrel{ \settowidth{\@tempdima}{$\scriptstyle\tau$} \settowidth{\@tempdimb}{$\scriptstyle\rho$} \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi \mathop{\vcenter{ \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr \rightarrowfill\cr\noalign{\kern.3ex} \leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}} \,\, {\mathsf{Univ}}(S,X).$$ We begin with the construction of the functor $$\tau\colon {\mathsf{Prin}}(S, X)\to {\mathsf{Univ}}(S,X).$$ Let $E\colon L(S) \to {\mathsf{Sh}}(X)$ be a principal bundle over $X$ and $x\in X$. We first describe the colimit sheaf $\widetilde{E}\in {\mathsf{Sh}}(X)$. By definition, for each $x\in X$ we have a filtered functor $f_x\colon L(S)\to {\mathsf{Sets}}$ obtained by restricting $E$ to the stalks over $x$. We now apply the functor $\Psi$ to each $f_x$. Proposition \[prop:ff\] ensures us that each $\Psi(f_x)$ is a universal $S$-set. Note that each of the sets $\Psi(f_x)$ is non-empty by (PB1). As the stalks of the colimit sheaf are colimits of stalks, we may set $$\widetilde{E}=\bigcup_{x\in X} \Psi(f_x)$$ to be a disjoint union of all the sets $\Psi(f_x)$. The projection map $$p\colon \widetilde{E}\to X$$ given by $y\mapsto x$ if $y\in \Psi(f_x)$. We may identify each space $E(e)$ with its image under the inclusion into $\widetilde{E}$. The colimit topology on $\widetilde{E}$ is the finest topology which makes all inclusion maps $E(e)\hookrightarrow \widetilde{E}$ continuous. The base of this topology is formed by the sets $A\subseteq \widetilde{E}$ such that $A\subseteq E(e)$ for some $e$. The structure of an $S$-set, $(s,y)\mapsto s*y$, where defined, on $\widetilde{E}$ is induced by the structures of $S$-sets on each on $\Psi(f_x)$, given by . We now prove that for each $s\in S$ the set $$D_s=\{y\in \widetilde{E}\colon s*y \text{ is defined}\}$$ is open. Clearly, $D_s=D_{{\mathbf d}(s)}$. Let $y\in D_{{\mathbf d}(s)}$ and $A$ be a neighbourhood of $y$ in $\widetilde{E}$. Since the inclusion map $i\colon E_{{\mathbf{d}}(s)}\hookrightarrow \widetilde{E}$ is open, we have that $ii^{-1}(A)$ is a neighbourhood of $y$ in $\widetilde{E}$ which is contained in $D_{{\mathbf d}(s)}$. This implies that the set $D_{{\mathbf d}(s)}$ is open. Using the fact that the translation maps are continuous, it is routine to verify that the partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous. We now define $\tau$ on morphisms. Let $E$ and $E'$ be principal $S$-bundles and $\varphi\colon E\to E'$ be a natural transformation. The family of continuous maps $E(e)\to E'(e)$ for each $e\in E(S)$ and the construction of $\widetilde{E}$ yield a continuous map $\tau(\varphi)\colon \widetilde{E}\to \widetilde{E'}$ which obviously satisfies the definition of a morphism of universal representations. We now turn to the construction of the functor $\rho\colon {\mathsf{Univ}}(S,X) \to {\mathsf{Prin}}(S, X)$. Let $p\colon \widetilde{E}\to X$ be an étale space and $(\widetilde{E},\mu)$, $(e,x)\mapsto e*x$, if defined, a structure of an $S$-set on $\widetilde{E}$ which satisfies (U1) – (U5). We fix $e\in E(S)$ and let $$E(e)=\{y\in \widetilde{E}\colon e*y\text{ is defined}\}.$$ Define $p_e\colon E(e)\to X$ to be the restriction of the map $p$ to $E(e)$. Clearly, $p_e$ is a local homeomorphism. By Proposition \[prop:ff\], for each $x\in X$, the restriction of $*$ to $E_x$ gives rise to a filtered functor $\Phi(\widetilde{E},\mu)_x\colon L(S)\to {\mathsf{Sets}}$ and it is routine to verify that these give rise to a filtered functor $\rho(p\colon \widetilde{E}\to X)\colon L(S)\to {\mathsf{Sh}}(X)$. To define $\rho$ on morphisms, we observe that a morphism $\psi \colon E\to E'$ of universal representations yields a family of maps $E(e)\to E'(e)$ for each $e\in E(S)$. By construction, these maps are continuous and are components of a natural transformation from $\rho(E)$ to $\rho(E')$. It follows that $S$-torsors in the topos ${\mathsf{Sh}}(X)$ can be defined as universal $S$-bundles. As a direct consequence of Theorem \[th:equiv\] and an analogue of Theorem \[th:filt\] for the topos ${\mathsf{Sh}}(X)$, we obtain the following result. The category of geometric morphisms ${\mathsf{Geom}}({\mathsf{Sh}}(X), {\mathcal{B}}(S))$ is equivalent to the category ${\mathsf{Univ}}(S,X)$ of universal $S$-bundles in the topos ${\mathsf{Sh}}(X)$. *Let $S$ be an inverse semigroup. A [*filter*]{} in $S$ is a filter with respect to the natural partial order in $S$, that is, a nonempty subset $F$ of $S$ such that* 1. $a\in F$ and $b\geq a$ imply that $b\in S$; 2. if $a,b\in F$ then there is $c\in F$ such that $c\leq a,b$. Let $E=E(S)$ and $\hat{E}$ denote the set of filters in $E$. A filter $F$ in $E$ defines a nonzero semilattice homomorphism, called a [*semi-character*]{}, $\varphi_F\colon E\to \{0,1\}$ such the inverse image of $1$ is $F$, and conversely, any nonzero semilattice homomorphism $\varphi\colon E\to \{0,1\}$ defines a filter $\varphi^{-1}(1)$ in $E$. These assignments are mutually inverse, so that the elements of $\hat{E}$ can be equivalently looked at as semi-characters. The space $\hat{E}$ is topologized as a subspace of the product space $\{0,1\}^E$ where $\{0,1\}$ is a discrete space. The space $\hat{E}$ is locally compact and is known as the [*filter space*]{} or the [*semi-character space*]{} of $S$. Let ${\mathcal G}$ denote the set of filters in $S$. For $s\in S$ let $M(s)=\{F\in {\mathcal G}\colon s\in F\}.$ The set ${\mathcal G}$ is topologized by letting the sets $M(s)\cap M(s_1)^c \cap \dots \cap M(s_n)^c$, where $s,s_1,\dots s_n$, $n\geq 0$, to be a base of the topology. The connection between filters in $S$ and filters in $E$ was studied in detail in [@LMS]. If $F\in {\mathcal G}$, the assignment $$\mathrm{d}(F)=\{\mathbf{d}(a)\colon a\in F\}\in \hat{E}$$ defines a map $\mathrm{d}\colon {\mathcal G}\to \hat{E}$ which is a local homeomorphism (in fact, this map is equivalent to the domain map of the [*groupoid of filters*]{} of $S$). We have an action of $S$ on each stalk ${\mathcal G}_F$ of the étale space $({\mathcal G}, {\mathrm{d}}, X)$ which is just the universal action of $S$ on the set of cosets with respect to the closed inverse subsemigroup $F^{\uparrow}$. It is routine to verify that these actions define on ${\mathcal G}$ the structure of a universal $S$-bundle which is natural to call the [*universal bundle associated to the domain map of the groupoid of filters of*]{} $S$. Towards actions of inverse semigroups in an arbitrary topos {#sec:fin} =========================================================== In [@FH Definition 2.14], Funk and Hofstra proposed a way to define a notion of a torsor for an arbitrary inverse semigroup $S$ in an arbitrary (Grothendieck) topos. Their definition [@FH Definition 2.14] is based on the concept of a semigroup $S$-set in an arbitrary topos: for an inverse semigroup $S$ they consider an internal semigroup $\Delta(S)$. In the topos of sets, a semigroup $S$-set $X$ is a (pre)homomorphism from $S$ to the partial transformation semigroup ${\mathcal{PT}}(X)$ on $X$. For $S$ inverse, only semigroup $S$-sets for which the action is by partial bijections should be considered. The diagrammatic definition of partial bijections is not written in [@FH], but can be done. Omitting the requirement of partial bijections leads to an incorrect claim in [@FH]. Section 6 of [@FH] discusses the actions of inverse semigroups in an arbitrary topos. However, the claim ‘If $T$ is inverse, then a $T$-set $T\to M(X)$ necessarily factors through $I(X)\subseteq M(X)$’ is incorrect (where $I(X)$ is ‘the object of partial bijections’). We now provide an example that, for the topos of sets where the meaning of ${\mathcal I}(X)$ is clear (the symmetric inverse semigroup on $X$), shows the claim to be incorrect. \[ex:counterexample\][*Let $S$ be a linearly ordered set considered as a semilattice and let $|S|>1$. The map $\mu\colon S \to {\mathcal{PT}}(S)$ given by $x\mapsto \varphi_x$, where $\varphi_x(y)= x\wedge y$, $y\in S$ is a homomorphism, but this is not an inverse semigroup $S$-set as the action is not by partial bijections. It is also easy to check that $\mu$ is free and transitive according to [@FH Definition 2.8] and [@FH Definition 2.14].* ]{} If $S$ is an inverse semigroup, then the internal semigroup $\Delta(S)$ can be readily endowed with the structure of an ‘internal inverse semigroup’ as $\Delta$ preserves the logic needed to express the fact of being an inverse semigroup (e.g. the varietal definition). Thus the approach taken in [@FH] of internalizing $S$ as a semigroup looks simpler than the possible internalizing it as an inverse semigroup. We, however, believe, that it is more natural, e.g., from the perspective of the cohomology theory [@Log], to keep the idempotents of $S$ and to internalize ${\mathcal H}$-classes of $S$ and the connection between them. This is the approach we outline below. Let ${\mathcal{E}}$ be a (Grothendieck) topos and $S$ an inverse semigroup. We define an action of an inverse semigroup in an arbitrary topos which arises from a functor $L(S)\to {\mathcal{E}}$. Then classes of functors $L(S)\to {\mathcal{E}}$ (such as pullback preserving functors, torsion-free functors or filtered functors) can be connected with respective classes of actions of $S$ in ${\mathcal{E}}$. In particular, actions connected to filtered functors, can be naturally called $S$-torsors. Let $e,f\in E(S)$ be such that $e\mathrel{\mathcal D} f$. By $H(e,f)$ be denote the ${\mathcal H}$-class of $S$ which consists of all $s\in S$ satisfying ${\mathbf{d}}(s)=f$ and ${\mathbf{r}}(s)=e$. Note that any ${\mathcal H}$-class of $S$ is of the form $H(e,f)$ for some $e\mathrel{\mathcal D} f$. We can bring all the sets $H(e,f)$ up to ${\mathcal{E}}$ by considering their images $\Delta H(e,f)$ under the constant sheaf functor $\Delta$. Let $A\colon L(S)\to {\mathcal{E}}$ be a functor. The colimit construction, given in [@FH] for the topos of sets, extends to ${\mathcal{E}}$, and we construct the colimit object ${\mathcal X}$ of the composite of the functors $$E(S)\to L(S)\stackrel{A}{\to} {\mathcal{E}}.$$ In particular, all objects $A(e)$ are subobjects of ${\mathcal X}$. Therefore, a morphism $A(e)\to A(f)$ in ${\mathcal{E}}$ can be thought of as a ‘partial’ morphism of ${\mathcal X}$. Note that if $S$ is a monoid with unit $1$ then we have ${\mathcal X}=A(1)$. The restriction of $A$ to $E(S)$ gives us a functor $E(S)\to {\mathcal{E}}$. If $S$ is a group, this functor just selects an objects in ${\mathcal{E}}$, in particular, the object ${\mathcal X}$. Reasoning similarly as in [@MM p. 432], we see that for each ${\mathcal H}$-class $H(e,f)$ the functor $A$ gives rise to a map $$\begin{gathered} \label{eq:manipulation} H(e,f)\to {\mathrm{Hom}}_{{\mathcal{E}}}(A(f),A(e)) \simeq {\mathrm{Hom}}_{{\mathcal{E}}}(1, A(e)^{A(f)})\simeq \\ {\mathrm{Hom}}_{{\mathcal{E}}}(\Delta 1, A(e)^{A(f)})\simeq {\mathrm{Hom}}_{\mathsf{Sets}}(1, \Gamma(A(e)^{A(f)}))\simeq \Gamma(A(e)^{A(f)})\end{gathered}$$ (here $1$ denotes the terminal object of ${\mathcal E}$). We obtain the map $$\label{eq:manip1} \Delta H(e,f) \to A(e)^{A(f)},$$ and applying the adjunction between product and exponentiation, $$\label{eq:manip2} \Delta H(e,f) \times A(f)\to A(e).$$ We recall that every morphism in $L(S)$ is a composition of some $({\mathbf{r}}(s),s)$ and some $(e,f)$, where $e,f\in E(S)$. Therefore, a functor $A\colon L(S)\to {\mathcal E}$ is determined by its restriction to $E(S)$ and by translations along isomorphisms $({\mathbf{r}}(s),s)$. The restriction of $A$ to $E(S)$ in the group case degenerates to selecting an object in ${\mathcal E}$, so it is natural to keep this restriction as a part of the definition of an $S$-set associated to $A$ in ${\mathcal E}$. The translations along isomorphisms are internalized using , and . Acknowledgements {#acknowledgements .unnumbered} ================ We thank Andrej Bauer and Alex Simpson for useful discussions. We are also grateful to Jonathon Funk and Pieter Hofstra for helpful communication, to the referee for their comments, as well as the editor for facilitating a fruitful discussion. [99]{} S. Awodey, [*Category theory*]{}, Oxford University Press, 2006. J. Funk, Semigroups and toposes, [*Semigroup Forum*]{} [**75**]{} (2007), no. 3, 481–520. J. Funk, P. Hofstra, Topos theoretic aspects of semigroups actions, [*Theory Appl. Cat.*]{} [**24**]{} (2010), no.7, 117–147. J. Funk, P. Hofstra, private communication. J. Funk, M. V. Lawson, B. Steinberg, Characterizations of Morita equivalent inverse semigroups, [*J. Pure Appl. Algebra*]{} [**215**]{} (2011), 2262–2279. J. Funk, B. Steinberg, The universal covering of an inverse semigroup, [*Appl. Categ. Structures*]{} [**18**]{} (2010), no. 2, 135–163. J. M. Howie, [*Fundamentals of semigroup theory*]{}, The Clarendon Press, Oxford University Press, New York, 1995. G. Kudryavtseva, M. V. Lawson. The classifying space of an inverse semigroup, [*Period. Math. Hungar.*]{}, [**70**]{} (1) (2015), 122–129. M. V. Lawson, [*Inverse semigroups: the theory of partial symmetries*]{}, World Scientific Publishing Co., Inc., NJ, 1998. M. V. Lawson, S. W. Margolis and B. Steinberg, The étale groupoid of an inverse semigroup as a groupoid of filters, [*J. Aust. Math. Soc.*]{} [**94**]{} (2014), 234–256. J. Leech, $\mathcal{H}$-coextensions of monoids, [*Mem. Amer. Math. Soc.*]{} [**1**]{} (2), no. 157 (1975), 1–66. M. Loganathan, Cohomology of inverse semigroups, [*J. Algebra*]{} [**70**]{} (1981), 375–393. S. Mac Lane, [*Categories for the working mathematician,*]{} Grad. Texts in Math., vol. 5, Springer-Verlag, 1998. S. Mac Lane, I. Moerdijk, [*Sheaves in geometry and logic. A first introduction to topos theory*]{}, Springer-Verlag, 1994. I. Moerdijk, [*Classifying spaces and classifying topoi*]{}, Lecture Notes in Math 1616, Springer-Verlag, 1995. B. M. Schein, Representations of generalized groups, [*Izv. Vyssh. Uchebn. Zaved Mat.*]{} [**28**]{} (3) (1962), 164–176 (in Russian). B. Steinberg, Strong Morita equivalence of inverse semigroups, [*Houston J. Math.*]{} [**37**]{} (2011), 895–927. [^1]: Note that in [@FH] non-strict $S$-sets are referred to as $S$-sets, and $S$-sets are referred to as strict $S$-sets. [^2]: We recall our convention that all $S$-sets are effective.

ArXiv

--- address: | Department of Mathematics, The University of Texas at San Antonio\ San Antonio, TX 78249, USA author: - Gelu Popescu date: 'December 27, 2005' title: ' Free holomorphic functions on the unit ball of $B({{\mathcal H}})^n$' --- [^1] Contents {#contents .unnumbered} ======== ** Introduction 1. Free holomorphic functions and Hausdorff derivations 2. Cauchy, Liouville, and Schwartz type results for free holomorphic functions 3. Algebras of free holomorphic functions 4. Free analytic functional calculus and noncommutative Cauchy transforms 5. Weierstrass and Montel theorems for free holomorphic functions 6. Free pluriharmonic functions and noncommutative Poisson transforms 7. Hardy spaces of free holomorphic functions References Introduction {#introduction .unnumbered} ============ The Shilov-Arens-Calderon theorem ([@S], [@AC]) states that if $a_1,\ldots, a_n$ are elements of a commutative Banach algebra $A$ with the joint spectrum included in a domain $\Omega\subset {{\mathbb C}}^n$, then the algebra homomorphism $${{\mathbb C}}[z_1,\ldots,z_n]\ni p\mapsto p(a_1,\ldots, a_n)\in A$$ extends to a continuous homomorphism from the algebra $Hol(\Omega)$, of holomorphic functions on $\Omega$, to the algebra $A$. This result was greatly improved by the pioneering work of J.L. Taylor ([@T1], [@T2], [@T3]) who introduced a “better” notion of joint spectrum for $n$-tuples of commuting operators, which is now called Taylor spectrum, and developed an analytic functional calculus. Stated for the open unit ball of ${{\mathbb C}}^n$, $${{\mathbb B}}_n:=\{(\lambda_1,\ldots, \lambda_n)\in {{\mathbb C}}^n: \ |\lambda|^2+\cdots +|\lambda_n|^2<1\},$$ his result states that if $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is an $n$-tuple of commuting bounded linear operators on a Hilbert space ${{\mathcal H}}$ with Taylor spectrum $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$, then there is a unique continuous unital algebra homomorphism $$Hol({{\mathbb B}}_n)\ni f\mapsto f(T_1,\ldots, T_n)\in B({{\mathcal H}})$$ such that $z_i\mapsto T_i$,  $i=1,\ldots,n$. Due to a result of V. M" uller [@M], the condition that $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$ is equivalent to the fact that the joint spectral radius $$r(T_1,\ldots, T_n):=\lim_{k\to\infty}\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2k}<1.$$ F.H. Vasilescu introduced and studied, in [@Va] and a joint paper with R.E. Curto [@CV], operator-valued Cauchy and Poisson transforms on the unit ball ${{\mathbb B}}_n$ associated with commuting operators with $r(T_1,\ldots, T_n)<1$, in connection with commutative multivariable dilation theory. In recent years, there has been exciting progress in noncommutative multivariable operator theory regarding noncommutative dilation theory ([@F], [@B], [@Po-models], [@Po-isometric], [@Po-charact], [@DKS], [@BB], [@BBD], [@Po-unitary], [@Po-varieties], etc.) and its applications concerning interpolation in several variables ([@Po-charact], [@Po-analytic], [@Po-interpo], [@ArPo2], [@DP], [@BTV], [@Po-entropy], etc.) and unitary invariants for $n$-tuples of operators ([@Po-charact], [@Arv], [@Arv2], [@Po-curvature], [@Kr], [@Po-similarity], [@BT], [@Po-entropy], [@Po-unitary], etc.). Our program to develop a [*free*]{} analogue of Sz.-Nagy–Foiaş theory [@SzF-book], for row contractions, fits perfectly the setting of the present paper, which includes that of free holomorphic functions on the open operatorial unit ball $$[B({{\mathcal H}})^n]_1:=\left\{ [X_1,\ldots, X_n]\in B({{\mathcal H}})^n: \ \|X_1X_n^*+\cdots + X_nX_n^*\|<1\right\}.$$ The present work is an attempt to develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary $n$-tuples of operators, and to introduce and study a free analytic functional calculus in connection with Hausdorff derivations, noncommutative Cauchy and Poisson transforms, and von Neumann inequalities. In Section 1, we introduce a notion of radius of convergence for formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and prove noncommutative multivariable analogues of Abel theorem and Hadamard formula from complex analysis ([@Co], [@R]). This enables us to define, in particular, the algebra $Hol(B({{\mathcal X}})^n_1)$  of free holomorphic functions on the open operatorial unit $n$-ball, as the set of all power series $\sum_{\alpha\in {{\mathbb F}}_n^+}a_\alpha Z_\alpha$ with radius of convergence $\geq 1$. When $n=1$, $Hol(B({{\mathcal X}})^1_1)$ coincides with the algebra of all analytic functions on the open unit disc  ${{\mathbb D}}:=\{z\in {{\mathbb C}}:\ |z|<1\}$. The algebra of free holomorphic functions  $Hol(B({{\mathcal X}})^n_1)$  has the following universal property. [*Any strictly contractive representation $\pi: {{\mathbb C}}[Z_1,\ldots, Z_n]\to B({{\mathcal H}})$, i.e., $\|[\pi(Z_1),\ldots, \pi(Z_n)]\|<1$, extends uniquely to a representation of $Hol(B({{\mathcal X}})^n_1)$.*]{} A free holomorphic function on the open operatorial unit ball  $[B({{\mathcal H}})^n]_1$  is the representation of an element $F\in Hol(B({{\mathcal X}})^n_1)$ on the Hilbert space ${{\mathcal H}}$, that is, the mapping $$[B({{\mathcal H}})^n]_1\ni (X_1,\ldots, X_n)\mapsto F(X_1,\ldots, X_n)\in B({{\mathcal H}}).$$ As expected, we prove that any free holomorphic function is continuous on $[B({{\mathcal H}})^n]_1$ in the operator norm topology. In the last part of this section, we show that the Hausdorff derivations $\frac{\partial}{\partial Z_i}$, $i=1,\ldots, n$, on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ ([@MKS], [@RSS]) can be extended to the algebra of free holomorphic functions. In Section 2, we obtain Cauchy type estimates for the coefficients of free holomorphic functions and a Liouville type theorem for free entire functions. Based on a noncommutative version of Gleason’s problem [@R2], which is obtained here, and the noncommutative von Neumann inequality [@Po-von], we provide a free analogue of Schwartz lemma from complex analysis ([@Co], [@R]). In particular, we prove that if $f$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$ such that $\|f\|_\infty\leq 1$ and $f(0)=0$, then $$\|f(X_1,\ldots, X_n)\|\leq \|[X_1,\ldots, X_n]\|,\qquad r(f(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n),$$ and  $\sum_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2\leq 1$. In Section 3, following the classical case ([@H], [@RR]), we introduce two Banach algebras of free holomorphic functions, $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$, and prove that, together with a natural operator space structure, they are completely isometrically isomorphic to the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively, which were introduced in [@Po-von] in connection with a multivariable von Neumann inequality. We recall that the algebra $F_n^\infty$ (resp. ${{\mathcal A}}_n$) is the weakly (resp. norm) closed algebra generated by the left creation operators $S_1,\ldots, S_n$ on the full Fock space with $n$ generators, $F^2(H_n)$, and the identity. These algebras have been intensively studied in recent years by many authors ([@Po-charact], [@Po-multi], [@Po-von], [@Po-funct], [@Po-analytic], [@Po-disc], [@Po-poisson], [@Po-curvature], [@Po-similarity], [@ArPo], [@ArPo2], [@DP1], [@DP2], [@DKP], [@PPoS], [@Po-unitary]). The results of this section are used to obtain a maximum principle for free holomorphic functions. In Section 4, we provide a free analytic functional calculus for $n$-tuples $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We show that there is a continuous unital algebra homomorphism $$\Phi_T:Hol(B({{\mathcal X}})^n_1)\to B({{\mathcal H}}), \quad \Phi_T(f)=f(T_1,\ldots, T_n),$$ which is uniquely determined by the mapping $z_i\mapsto T_i$,  $i=1,\ldots,n$. (The continuity and uniqueness of $\Phi_T$ are proved in Section 5.) We introduce a noncommutative Cauchy transform ${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$ associated with any $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. The definition is based on the [*reconstruction operator*]{} $$S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*,$$ which has played an important role in noncommutative multivariable operator theory ([@Po-entropy], [@Po-unitary], [@Po-varieties]). We prove that $$f(T_1,\ldots, T_n)=C_T(f(S_1,\ldots, S_n)),\quad f\in H^\infty (B({{\mathcal X}})^n_1),$$ where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Hence, we deduce that $$\|f(T_1,\ldots, T_n)\|\leq M \|f\|_\infty$$ where $M:=\sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}$. Similar Cauchy representations are obtained for the $k$-order Hausdorff derivations of $f$. Finally, we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}({{\mathbb C}}^n)$. More precisely, we prove that $${{\mathcal C}}_T(\beta_U(f))={{\mathcal C}}_{\beta_U(T)}(f)\quad \text{ for any } \ U\in {{\mathcal U}}({{\mathbb C}}^n), \, f\in {{\mathcal A}}_n,$$ where $\beta_U$ denotes a natural isometric automorphism (generated by $U$) of the noncommutative disc algebra ${{\mathcal A}}_n$, or the open unit ball $[B({{\mathcal H}})^n]_1$. In Section 5, we obtain Weierstrass and Montel type theorems [@Co] for the algebra of free holomorphic functions on the open operatorial unit $n$-ball. This enables us to introduce a metric on $Hol(B({{\mathcal X}})^n_1)$ with respect to which it becomes a complete metric space, and the Hausdorff derivations $$\frac{\partial}{\partial Z_i}:Hol(B({{\mathcal X}})^n_1)\to Hol(B({{\mathcal X}})^n_1),\quad i=1,\ldots,n,$$ are continuous. In the end of this section, we prove the continuity and uniqueness of the free functional calculus for $n$-tuples of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Connections with the $F_n^\infty$-functional calculus for row contractions [@Po-funct] and, in the commutative case, with Taylor’s functional calculus [@T2] are also discussed. Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform [@Po-poisson] generates a function $$P[A]: [B({{\mathcal H}})^n]_1\to B({{\mathcal H}}).$$ In Section 6, we provide classes of operators $A\in B(F^2(H_n))$ such that $P[A]$ is a free holomorphic (resp. pluriharmonic) function on $[B({{\mathcal H}})^n]_1$. In this case, the operator $A$ can be regarded as the boundary function of the Poisson extension $P[A]$. Using some results from [@Po-von], [@Po-funct], and [@Po-poisson], we characterize the free holomorphic functions $u$ on the open unit ball $[B({{\mathcal H}})^n]_1$ such that $u=P[f]$ for some boundary function $f$ in the noncommutative analytic Toeplitz algebra $F_n^\infty$, or the noncommutative disc algebra ${{\mathcal A}}_n$. For example, we prove that there exists $f\in F_n^\infty$ such that $u=P[f]$ if and only if $$\sup_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|<\infty.$$ We also obtain noncommutative multivariable versions of Herglotz theorem and Dirichlet extension problem ([@Co], [@H]) for free pluriharmonic functions. In Section 7, we define the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$, as the set of all free holomorphic function $F$ such that $$\|F\|_p:=\left( \int_0^1\|F(rS_1,\ldots, rS_n)\|^p dr \right)^{1/p}<\infty,$$ and prove that it is a Banach space. Moreover, we show that $$\|f(T_1,\ldots, T_n)\|\leq \frac{1}{(1-\|[T_1,\ldots, T_n]\|)^{1/p}} \|f\|_p$$ for any $[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1$ and $f\in H^p(B({{\mathcal X}})^n_1)$. Finally, we introduce the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$ as the set of all holomorphic function on ${{\mathbb B}}_n$ such that $ \|f\|_{\text{\rm sym}}:= \|f_{\text{\rm sym}}\|_\infty<\infty, $ where $f_{\text{\rm sym}}\in Hol(B({{\mathcal X}})^n_1)$ is the symmetrized functional calculus of $f\in Hol({{\mathbb B}}_n)$. We prove that $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$ is a Banach space and $$\|f(T_1,\ldots, T_n)\|\leq M \|f_{\text{\rm sym}}\|_\infty,$$ for any commuting $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$. Several classical results from complex analysis are extended to our noncommutative multivariable setting. The present paper exhibits, in particular, a “very good” free analogue of the algebra of analytic functions on the open unit disc ${{\mathbb D}}$. This claim is also supported by the fact that numerous results in noncommutative multivariable operator theory ([@Po-von], [@Po-funct], [@Po-disc], [@Po-poisson], [@Po-unitary]) fit perfectly our setting and can be seen in a new light. We strongly believe that many other results in the theory of analytic functions have free analogues in our noncommutative multivariable setting. In a forthcoming paper [@Po-Bohr], we consider operator-valued Wiener and Bohr type inequalities for free holomorphic (resp. pluriharmonic) functions on the open operatorial unit $n$-ball. As consequences, we obtain operator-valued Bohr inequalities for the noncommutative Hardy algebra $H^\infty(B({{\mathcal X}})^n_1)$ and the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$. Free holomorphic functions {#free holomorphic} =========================== We introduce a notion of radius of convergence for formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and prove noncommutative multivariable analogues of Abel theorem and Hadamard formula. This enables us to define algebras of free holomorphic functions on open operatorial $n$-balls. We show that the Hausdorff derivations $\frac{\partial}{\partial Z_i}$, $i=1,\ldots, n$, on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ (see [@MKS], [@RSS]) can be extended to algebras of free holomorphic functions. Let ${{\mathbb F}}_n^+$ be the unital free semigroup on $n$ generators $g_1,\ldots, g_n$ and the identity $g_0$. The length of $\alpha\in {{\mathbb F}}_n^+$ is defined by $|\alpha|=0$ if $\alpha=g_0$ and $|\alpha|:=k$ if $\alpha=g_{i_1}\cdots g_{i_k}$, where $i_1,\ldots, i_k\in \{1,\ldots, n\}$. We consider formal power series in $n$ noncommuting indeterminates $Z_1,\ldots, Z_n$ and coefficients in $B({{\mathcal K}})$, the algebra of all bounded linear operators on the Hilbert space ${{\mathcal K}}$, of the form $$\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha,\quad A_{(\alpha)}\in B({{\mathcal K}}),$$ where $Z_\alpha:=Z_{i_1}\cdots Z_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$ and $Z_{g_0}:=I$. If $F=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha $ and $G=\sum_{\alpha\in {{\mathbb F}}_n^+} B_{(\alpha)}\otimes Z_\alpha $ are such formal power series, we define their sum and product by setting $$F+G:=\sum_{\alpha\in {{\mathbb F}}_n^+} (A_{(\alpha)}+B_{(\alpha)})\otimes Z_\alpha \quad \text{ and }\quad FG:=\sum_{\alpha\in {{\mathbb F}}_n^+} C_{(\alpha)}\otimes Z_\alpha,$$ respectively, where $C_{(\alpha)}:=\sum\limits_{\sigma, \beta\in {{\mathbb F}}_n^+:\ \alpha=\sigma \beta} A_{(\sigma)} B_{(\beta)}$. By abuse of notation, throughout this paper, we will denote by $[T_1,\ldots,T_n]$ either the $n$-tuple of operators $(T_1,\ldots, T_n)\in B({{\mathcal H}})^n$ or the row operator matrix $[T_1\,\cdots \,T_n]\in B({{\mathcal H}}^{(n)}, {{\mathcal H}})$ acting as an operator from ${{\mathcal H}}^{(n)}$, the direct sum of $n$ copies of the Hilbert space ${{\mathcal H}}$, to ${{\mathcal H}}$. We also denote by $[T_\alpha:\ |\alpha|=k]$ the row operator matrix acting from ${{\mathcal H}}^{n^k}$ to ${{\mathcal H}}$, where the entries are arranged in the lexicographic order of the free semigroup ${{\mathbb F}}_n^+$. In what follows we show that given a sequence of operators $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, there is a unique $R\in [0,\infty]$ such that the series $$\sum_{k=0}^\infty\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ converges in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$ (${{\mathcal K}}\otimes {{\mathcal H}}$ is the Hilbert tensor product) for any Hilbert space ${{\mathcal H}}$ and any $n$-tuple $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with $\|[X_1,\ldots, X_n]\|<R$, and it is divergent for some $n$-tuples $[Y_1,\ldots, Y_n]$ of operators with $\|[Y_1,\ldots, Y_n]\|>R$. The result can be regarded as a noncommutative multivariable analogue of Abel theorem and Hadamard’s formula from complex analysis. \[Abel\] Let ${{\mathcal H}}$, ${{\mathcal K}}$ be Hilbert spaces and let $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, be a sequence of operators. Define $R\in [0,\infty]$ by setting $$\frac {1} {R}:= \limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} A^*_{(\alpha)} A_{(\alpha)}\right\|^{\frac{1} {2k}}.$$ Then the following properties hold: 1. For any $n$-tuple of operators $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$, the series $\sum\limits_{k=0}^\infty \left\| \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right\| $ converges if  $\|[X_1,\ldots,X_n]\|<R$. Moreover, if $0\leq \rho<R$, then the convergence is uniform for $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots,X_n]\|\leq \rho$. 2. If $R<R'<\infty$ and ${{\mathcal H}}$ is infinite dimensional, then there is an $n$-tuple $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ of operators with $$\|X_1X_1^*+\cdots +X_nX_n^*\|^{1/2}=R'$$ such that $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ is divergent in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Moreover, the number $R$ satisfying properties (i) and (ii) is unique. Assume that $R>0$ and $[X_1,\ldots, X_n]$ is an $n$-tuple of operators on ${{\mathcal H}}$ such that $\|[X_1,\ldots,X_n]\|<R$. Let $\rho',\rho>0$ be such that $\|[X_1,\ldots,X_n]\|<\rho'<\rho<R$. Since $\frac{1}{\rho}> \frac{1}{R}$, we can find $m_0\in{{\mathbb N}}:=\{1,2,\ldots\}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2k}< \frac{1}{\rho}\quad \text{ for any }\ k\geq m_0.$$ Hence, we deduce that $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&= \left\|\left[ I\otimes X_\alpha:\ |\alpha|=k\right] \left[\begin{matrix} A_{(\alpha)}\otimes I\\ :\\|\alpha|=k \end{matrix}\right]\right\|\\ &=\left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}\\ &\leq \left\|\sum_{i=1}^nX_iX_i^*\right\|^{k/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}\\ &\leq \left(\frac{\rho'}{\rho}\right)^k \end{split}$$ for any $k\geq m_0$. This proves the convergence of the series $\sum\limits_{k=0}^\infty \left\| \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right\|$. Assume now that $0\leq \rho<R$ and $\|[X_1,\ldots, X_n]\|\leq \rho$. Choose $\gamma$ such that $0\leq \rho< \gamma<R$ and notice that, due to similar calculations as above, there exists $n_0\in {{\mathbb N}}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\| \leq \left(\frac{\rho}{\gamma}\right)^k$$ for any $(X_1,\ldots, X_n)$ with $\|[X_1,\ldots, X_n]\|\leq \rho$, and $k\geq n_0$, which proves the uniform convergence of the above series. The case $R=\infty$, can be treated in a similar manner. To prove part (ii), assume that $R<\infty$ and ${{\mathcal H}}$ is infinite dimensional. Let $R', \rho>0$ be such that $R<\rho< R'$ and define the operators $X_i:= R' V_i$,  $i=1,\ldots, n$, where $V_1,\ldots, V_n$ are isometries with orthogonal ranges. Notice that $\|[X_1,\ldots, X_n]\|=R'$ and $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&={R'}^k \left\|\left(\sum\limits_{|\alpha|=k}A_{(\alpha)}^*\otimes V_\alpha^*\right) \left(\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes V_\alpha\right)\right\|^{1/2}\\ &= {R'}^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^*A_{(\alpha)} \otimes I\right\|^{1/2}\\ &= {R'}^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}. \end{split}$$ On the other hand, since $\frac{1}{\rho}<\frac{1}{R}$, there are arbitrarily large $k\in {{\mathbb N}}$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}>\left(\frac{1}{\rho}\right)^k.$$ Consequently, we deduce that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|>\left(\frac{R'}{\rho}\right)^k,$$ which proves that the series $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ is divergent in the operator norm. The uniqueness of the number $R$ satisfying properties (i) and (ii) is now obvious. As expected, the number $R$ in the above theorem is called the radius of convergence of the power series $\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha.$ Let us consider the full Fock space $$F^2(H_n)={{\mathbb C}}1\oplus\ \oplus_{m\ge1}H_n^{\otimes m}$$ where $H_n$ is an $n$-dimensional complex Hilbert space with orthonormal basis $\{e_1,\dots,e_n\}$. Setting $e_\alpha:=e_{i_1}\otimes\cdots e_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$, and $e_{g_0}=1$, it is clear that $\{ e_\alpha:\ \alpha\in {{\mathbb F}}_n^+\}$ is an orthonormal basis of the full Fock space $F^2(H_n)$. For each $i=1,2,\dots$, we define the left creation operator $\ S_i\in B(F^2(H_n))$ by $$S_i\xi=e_i\otimes\xi,\qquad \xi\in F^2(H_n).$$ We can now obtain the following characterization of the radius of convergence, which will be useful later. \[Cs\] Let $\sum\limits_{\alpha\in {{\mathbb F}}_n^+}A_{(\alpha)}\otimes Z_\alpha$ be a formal power series with radius of convergence $R$. 1. If $R>0$ and $0<r<R$, then there exists $C>0$ such that $$\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2}\leq \frac {C}{r^k}\quad \text{for any } \ k=0,1,\ldots.$$ 2. The radius of convergence of the power series satisfies the relations $$R=\sup\left\{ r\geq 0:\ \text{ the sequence }\ \left\{r^k\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2} \right\}_{k=0}^\infty \text{ is bounded }\right\}$$ and $$R=\sup\left\{ r\geq 0: \ \sum_{k=0}^\infty \sum_{|\alpha|=k}r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha\ \text{ is convergent in the operator norm }\right\}.$$ Setting $X_i:= rS_i$,  $i=1,\ldots, n$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space, we have $\|[X_1,\ldots, X_n]\|=r<R$. According to Theorem \[Abel\], the series $ \sum_{k=0}^\infty \left\|r^k\sum_{|\alpha|=k} A_{(\alpha)} \otimes S_\alpha\right\|$ is convergent. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, the above series is equal to $\sum_{k=0}^\infty r^k \left\|\sum_{|\alpha|=k}A_{(\alpha)}^* A_\alpha \right\|^{1/2}. $ Consequently, there is a constant $C>0$ such that $$r^k \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}^* A_\alpha \right\|^{1/2}\leq C\ \text{ for any } k=0,1,\ldots.$$ Now, the second part of this corollary follows easily from part (i) and Theorem \[Abel\]. This completes the proof. We establish terminology which will be used throughout the paper. Denote by $[B({{\mathcal H}})^n]_{\gamma}$ the open ball of $B({{\mathcal H}})^n$ of radius $\gamma> 0$, i.e., $$[B({{\mathcal H}})^n]_{\gamma}:=\{[X_1,\ldots, X_n]:\ \|X_1X_1^*+\cdots +X_nX_n^*\|^{1/2}<\gamma\}.$$ We also use the notation $[B({{\mathcal H}})^n]_1^-$ for the closed ball. A formal power series $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$ with coefficients in $B({{\mathcal K}})$, if for any Hilbert space ${{\mathcal H}}$ and any representation $$\pi:{{\mathbb C}}[Z_1,\ldots, Z_n]\to B({{\mathcal H}})\quad \text{ such that } \quad [\pi(Z_1),\ldots, \pi(Z_n)]\in [B({{\mathcal H}})^n]_{\gamma}$$ the series $$F(\pi(Z_1),\ldots, \pi(Z_n)):=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} A_{(\alpha)} \otimes \pi(Z_\alpha)$$ converges in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Due to Theorem \[Abel\], we must have $\gamma\leq R$, where $R$ is the radius of convergence of $F$. The mapping $$[B({{\mathcal H}})^n]_{\gamma}\ni [X_1,\ldots, X_n]\mapsto F(X_1,\ldots X_n)\in B({{\mathcal K}}\otimes {{\mathcal H}}).$$ is called the representation of $F$ on the Hilbert space ${{\mathcal H}}$. Given a Hilbert space ${{\mathcal H}}$, we say that a function $G:[B({{\mathcal H}})^n]_{\gamma}\to B({{\mathcal K}}\otimes {{\mathcal H}})$ is a [*free holomorphic function*]{} on $[B({{\mathcal H}})^n]_{\gamma }$ with coefficients in $B({{\mathcal K}})$ if there exist operators $A_{(\alpha)}\in B({{\mathcal K}})$, $\alpha\in {{\mathbb F}}_n^+$, such that the power series $\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ has radius of convergence $\geq \gamma$ and $$G(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right),$$ where the series converges in the operator norm for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{\gamma}$. We remark that the coefficients of a free holomorphic function are uniquely determined by its representation on an infinite dimensional Hilbert space. Indeed, let $0<r<\gamma$ and assume $F(rS_1,\ldots, rS_n)=0$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$. Taking into account that $S_i^* S_j=\delta_{ij} I$, we have $$\left< F(rS_1,\ldots, rS_n)(x\otimes 1), (I_{{\mathcal K}}\otimes S_\alpha)(y\otimes 1)\right>=\left<A_{(\alpha)}x,y\right>=0$$ for any $x,y\in {{\mathcal K}}$ and $\alpha\in {{\mathbb F}}_n^+$. Therefore $A_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$. We establish now the continuity of free holomorphic functions on the open ball $[B({{\mathcal H}})^n]_{\gamma}$. \[continuous\] Let $ f(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right) $ be a free holomorphic function on $[B({{\mathcal H}})^n]_{\gamma}$ with coefficients in $B({{\mathcal K}})$. If $X:=[X_1,\ldots, X_n]$, $Y:=[Y_1,\ldots, Y_n]$ are in the closed ball $[B({{\mathcal H}})^n]_r^-$, $0<r<\gamma$, then $$\|f(X)-f(Y)\|\leq \|X-Y\|\sum _{k=1}^\infty kr^{k-1}\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}.$$ In particular, $f$ is continuous on $[B({{\mathcal H}})^n]_{\gamma}$ and uniformly continuous on $[B({{\mathcal H}})^n]_r^-$ in the operator norm topology. Let $X^{[k]}:=[X_\alpha:\ \alpha\in {{\mathbb F}}_n^+, \ |\alpha|=k]$,  $k=1,2,\ldots$, be the row operator matrix with entries arranged in the lexicographic order of the free semigroup ${{\mathbb F}}_n^+$. First, we prove that if $\|X\|\neq \|Y\|$, then $$\label{[k]} \frac{\|X^{[k]}-Y^{[k]}\|}{\|X-Y\|}\leq \frac{\|X\|^k-\|Y\|^k}{\|X\|-\|Y\|}.$$ Notice that $$\begin{split} X^{[k]}-Y^{[k]}&= \left[(X_1-Y_1)X^{[k-1]},\ldots, (X_n-Y_n) X^{[k-1]}\right]\\ &\qquad + \left[ Y_1(X^{[k-1]}-Y^{[k-1]}),\ldots, Y_n(X^{[k-1]}-Y^{[k-1]})\right]\\ &= (X-Y)\text{\rm diag}_n(X^{[k-1]})+Y\text{\rm diag}_n(X^{[k-1]}-Y^{[k-1]}), \end{split}$$ where $\text{\rm diag}_n(A)$ is the $n\times n$ block diagonal operator matrix with $A$ on the diagonal and $0$ otherwise. Hence, we deduce that $$\|X^{[k]}-Y^{[k]}\|\leq \|X-Y\|\|X^{[k-1]}\|+\|Y\| \|X^{[k-1]}-Y^{[k-1]}\|$$ for any $k\geq 2$. Iterating this relation and taking into account that $\|X^{[k]}\|\leq \|X\|^k$ for $k=1,2,\ldots$, we obtain $$\begin{split} \|X^{[k]}-Y^{[k]}\|&\leq \|X-Y\|\left(\|X^{[k-1]}\| +\|\|X^{[k-2]}\|\|Y^{[1]}\|+\cdots + \|Y^{[k-1]}\|\right)\\ &\leq \|X-Y\|\left(\|X\|^{k-1}+\|X\|^{k-2}\|Y\|+\cdots+ \|Y\|^{k-1}\right), \end{split}$$ which proves inequality . Assuming that $\|X\|\leq r$ and $\|Y\|\leq r$, we deduce that $$\|X^{[k]}-Y^{[k]}\|\leq kr^{k-1}\|X-Y\|,\quad k=1,2,\ldots.$$ Hence, we obtain $$\begin{split} \|f(X)-f(Y)\|&\leq \sum_{k=1}^\infty\left\|\sum_{|\alpha|=k} A_{(\alpha)}\otimes (X_\alpha-Y_\alpha)\right\|\\ &\leq \sum_{k=1}^\infty\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2} \|X^{[k]}-Y^{[k]}\|\\ &\leq \|X-Y\| \sum_{k=1}^\infty kr^{k-1} \left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}. \end{split}$$ Let $\rho$ be a constant such that $r<\rho<\gamma$. Since $\gamma\leq R$ ($R$ is the radius of convergence of $f$) and $\frac{1}{\rho}>\frac{1}{\gamma}\geq \frac{1}{R}$, we can find $m_0\in {{\mathbb N}}$, such that $$\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2k}<\frac{1}{\rho}\quad \text{ for any } \ k\geq m_0.$$ Combining this with the above inequality, we deduce that $$\|f(X)-f(Y)\|\leq \|X-Y\|\left(\sum_{k=1}^{m_0-1} kr^{k-1} \left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}+ \sum_{k=m_0}^\infty \frac{k}{r} \left(\frac{r}{\rho}\right)^k\right).$$ Since $r<\rho$, the above series is convergent. Consequently, there exists a constant $M>0$ such that $$\|f(X)-f(Y)\|\leq M \|X-Y\|\qquad \text{ for any }\ X,Y\in [B({{\mathcal H}})^n]_r^-.$$ This implies the uniform continuity of $f$ on any closed ball $[B({{\mathcal H}})^n]_r^-$, $0<r<\gamma$, in the norm topology and, consequently, the continuity of $f$ on $[B({{\mathcal H}})^n]_{\gamma}$. \[operations\] Let $F$ and $G$ be formal power series such that $$\begin{split} F(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)\text{ and }\\ G(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}B_{(\alpha)}\otimes X_\alpha\right) \end{split}$$ are free holomorphic functions on $[B({{\mathcal H}})^n]_{\gamma}$, and let $a, b\in {{\mathbb C}}$. Then the power series $aF+bG$, and $FG$ generate free holomorphic functions on $[B({{\mathcal H}})^n]_{\gamma}$. Moreover, $$\begin{split} aF(X_1,\ldots, X_n)+bG(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}(aA_{(\alpha)}+bB_{(\alpha)})\otimes X_\alpha\right) \text{ and }\\ F(X_1,\ldots, X_n)G(X_1,\ldots, X_n)&=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k} C_{(\alpha)}\otimes X_\alpha \right) \end{split}$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_\gamma$, where $C_{(\alpha)}:= \sum\limits_{\alpha=\sigma\beta}A_{(\sigma)} B_{(\beta)}$,  $\alpha\in {{\mathbb F}}_n^+$. According to the hypotheses, both power series $F$ and $G$ have radius of convergence $\geq \gamma$. Due to Theorem \[Abel\], we deduce that, given any $\epsilon>0$, there exists $k_0\in {{\mathbb N}}$ such that $$\left\|\sum_{|\alpha|=k} A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2k}\leq \frac{1}{\gamma} +\epsilon \ \text{ and }\ \left\|\sum_{|\alpha|=k} B_{(\alpha)}^* B_{(\alpha)}\right\|^{1/2k}\leq \frac{1}{\gamma} +\epsilon$$ for any $k\geq k_0$. Assume that $|a|+|b|\neq 0$. Since the left creation operators $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\begin{split} \Biggl\|\sum_{|\alpha|=k} (aA_{(\alpha)}+bB_{(\alpha)})^* &(aA_{(\alpha)}+ bB_{(\alpha)})\Biggr\|^{1/2}\\ &= \left\|\sum_{|\alpha|=k} (aA_{(\alpha)} +bB_{(\alpha)})\otimes S_\alpha \right\|\\ &\leq \left\|\sum_{|\alpha|=k} aA_{(\alpha)} \otimes S_\alpha \right\|+\left\|\sum_{|\alpha|=k} bB_{(\alpha)}\otimes S_\alpha\right\|\\ &=\left\|\sum_{|\alpha|=k} |a|^2A_{(\alpha)}^* A_{(\alpha)}\right\|^{1/2}+\left\|\sum_{|\alpha|=k} |b|^2B_{(\alpha)}^* B_{(\alpha)}\right\|^{1/2}\\ &=(|a|+|b|)\left( \frac{1}{\gamma}+\epsilon\right)^k \end{split}$$ for any $k\geq k_0$. Hence, we deduce that $$\limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} (aA_{(\alpha)}+bB_{(\alpha)})^* (aA_{(\alpha)}+ bB_{(\alpha)})\right\|^{1/2k}\leq \frac{1}{\gamma}+\epsilon$$ for any $\epsilon>0$. Taking $\epsilon\to 0$, we deduce that the power series $aF+bG$ has the radius of convergence $\geq \gamma$. Now, we prove that the power series $FG$ has radius of convergence $\geq \gamma$. If $0<r<\gamma$, then, due to Corollary \[Cs\], there is a constant $M>0$ such that $$\begin{split} \left\|\sum_{|\sigma|=k} C_{(\sigma)}^* C_{(\sigma)}\right\|^{1/2}&= \left\| \sum_{|\sigma|=k}C_{(\sigma)}\otimes S_\sigma\right\|\\ &= \left\| \sum_{p+q=k} \left( \sum_{|\alpha|=p}A_{(\alpha)}\otimes S_\alpha\right) \left( \sum_{|\beta|=q}B_{(\beta)}\otimes S_\beta\right)\right\|\\ &\leq \sum_{p+q=k}\left\| \sum_{|\alpha|=p}A_{(\alpha)}^* A_{(\alpha)} \right\|^{1/2} \left\|\sum_{|\beta|=q}B_{(\beta)}^*B_{(\beta)} \right\|^{1/2}\\ &\leq \sum_{p+q=k} \frac{M}{r^p}\cdot\frac{M}{r^q}\\ &= (k+1) \frac{M^2}{r^k} \end{split}$$ for any $k=0,1,\ldots$. Hence, we obtain $$\limsup_{k\to\infty} \left\|\sum_{|\sigma|=k} C_{(\sigma)}^* C_{(\sigma)}\right\|^{1/2k}\leq \frac {1}{r}$$ for any $r$ such that $0<r<\gamma$. Consequently, the radius of convergence of the power series $FG$ is $\geq \gamma$. The last part of the theorem follows easily using Theorem \[Abel\]. We are in position to give a characterization as well as models for free holomorphic functions on the open operatorial $n$-ball of radius $\gamma$. \[caract-shifts\] A power series $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$ with coefficients in $B({{\mathcal K}})$ if and only if the series $$\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha$$ is convergent for any $r\in [0,\gamma)$, where $S_1,\ldots, S_n$ are the left creation operators on the Fock space $F^2(H_n)$. Moreover, in this case, the series $$\label{cre-seri} \sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha \right\|=\sum_{k=0}^\infty r^k\left\| \sum_{|\alpha|=k} A_{(\alpha)}^*A_{(\alpha)}\right\|^{1/2}$$ are convergent for any $r\in [0,\gamma)$. Assume that $F$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. According to Theorem \[Abel\], $\gamma\leq R$, where $R$ is the radius of convergence of $F$, and $\sum\limits_{k=0}^\infty \left\|\sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right\|$ converges for any $n$-tuple $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|=r<\gamma$. Since $\|[rS_1,\ldots, rS_n]\|=r<\gamma$, we deduce that the series is convergent for any $r\in [0,\gamma)$. Now, assume that the series is convergent for any $r\in [0,\gamma)$. According to the noncommutative von Neumann inequality [@Po-von], we have $$\sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes T_\alpha \right\|\leq \sum\limits_{k=0}^\infty\left\| \sum\limits_{|\alpha|=k} r^{|\alpha|} A_{(\alpha)}\otimes S_\alpha \right\|$$ for any $n$-tuple $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ with $T_1T_1^*+\cdots T_nT_n^*\leq I$ and any $r\in [0,\gamma)$. Hence, we deduce that the series $$\sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right\|$$ converges for any $n$-tuple of operators $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|<\gamma$. Due to Theorem \[Abel\], the power series $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. This completes the proof. Let $\{a_k\}_{k=0}^\infty$ be a sequence of complex numbers. Then the following statements are equivalent: 1. $f(z):=\sum_{k=0}^\infty a_k z^k$ is an analytic function on the open unit disc ${{\mathbb D}}:=\{z\in {{\mathbb C}}:\ |z|<1\}$. 2. $f_r(S):=\sum_{k=0}^\infty r^ka_k S^k$ is convergent in the operator norm for each $r\in [0,1)$, where $S$ is the unilateral shift on the Hardy space $H^2$. 3. $f(Z):=\sum_{k=0}^\infty a_k Z^k$ is a free holomorphic function on the open operatorial unit $1$-ball. If $f(z)=\sum\limits_{k=0}^\infty a_k z^k$ is an analytic function on the open unit disc, then Hadamard’s theorem implies $\limsup\limits_{k\to\infty} |a_k|^{1/k}\leq 1$. Hence $\sum\limits_{k=0}^\infty r^k|a_k|<\infty$ for any $r\in [0,1)$ and, consequently, the series $\sum\limits_{k=0}^\infty r^k a_k S^k$ is convergent in the operator norm. Conversely, if the latter series is norm convergent, then, due to von Neumann inequality [@vN], the series $\sum\limits_{k=0}^\infty r^k a_k z$ converges for any $r\in[0,1)$ and $z\in {{\mathbb D}}$. Hence, we deduce (i). The equivalence (ii)$\Longleftrightarrow$ (iii) is a particular case of Theorem \[caract-shifts\]. If $\lambda:=(\lambda_1,\ldots, \lambda_n)\in{{\mathbb C}}^n$ and $\alpha=g_{i_1}\cdots g_{i_k}\in {{\mathbb F}}_n^+$, then we set $\lambda_\alpha:=\lambda_{i_1}\cdots \lambda_{i_k}$ and $\lambda_0=1$. \[part-case\] If $f =\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$, $a_\alpha\in {{\mathbb C}}$, is a free holomorphic function on the open operatorial unit $n$-ball, then its representation on ${{\mathbb C}}$, $$f(\lambda_1,\ldots, \lambda_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha \lambda_\alpha,$$ is a holomorphic function on ${{\mathbb B}}_n$, the open unit ball of ${{\mathbb C}}^n$. Due to Theorem \[caract-shifts\], we have $$\begin{split} \sum_{k=0}^\infty\sum_{|\alpha|=k}|a_\alpha||\lambda_\alpha|&\leq \sum_{k=0}^\infty\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\left(\sum_{|\alpha|=k}|\lambda_\alpha|^2\right)^{1/2}\\ &\leq \sum_{k=0}^\infty\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\left( \sum_{i=1}^n |\lambda_i|^2\right)^{k/2}<\infty \end{split}$$ for any $(\lambda_1,\ldots, \lambda_n)\in {{\mathbb B}}_n$. Hence, the result follows. In the last part of this section, we show that the Hausdorff derivations on the algebra of noncommutative polynomials ${{\mathbb C}}[Z_1,\ldots, Z_n]$ (see [@MKS], [@RSS]) can be extended to the algebra of free holomorphic functions. For each $i=1,\ldots, n$, we define the free partial derivation $\frac{\partial } {\partial Z_i}$ on ${{\mathbb C}}[Z_1,\ldots, Z_n]$ as the unique linear operator on this algebra, satisfying the conditions $$\frac{\partial I} {\partial Z_i}=0, \quad \frac{\partial Z_i} {\partial Z_i}=I, \quad \frac{\partial Z_j} {\partial Z_i}=0\ \text{ if } \ i\neq j,$$ and $$\frac{\partial (fg)} {\partial Z_i}=\frac{\partial f} {\partial Z_i} g +f\frac{\partial g} {\partial Z_i}$$ for any $f,g\in {{\mathbb C}}[Z_1,\ldots, Z_n]$ and $i,j=1,\ldots n$. The same definition extends to formal power series in the noncommuting indeterminates $Z_1,\ldots, Z_n$. Notice that if $\alpha=g_{i_1}\cdots g_{i_p}$, $|\alpha|=p$, and $q$ of the $g_{i_1},\ldots, g_{i_p}$ are equal to $g_j$, then $\frac{\partial Z_\alpha} {\partial Z_j}$ is the sum of the $q$ words obtained by deleting each occurence of $Z_j$ in $Z_\alpha:=Z_{i_1}\cdots Z_{i_p}$. For example, $$\frac{\partial (Z_1 Z_2 Z_1^2)} {\partial Z_1}= Z_2 Z_1^2+ Z_1Z_2Z_1+ Z_1Z_2Z_1.$$ One can easily show that $\frac{\partial } {\partial Z_i}$ coincides with the Hausdorff derivative. If $\beta:=g_{i_1}\cdots g_{i_k}\in {{\mathbb F}}_n^+$, $i_1,\ldots, i_k\in \{1,2,\ldots, n\}$, we denote $Z_\beta:=Z_{i_1}\cdots Z_{i_k}$ and define the $k$-order free partial derivative of $G\in {{\mathbb C}}[Z_1,\ldots, Z_n]$ with respect to $Z_{i_1},\ldots, Z_{i_k}$ by $$\frac {\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}:= \frac{\partial} {\partial Z_{i_1}}\left(\frac{\partial} {\partial Z_{i_2}}\cdots \left( \frac{\partial G} {\partial Z_{i_k}}\right)\cdots \right).$$ These definitions can easily be extended to formal power series. If $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes Z_\alpha$ is a power series with operator-valued coefficients, then we define the $k$-order free partial derivative of $F$ with respect to $Z_{i_1}, \ldots, Z_{i_k}$ to be the power series $$\frac {\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}} := \sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes \frac {\partial^k Z_\alpha}{\partial Z_{i_1}\cdots \partial Z_{i_k}}.$$ \[deriv-comu\] If $i,j\in \{1,\ldots, n\}$, then $$\frac{\partial^2 F}{\partial Z_i \partial Z_j}= \frac{\partial^2 F}{\partial Z_j \partial Z_i}$$ for any formal power series $F$. Due to linearity, it is enough to prove the result for monomials. Let $\alpha:=g_{i_1}\cdots g_{i_k}$ be a word in $ {{\mathbb F}}_n^+$ and $Z_\alpha:=Z_{i_1}\cdots Z_{i_k}$. Let $i,j\in \{1,\ldots, n\}$ be such that $i\neq j$. Assume that $Z_i$ occurs $q$ times in $Z_\alpha$, and $Z_j$ occurs $p$ times in $Z_\alpha$. Then $\frac{\partial Z_\alpha}{\partial Z_i}$ is the sum of the $q$ words obtained by deleting each occurence of $Z_i$ in $Z_\alpha$. Notice that $Z_j$ occurs $p$ times in each of these $q$ words. Therefore, $\frac{\partial ^2 Z_\alpha}{\partial Z_j \partial Z_i}$ is the sum of the $qp$ words obtained by deleting each occurence of $Z_i$ in $Z_\alpha$ and then deleting each occurence of $Z_j$ in the resulting words. Similarly, $\frac{\partial ^2 Z_\alpha}{\partial Z_i \partial Z_j}$ is the sum of the $qp$ words obtained by deleting each occurence of $Z_j$ in $Z_\alpha$ and then deleting each occurence of $Z_i$ in the resulting words. Hence, it is clear that $$\frac{\partial^2 Z_\alpha}{\partial Z_i \partial Z_j}= \frac{\partial^2 Z_\alpha}{\partial Z_j \partial Z_i}.$$ This completes the proof. \[derivation\] Let $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)} \otimes Z_\alpha$ be a power series with radius of convergence $R$ and let $R'$ be the radius of convergence of the power series $ \frac{\partial^k F} {\partial Z_{j_1}\cdots \partial Z_{j_k}}$, where $j_1,\ldots, j_k\in \{1,\ldots, n\}$. Then $R'\geq R$ and, in general, the inequality is strict. It is enough to prove the result for first order free partial derivatives. For any word $\omega:=g_{i_1}\cdots g_{i_k}$, $|\omega|=k\geq 1$, and $0\leq m\leq k$, we define the insertion mapping of $g_j$, $j=1,\ldots, n$, on the $m$ position of $\omega$ by setting $$\chi(g_j, m,\omega):= \begin{cases} g_j\omega & \text{ if } m=0,\\ g_{i_1}\cdots g_{i_m} g_j g_{i_{m+1}}\cdots g_{i_k} & \text{ if } 1\leq m\leq k-1,\\ \omega g_j & \text{ if } m=|\omega|=k, \end{cases}$$ and $\chi(g_j, 0, g_0):=g_j$. Let $$\frac{\partial F} {\partial Z_j}= \sum_{\beta\in {{\mathbb F}}_n^+} B_{(\beta)} \otimes Z_\beta.$$ Using the definition of the Hausdorff derivation and the insertion mapping, we deduce that $$B_{(\beta)}=\sum_{m=0}^k A_{(\chi(g_j,m,\beta))}$$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$. This is the case, since the monomial $Z_\beta$ comes from free differentiation with respect to $Z_j$ of the monomials $Z_{\chi(g_j,m,\beta)}$, $m=0,1,\ldots, |\beta|$. Therefore, we have $$\begin{split} \sum_{|\beta|=k} B_{(\beta)}^* B_{(\beta)} &= \sum_{|\beta|=k} \left( \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}^*\right)\left( \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}\right)\\ &\leq (k+1)\sum_{|\beta|=k} \sum_{m=0}^k A_{(\chi(g_j,m,\beta))}^*A_{(\chi(g_j,m,\beta))} \\ &\leq (k+1)^2 \sum_{|\alpha|=k+1} A_{(\alpha)}^* A_{(\alpha)}. \end{split}$$ The last inequality holds since, for each $j=1,\ldots,n$, each $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|=k+1$, and each $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$, the cardinal of the set $$\{(g_j,m,\beta):\ \chi(g_j,m,\beta)=\alpha, \text{ where } m=0,1,\ldots, k\}$$ is $\leq k+1$. Hence, we deduce that $$\left(\sum_{|\beta|=k} B_{(\beta)}^* B_{(\beta)} \right)^{1/2k}\leq (k+1)^{1/k} \left( \sum_{|\alpha|=k+1} A_{(\alpha)}^* A_{(\alpha)}\right)^{1/2k}.$$ Consequently, due to Theorem \[Abel\], we have $\frac {1}{R'}\leq \frac {1}{R}$. Therefore, $R'\geq R$. To prove the last part of the theorem, let $R_1, R_2>0$ be such that $R_1<R_2$. Let us consider two power series $$F=\sum_{k=0}^\infty a_k Z_1^k \ \text { and } \ G=\sum_{k=0}^\infty b_k Z_2^k$$ with radius of convergence $R_1$ and $R_2$, respectively. We shall show that the power series $$F+G=\sum_{k=0}^\infty (a_kZ_1^k+b_k Z_2^k)$$ has the radius of convergence equal to $R_1$. First, since $$\sup_k\left( |a_k|^2+|b_k|^2\right)^{1/2k}\geq \sup|a_k|^{1/k}=\frac{1}{R_1},$$ we deduce that the radius of convergence of $F+G$ is $\leq R_1$. On the other hand, if $r<R_1$, Corollary \[Cs\] shows that both sequences $\{r^k |a_k|\}_{k=0}^\infty$ and $\{r^k |b_k|\}_{k=0}^\infty$ are bounded. This implies that the sequence $\{r^k \left(|a_k|^2+|b_k|^2\right)^{1/2}\}_{k=0}^\infty$ is bounded. Applying again Corollary \[Cs\], we can conclude that $F+G$ has radius of convergence $R_1$. Since $$\frac{\partial (F+G)}{\partial Z_2}=\sum_{k=1}^\infty k b_kZ_2^{k-1},$$ the power series $ \frac{\partial (F+G)}{\partial Z_2}$ has radius of convergence $R_2$, which is strictly larger than the radius of convergence of $F+G$. This completes the proof. Cauchy, Liouville, and Schwartz type results for free holomorphic functions {#Liouville} =========================================================================== In this section , we obtain Cauchy type estimates for the coefficients of free holomorphic functions and a Liouville type theorem for free entire functions. Based on a noncommutative version of Gleason’s problem [@R2] and the noncommutative von Neumann inequality [@Po-von], we provide a free analogue of Schwartz lemma. First, we obtain Cauchy type estimates for the coefficients of free holomorphic functions on the open ball $[B({{\mathcal H}})^n]_{\gamma}$ with coefficients in $B({{\mathcal K}})$. \[Cauchy-est\] Let $F:[B({{\mathcal H}})^n]_{\gamma}\to B({{\mathcal K}})\bar\otimes B({{\mathcal H}})$ be a free holomorphic function on $[B({{\mathcal H}})^n]_{\gamma}$ with the representation $$F(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha\right),$$ and define $$M(\rho):= \|F(\rho S_1,\ldots, \rho S_n)\|\quad \text{for any } \ \rho\in (0,\gamma),$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space. Then, for each $k=0,1,\ldots,$ $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq \frac{1} {\rho^k} M(\rho).$$ Let $\{Y_{(\alpha)}\}_{|\alpha|=k}$ be an arbitrary sequence of operators in $B({{\mathcal K}})$. Using Theorem \[caract-shifts\], we have $$\begin{split} \left|\left<\left(\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\right)F(\rho S_1,\ldots, \rho S_n) h\otimes 1, h\otimes 1\right>\right| &\leq \left\|\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\right\| M(\rho) \|h\|^2\\ &=\left\|\sum_{|\alpha|=k} Y_{(\alpha)}^* Y_{(\alpha)}\right\|^{1/2} M(\rho) \|h\|^2 \end{split}$$ for any $h\in {{\mathcal K}}$. On the other hand, since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\begin{split} \Bigl|\Bigl<\Bigl(\sum_{|\alpha|=k} Y_{(\alpha)}^*\otimes S_\alpha^*\Bigr)F(\rho S_1,\ldots, &\rho S_n) h\otimes 1, h\otimes 1\Bigr>\Bigr|\\ &= \rho^k\left|\left<\left(\sum_{|\alpha|=k} Y_{(\alpha)}^* A_{(\alpha)}\otimes I\right) h\otimes 1, h\otimes 1\right>\right|\\ &= \rho^k\left|\left<[Y_{(\alpha)}^*: |\alpha|=k] \left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]h,h\right>\right|. \end{split}$$ Combining these relations and taking $Y_{(\alpha)}:=A_{(\alpha)}$, $|\alpha|=k$, we deduce that $$\rho^k\left\|\left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]h\right\|^2\leq \left\|\left[\begin{matrix}A_{(\alpha)}\\:\\|\alpha|=k\end{matrix} \right]\right\| M(\rho) \|h\|^2$$ for any $h\in {{\mathcal K}}$. Therefore, $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}=\|[A_{(\alpha)}^*:\ |\alpha|=k]\|\leq \frac{1} {\rho^k} M(\rho),$$ which completes the proof. A free holomorphic function with radius of convergence $R=\infty$ is called free entire function. We can prove now the following noncommutative multivariable generalization of Liouville’s theorem. \[Liou\] Let $F$ be an entire function and let $$F(X_1,\ldots, X_n)=\sum_{k=0}^\infty\sum_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ be its representation on an infinite dimensional Hilbert space ${{\mathcal H}}$. Then $F$ is a polynomial of degree $\leq m$, $m=0, 1, \ldots$, if and only if there are constants $M>0$ and $C>1$ such that $$\|F(X_1,\ldots, X_n)\|\leq M\|[X_1,\ldots, X_n]\|^m$$ for any $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ such that $\|[X_1,\ldots, X_n]\|\geq C$. If $F=\sum_{|\alpha|\leq m} A_{(\alpha)}\otimes X_\alpha$ is a polynomial, then $$\begin{split} \|F\|&\leq \sum_{k=0}^m \left\|\sum_{|\alpha|=m} A_{(\alpha)}\otimes X_\alpha \right\|\\ &\leq \sum_{k=0}^m \left\| \sum_{|\alpha|=k} A_{(\alpha)}^* A_{\alpha)} \right\|^{1/2}\|[X_1,\ldots, X_n]\|^k \end{split}$$ if $\|[X_1,\ldots, X_n]\|\geq 1$. Therefore, there exists $M>0$ and $R>1$ such that $$\label{f-norm} \|F(X_1,\ldots, X_n)\|\leq M \|[X_1,\ldots, X_n]\|^k$$ for any $n$-tuple of operators $[X_1,\ldots, X_n]$ with $\|[X_1,\ldots, X_n]\|\geq R$. Conversely, if the inequality holds, then $$\|F(\rho S_1,\ldots, \rho S_n)\|\leq M \rho^m,\quad \text{ as }\ \rho\to\infty.$$ According to Theorem \[Cauchy-est\], we have $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq \frac{1} {\rho^k} M(\rho),$$ where $M(\rho):=\|F(\rho S_1,\ldots, \rho S_n)\|$. Combining these inequalities, we deduce that $$\left\|\sum_{|\alpha|=k} A_\alpha^* A_\alpha\right\|^{1/2}\leq M\frac{1} {\rho^{k-m}}.$$ Consequently, if $k>m$ and $\rho\to\infty$, we obtain $\sum_{|\alpha|=k} A_\alpha^* A_\alpha=0$. This shows that $A_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|>m$. We say that a free holomorphic function $F$ on the open operatorial $n$-ball of radius $\gamma$ is bounded if $$\|F\|_\infty:=\sup \|F(X_1,\ldots, X_n)\|<\infty,$$ where the supremum is taken over all $n$-tuples of operators $[X_1,\ldots, X_n]\in (B({{\mathcal H}})^n)_\gamma$ and any Hilbert space ${{\mathcal H}}$. In the particular case when $m=0$, Theorem \[Liou\] implies the following free analogue of Liouville’s theorem from complex analysis (see [@R], [@Co]). If $F$ is a bounded free entire function, then it is constant. We recall that the joint spectral radius of an $n$-tuple of operators $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$, $$r(T_1,\ldots,T_n):=\lim_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{\frac{1} {2k}},$$ is also equal to the spectral radius of the reconstruction operator $S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*$ (see [@Po-unitary]). Consequently, $r(T_1,\ldots,T_n)<1$ if and only if $$\sigma(S_1\otimes T_1^*+\cdots + S_n\otimes T_n^*)\subset {{\mathbb D}}.$$ Moreover, the joint right spectrum  $\sigma_r(T_1,\ldots, T_n)$  is included in the closed ball of ${{\mathbb C}}^n$ of radius equal to $r(T_1,\ldots,T_n)$. We recall that $\sigma_r(T_1,\ldots, T_n)$ is the set of all $n$-tuples $(\lambda_1,\ldots, \lambda_n)\in {{\mathbb C}}^n$ such that the right ideal of $B({{\mathcal H}})$ generated by $\lambda_1 I-T_1,\ldots, \lambda_n I-T_n$ does not contain the identity. Now, we prove an analogue of Schwartz lemma, in our multivariable operatorial setting. \[Schwartz\] Let $F(X_1,\ldots, X_n)=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes X_\alpha$,  $A_{(\alpha)}\in B({{\mathcal K}})$, be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ with the properties: 1. $\|F\|_\infty\leq 1$ and 2. $A_{(\beta)}=0$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m-1$, where $m=1,2,\ldots$. Then $$\|F(X_1,\ldots, X_n)\| \leq \|[X_1,\ldots, X_n]\|^m \quad \text{ and }\quad r(F(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n)^m$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Moreover, $$\left\|\sum_{|\alpha|=k} A_\alpha A_\alpha^*\right\|^{1/2} \leq 1\quad \text{ for any } \ k\geq m.$$ For each $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m$, define the formal power series $$\Phi_{(\beta)}(Z_1,\ldots, Z_n):=\sum_{\alpha\in {{\mathbb F}}_n^+} A_{(\beta\alpha)}\otimes Z_\alpha.$$ Since $$\left\|\sum_{|\alpha|=k} A_{(\beta \alpha)}^* A_{(\beta\alpha)}^*\right\|\leq \left\| \sum_{|\gamma|=m+k} A_{(\gamma)}^* A_{(\gamma)} \right\|,$$ we deduce that $$\limsup_{k\to\infty}\left\|\sum_{|\alpha|=k} A_{(\beta \alpha)}^* A_{(\beta\alpha)}^*\right\|^{1/2k} \leq \limsup_{k\to\infty} \left\| \sum_{|\gamma|=m+k} A_{(\gamma)}^* A_{(\gamma)} \right\|^{\frac{1}{2(m+k)}}.$$ Consequently, due to Theorem \[Abel\], the radius of convergence of $\Phi_{(\beta)}$ is greater than the radius of convergence of $F$. Therefore, $\Phi_{(\beta)}$ represents a free holomorphic function on the open operatorial unit $n$-ball. Since $A_{(\beta)}=0$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|\leq m-1$, and due to Theorem \[operations\], we have the following Gleason type decomposition $$F(Z_1,\ldots, Z_n)=\sum_{|\beta|=m}\left[(I_{{\mathcal K}}\otimes Z_\beta)\sum_{\alpha\in {{\mathbb F}}_n^+} A_{\beta \alpha)} \otimes Z_\alpha\right]= \sum_{|\beta|=m}(I_{{\mathcal K}}\otimes Z_\beta)\Phi_{(\beta)}(Z_1,\ldots, Z_n).$$ Therefore, $$\label{F-Phi} F(rS_1,\ldots, rS_n) =\sum_{|\beta|=m} (I_{{\mathcal K}}\otimes r^{|\beta|} S_\beta )\Phi_{(\beta)}(rS_1,\ldots, rS_n)$$ for any $r\in [0,1)$. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, $S_\beta$, $|\beta|=m$, are also isometries with orthogonal ranges and we have $$F(rS_1,\ldots, rS_n)^*F(rS_1,\ldots, rS_n) =r^{2m}\sum_{|\beta|=m} \Phi_{(\beta)}(rS_1,\ldots, rS_n)^*\Phi_{(\beta)}(rS_1,\ldots, rS_n).$$ Now, due to the noncommutative von Neumann inequality [@Po-von] and Theorem \[caract-shifts\], we deduce that $$\label{M-P} \left\|\left[\begin{matrix} \Phi_{(\beta)}(rX_1,\ldots, rX_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\|\leq \left\|\left[\begin{matrix} \Phi_{(\beta)}(rS_1,\ldots, rS_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\|$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Consequently, using relations and , we obtain $$\begin{split} \|F(rX_1,\ldots, rX_n)\| &= \left\|\sum_{|\beta|=m} (I_{{\mathcal K}}\otimes r^{|\beta|} X_\beta )\Phi_{(\beta)}(rX_1,\ldots, rX_n)\right\| \\ &\leq \left\|[r^m X_\beta:\ |\beta|=m]\right\| \left\|\left[\begin{matrix} \Phi_{(\beta)}(rX_1,\ldots, rX_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\| \\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2}\left\|\left[\begin{matrix} \Phi_{(\beta)}(rS_1,\ldots, rS_n)\\ :\\ |\beta|=m \end{matrix} \right]\right\| \\ &= r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \left\| \sum_{|\beta|=m} \Phi_{(\beta)}(rS_1,\ldots, rS_n)^*\Phi_{(\beta)}(rS_1,\ldots, rS_n) \right\|^{1/2}\\ &= r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \left\|F(rS_1,\ldots, rS_n)^*F(rS_1,\ldots, rS_n)\right\|\\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \|F\|_\infty\\ &\leq r^m\left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \leq r^m\|[X_1,\ldots, X_n]\|^m. \end{split}$$ Taking $r\to 1$ and using the continuity of the free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$ (see Theorem \[continuous\]), we infer that $$\|F(X_1,\ldots, X_n)\|\leq \left\|\sum_{|\beta|=m} X_\beta X_\beta^*\right\|^{1/2} \leq \|[X_1,\ldots, X_n]\|^m$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Due to Theorem \[operations\], the power series $F^k =\sum_{\alpha\in {{\mathbb F}}_n^+} B_{(\alpha)}\otimes Z_\alpha$ represents a free holomorphic function on the open operatorial unit $n$-ball, with $B_{(\alpha)}=0$ for any $\alpha\in {{\mathbb F}}_n^+$ with $|\alpha|\leq mk$. Applying the above inequality to $F^k$, we obtain $$\|F(X_1,\ldots, X_n)^k\|\leq \left\|\sum_{|\beta|=mk} X_\beta X_\beta^*\right\|^{1/2}\leq \left\|\sum_{|\beta|=k} X_\beta X_\beta^*\right\|^{m/2}.$$ Hence, and using the definition of the joint spectral radius, we deduce that $r(F(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n)^m. $ To prove the last part of the theorem, notice that, according to Theorem \[Cauchy-est\], we have $$\left\|\sum_{|\alpha|=k} A_{(\alpha)} A_{(\alpha)}^*\right\|^{1/2}\leq \frac{1}{\rho^k} M(\rho)$$ for any $\rho\in (0,1)$, where $M(\rho)=\|F(\rho S_1,\ldots, \rho S_n)\|$. Since $M(\rho)\leq \|F\|_\infty\leq 1$, we take $\rho\to 1$ and deduce that $\left\|\sum_{|\alpha|=k} A_{(\alpha)} A_{(\alpha)}^*\right\|^{1/2}\leq 1 $ for any $k\geq m$. The proof is complete. In the scalar case we get a little bit more. Let $f(X_1,\ldots, X_n)=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha X_\alpha$, $a_\alpha\in {{\mathbb C}}$, be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ with scalar coefficients and the properties: 1. $\|f\|_\infty\leq 1$ and 2. $f(0)=0$. Then 1. $\|f(X_1,\ldots, X_n)\|\leq \left\|[X_1,\ldots, X_n]\right\| $ and  $r(f(X_1,\ldots, X_n))\leq r(X_1,\ldots, X_n) $ for any $n$-tuple $ \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$; 2. $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2\leq 1. $ Moreover, if $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2= 1, $ then $\|f\|_\infty=1$. The first part of this corollary is a particular case of Theorem \[Schwartz\], when $m=1$ and ${{\mathcal K}}={{\mathbb C}}$. To prove the second part, assume that $ \sum\limits_{i=1}^n \left|\frac{\partial f}{\partial X_i}(0)\right|^2= 1 $ Consequently, we have $\sum_{i=1}^n |a_i|^2=1$. Hence, and due to Theorem \[Cauchy-est\], we have $$1\leq \sum_{i=1}^n |a_i|^2\leq \frac{1}{\rho} \|f\|_\infty$$ for any $0<\rho<1$. Therefore, $\|f\|_\infty=1$. This completes the proof. Algebras of free holomorphic functions {#algebras} ======================================== In this section, we introduce two Banach algebras of free holomorphic functions, $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$, and prove that they are isometrically isomorphic to the the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively. The results of this section are used to obtain a maximum principle for free holomorphic functions. We denote by $Hol(B({{\mathcal X}})^n_\gamma)$ the set of all free holomorphic functions with scalar coefficients on the open operatorial $n$-ball of radius $\gamma$. Due to Theorem \[operations\] and Theorem \[Abel\], $Hol(B({{\mathcal X}})^n_\gamma)$ is an algebra and an element $F=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ is in $Hol(B({{\mathcal X}})^n_\gamma)$ if and only if $$\limsup_{k\to\infty}\left(\sum_{|\alpha|=k} |a_\alpha|^2\right)^{1/2k}\leq 1.$$ Let $H^\infty(B({{\mathcal X}})^n_1)$ denote the set of all elements $F$ in $Hol(B({{\mathcal X}})^n_1)$ such that $$\|F\|_\infty:= \sup \|F(X_1,\ldots, X_n)\|<\infty,$$ where the supremum is taken over all $n$-tuples $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and any Hilbert space ${{\mathcal H}}$. We denote by $A(B({{\mathcal X}})^n_1)$ be the set of all elements $F$ in $Hol(B({{\mathcal X}})^n_1)$ such that, for any Hilbert space ${{\mathcal H}}$, the mapping $$[B({{\mathcal H}})^n]_1\ni (X_1,\ldots, X_n)\mapsto F(X_1,\ldots, X_n)\in B({{\mathcal H}})$$ has a continuous extension to the closed unit ball $[B({{\mathcal H}})^n]^-_1$. In this section, we will show that $H^\infty(B({{\mathcal X}})^n_1)$ and $A(B({{\mathcal X}})^n_1)$ are Banach algebras under pointwise multiplication and the norm $\|\cdot \|_\infty$, which can be identified with the noncommutative analytic Toeplitz algebra $F_n^\infty$ and the noncommutative disc algebra ${{\mathcal A}}_n$, respectively. Let us recall (see [@Po-von], [@Po-funct], [@Po-disc], [@Po-poisson]) a few facts about the Banach algebras ${{\mathcal A}}_n$ and $F_n^\infty$. Any element $~f$ in the full Fock space $ F^2(H_n)$ has the form $$f=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha,\quad\text{\ with\ }a_\alpha\in{{\mathbb C}}, \quad\text {\ such that\ }\ \|f\|_2:=\left(\sum\limits_{\alpha\in{{\mathbb F}}_n^+}|a_\alpha|^2\right)^{1/2}<\infty.$$ If $g=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} b_\alpha e_\alpha\in F^2(H_n)$, we define the product $f\otimes g$ to be the formal power series $$f\otimes g:=\sum_{\gamma\in {{\mathbb F}}_n^+} c_\gamma e_\gamma,\quad \text{ where }\quad c_\gamma :=\sum_{\stackrel{\alpha,\beta\in {{\mathbb F}}_n^+}{ \alpha\beta=\gamma}} a_\alpha b_\beta, \quad \gamma\in {{\mathbb F}}_n^+.$$ We also make the natural identification of $e_\alpha\otimes 1$ and $1\otimes e_\alpha$ with $e_\alpha$. Let ${{\mathcal P}}$ denote the set of all polynomials $p\in F^2(H_n)$, i.e., elements of the form $p=\sum_{|\alpha|\leq m} a_\alpha e_\alpha$, where $m=0,1,\ldots$. In [@Po-von], we introduced the noncommutative Hardy algebra $F_n^\infty$ as the set of all $f\in F^2(H_n)$ such that $$\label{norm} \|f\|_\infty:=\sup\{\|f\otimes p\|_2:p\in{{\mathcal P}}, \ \|p\|_2\le 1\}<\infty.$$ If $f\in F^2(H_n)$, then $f\in F_n^\infty$ if and only if $f\otimes g\in F^2(H_n)$ for any $g\in F^2(H_n)$. Moreover, if $f\in F_n^\infty$, then the left multiplication mapping $L_f:F^2(H_n)\to F^2(H_n)$ defined by $$L_fg:=f\otimes g, \quad g\in F^2(H_n),$$ is a bounded linear operator with $\|L_f\|=\|f\|_\infty$. The noncommutative Hardy algebra $F_n^\infty$ is isometrically isomorphic to the left multiplier algebra of the full Fock space $F^2(H_n)$, which is also called the noncommutative Toeplitz algebra. Under this identification, $F_n^\infty$ is the weakly closed algebra generated by the left creation operators $S_1,\ldots, S_n$ and the identity. The noncommutative disc algebra ${{\mathcal A}}_n$ was introduced in [@Po-von] as is the norm closed algebra generated by the left creation operators and the identity. Let $ f=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ be an element in $F^2(H_n)$ and define $$f_r:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} r^{|\alpha|} a_\alpha e_\alpha \quad \text{ for } \ ~0<r<1.$$ In [@Po-funct], [@Po-poisson], we proved that if $~f\in F_n^\infty~$ then $\|f_r\|_\infty\leq \|f\|_\infty$ for $ 0\leq r<1$, and $$\label{So} L_f=\text{\rm{SOT-}}\lim\limits_{r\to1}f_r(S_1,\dots,S_n),$$ where $f_r(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$. Moreover, if $f\in {{\mathcal A}}_n$ then the above limit exists in the operator norm topology. We identify $M_m(B({{\mathcal H}}))$, the set of $m\times m$ matrices with entries from $B({{\mathcal H}})$, with $B( {{\mathcal H}}^{(m)})$, where ${{\mathcal H}}^{(m)}$ is the direct sum of $m$ copies of ${{\mathcal H}}$. Thus we have a natural $C^*$-norm on $M_m(B({{\mathcal H}}))$. If $X$ is an operator space, i.e., a closed subspace of $B({{\mathcal H}})$, we consider $M_m(X)$ as a subspace of $M_m(B({{\mathcal H}}))$ with the induced norm. Let $X, Y$ be operator spaces and $u:X\to Y$ be a linear map. Define the map $u_m:M_m(X)\to M_m(Y)$ by $$u_m ([x_{ij}]):=[u(x_{ij})].$$ We say that $u$ is completely bounded ($cb$ in short) if $$\|u\|_{cb}:=\sup_{m\ge1}\|u_m\|<\infty.$$ If $\|u\|_{cb}\leq1$ (resp. $u_m$ is an isometry for any $m\geq1$) then $u$ is completely contractive (resp. isometric), and if $u_m$ is positive for all $m$, then $u$ is called completely positive. For each $m=1,2,\ldots$, we define the norms $\|\cdot \|_m:M_m\left(H^\infty(B({{\mathcal X}})^n_1)\right)\to [0,\infty)$ by setting $$\|[F_{ij}]_m\|_m:= \sup \|[F_{ij}(X_1,\ldots, X_n)]_m\|,$$ where the supremum is taken over all $n$-tuples $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and any Hilbert space ${{\mathcal H}}$. It is easy to see that the norms $\|\cdot\|_m$, $m=1,2,\ldots$, determine an operator space structure on $H^\infty(B({{\mathcal X}})^n_1)$, in the sense of Ruan (see [@ER]). \[f-infty\] Let $F:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ be a free holomorphic function on the open operatorial unit $n$-ball. Then the following statements are equivalent: 1. $F$ is in $H^\infty(B({{\mathcal X}})^n_1)$; 2. $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F_n^\infty$; 3. $\sup\limits_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|<\infty$; 4. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $$\varphi(r):=F(rS_1,\ldots, rS_n) \quad \text{ for any } \ r\in [0,1)$$ has a continuous extension to $[0,1]$ with respect to the strong operator topology of $B(F^2(H_n))$. In this case, we have $$\label{many eq} \|L_f\|=\|f\|_\infty=\sup_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|= \lim_{r\to 1}\|F(rS_1,\ldots, rS_n)\|=\|F\|_\infty.$$ Moreover, the map $$\Phi:H^\infty((B({{\mathcal X}})^n_1)\to F_n^\infty\quad \text{ defined by } \quad \Phi(F):=f$$ is a completely isometric isomorphism of operator algebras. Assume (ii) holds. Since $f\in F_n^\infty$, we have $$\label{f-inf} \|F(rS_1,\ldots, rS_n)\|=\|f(rS_1,\ldots, rS_n)\|=\|L_{f_r}\|=\|f_r\|\leq \|f\|_\infty$$ for any $r\in [0,1)$. Therefore, (ii)$\implies$(iii). To prove that (iii)$\implies$(ii), assume that (iii) holds. Consequently, we have $$\begin{split} \sum_{\alpha\in {{\mathbb F}}_n^+} r^{2|\alpha|} |a_\alpha|^2&= \left\|\sum_{\alpha\in {{\mathbb F}}_n^+} r^{|\alpha|} a_\alpha S_\alpha(1)\right\|\\ &\leq \sup\limits_{0\leq r<1}\|F(rS_1,\ldots, rS_n)\|<\infty \end{split}$$ for any $0\leq r<1$. Hence, $\sum_{\alpha\in {{\mathbb F}}_n^+} |a_\alpha|^2<\infty$, which shows that $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F^2(H_n)$. Now assume that $f\notin F_n^\infty$. Due to the definition of $F_n^\infty$, given an arbitrary positive number $M$, there exists a polynomial $q\in {{\mathcal P}}$ with $\|q\|_2=1$ such that $$\|f\otimes q\|_2>M.$$ Since $\|f_r-f\|_2\to 0$ as $r\to 1$, we have $$\|f\otimes q-f_r\otimes q\|_2=\|(f-f_r)\otimes q\|_2\to 0, \quad \text{ as }\ r\to 1.$$ Therefore, there is $r_0\in (0,1)$ such that $ \|f_{r_0}\otimes q\|_2> M. $ Hence, $$\|f_{r_0}(S_1,\ldots, S_n)\|=\|L_{f_{r_0}}\|=\|f_{r_0}\|_\infty>M.$$ Since $M>0$ is arbitrary, we deduce that $$\sup_{0\leq r<1}\|f(rS_1,\ldots, rS_n)\|=\infty,$$ which is a contradiction. Consequently, (ii)$\Longleftrightarrow$(iii). Now, let us prove that (ii)$\implies$(iv). Assume (ii) and define the map $\tilde\varphi:[0,1]\to B(F^2(H_n)$ by setting $$\tilde\varphi(r):= \begin{cases} F(rS_1,\ldots, rS_n) &\quad \text{if } 0\leq r<1\\ L_f &\quad \text{if } r=1. \end{cases}$$ Since $f(rS_1,\ldots, rS_n)=F(rS_1,\ldots, rS_n)$,  $0\leq r<1$, the SOT-continuity of $\tilde\varphi$ at $r=1$ is due to relation , while the continuity of $\tilde\varphi$ on $[0,1)$ is a consequence of Theorem \[continuous\]. Therefore, the item (iv) holds. Assume now that (iv) holds. For each $x\in F^2(H_n)$, the map $[0,1)\ni \mapsto \|\varphi(r)x\|\in {{\mathbb R}}^+$ is bounded, i.e., $\sup\limits_{0\leq r<1}\|\varphi(r)x\|<\infty$. Due to the principle of uniform boundedness, we deduce condition (iii). The implication (i)$\implies$(iii) is obvious, and the implication (iii)$\implies$(i) is due to Theorem \[Abel\] and the noncommutative von Neumann inequality. Indeed, if $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$, ${{\mathcal H}}$ is an arbitrary Hilbert space, and $\|[X_1,\ldots, X_n]\|=r<1$, then $$\left\|\sum_{k=0}^m \sum_{|\alpha|=k} a_\alpha X_\alpha \right\|\leq \left\|\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|}a_\alpha S_\alpha \right\|, \quad m=1,2,\ldots.$$ Hence, and taking into account Theorem \[Abel\], we deduce that $$\|F(X_1,\ldots, X_n)\|\leq \|F(rS_1, \ldots, rS_n)\|,\quad \text{ for any } \ r\in [0,1).$$ Consequently, $$\label{supsup} \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1} \|F(X_1,\ldots, X_n)\|\leq \sup_{0\leq r<1} \|F(rS_1, \ldots, rS_n)\|<\infty,$$ whence (i) holds. We prove now the last part of the theorem. If $f\in F_n^\infty$ and $\epsilon>0$, then there exists a polynomial $q\in {{\mathcal P}}$ with $\|q\|_2=1$ such that $$\|f\otimes q\|_2>\|f\|_\infty-\epsilon.$$ Due to relation , there exists $r_0\in (0,1)$ such that $\|f_{r_0}(S_1,\ldots, S_n)q\|>\|f\|_\infty-\epsilon$. Using now relation , we deduce that $$\sup_{0\leq r<1}\|f(rS_1,\ldots, rS_n)\|=\|f\|_\infty.$$ Now, let $r_1,r_2\in [0,1)$ with $r_1<r_2$ and let $f:=\sum_{\alpha\in {{\mathbb F}}_n^+}a_\alpha e_\alpha $. Since $g:=\sum_{\alpha\in {{\mathbb F}}_n^+} r_2^{|\alpha|}a_\alpha e_\alpha $ is in the noncommutative disc algebra ${{\mathcal A}}_n$, we have $\|g_r\|_\infty\leq \|g\|_\infty$ for any $0\leq r<1$. In particular, when $r:=\frac {r_1}{r_2}$, we deduce that $$\|f_{r_1}(S_1,\ldots, S_n)\|\leq \|f_{r_2}(S_1,\ldots, S_n)\|.$$ Consequently, the function $[0,1]\ni r\to \|f(rS_1,\ldots, rS_n)\|\in {{\mathbb R}}^+$ is increasing. Hence, and using relation , we deduce . Using the same techniques, one can prove a matrix form of relation . In particular, we have $\|[F_{ij}]_m\|_m=\|[L_{f_{ij}}]_m\|$ for any $[F_{ij}]_m\in M_m\left(H^\infty(B({{\mathcal X}})^n_1)\right)$ and $m=1,2,\ldots$. Hence, we deduce that $\Phi$ is a complete isometry of $ H^\infty(B({{\mathcal X}})^n_1)$ onto $F_n^\infty$. The proof is complete. \[A-infty\] Let $F:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$ be a free holomorphic function on the open operatorial unit $n$-ball. Then the following statements are equivalent: 1. $F$ is in $A(B({{\mathcal X}})^n_1)$; 2. $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in ${{\mathcal A}}_n$; 3. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $$\varphi(r):=F(rS_1,\ldots, rS_n)$$ has a continuous extension to $[0,1]$, with respect to the operator norm topology of $B(F^2(H_n))$. Moreover, the map $$\Psi:A((B({{\mathcal X}})^n_1)\to {{\mathcal A}}_n\quad \text{ defined by } \quad \Psi(F):=f$$ is a completely isometric isomorphism of operator algebras. The implication (i)$\implies$(iii) is due to the definition of $A(B({{\mathcal X}})^n_1)$. Assume that item (ii) holds, i.e., $f\in {{\mathcal A}}_n$. The norm continuity of $\varphi$ on \[0,1) is due to Theorem \[continuous\], while the continuity of $\varphi$ at $r=1$ is due to the fact that $\lim_{r\to 1} f_r(S_1,\dots, S_n)=L_f$ in the operator norm for any $f\in {{\mathcal A}}_n$ (see the remarks preceeding this theorem). Therefore, the implication (ii)$\implies$(iii) is true. Conversely, assume item (iii) holds. Then $\lim_{r\to \infty} F(rS_1,\ldots, rS_n) $ exists in the operator norm. Since $F(rS_1,\ldots, rS_n)\in {{\mathcal A}}_n$ and ${{\mathcal A}}_n$ is a Banach algebra, there exists $g\in {{\mathcal A}}_n$ such that $L_g=\lim_{r\to \infty} F(rS_1,\ldots, rS_n)$ in the operator norm. On the other hand, due to Theorem \[f-infty\], we deduce that $f:=\sum_{\alpha\in {{\mathbb F}}_n} a_\alpha e_\alpha\in F_n^\infty$. Since $f(rS_1,\ldots, rS_n)=F(rS_1,\ldots, rS_n)$,  $0\leq r<1$, and $L_f=\text{\rm SOT}-\lim_{r\to \infty} f(rS_1,\ldots, rS_n)$, we conclude that $L_f=L_g$, i.e., $f=g$. Therefore, condition (ii) holds. It remains to prove that (ii)$\implies$(i). According to [@Po-funct] (see also [@Po-poisson]), if $f\in{{\mathcal A}}_n$ then, for any $n$-tuple $[Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-$, $$\tilde F(Y_1,\ldots, Y_n):= \lim_{r\to 1} f(rY_1,\ldots, rY_n),$$ exists in the operator norm, and $$\|\tilde F(Y_1,\ldots, Y_n)\|\leq \|f\|_\infty\quad \text{ for any } \ [Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-.$$ Notice also that $\tilde F$ is an extension of the free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$. Indeed, if $ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$, then $$\begin{split} \tilde F(X_1,\ldots, X_n)&=\lim_{r\to 1}f(rX_1,\ldots, rX_n)\\ &=\lim_{r\to 1}F(rX_1,\ldots, rX_n)=F(X_1,\ldots, X_n). \end{split}$$ The last equality is due to Theorem \[continuous\]. Let us prove that $\tilde F:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is continuous. Since $f\in {{\mathcal A}}_n$, for any $\epsilon>0$ there exists $r_0\in [0,1)$ such that $\|L_f-f(r_0S_1,\ldots, r_0 S_n)\|<\epsilon$. Applying the above mentioned result from [@Po-poisson] to $ f-\ f_{r_0}\in {{\mathcal A}}_n$, we deduce that $$\label{tild-f} \|\tilde F(T_1,\ldots, T_n)-f_{r_0}(T_1,\ldots, T_n)\|\leq \|L_f-L_{f_{r_0}} \|< \frac{\epsilon}{3}$$ for any $[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1^-$. Due to Theorem \[continuous\], $F$ is a continuous function on $[B({{\mathcal H}})^n]_1$. Therefore, there exists $\delta>0$ such that $$\|F_{r_0}(T_1,\ldots, T_n)-F_{r_0}(Y_1,\ldots, Y_n)\|<\frac{\epsilon}{3}$$ for any $n$-tuples $[T_1,\ldots, T_n]$ and $[Y_1,\ldots, Y_n]$ in $[B({{\mathcal H}})^n]_1^-$ such that $\|[T_1-Y_1,\ldots, T_n-Y_n]\|<\delta$. Hence, and using , we have $$\begin{split} \|\tilde F(T_1,\ldots, T_n)-\tilde F(Y_1,\ldots, Y_n)\| &\leq \|\tilde F(T_1,\ldots, T_n)-f_{r_0}(T_1,\ldots, T_n)\|\\ &\qquad + \| f_{r_0}(T_1,\ldots, T_n)- f_{r_0}(Y_1,\ldots, Y_n)\|\\ &\qquad + \|f_{r_0}(Y_1,\ldots, Y_n)-\tilde F(Y_1,\ldots, Y_n)\| <\epsilon, \end{split}$$ whenever $\|[T_1-Y_1,\ldots, T_n-Y_n]\|<\delta$. This proves the continuity of $\tilde F$ on $[B({{\mathcal H}})^n]_1^-$. Therefore, $F\in A(B({{\mathcal X}})^n_1)$. To prove the last part of the theorem, notice that if $f\in{{\mathcal A}}_n\subset F_n^\infty$, then by Theorem \[f-infty\] (see relation and its matrix form), we have $\|[L_{f_{ij}}]_m\|=\|[F_{ij}]_m\|_m$. Since ${{\mathcal A}}_n\subset B(F^2(H_n))$ is an operator algebra, we deduce that $\Psi$ is a completely isometric isomorphism of operator algebras. This completes the proof. Here is our version of the maximum principle for free holomorphic functions. \[max-mod1\] Let ${{\mathcal H}}$ be an infinite dimensional Hilbert space. Assume that $f:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is a continuous function in the operator norm, and it is free holomorphic on $[B({{\mathcal H}})^n]_1$. Then $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq 1\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= 1\}. \end{split}$$ Due to the continuity of $f$, for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1^-$, $$\|f(X_1,\ldots, X_n)\|=\lim_{r\to 1} \|f(rX_1,\ldots, rX_n)\|.$$ On the other hand, the noncommutative von Neumann inequality implies $$\|f(rX_1,\ldots, rX_n)\|\leq \|f(rS_1,\ldots, rS_n)\| \quad \text{ for } \ 0\leq r<1.$$ By Theorem \[A-infty\], $f\in {{\mathcal A}}_n$ and, consequently, $$\lim_{r\to 1} \|f(rS_1,\ldots, rS_n)\|=\|L_f\|=\|f\|_\infty.$$ Combining these relations, we deduce that $$\label{ff} \|f(X_1,\ldots, X_n)\|\leq \|f\|_\infty\quad \text{ for any } \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Since ${{\mathcal H}}$ is infinite dimensional, there exists a subspace ${{\mathcal K}}\subset {{\mathcal H}}$ and a unitary operator $U:F^2(H_n)\to {{\mathcal K}}$. Define the operators $$V_i:=\left(\begin{matrix} US_iU^*&0\\ 0&0 \end{matrix}\right), \quad i=1,\ldots,n,$$ with respect to the orthogonal decomposition ${{\mathcal H}}={{\mathcal K}}\oplus {{\mathcal K}}^{\perp}$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$. Notice that $\|[V_1,\ldots, V_n]\|=1$ and $$f(V_1,\ldots, V_n)=\lim_{r\to 1} \left(\begin{matrix} Uf_r(S_1,\ldots, S_n)U^*&0\\ 0&0 \end{matrix}\right)$$ in the operator norm. Consequently, $$\|f(V_1,\ldots, V_n)\|=\lim_{r\to 1}\|f_r(S_1,\ldots, S_n)\|=\|f\|_\infty.$$ Hence, and using inequality , we deduce that $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq 1\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= 1\}\\ &=\|f\|_\infty. \end{split}$$ This completes the proof. \[max-mod2\] Let $f$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space, and let $r\in [0,1)$. Then $$\begin{split} \max\{\|f(X_1,\ldots, X_n)\|&:\ \|[X_1,\ldots, X_n]\|\leq r\}\\ &= \max\{\|f(X_1,\ldots, X_n)\|:\ \|[X_1,\ldots, X_n]\|= r\}\\ &=\|f(rS_1,\ldots, rS_n)\|. \end{split}$$ In a forthcoming paper [@Po-Bohr], we obtain operator-valued multivariable Bohr type inequalities for free holomorphic functions on the open operatorial unit $n$-ball. As consequences, we obtain operator-valued Bohr inequalities for the noncommutative disc algebra ${{\mathcal A}}_n$ and the noncommutative analytic Toeplitz algebra $F_n^\infty$. Free analytic functional calculus and noncommutative Cauchy transforms {#free analytic} ======================================================================= In this section, we introduce a free analytic functional calculus for $n$-tuples $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We introduce a noncommutative Cauchy transform ${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$ associated with any such $n$-tuple of operators and prove that $$f(T_1,\ldots, T_n)=C_T(f(S_1,\ldots, S_n)),\quad f\in H^\infty (B({{\mathcal X}})^n_1),$$ where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Similar Cauchy representations are obtained for the $k$-order Hausdorff derivations of $f$. Finally, we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}({{\mathbb C}}^n)$. \[abel\] Let $F:=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} A_{(\alpha)}\otimes Z_\alpha$ be a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$. Then, for any Hilbert space ${{\mathcal H}}$ and any $n$-tuple of operators $[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with $r(X_1,\ldots,X_n)<\gamma$, the series $$F(X_1,\ldots, X_n)=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} A_{(\alpha)}\otimes X_\alpha$$ is convergent in the operator norm of $B({{\mathcal K}}\otimes {{\mathcal H}})$. Moreover, if   $0<r<1$, then $$\label{lim-Fr} \lim_{r\to 1}F_r(X_1,\ldots, X_n)=F(X_1,\ldots, X_n)$$ and $$\label{lim-PFr} \lim_{r\to 1}\left(\frac{\partial^k F_r}{\partial Z_{i_1}\cdots Z_{i_k}}\right)(X_1,\ldots, X_n)=\left(\frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}\right)(X_1,\ldots, X_n)$$ for $i_1,\ldots, i_k\in \{1,\ldots, n\}$, where the limits are in the operator norm. . Assume that $[X_1,\ldots, X_n]$ is an $n$-tuple of operators on ${{\mathcal H}}$ such that $r(X_1,\ldots,X_n)<R$, where $R$ is the radius of convergence of $F$. Let $\rho',\rho>0$ be such that $r(X_1,\ldots,X_n)<\rho'<\rho<R$. Due to the definition of $r(X_1,\ldots,X_n)$, there exists $k_0\in {{\mathbb N}}$ such that $$\label{ro'} \left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2k}< \rho'\quad \text{ for any }\ k\geq k_0.$$ Since $\frac{1}{\rho}> \frac{1}{R}$, we can find $m_0$ such that $$\label{ro} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}A_{(\alpha)}^*\right\|^{1/2k}< \frac{1}{\rho}\quad \text{ for any }\ k\geq m_0.$$ If $k\geq \max\{k_0,m_0\}$, then relations and imply $$\begin{split} \left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha \right\|&= \left\|\left[ I\otimes X_\alpha:\ |\alpha|=k\right] \left[\begin{matrix} A_{(\alpha)}\otimes I\\ :\\|\alpha|=k \end{matrix}\right]\right\|\\ &=\left\|\sum\limits_{|\alpha|=k}X_{\alpha}X_\alpha^*\right\|^{1/2}\left\|\sum\limits_{|\alpha|=k}A_{(\alpha)}A_{(\alpha)}^*\right\|^{1/2}\\ &\leq \left(\frac{\rho'}{\rho}\right)^k. \end{split}$$ This proves the convergence of the series $\sum\limits_{k=0}^\infty \left( \sum\limits_{|\alpha|=k}A_{(\alpha)}\otimes X_\alpha\right)$ in the operator norm. Now, using the above inequalities, we obtain $$\begin{split} \left\|\sum_{k=0}^\infty \sum_{|\alpha|=k} (r^{|\alpha|}-1)A_\alpha\otimes X_\alpha\right\|&\leq \sum_{k=1}^\infty (r^k-1)\left\|\sum_{|\alpha |=k}A_\alpha \otimes X_\alpha\right\|\\ &\leq \sum_{k=1}^\infty (r^k-1) \left(\frac{\rho'}{\rho}\right)^k\\ &\leq (r-1)\sum_{k=1}^\infty k \left(\frac{\rho'}{\rho}\right)^k. \end{split}$$ Since $\rho'<\rho$, the latter series is convergent and therefore relation holds. Due to Theorem \[derivation\], $\frac{\partial F}{\partial Z_i}$ is a free holomorphic function on the open operatorial $n$-ball of radius $\gamma$, and $$\frac{\partial^k F_r}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(X_1,\ldots, X_n)=r^k \frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(rX_1,\ldots, rX_n),\quad 0<r<1.$$ Applying relation to $\frac{\partial^k F}{\partial Z_{i_1}\cdots \partial Z_{i_k}}$, we deduce . The proof is complete. Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. We introduce the [*Cauchy kernel*]{} associated with $T$ to be the operator $C_T(S_1,\ldots, S_n)\in B(F^2(H_n)\otimes {{\mathcal H}})$ defined by $$\label{Cauc} C_T(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde\alpha}^*,$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$, and $\tilde\alpha$ is the reverse of $\alpha$, i.e., $\tilde \alpha= g_{i_k}\cdots g_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$. Applying Theorem \[Abel\], when $A_{(\alpha)}:=T_\alpha^*$, $\alpha\in {{\mathbb F}}_n^+$ and $X_i:=S_i$, $i=1,\ldots, n$, we deduce that $$\frac{1}{R}=\lim_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2k}=r(T_1,\ldots, T_n)<1$$ and $\|[S_1,\ldots, S_n]\|=1<R$. Consequently, the series in is convergent in the operator norm and $C_T(S_1,\ldots, S_n)\in {{\mathcal A}}_n\bar\otimes B({{\mathcal H}})\subset B(F^2(H_n)\otimes {{\mathcal H}})$. Now, one can easily see that $$\label{Cauc-inv} C_T(S_1,\ldots, S_n)=\left( I-S_1\otimes T_1^*-\cdots -S_n\otimes T_n^*\right)^{-1}.$$ We call the operator $$S_1\otimes T_1^*+\cdots +S_n\otimes T_n^*$$ the [*reconstruction operator*]{} associated with the $n$-tuple $[T_1,\ldots, T_n]$. We should mention that this operator plays an important role in noncommutative multivariable operator theory (see [@Po-varieties], [@Po-unitary]). We remark that if $1$ is not in the spectrum of the reconstruction operator, then the Cauchy kernel defined by makes sense. In this case, $C_T(S_1,\ldots, S_n)$ is in $F_n^\infty\bar\otimes B({{\mathcal H}})$, the $WOT$-closed operator algebra generated by the spatial tensor product, and not necessarily in ${{\mathcal A}}_n\bar\otimes B({{\mathcal H}})$. Morever, we can think of the series $\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde \alpha}^*$ as the Fourier representation of the Cauchy kernel. In what follows we also use the notation $C_T:=C_T(S_1,\ldots, S_n)$. \[Prop-Cauc\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Then: 1. $\|C_T\|\leq \sum\limits_{k=0}^\infty\left\|\sum\limits_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}$. In particular, if $T:=[T_1,\ldots, T_n]\in[B({{\mathcal H}})^n]_1$, then $\|C_T\|\leq \frac{1}{1-\|T\|}$. 2. $C_T-C_X=C_T\left[\sum\limits_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right] C_X$ and $$\|C_T-C_X\|\leq \|C_T\| \|C_X\|\|[T_1-X_1,\ldots, T_n-X_n]\|$$ for any $n$-tuple $X:=[X_1,\ldots, X_n]\in B({{\mathcal H}})^n$ with joint spectral radius $r(X_1,\ldots, X_n)<1$. Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we have $$\|C_T\|\leq \sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} S_\alpha\otimes T_{\tilde \alpha}^*\right\| = \sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} T_\alpha T_{\alpha}^*\right\|^{1/2}.$$ If $\|[T_1,\ldots, T_n]\|<1$, then $$\sum_{k=0}^\infty\left\| \sum_{|\alpha|=k} T_\alpha T_{\alpha}^*\right\|^{1/2} \leq \sum_{k=0}^\infty\left\| \sum_{i=1}^n T_iT_i^*\right\|^{k/2} =\frac{1}{1-\|T\|}.$$ To prove (ii), notice that $$\begin{split} C_T-C_X&= \left(I-\sum_{i=1}^n S_i\otimes T_i^*\right)^{-1} \left[ I-\sum_{i=1}^n S_i\otimes X_i^*-\left(I-\sum_{i=1}^n S_i\otimes T_i^*\right)\right]\left(I-\sum_{i=1}^n S_i\otimes X_i^*\right)^{-1}\\ &=C_T\left[\sum\limits_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right] C_X, \end{split}$$ and $$\begin{split} \|C_T-C_X\|&\leq \|C_T\| \|C_X\|\left\| \sum_{i=1}^n S_i\otimes (T_i^*-X_i^*)\right\|\\ &=\|C_T\| \|C_X\|\left\|\sum_{i=1}^n (T_i-X_i)(T_i-X_i)^*\right\|^{1/2}, \end{split}$$ which completes the proof. The [*Cauchy transform*]{} at $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is the mapping $${{\mathcal C}}_T:B(F^2(H_n))\to B({{\mathcal H}})$$ defined by $$\left< {{\mathcal C}}_T(A)x,y\right>:= \left<(A\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>$$ for any $x,y\in {{\mathcal H}}$, where $R_1,\ldots, R_n$ are the right creation operators on the full Fock space $F^2(H_n)$. The operator ${{\mathcal C}}_T(A)$ is called the Cauchy transform of $A$ at $T$. Given $A\in B(F^2(H_n))$, the Cauchy transform generates a function (the Cauchy transform of $A$) $${{\mathcal C}}[A]:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$$ by setting $${{\mathcal C}}[A](X_1,\ldots, X_n):={{\mathcal C}}_X(A) \quad \text{ for any } \ X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Indeed, it is enough to see that $r(X_1,\ldots, X_n)\leq \|[X_1,\ldots, X_n]\|<1$, and therefore ${{\mathcal C}}_X(A)$ is well-defined. This gives rise to an important question: when is ${{\mathcal C}}[A]$ a free holomorphic function on $[B({{\mathcal H}})^n]_1$. Due to Theorem \[abel\], if $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$ then, we can define a bounded linear operator $$f(T_1,\ldots, T_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha T_\alpha,$$ where the series converges in norm. This provides the [*free analytic functional calculus*]{}. If $F=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha $ is in the Hardy algebra $H^\infty(B({{\mathcal X}})^n_1)$, we denote by $F(S_1,\ldots, S_n)$ the boundary function of $F$, i.e., $F(S_1,\ldots, S_n):=L_f\in B(F^2(H_n))$, where $f:= \sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$. \[an=cauch\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. Then, for any $f\in H^\infty(B({{\mathcal X}})^n_1)$, $$f(T_1,\ldots, T_n)={{\mathcal C}}_T(f(S_1,\ldots, S_n)),$$ where $f(T_1,\ldots, T_n)$ is defined by the free analytic functional calculus, and $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Moreover, $$\|f(T_1,\ldots, T_n)\|\leq \left(\sum\limits_{k=0}^\infty\left\|\sum\limits_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\right) \|f\|_\infty.$$ First, we prove the above equality for monomials. Notice that $$\begin{split} \left<{{\mathcal C}}_T(S_\alpha)x,y\right>&= \left< (S_\alpha\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>\\ &=\left< e_\alpha\otimes x, \left(\sum_{\beta\in {{\mathbb F}}_n^+} R_\beta\otimes T_{\tilde \beta}^*\right)(1\otimes y)\right>\\ &= \left< e_\alpha\otimes x, \sum_{\beta\in {{\mathbb F}}_n^+} e_{\tilde \beta}\otimes T_{\tilde \beta}^*y\right>\\ &=\left<T_\alpha x,y\right> \end{split}$$ for any $ x,y\in {{\mathcal H}}$. Now, assume that $f:=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha Z_\alpha$ is in $H^\infty(B({{\mathcal X}})^n_1)$ and $0<r<1$. Then, due to Theorem \[abel\], we have $$\lim_{m\to\infty}\sum_{k=0}^m r^k \sum_{|\alpha|=k} a_\alpha S_\alpha=f_r(S_1,\ldots, S_n)\in {{\mathcal A}}_n$$ in the operator norm of $B(F^2(H_n))$, and $$\lim_{m\to\infty}\sum_{k=0}^m r^k \sum_{|\alpha|=k} a_\alpha T_\alpha=f_r(T_1,\ldots, T_n)$$ in the operator norm of $B({{\mathcal H}})$. Now, due to the continuity of the noncommutative Cauchy transform in the operator norm, we deduce that $$\label{f_r-C} f_r(T_1,\ldots, T_n)={{\mathcal C}}_T(f_r(S_1,\ldots, S_n)).$$ Since $f(S_1,\ldots, S_n)\in F_n^\infty$, we know that $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=f(S_1,\ldots, S_n)$ in the strong operator topology. Since $\|f_r(S_1,\ldots, S_n)\|\leq \|f\|_\infty$, we deduce that $$\text{\rm SOT}-\lim\limits_{r\to 1}f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}=f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}.$$ On the other hand, by Theorem \[abel\], $\lim\limits_{r\to 1} f_r(T_1,\ldots, T_n)=f(T_1,\ldots, T_n)$ in the operator norm. Passing to the limit, as $r\to 1$, in the equality $$\left<f_r(T_1,\ldots, T_n)x,y\right>=\left<(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), C_T(R_1,\ldots, R_n)(1\otimes y)\right>, \quad x, y\in {{\mathcal H}},$$ we obtain $f(T_1,\ldots, T_n)={{\mathcal C}}_T(f(S_1,\ldots, S_n))$, which proves the first part of the theorem. Now, we can deduce the second part of the theorem using Proposition \[Prop-Cauc\]. This completes the proof. Using the Cauchy representation provided by Theorem \[an=cauch\], one can deduce the following result. \[conv-u-w\*\] Let $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$. 1. If $\{f_k\}_{k=1}^\infty$ and $f$ are free holomorphic functions in $Hol(B({{\mathcal X}})^n_1)$ such that $\|f_k-f\|_\infty\to 0$, as $k\to \infty$, then $f_k(T_1,\ldots, T_n)\to f(T_1,\ldots, T_n)$ in the operator norm of $B({{\mathcal H}})$. 2. If $\{f_k\}_{k=1}^\infty$ and $f$ are in the algebra $H^\infty (B({{\mathcal X}})^n_1)$ such that $f_k(S_1,\ldots, S_n)\to f(S_1,\ldots, S_n)$ in the $w^*$-topology (or strong operator topology) and $\|f_k\|_\infty\leq M$ for any $k=1,2,\ldots$, then $f_k(T_1,\ldots, T_n)\to f(T_1,\ldots, T_n)$ in the weak operator topology. We can extend Theorem \[an=cauch\] and obtain Cauchy representations for the $k$-order Hausdorff derivations of bounded free holomorphic functions. \[cauc-dif\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with the joint spectral radius $r(T_1,\ldots, T_n)<1$ and let $f \in H^\infty(B({{\mathcal X}})^n_1)$. Then $$\label{deriv-Cau} \begin{split} \Bigl<\left(\frac{\partial^k f}{\partial Z_{i_1}\cdots \partial Z_{i_k}}\right)&(T_1,\ldots, T_n) x,y\Bigr>\\ &= \left<\left[\frac{\partial^k \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i_1}\cdots \partial T_{i_k}}\right](f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right> \end{split}$$ for any  $i_1,\ldots, i_k\in \{ 1,\ldots, n\}$ and $x,y\in {{\mathcal H}}$, where $f(S_1,\ldots, S_n)$ is the boundary function of $f$. Moreover, $$\label{deriv-est} \left\| \left(\frac{\partial f}{\partial Z_i}\right)(T_1,\ldots, T_n)\right\|\leq \|f\|_\infty \sum_{k=1}^\infty k^{3/2}\left\|\sum_{|\beta|=k-1}T_\beta T_\beta\right\|^{1/2}, \quad i=1,\ldots,n.$$ First, notice that $$C_X(R_1,\ldots, R_n)^*=\sum_{k=0}^\infty \sum_{|\alpha|=k} R_{\tilde \alpha}^*\otimes X_\alpha,$$ where the series is convergent in norm for each $n$-tuple $[X_1,\ldots, X_n] $ with $r(X_1,\ldots, X_n)<1$. Therefore, $$G:=\sum_{k=0}^\infty \sum_{|\alpha|=k} R_{\tilde \alpha}^*\otimes Z_\alpha$$ is a free holomorphic function on the open operatorial unit $n$-ball. Due to Theorem \[derivation\], $\frac{\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}$ is also a free holomorphic function. By Theorem \[abel\], $\frac{\partial^k G}{\partial Z_{i_1}\cdots \partial Z_{i_k}}(X_1,\ldots, X_n)$ is a bounded operator for any $n$-tuple $[X_1,\ldots, X_n]$ with spectral radius $r(X_1,\ldots, X_n)<1$. Now, notice that, for each $\alpha\in {{\mathbb F}}_n^+$, $i=1,\ldots, n$, and $ x,y\in {{\mathcal H}}$, we have $$\begin{split} \Bigl<\Bigl[\frac{\partial \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i}}\Bigr]&(S_\alpha\otimes I_{{\mathcal H}}) (1\otimes x), 1\otimes y\Bigr>\\ &= \left< \left(\sum_{k=0}^\infty \sum_{|\beta|=k}R_\beta^*\otimes \frac{\partial T_{\tilde \beta}}{\partial T_i}\right)(S_\alpha\otimes I_{{\mathcal H}}) (1\otimes x), 1\otimes y\right>\\ &= \left<e_\alpha\otimes x, \sum_{k=0}^\infty \sum_{|\beta|=k} e_{\tilde \beta}\otimes \left(\frac{\partial T_{\tilde \beta}}{\partial T_i}\right)^*y\right>\\ &= \left< \frac{\partial T_\alpha}{\partial T_i}x,y\right> =\left<\left(\frac{\partial Z_\alpha}{\partial Z_i}\right)(T_1,\ldots, T_n) x,y\right>. \end{split}$$ Hence, we deduce relation for polynomials. Let $f=\sum_{k=1}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ be in $H^\infty(B({{\mathcal X}})^n_1)$. Due to Theorem \[abel\], we have $$\left(\frac{\partial f_r}{\partial Z_i}\right)(T_1,\ldots, T_n) =\lim_{m\to\infty}\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha \left(\frac{\partial Z_\alpha}{\partial Z_i}\right)(T_1,\ldots, T_n),$$ where the convergence is in the operator norm of $B({{\mathcal H}})$, and $$f_r(S_1,\ldots, S_n)=\lim_{m\to\infty}\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha\in {{\mathcal A}}_n$$ where the convergence is in the operator norm of $B(F^2(H_n))$. Since holds for polynomials, the last two relations imply $$\begin{split} \Bigl<\left(\frac{\partial f_r}{\partial Z_{i}}\right)&(T_1,\ldots, T_n) x,y\Bigr>\\ &= \left<\left[\frac{\partial \left(C_T(R_1,\ldots, R_n)^*\right)}{\partial T_{i} }\right](f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right> \end{split}$$ for any $x,y\in {{\mathcal H}}$ and $0<r<1$. Using again Theorem \[abel\], we have $$\lim_{r\to 1} \left(\frac{\partial f_r}{\partial Z_{i}}\right)(T_1,\ldots, T_n)= \left(\frac{\partial f}{\partial Z_{i}}\right)(T_1,\ldots, T_n)$$ in the operator norm. Since $f(S_1,\ldots, S_n)\in F_n^\infty$ (see Theorem \[f-infty\]), as in the proof of Theorem \[an=cauch\], we deduce that $$\text{\rm SOT}-\lim_{r\to\infty} f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}=f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}.$$ Passing to the limit, as $r\to\infty$, in the above equality, we deduce relation in the particular case when $k=1$. Repeating this argument, one can prove the general case when $\frac{\partial}{\partial T_i}$ is replaced by $\frac{\partial^k}{\partial T_{i_1}\cdots \partial T_{i_k}}$. Now, we prove the second part of the theorem. Notice that $$\begin{split} \left\|\frac{\partial G}{\partial Z_i}(X_1,\ldots, X_n)\right\| &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}R_{\tilde \alpha}\otimes \left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|\\ &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k} \left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|^{1/2}. \end{split}$$ For each $\alpha\in {{\mathbb F}}_n^+$, $|\alpha|=k$, we can prove that $$\label{X_ga} \left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\leq k^2\sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*,$$ where the sum is taken over all distinct words $\gamma$ obtained by deleting each occurence of $g_i$ in $\alpha$. Indeed, notice first that $\frac{\partial X_\alpha}{\partial X_i}=\sideset{}{^\alpha}\sum\limits_\beta X_\beta$, where the sum is taken over all words $\beta$ obtained by deleting each occurence of $g_i$ in $\alpha$. Since the above some contains at most $k$ terms, one can show that $$\left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\leq k\sideset{}{^\alpha}\sum_\beta X_\beta X_\beta^*.$$ Indeed, it enough to use the following result which is an easy consequence of the classical Cauchy inequality: if $A_1,\ldots, A_k\in B({{\mathcal H}})$, then $$\left(\sum_{i=1}^k A_i\right)\left(\sum_{i=1}^k A_i^*\right) \leq k\sum_{i=1}^k A_iA_i^*.$$ Now, the $X_\beta$’s in the above sum are not necessarily distinct but each of them can occur at most $k$ times. Consequently, $$\sideset{}{^\alpha}\sum\limits_\beta X_\beta X_\beta^*\leq k \sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*.$$ Combining these inequalities, we deduce . (We remark that the inequality is sharp and the equality occurs, for example, when $\alpha=g_i^k$.) Therefore, we have $$\begin{split} \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}\left(\frac{\partial X_\alpha}{\partial X_i}\right)\left(\frac{\partial X_\alpha}{\partial X_i}\right)^*\right\|^{1/2} &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k} k^2\sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*\right\|^{1/2}. \end{split}$$ We remark that if $\beta\in {{\mathbb F}}_n^+$, $|\beta|=k-1$, then $X_\beta$ can come from free differentiation with respect to $X_i$ of the monomials $X_{\chi(g_i,m,\beta)}$, $m=0,1,\ldots, k-1$, where $\chi(g_i,m,\beta)$ is the insertion mapping of $g_i$ on the $m$ position of $\beta$ (see the proof of Theorem \[derivation\]). Consequently, we have $$\sum_{|\alpha|=k} \sideset{}{^\alpha_d}\sum_\gamma X_\gamma X_\gamma^*\leq k \sum_{|\beta|=k-1} X_\beta X_\beta^*.$$ Using the above inequalities, we obtain $$\left\|\frac{\partial G}{\partial Z_i}(X_1,\ldots, X_n)\right\|\leq \sum_{k=1}^\infty k^{3/2}\left\|\sum_{|\beta|=k-1}X_\beta X_\beta\right\|^{1/2}.$$ Hence, and due to relation , we deduce inequality . The proof is complete. We remark that inequalities of type can be obtained for $k$-order Housdorff derivations. On the other hand, a similar result to Corollary \[conv-u-w\*\] can be obtain for $k$-order Housdorff derivations, if one uses Theorem \[cauc-dif\]. In the last part of this section, we show that the noncommutative Cauchy transform commutes with certain classes of automorphisms. Let ${{\mathcal U}}(H_n)$ be the group of all unitaries on $H_n$ and let $U\in {{\mathcal U}}(H_n)$. If $U:=\left[\lambda_{ij}\right]_{i,j=1}^n$ and $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$, we define $$\beta_U(T_j):=\sum_{i=1}^n \lambda_{ij} T_i,\quad j=1,\ldots, n,$$ and the map $\beta_U: B({{\mathcal H}})^n\to B({{\mathcal H}})^n$ by setting $\beta_U(T):=[\beta_U(T_1),\ldots, \beta_U(T_n)]. $ \[auto\] If $U\in {{\mathcal U}}(H_n)$,  $U:=\left[\lambda_{ij}\right]_{i,j=1}^n$, then the map $\beta_U $ is an isometric automorphism of the open unit ball $[B({{\mathcal H}})^n]_1$ and also of the ball $$\{[T_1,\ldots, T_n]\in B({{\mathcal H}})^n:\ r(T_1,\ldots, T_n)<1\}.$$ Moreover, there is a unique completely isometric automorphism of the noncommutative disc algebra ${{\mathcal A}}_n$, denoted also by $\beta_U$, such that $$\beta_U(S_j):= \sum_{i=1}^n \lambda_{ij} S_i,\quad j=1,\ldots, n,$$ where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space. For each $j=1,\ldots, n$, we define the operators $${\bf U}_j:=\left[\begin{matrix} \lambda_{1j}I_{{\mathcal H}}\\ \vdots\\ \lambda_{nj}I_{{\mathcal H}}\end{matrix}\right]:{{\mathcal H}}\to {{\mathcal H}}^{(n)},$$ where ${{\mathcal H}}^{(n)}$ is the direct sum of $n$ copies of ${{\mathcal H}}$. Notice that $$\label{Cuntz} {\bf U}_i^*{\bf U}_j=\delta_{ij} I_{{{\mathcal H}}^{(n)}}\quad \text{ and } \quad \sum_{i=1}^n{\bf U}_i{\bf U}_i^*=I_{{{\mathcal H}}^{(n)}}.$$ We have $\beta_U(T)=[B_1,\ldots,B_n]$, where $B_i:=T{\bf U}_i$,  $i=1,\ldots, n$ and $T:=[T_1,\ldots, T_n]$. Now, it is clear that $\sum_{i=1}^n B_iB_i^*=\sum_{i=1}^n T_iT_i$. If $A\in B({{\mathcal H}})$ then $${\bf U}_iA=\text{diag}_n (A) {\bf U}_i,\qquad i=1,\ldots, n,$$ where $\text{diag}_n (A)$ is the $n\times n$ block diagonal operator matrix having $A$ on the diagonal and $0$ otherwise. Using this relation and , we deduce that $$\begin{split} \sum_{|\alpha|=2} B_\alpha B_\alpha^*&=\sum_{i=1}^n B_i\left(\sum_{|\alpha|=1} B_\alpha B_\alpha^*\right) B_i\\ &= T\left[ \sum_{i=1}^n {\bf U}_i(TT^*){\bf U}_i^*\right] T^*\\ &= T\text{diag}_n (TT^*) \left( \sum_{i=1}^n{\bf U}_i{\bf U}_i^* \right) T^*\\ &=T\text{diag}_n (TT^*)T=\sum_{|\alpha|=2} T_\alpha T_\alpha^*. \end{split}$$ By induction over $k$, one can similarly prove that $$\label{B-T} \sum_{|\alpha|=k} B_\alpha B_\alpha^*=\sum_{|\alpha|=k} T_\alpha T_\alpha^*, \quad k=1,2,\ldots.$$ Consequently, we have $$\|\beta_U(T)\|=\|T\|\quad \text{ and }\quad r(\beta_U(T))=r(T).$$ Hence, and since $\beta_U(T)=T{\bf U}$, where ${\bf U}:=[{\bf U}_1,\ldots, {\bf U}_n]$ is a unitary operator, we deduce that the map $\beta_U:[B({{\mathcal H}})^n]_1\to [B({{\mathcal H}})^n]_1$ is an isometric authomorphism of the open unit ball of $B({{\mathcal H}})^n$ and $$\beta_U^{-1}(Y)=Y{\bf U}^*,\quad Y\in [B({{\mathcal H}})^n]_1.$$ Moreover, $\beta_U$ is an isometric automorphism of the operatorial ball $$\{[T_1,\ldots, T_n]\in B({{\mathcal H}})^n:\ r(T_1,\ldots, T_n)<1\}.$$ Now, let us prove the second part of the theorem. Using the same notation for the unitary operator ${\bf U}$, when ${{\mathcal H}}:=F^2(H_n)$, we deduce that $[\beta_U(S_1),\ldots, \beta_U(S_n)]=S{\bf U}$, where $S:=[S_1,\ldots, S_n]$. Setting $V_i:=\beta_U(S_i)$, $i=1,\ldots, n$, one can easily see that $V_1,\ldots, V_n$ are isometries with orthogonal ranges. For any polynomial $p(S_1,\ldots, S_n)$ in the noncommutative disc algebra ${{\mathcal A}}_n$, we have $\beta_U(p(S_1,\ldots, S_n))=p(V_1,\ldots, V_n)$. According to [@Po-disc], we have $$\|[p_{ij}(S_1,\ldots, S_n)]_m\|=\|[p_{ij}(V_1,\ldots, V_n)]_m\|.$$ Since ${{\mathcal A}}_n$ is the norm closure of all polynomials in $S_1,\ldots, S_n$ and the identity, $\beta_U$ can be uniquely extended to a completely isometric homomorphism from ${{\mathcal A}}_n$ to ${{\mathcal A}}_n$. Define the $n$-tuple $[X_1,\ldots, X_n]:=[S_1,\ldots,S_n]{\bf U}^*$ and notice that each entry $X_i$ is a homogenous polynomial of degree one in $S_1,\ldots, S_n$. Since $$[\beta_U(X_1),\ldots, \beta_U(X_n)]= [X_1,\ldots, X_n] {\bf U}= [S_1,\ldots,S_n],$$ we deduce that $\beta_U(X_i)=S_i$,  $i=1,\ldots, n$, and consequently, $\beta_U(X_\alpha)=S_\alpha$,  $\alpha\in {{\mathbb F}}_n^+$. Hence, the range of $\beta_U:{{\mathcal A}}_n\to {{\mathcal A}}_n$ contains all polynomials in ${{\mathcal A}}_n$. Using again the norm density of polynomials in ${{\mathcal A}}_n$, we conclude that $\beta_U$ is a completely isometric automorphism of ${{\mathcal A}}_n$. In what follows we show that the noncommutative Cauchy transform commutes with the action of the unitary group ${{\mathcal U}}(H_n)$. \[Cau-inv\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ and $U\in {{\mathcal U}}(H_n)$. Then $${{\mathcal C}}_T(\beta_U(f))={{\mathcal C}}_{\beta_U(T)}(f), \quad f\in {{\mathcal A}}_n,$$ where $\beta_U$ is the canonical automorphism generated by $U$. Remember that ${{\mathcal A}}_n$ is the norm closure of the polynomials in $S_1,\ldots, S_n$ and the identity. Due to the continuity of the noncommutative Cauchy transform in the operator norm, it is enough to prove the above relation for $f:=S_\alpha$, $\alpha\in {{\mathbb F}}_n^+$. By Theorem \[an=cauch\], we have $$\begin{split} \left<{{\mathcal C}}_T(\beta_U(S_\alpha))x,y\right>&=\left<C_T(R_1,\ldots, R_n)^*(\beta_U(S_\alpha)\otimes I_{{\mathcal H}})(1\otimes x),1\otimes y)\right>\\ &= \left<B_\alpha x,y\right> \end{split}$$ for any $x,y\in {{\mathcal H}}$, where $[B_1,\ldots, B_n]:=\beta_U(T)$. On the other hand, due to Theorem \[auto\], we have $r(\beta_U(T))<1$. Applying again Theorem \[an=cauch\], we obtain $$\begin{split} \left<{{\mathcal C}}_{\beta_U(T)}(S_\alpha) x,y\right>&=\left<C_{\beta_U(T)}(R_1,\ldots, R_n)^*(S_\alpha\otimes I_{{\mathcal H}})(1\otimes x),1\otimes y)\right>\\ &= \left<B_\alpha x,y\right>. \end{split}$$ Hence, ${{\mathcal C}}_T(\beta_U(S_\alpha))={{\mathcal C}}_{\beta_U(T)}(S_\alpha)$, and the result follows. The continuity and the uniqueness of the free analytic functional calculus for $n$-tuples of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ will be proved in the next section. Weierstrass and Montel theorems for free holomorphic functions {#Weierstrass and Montel} ============================================================== In this section, we obtain Weierstrass and Montel type theorems for the algebra of free holomorphic functions with scalar coefficients on the open operatorial unit $n$-ball. This enables us to introduce a metric on $Hol(B({{\mathcal X}})^n_1)$ with respect to which it becomes a complete metric space, and the Hausdorff derivations are continuous. In the end of this section, we prove the continuity and uniqueness of the free functional calculus. Connections with the $F_n^\infty$-functional calculus for row contractions [@Po-funct] and, in the commutative case, with Taylor’s functional calculus [@T2] are also discussed. We say that a sequence $\{F_m\}_{m=1}^\infty\subset Hol(B({{\mathcal X}})^n_1)$ of free holomorphic functions converges uniformly on the closed operatorial $n$-ball of radius $r\in [0,1)$ if it converges uniformly on the closed ball $$[B({{\mathcal H}})^n]_{ r}^{-}:=\{ [X_1,\ldots, X_n]\in B({{\mathcal H}})^n:\ \|X_1X_1^*+\cdots+X_nX_n^*\|\leq r\},$$ where ${{\mathcal H}}$ is an infinite dimensional Hilbert space. According to the maximum principle of Corollary \[max-mod2\], this is equivalent to the fact that the sequence $\{F_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in the operator norm topology of $B(F^2(H_n))$. The first result of this section is a multivariable operatorial version of Weierstrass theorem ([@Co]). \[Weierstrass\] Let $\{F_m\}_{m=1}^\infty\subset Hol(B({{\mathcal X}})^n_1)$ be a sequence of free holomorphic functions which is uniformly convergent on any closed operatorial $n$-ball of radius $r\in [0,1)$. Then there is a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ such that $F_m$ converges to $F$ on any closed operatorial $n$-ball of radius $r\in [0,1)$. Moreover, given $i_1,\ldots, i_k\in \{1,\ldots, n\}$, the sequence $\left\{\frac{\partial^k F_m} {\partial Z_{i_1}\cdots \partial Z_{i_k}}\right\}_{m=1}^\infty $ is uniformly convergent to $\frac{\partial^k F} {\partial Z_{i_1}\cdots \partial Z_{i_k}}$ on any closed operatorial $n$-ball of radius $r\in [0,1)$, where $\frac{\partial^k} {\partial Z_{i_1}\cdots \partial Z_{i_k}}$ is the $k$-order Hausdorff derivation. Let $F_m:=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} a_{\alpha}^{(m)} Z_\alpha$ and fix $r\in (0,1)$. Then, due to Theorem \[caract-shifts\], $$F_m(rS_1,\ldots, rS_n)=\sum\limits_{k=0}^\infty\sum\limits_{|\alpha|=k} r^{|\alpha|} a_{\alpha}^{(m)} S_\alpha$$ is in the noncommutative disc algebra ${{\mathcal A}}_n$. Since $\{F_m\}_{m=1}^\infty$ is uniformly convergent on the closed operatorial $n$-ball of radius $r$, the sequence $\{F_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in the operator norm of $B(F^2(H_n))$. On the other hand, since the noncommutative disc algebra ${{\mathcal A}}_n$ is closed in the operator norm, there exists $g\in {{\mathcal A}}_n$ such that $$\label{Fm-to} F_m(rS_1,\ldots, rS_n)\to L_g, \quad \text{ as } \ m\to\infty.$$ Assume $g=\sum\limits_{\alpha\in {{\mathbb F}}_n^+} b_\alpha(r) e_\alpha$, and notice also that $$b_\alpha(r)=\left< S_\alpha^* L_g(1),1\right>, \quad \alpha\in {{\mathbb F}}_n^+.$$ If $\lambda_{(\beta)}\in {{\mathbb C}}$ for $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$, we have $$\left|\left<\sum_{|\beta|=k} \lambda_{(\beta)} S_\beta^*(F_m(rS_1,\ldots, rS_n)-L_g)1,1\right>\right| \leq \|F_m(rS_1,\ldots, rS_n)-L_g\| \left\|\sum_{|\beta|=k} \lambda_{(\beta)} S_\beta^*\right\|.$$ Since $S_1,\ldots, S_n$ are isometries with orthogonal ranges, we deduce that $$\left|\sum_{|\beta|=k}(r^ka_\beta^{(m)}-b_\beta(r))\lambda_{(\beta)}\right|\leq \|F_m(rS_1,\ldots, rS_n)-L_g\| \left(\sum_{|\beta|=k} |\lambda_{(\beta)}|^2\right)^{1/2}.$$ for any $\lambda_{(\beta)}\in {{\mathbb C}}$ with $|\beta|=k$. Consequently, we have $$\left(\sum_{|\beta|=k}|r^ka_\beta^{(m)}-b_\beta(r)|^2\right)^{1/2}\leq \|F_m(rS_1,\ldots, rS_n)-L_g\|$$ for any $k=0,1,\ldots$. Since $\|F_m(rS_1,\ldots, rS_n)-L_g\| \to 0$, as $m\to\infty$, we deduce that $r^ka_\beta^{(m)}\to b_\beta(r)$, as $m\to\infty$, for any $|\beta|=k$ and $k=0,1,\ldots$. Hence, $a_\beta:=\lim\limits_{m\to\infty}a_\beta^{(m)}$ exists and $b_\beta(r)=r^k a_\beta$ for any $\beta\in {{\mathbb F}}_n^+$ with $|\beta|=k$ and $k=0,1,\ldots$. Consider the formal power series $F:= \sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha Z_\alpha$. We show now that $F$ is a free holomorphic function on the open operatorial unit $n$-ball. Due to the above calculations, we have $$r^k \left|\left( \sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2}- \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}\right| \leq \|F_m(rS_1,\ldots, rS_n)-L_g\|.$$ Therefore, $$\label{conv-coef} \sum_{|\beta|=k}|a_\beta^{(m)}|^2 \to \sum_{|\beta|=k}|a_\beta|^2,\quad \text{ as } \ m\to\infty,$$ uniformly with respect to $k=0,1,\ldots$. Let us show that the radius of convergence of $F$ is $\geq 1$. To this end, assume that $\gamma>1$ and $$\limsup_{k\to\infty} \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2k}>\gamma.$$ Then there is $k\in {{\mathbb N}}$ as large as we want such that $$\label{sup-ga} \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}>\gamma^k.$$ Choose $\lambda$ such that $1<\lambda< \gamma$ and let $\epsilon>0$ be such that $\epsilon<\gamma-\lambda$. Notice that $\epsilon<\gamma^k-\lambda^k$ for any $k=1,2,\ldots$. Now, due to relation , there exists $N_\epsilon\in {{\mathbb N}}$ such that $$\left|\left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2} - \left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2}\right|< \epsilon$$ for any $m>N_\epsilon$ and any $k=0,1,\ldots$. Hence, and using inequality , we deduce that $$\left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2}\geq \gamma^k-\epsilon>\lambda^k$$ for any $m>N_\epsilon$ and some $k$ as large as we want. Consequently, we have $$\limsup_{k\to \infty} \left(\sum_{|\beta|=k}|a_\beta^{(m)}|^2\right)^{1/2k}\geq \lambda>1$$ for $m\geq N_\epsilon$. Due to Theorem \[Abel\], this shows that the radius of convergence of $F_m$ is $<1$, which contradicts the fact that $F_m$ is a free holomorphic function with radius of convergence $\geq 1$. Therefore, $$\limsup_{k\to\infty}\left(\sum_{|\beta|=k}|a_\beta|^2\right)^{1/2k}\leq 1$$ and, consequently, Theorem \[Abel\] shows that $F$ is a free holomorphic function on the open operatorial unit ball. The same theorem implies that $F(rS_1,\ldots, rS_n)=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ is convergent in norm. Since $L_g$ and $F(rS_1,\ldots, rS_n)$ have the same Fourier coefficients, we must have $L_g=F(rS_1,\ldots, rS_n)$. Due to relation , we have $$\|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|\to 0, \quad \text{ as }\ m\to \infty.$$ If $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{1}$ and $\|[X_1,\ldots, X_n]\|=r<1$, the noncommutative von Neumann inequality implies $$\|F_m(X_1,\ldots, X_n)-F(X_1,\ldots, X_n)\|\leq \|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|.$$ Taking $m\to\infty$, we deduce that $F_m$ converges to $F$ on any closed operatorial $n$-ball of radius $r\in [0,1)$. Now, we show that for each $\gamma\in (0,1)$ $$\label{co-de} \left(\frac{\partial F_m}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n)\to \left(\frac{\partial F}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n)$$ in the operator norm, as $m\to \infty$. Let $r,r'\in (0,1)$ such that $\gamma=r r'$. Since $(F_m)_r$ and $ F_r\in {{\mathcal A}}_n$ are in the noncommutative disc algebra ${{\mathcal A}}_n$, we can apply Theorem \[cauc-dif\] (see inequality ) and obtain $$\left\|\left(\frac{\partial ((F_m)_r-F_r)}{\partial Z_i}\right)(r' S_1,\ldots, r' S_n)\right\| \leq M\|(F_m)_r-F_r\|_\infty,$$ where $M$ is an appropriate constant which does not depend on $m$. Since $\|(F_m)_r-F_r\|_\infty\to 0$ as $m\to\infty$ and $$\left(\frac{\partial ((F_m)_r-F_r)}{\partial Z_i}\right)(r' S_1,\ldots, r' S_n)=r\left(\frac{\partial (F_m-F)}{\partial Z_i}\right)(\gamma S_1,\ldots, \gamma S_n),$$ we deduce relation . Using the result for $\frac{\partial}{\partial Z_i}$, one can obtain the general case for $k$-order Housdorff partial derivations. The proof is complete. We say that a set ${{\mathcal F}}\subset Hol(B({{\mathcal X}})^n_1)$ is normal if each sequence in ${{\mathcal F}}$ has a subsequence which converges to a function in $Hol(B({{\mathcal X}})^n_1)$ uniformly on any closed operatorial ball of radius $r\in [0,1)$. The set ${{\mathcal F}}$ is called locally bounded if, for any $r\in[0,1)$, there exists $M>0$ such that $\|f(X_1,\ldots, X_n)\|\leq M$ for any $f\in {{\mathcal F}}$ and $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_r$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space. We can prove now the following noncommutative version of Montel theorem (see [@Co]). \[Montel\] Let ${{\mathcal F}}\subset Hol(B({{\mathcal X}})^n_1)$ be a family of free holomorphic functions. Then the following statements are equivalent: 1. $ \sup_{f\in {{\mathcal F}}}\|f(rS_1,\ldots, rS_n)\|<\infty $ for each $r\in [0,1)$. 2. ${{\mathcal F}}$ is a normal set. 3. ${{\mathcal F}}$ is locally bounded. Assume that condition (i) holds. For each $f\in {{\mathcal F}}$, let $\{a_\alpha(f)\}_{\alpha\in {{\mathbb F}}_n^+}$ be the sequence of coefficients. Due to (i), for each $r\in [0,1)$, there exists $M_r>0$ such that $$\label{FrMr} \|f(rS_1,\ldots, rS_n)\|\leq M_r \quad \text{ for any } \ f\in {{\mathcal F}}. $$ By the Cauchy type estimate of Theorem \[Cauchy-est\], if $r\in(0,1)$, then $$\label{Cau-est} \left( \sum_{|\alpha|=k} |a_\alpha(f)|^2\right)^{1/2}\leq \frac{1}{r^k} M_r\quad \text{ for any } \ f\in {{\mathcal F}}, k=0,1,\ldots.$$ Let $\{F_m\}_{m=1}^\infty$ be a sequence of elements in ${{\mathcal F}}$. Then, relation implies $$|a_0(F_m)|\leq M_0\quad \text{ for any } \ m=1,2,\ldots.$$ Due to the classical Bolzano-Weierstrass theorem for bounded sequences of complex numbers, there is a subsequence $\{F_{m_k^{(0)}}\}_{k=1}^\infty$ of $ \{F_m\}_{m=1}^\infty$ such that the scalar sequence $\{a_0(F_{m_k^{(0)}})\}_{k=1}^\infty$ is convergent in ${{\mathbb C}}$, as $k\to\infty$. Inductively, using relation , we find, for each $\alpha\in {{\mathbb F}}_n^+$, $|\alpha|\geq 1$, a subsequence $\{F_{m_k^{(\alpha)}}\}_{k=1}^\infty$ of $\{F_{m_k^{(\beta)}}\}_{k=1}^\infty$, where $\alpha$ is the succesor of $\beta$ in the lexicographic order of ${{\mathbb F}}_n^+$, such that the sequence $\{a_\alpha(F_{m_k^{(\alpha)}})\}_{k=1}^\infty$ is convergent in ${{\mathbb C}}$, as $k\to\infty$. Using the diagonal process, we find a subsequence $\{F_{p_k}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that $\{a_\alpha(F_{p_k})\}_{k=1}^\infty$ converges in ${{\mathbb C}}$ as $k\to\infty$, for any $\alpha\in {{\mathbb F}}_n^+$. Now let us prove that, if $\gamma>1$, then $\{F_{p_k}(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n)\}_{k=1}^\infty $ converges in the norm topology of $B(F^2(H_n))$. Indeed, if $N\in {{\mathbb N}}$, then relation implies $$\begin{split} &\Bigl\|F_{p_k}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)-F_{p_s}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)\Bigr\|\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} + \sum_{j=N+1} \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2}\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} +\sum_{j=N+1}^\infty \frac{r^j}{\gamma^j} \frac{2M_r}{r^j}\\ &\leq \sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2} +\frac{2M_r}{\gamma^N(\gamma-1)}. \end{split}$$ Given $\epsilon>0$, we choose $N\in {{\mathbb N}}$ such that $\frac{2M_r}{\gamma^N}<\frac{\epsilon}{2}$. On the other hand, since $\{a_\alpha(F_{p_k})\}_{k=1}^\infty$ is a Cauchy sequence in ${{\mathbb C}}$, there is $k_0\in {{\mathbb N}}$ such that $$\sum_{j=1}^N \frac{r^j}{\gamma^j}\left(\sum_{|\alpha|=j} |a_\alpha(F_{p_k})-a_\alpha(F_{p_s})|^2\right)^{1/2}<\frac{\epsilon}{2}\quad \text{ for any } \ k,s\geq k_0.$$ Summing up the above results, we deduce that $$\Bigl\|F_{p_k}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)-F_{p_s}\Bigl(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n\Bigr)\Bigr\| <\epsilon \quad \text{ for any } \ k,s\geq k_0.$$ This proves that the the sequence $\{F_{p_k}(\frac{r}{\gamma}S_1,\ldots, \frac{r}{\gamma}S_n)\}_{k=1}^\infty $ converges in the norm topology of $B(F^2(H_n))$, for any $r\in[0,1)$ and $\gamma>1$. Since the set $A:=\{\frac{r}{\gamma}:\ 0\leq r<1, \gamma>1\}$ is equal to $[0,1)$, one can choose an increasing sequence $\{t_q\}_{q=1}^\infty$ such that $t_q\in A$ and $t_q\to 1$ as $q\to\infty$. Now, if $\{F_m\}_{m=1}^\infty\subset {{\mathcal F}}$, then, using the above result, there is a subsequence $\{F_{n_k^{(1)}}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that $\{ F_{n_k^{(1)}}(t_1S_1,\ldots, t_1S_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Inductively, for each $q=2,3,\ldots$, we find a subsequence $\{F_{n_k^{(q)}}\}_{k=1}^\infty$ of $\{F_{n_k^{(q-1)}}\}_{k=1}^\infty$ such that $\{ F_{n_k^{(q)}}(t_qS_1,\ldots, t_qS_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Using again the diagonal process, we find a subsequence $\{F_{m_k}\}_{k=1}^\infty$ of $\{F_m\}_{m=1}^\infty$ such that, for each $r\in [0,1)$, the subsequence $\{F_{m_k}(rS_1,\ldots, rS_n)\}$ is convergent in the norm topology of $B(F^2(H_n))$, as $k\to\infty$. Applying Theorem \[Weierstrass\], we deduce that ${{\mathcal F}}$ is a normal set. Therefore, the implication $(i)\implies (ii)$ is true. To prove the converse, assume that there is $r_0\in (0,1)$ such that $$\sup_{f\in {{\mathcal F}}}\|f(r_0S_1,\ldots, r_0S_n)\|=\infty.$$ Let $\{f_m\}_{m=1}^\infty\subset {{\mathcal F}}$ be such that $$\label{r_0} \|f_m(r_0S_1,\ldots, r_0S_n)\|\to\infty \quad \text{ as } \ m\to \infty.$$ Since (ii) holds, there exists a subsequence $\{f_{m_k}\}_{k=1}^\infty$ such that $\{f_{m_k}(rS_1,\ldots, r S_n)\}_{k=1}^\infty$ is convergent for any $r\in [0,1)$. This contradicts relation . The equivalence (i)$\Longleftrightarrow$(ii) follows from Corollary \[max-mod2\]. The proof is complete. Now, we can obtain the following Vitali type result in our setting. \[Vitali\] Let $\{F_m\}_{m=1}^\infty$ be a sequence of free holomorphic functions on $[B({{\mathcal H}})^n]_{1}$ with scalar coefficients such that, for each $r\in [0,1)$, $$\sup_{m} \|F_m(rS_1,\ldots, rS_n)\|<\infty.$$ If there exists $0<\gamma<1$ such that $F_m(\gamma S_1,\ldots, \gamma S_n)$ converges in norm as $m\to \infty$, then $F_m$ converges uniformly on $[B({{\mathcal H}})^n]_{ r}^{-}$ for any $r\in [0, 1)$. Suppose that $\{F_m\}_{m=1}^\infty$ does not converge uniformly on $[B({{\mathcal H}})^n]_{r_0}^-$ for some $r_0\in (0,1)$. Then there exist $\delta>0$, subsequences $\{F_{m_k}\}_{k=1}^\infty$ and $\{F_{n_k}\}_{k=1}^\infty$ of $\{F_{m}\}_{m=1}^\infty$, and $n$-tuples of operators $[X_1^{(k)},\ldots, X_n^{(k)}]\in [B({{\mathcal H}})^n]_{r_0}^-$ such that $$\label{Fnkmk} \|F_{n_k}(X_1^{(k)},\ldots, X_n^{(k)})- F_{m_k}(X_1^{(k)},\ldots, X_n^{(k)}\|\geq \delta$$ for any $k=1,2,\ldots$. By Theorem \[Montel\], we find a subsequence $\{k_p\}_{p=1}^\infty$ of $\{k\}_{k=1}^\infty$ such that $\{F_{m_{k_p}}\}_{k=1}^\infty$ and $\{F_{n_{k_p}}\}_{k=1}^\infty$ are uniformly convergent to $f$ and $g$, respectively, on any closed operatorial $n$-ball of radius $r\in [0,1)$. Using Theorem \[Weierstrass\], we deduce that $f,g$ are free holomorphic functions on $[B({{\mathcal H}})^n]_1$ Now, the inequality and the noncommutative von Neumann inequality imply $$\|F_{n_{k_p}}(r_0 S_1,\ldots, r_0S_n)-F_{m_{k_p}}(r_0 S_1,\ldots, r_0S_n)\|\geq \delta>0$$ for any $k=1,2,\ldots$. Consequently, we have $$\label{fr0gr0} \|f(r_0 S_1,\ldots, r_0S_n)-g(r_0 S_1,\ldots, r_0S_n)\|\geq \delta>0.$$ On the other hand, since $\{F_m(\gamma S_1,\ldots, \gamma S_n)\}_{m=1}^\infty$ converges in norm as $m\to\infty$, we must have $$f(\gamma S_1,\ldots, \gamma S_n)=g(\gamma S_1,\ldots, \gamma S_n).$$ Since $0<\gamma<1$ and $f,g$ are free holomorphic functions on $[B({{\mathcal H}})^n]_1$, we deduce that $f=g$, which contradicts inequality . The proof is complete. Let ${{\mathcal H}}$ be a Hilbert space and let $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ be the vector space of all continuous functions from the open operatorial unit ball $[B({{\mathcal H}})^n]_1$ to $B({{\mathcal H}})$. If $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ and $0<r<1$, we define $$\rho_r(f,g):=\sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_r^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|.$$ Let $0<r_m<1$ be such that $\{r_m\}_{m=1}^\infty$ is an increasing sequence convergent to $1$. For any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$, we define $$\rho (f,g):=\sum_{m=1}^\infty \left(\frac{1}{2}\right)^m \frac{\rho_{r_m}(f,g)}{1+\rho_{r_m}(f,g)}.$$ Based on standard arguments, one can prove that $\rho$ is a metric on $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$. Following the corresponding result (see [@Co]) for the set of all continuous functions from a set $G\subset {{\mathbb C}}$ to a metric space $\Omega$, one can easily obtain the following operator version. We leave the proof to the reader. \[Conway\] If $\epsilon>0$, then there exists $\delta>0$ and $m\in {{\mathbb N}}$ such that for any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ $$\sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r_m}^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|<\delta\implies \rho(f,g)<\epsilon.$$ Conversely, if $\delta>0$ and $m\in {{\mathbb N}}$ are fixed, then there is $\epsilon>0$ such that for any $f,g\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$ $$\rho(f,g)<\epsilon \implies \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r_m}^-} \|f(X_1,\ldots, X_n)-g(X_1,\ldots, X_n)\|<\delta.$$ An immediate consequence of Lemma \[Conway\] is the following: if $\{f_m\}_{k=1}^\infty$ and $f$ are in $C(B({{\mathcal H}})^n_1, B({{\mathcal H}}))$, then $f_k$ is convergent to $f$ in the metric $\rho$ if and only if $f_m\to f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$,  $m=1,2,\ldots$. This result is needed to prove the following. \[cont-comp\] $\left( C(B({{\mathcal H}})^n_1, B({{\mathcal H}})), \rho\right)$ is a complete metric space. Suppose that $\{f_k\}_{k=1}^\infty$ is a Cauchy sequence in $\left(C(B({{\mathcal H}})^n_1, B({{\mathcal H}})),\rho\right)$. Due to Lemma \[Conway\], the sequence $\left\{f_k|_{[B({{\mathcal H}})^n]_{r}^-}\right\}_{k=1}^\infty$ is Cauchy in $C([B({{\mathcal H}})^n]_r^-, B({{\mathcal H}}))$. Consequently, for any $\epsilon>0$, there exists $N\in {{\mathbb N}}$, such that $$\label{unif} \sup_{[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-} \|f_m(X_1,\ldots, X_n)-f_k(X_1,\ldots, X_n)\|<\epsilon\quad \text{ for any } k,m\geq N.$$ In particular, $\{f_k(X_1,\ldots, X_n)\}_{k=1}^\infty$ is a Cauchy sequence in the operator norm of $B({{\mathcal H}})$. Therefore, there is an operator $f(X_1,\ldots, X_n)\in B({{\mathcal H}})$ such that $$\label{li} f(X_1,\ldots, X_n)=\lim_{k\to\infty} f_k(X_1,\ldots, X_n)$$ in the operator norm. This gives rise to a function $f:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$. We need to show that $\rho(f_k,f)\to 0$, as $k\to\infty$, and that $f$ is continuous. If $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-$, then, due to relations and , there exists $m\geq N$ such that $$\|f(X_1,\ldots, X_n)-f_m(X_1,\ldots, X_n)\|<\epsilon \quad \text{ and } \quad \|f(X_1,\ldots, X_n)-f_k(X_1,\ldots, X_n)\|<\epsilon$$ for any $k\geq N$. Since $N$ does not depend on $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_{r}^-$, we deduce that $\{f_k\}_{k=1}^\infty$ converges to $f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r}^-$. Due to Lemma \[Conway\], this shows that $\rho(f_k,f)\to 0$, as $k\to\infty$. The continuity of $f$ can be proved using standard arguments in the theory of metric spaces. We leave it to the reader. Let ${{\mathcal H}}$ be an infinite dimensional Hilbert space and denote by $Hol(B({{\mathcal H}})^n_1)$ the algebra of free holomorphic functions on $[B({{\mathcal H}})^n]_1$. \[complete-metric\] $\left(Hol(B({{\mathcal H}})^n_1), \rho\right)$ is a complete metric space and the Hausdorff derivations $$\frac {\partial}{\partial Z_i}: \left(Hol(B({{\mathcal H}})^n_1), \rho\right) \to \left(Hol(B({{\mathcal H}})^n_1), \rho\right),\quad i=1,\ldots, n,$$ are continuous. First, note that Theorem \[continuous\] implies that $Hol(B({{\mathcal H}})^n_1)\subset C(B({{\mathcal H}})^n_1, B({{\mathcal H}})$. Due to Theorem \[cont-comp\], it is enough to show that $\left({{\mathcal H}}ol(B({{\mathcal H}})^n_1), \rho\right)$ is closed in $\left( C(B({{\mathcal H}})^n_1, B({{\mathcal H}})), \rho\right)$. Let $\{f_m\}_{m=1}^\infty\subset Hol(B({{\mathcal H}})^n_1)$ and $f\in C(B({{\mathcal H}})^n_1, B({{\mathcal H}})$ be such that $\rho(f_m,f)\to 0$, as $m\to\infty$. Due to Lemma \[Conway\], $f_m\to f$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$,  $m=1,2,\ldots$. Applying now Theorem \[Weierstrass\], we deduce that $f\in Hol(B({{\mathcal H}})^n_1)$ and that $$\frac{\partial f_m}{\partial Z_i}\to \frac{\partial f}{\partial Z_i}$$ uniformly on any closed ball $[B({{\mathcal H}})^n]_{r_m}^-$ and, therefore, in the metric $\rho$. This completes the proof of the theorem. Now, Theorem \[Montel\] implies the following compactness criterion for subsets of $Hol(B({{\mathcal H}})^n_1)$. A subset ${{\mathcal F}}$ of $(Hol(B({{\mathcal H}})^n_1), \rho)$ is compact if and only if it is closed and locally bounded. We return now to the setting of Section \[free analytic\], where we showed that if $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with $r(T_1,\ldots, T_n)<1$, then we can define the bounded linear operator $$f(T_1,\ldots, T_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha T_\alpha,$$ where the series converges in norm. This provides a [*free analytic functional calculus*]{}, which now turns out to be continuous and unique. If $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is any $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$ then the mapping $\Phi_T: Hol(B({{\mathcal X}})^n_1) \to B({{\mathcal H}})$ defined by $$\Phi_T(f):=f(T_1,\ldots, T_n)$$ is a continuous unital algebra homomorphism. Moreover, the free analytic functional calculus is uniquely determined by the mapping $$Z_i\mapsto T_i,\qquad i=1,\ldots,n.$$ Due to Theorem \[abel\] and Theorem \[operations\], we deduce that $\Phi_T$ is a well-defined unital algebra homomorphism. To prove the continuity of $\Phi_T$, let $f_m$ and $f$ be in $Hol(B({{\mathcal X}})^n_1)$ such that $f_m\to f$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$, as $m\to\infty$. Due to Lemma \[Conway\] and Corollary \[max-mod2\], this is equivalent to the fact that, for each $r\in [0,1)$, $$\label{conv-S} f_m(rS_1,\ldots, rS_n)\to f(rS_1,\ldots, rS_n),\quad \text{ as }\ m\to\infty,$$ where the convergence is in the operator norm of $B(F^2(H_n))$. We shall prove that $$\label{conv-f_m} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|\to 0, \quad \text{ as }\ m\to\infty.$$ Let $f:=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ and $f_m:=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha^{(m)} Z_\alpha$. Due to Theorem \[abel\], the series defining $f_m(T_1,\ldots, T_n)$ and $f(T_1,\ldots, T_n)$ are norm convergent. Notice that $$\begin{split} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|&= \left\|\sum_{k=0}^\infty \sum_{|\alpha|=k}(a_\alpha^{(m)}-a_\alpha)T_\alpha\right\| \\ &\leq \sum_{k=0}^\infty\left\|\sum_{|\alpha|=k}(a_\alpha^{(m)}-a_\alpha)T_\alpha\right\| \\ &\leq \sum_{k=0}^\infty \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\left(\sum_{|\alpha|=k}|a_\alpha^{m)}-a_\alpha|^2\right)^{1/2}. \end{split}$$ If $r(T_1,\ldots, T_n)<\rho<r<1$, then there exists $k_0\in {{\mathbb N}}$ such that $$\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\leq \rho^k \quad \text{ for any }\ k\geq k_0.$$ According to Theorem \[Cauchy-est\], we have $$\left(\sum_{|\alpha|=k}|a_\alpha^{m)}-a_\alpha|^2\right)^{1/2}\leq \frac{1}{r^k}\|f_m(rS_1,\ldots, rS_n)-f(rS_1,\ldots, rS_n)\|.$$ Combining this with the above inequalities, we obtain $$\begin{split} \|f_m(T_1,\ldots, T_n)-f(T_1,\ldots, T_n)\|&\leq M(T,\rho,r) \|f_m(rS_1,\ldots, rS_n)-f(rS_1,\ldots, rS_n)\|, \end{split}$$ where $$M(T,\rho,r):=\sum_{k=0}^{k_0}\left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*\right\|^{1/2}\frac{1}{r^k}+\sum_{k=k_0+1}^\infty\left(\frac{\rho}{r}\right)^{k}.$$ Now, using relation , we deduce , which proves the continuity of $\Phi_T$. To prove the uniqueness of the free analytic functional calculus, let $\Phi:Hol(B({{\mathcal X}})_1^n)\to B({{\mathcal H}})$ be a continuous unital algebra homomorphism such that $\Phi(Z_i)=T_i$,  $i=1,\ldots, n$. Hence, we deduce that $$\label{pol2} \Phi_T(p(Z_1,\ldots, Z_n))=\Phi(p(Z_1,\ldots, Z_n))$$ for any polynomial $p(Z_1,\ldots, Z_n)$ in $Hol(B({{\mathcal X}})_1^n)$. Let $f=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha Z_\alpha$ be an element in $Hol(B({{\mathcal X}})_1^n)$ and let $p_m:=\sum_{k=0}^m\sum_{|\alpha|=k} a_\alpha Z_\alpha$,   $m=1,2,\ldots$. Since $$f(rS_1,\ldots, rS_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^k a_\alpha S_\alpha$$ and the series $\sum_{k=0}^\infty r^k\left\|\sum_{|\alpha|=k} a_\alpha S_\alpha\right\|$ converges due to Theorem \[caract-shifts\], we deduce that $$p_m(rS_1,\ldots, rS_n)\to f(rS_1,\ldots, rS_n)$$ in the operator norm, as $m\to\infty$. Therefore, $p_m\to f$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$. Hence, using and the continuity of $\Phi$ and $\Phi_T$, we deduce that $\Phi=\Phi_T$. This completes the proof. Using Theorem \[f-infty\], Theorem \[abel\], and the results from [@Po-funct] concerning the $F_n^\infty$ functional calculus for row contractions, one can make the following observation. For strict row contractions, i.e. $\|[T_1,\ldots, T_n]\|<1$, and $F\in H^\infty(B({{\mathcal X}})^n_1)$, the free analytic functional calculus $F(T_1,\ldots, T_n)$ coincides with the $F_n^\infty$-functional calculus for row contractions. Let $\{F_m\}_{m=1}^\infty$ and $F$ be in $Hol(B({{\mathcal X}})^n_1)$ and let $\{f_m\}_{m=1}^\infty$ and $f$ be the corresponding representations on ${{\mathbb C}}$, respectively (see Corollary \[part-case\]). Due to the noncommuting von Neumann inequality, we have $$\sup_{|\lambda_1|^2+\cdots +|\lambda_n|^2\leq r^2} |f_m(\lambda_1,\ldots, \lambda_n)-f(\lambda_1,\ldots, \lambda_n)|\leq \|F_m(rS_1,\ldots, rS_n)-F(rS_1,\ldots, rS_n)\|$$ for any $r\in [0,1)$. Hence, we deduce that if $F_m\to F$ in the metric $\rho$ of $Hol(B({{\mathcal X}})^n_1)$, then $f_m\to f$ uniformly on compact subsets of ${{\mathbb B}}_n$. Since there is a sequence of polynomials $\{p_m\}_{m=1}^\infty$ such that $p_m\to F$ in the metric $\rho$, one can use the continuity of Taylor’s functional calculus and the continuity of the free analytic functional calculus as well as the fact that they coincide on polynomials, to deduce the following result. If $f$ is the representation of a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ on ${{\mathbb C}}$ and $[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is an $n$-tuple of commuting operators with Taylor spectrum $\sigma(T_1,\ldots, T_n)\subset {{\mathbb B}}_n$, then the free analytic calculus $F(T_1,\ldots, T_n)$ coincides with Taylor’s functional calculus $f(T_1,\ldots, T_n)$. Free pluriharmonic functions and noncommutative Poisson transforms {#free harmonic} ================================================================== Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform [@Po-poisson] generates a function $$P[A]: [B({{\mathcal H}})^n]_1\to B({{\mathcal H}}).$$ In this section, we provide classes of operators $A\in B(F^2(H_n))$ such that $P[A]$ is a free holomorphic (resp. pluriharmonic) function on $[B({{\mathcal H}})^n]_1$. We characterize the free holomorphic functions $u$ on $[B({{\mathcal H}})^n]_1$ such that $u=P[f]$ for some boundary function $f$ in the noncommutative analytic Toeplitz algebra $F_n^\infty$, or the noncommutative disc algebra ${{\mathcal A}}_n$. We also obtain noncommutative multivariable versions of Herglotz theorem and Dirichlet extension problem (see [@Co], [@H]), for free pluriharmonic functions. We define the operator $K_T(S_1,\ldots, S_n)\in B(F^2(H_n)\otimes {{\mathcal H}})$ associated with a row contraction $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ by setting $$K_T(S_1,\ldots, S_n):=\sum_{k=0}^\infty\sum_{|\alpha|=k} S_\alpha \otimes \Delta_T T_\alpha^*,$$ where $\Delta_T:=(I_{{\mathcal H}}-\sum_{i=1}^n T_iT_i^*)^{1/2}$. Due to Theorem \[Abel\], when $A_{(\alpha)}:=\Delta_T T_\alpha^*$ and $X_i:=S_i$, $i=1,\ldots, n$, the above series is convergent in the operator norm if $$\label{cond-conv} \limsup_{k\to\infty} \left\|\sum_{|\alpha|=k} T_\alpha T_\alpha^*-\sum_{|\alpha|=k+1} T_\alpha T_\alpha^* \right\|^{1/2k}<1.$$ In particular, if $\|[T_1,\ldots, T_n]\|<1$, then relation holds and the operator $K_T(S_1,\ldots, S_n)$ is in ${{\mathcal A}}_n\bar \otimes B({{\mathcal H}})$. Notice also that $$(S_\alpha^*\otimes I_{{\mathcal H}})K_T(S_1,\ldots, S_n)=K_T(S_1,\ldots, S_n) (I_{F^2(H_n)} \otimes T_\alpha^*),\qquad \alpha\in {{\mathbb F}}_n^+.$$ Introduced in [@Po-poisson], the noncommutative Poisson transform at $T:=[T_1,\ldots, T_n]$ is the map $P_T:B(F^2(H_n))\to B({{\mathcal H}})$ defined by $$\begin{split} \left<P_T(A)x,y\right>&:= \left<K_T(S_1,\ldots, S_n)^* ( A\otimes I_{{\mathcal H}}) K_T(S_1,\ldots, S_n) (1\otimes x),1\otimes y\right>\\ &:=\left< K_T^*( A\otimes I_{{\mathcal H}}) K_Tx,y\right> \end{split}$$ for any $x,y\in B({{\mathcal H}})$, where $K_T:=K_T(S_1,\ldots, S_n)|_{1\otimes {{\mathcal H}}}:{{\mathcal H}}\to F^2(H_n)\otimes {{\mathcal H}}$. We recall that the Poisson kernel $K_T$ is an isometry if $\|T\|<1$, and $$\label{pol} p(T_1,\ldots, T_n)=K_T^*(p(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})K_T$$ for any polynomial $p$. We refer to [@Po-poisson], [@Po-curvature], [@Po-similarity], and [@Po-unitary] for more on noncommutative Poisson transforms on $C^*$-algebras generated by isometries. Given an operator $A\in B(F^2(H_n))$, the noncommutative Poisson transform generates a function $$P[A]:[B({{\mathcal H}})^n]_1\to B({{\mathcal H}})$$ by setting $$P[A](X_1,\ldots, X_n):=P_X(A)\quad \text{ for }\ X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ In what follows, we provide classes of operators $A\in B(F^2(H_n))$ such that the mapping $P[A]$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. In this case, the operator $A$ can be seen as the boundary function of the Poisson transform $P[A]$. As in the previous sections, we identify $f\in F_n^\infty$ with the multiplication operator $L_f\in B(F^2(H_n))$. \[behave\] Let ${{\mathcal H}}$ be a Hilbert space and $u$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$. 1. There exists $f\in F_n^\infty$ with $u=P[f]$ if and only if $\sup\limits_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|<\infty$. In this case, $u(rS_1,\ldots, rS_n)\to f$, as $ r\to 1$, in the $w^*$-topology (or strong operator topology). 2. There exists $f\in {{\mathcal A}}_n$ with $u=P[f]$ if and only if $\{u(rS_1,\ldots, rS_n)\}_{0\leq r<1}$ is convergent in norm as, $r\to 1$. In this case, $u(rS_1,\ldots, rS_n)\to f$ in the operator norm, as $r\to 1$. To prove (i), assume that $f\in F_n^\infty$ and $u=P[f]$, where $f$ is identified with the multiplication operator $L_f\in B(F^2(H_n)$. Then $$u(X_1,\ldots, X_n)=K_X^*(L_f\otimes I_{{\mathcal H}})K_X,\quad [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$$ and $\|u(X_1,\ldots, X_n)\|\leq \|L_f\|=\|f\|_\infty$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. In particular, $$\label{sup-u}\sup\limits_{0\leq r<1}\|u(rS_1,\ldots, rS_n)\|\leq \|f\|_\infty<\infty.$$ Conversely, assume that $u(X_1,\ldots, X_n):=\sum_{k=0}\sum_{|\alpha|=k} a_\alpha X_\alpha$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$ such that holds. By Theorem \[f-infty\], $f:=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in $F_n^\infty$. Due to Theorem \[Abel\], we have that $u_r(X_1,\ldots, X_n):=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha X_\alpha$ is convergent in norm for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$ and $r\in [0,1]$. Similarly, we have that $f_r(S_1,\ldots, S_n):=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ is convergent in norm for any $r\in [0,1)$. Using relation , we deduce that $$\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha X_\alpha =K_X^*\left(\sum_{k=0}^m \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha\otimes I_{{\mathcal H}}\right)K_X.$$ Taking $m\to \infty$ and using the above convergences, we get $$\label{u_r-f_r} u_r(X_1,\ldots, X_n)=K_X^*(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}) K_X,\qquad r\in [0,1).$$ By Theorem \[continuous\], we have $$\lim_{r\to 1} u_r(X_1,\ldots, X_n)=u(X_1,\ldots, X_n)$$ in the operator norm. On the other hand, due to relation , we have $$\label{So2} \text{\rm SOT-}\lim_{r\to 1} f_r(S_1,\ldots, S_n)=L_f.$$ Since $\|f_r(S_1,\ldots, S_n)\|\leq \|f\|_\infty$ and the map $A\mapsto A\otimes I_{{\mathcal H}}$ is SOT-continuous on bounded subsets of $B(F^2(H_n))$, we take $r\to 1$ in relation and deduce that $u(X_1,\ldots, X_n)=P_X(f)$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Since $u_r(S_1,\ldots, S_n)=f(rS_1,\ldots, rS_n)$ and the strong operator topology coincides with the $w^*$-topology on $F_n^\infty$ (see [@DP1]), one can use to complete the proof of part (i). To prove (ii), assume that $f=\sum_{\alpha\in {{\mathbb F}}_n^+} a_\alpha e_\alpha$ is in ${{\mathcal A}}_n$ and $u=P[f]$, i.e., $$u(X_1,\ldots, X_n)= K_X^* (L_f\otimes I_{{\mathcal H}}) K_X$$ for any $X=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Due to Theorem \[A-infty\], we have $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=L_f$ in the operator norm. Hence, using relation and Theorem \[continuous\], we deduce that $$\begin{split} K_X^* (L_f\otimes I_{{\mathcal H}}) K_X&=\lim_{r\to 1}K_X (f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}}) K_X\\ &=\lim_{r\to 1} f(rX_1,\ldots, rX_n)=f(X_1,\ldots, X_n). \end{split}$$ This proves that $u(X_1,\ldots, X_n)=f(X_1,\ldots, X_n)$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. In particular, we deduce that $$u(rS_1,\ldots, rS_n)=f_r(S_1,\ldots, S_n)\to L_f, \quad \text{ as } \ r\to1,$$ in the operator norm. Conversely, assume that $u:=\sum_{k=0}\sum_{|\alpha|=k} a_\alpha Z_\alpha$ is a free holomorphic function on the open operatorial unit $n$-ball, such that $\{u(rS_1,\ldots, rS_n)\}_{0\leq r<1}$ is convergent in norm, as $r\to 1$. By Theorem \[caract-shifts\], we have that $u(rS_1,\ldots, rS_n)\in {{\mathcal A}}_n$. Since ${{\mathcal A}}_n$ is a Banach algebra, there exists $f\in {{\mathcal A}}_n$ such that $ u(rS_1,\ldots, rS_n)\to f$ in norm, as $r\to 1$. Due to Theorem \[A-infty\], we must have $f=\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha e_\alpha$. As in the proof of part (i), we have $$u(X_1,\ldots, X_n)=\lim_{r\to 1} f_r(X_1,\ldots, X_n)= \lim_{r\to 1} K_X^*(f_r(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})K_X$$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Now, since $\lim\limits_{r\to 1} f_r(S_1,\ldots, S_n)=L_f$ in norm, we deduce that $u=P[f]$. This completes the proof. We now turn our attention to a noncommutative generalization of the harmonic functions on the open unit disc ${{\mathbb D}}$. We say that $G$ is a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ if there exists a free holomorphic function $F$ on $[B({{\mathcal H}})^n]_1$ such that $$G(X_1,\ldots, X_n)=\text{\rm Re}\,F(X_1,\ldots, X_n):=\frac{1}{2}\left(F(X_1,\ldots, X_n)+ F(X_1,\ldots, X_n)^*\right)$$ We remark that if ${{\mathcal H}}$ be an infinite dimensional Hilbert space, then $G$ determines $F$ up to an imaginary complex number. Indeed, if we assume that $\text{\rm Re}\, F=0$ and take the representation on the full Fock space $F^2(H_n)$, we obtain $F(rS_1,\ldots, rS_n)=-F(rS_1,\ldots, rS_n)^*$, $0<r<1$. If $F(rS_1,\ldots, rS_n)$ has the representation $\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$, $a_\alpha\in {{\mathbb C}}$, the above relation implies $$\sum_{k=0}^\infty\sum_{|\alpha|=k} r^{|\alpha|} a_\alpha e_\alpha=F(rS_1,\ldots, rS_n)1=-F(rS_1,\ldots, rS_n)^*1=-\overline{a}_0.$$ Hence, $a_\alpha=0$ if $|\alpha|\geq 1$ and $a_0+\overline{a}_0=0$. Therefore, $F=a_0$, where $a_0$ is an imaginary complex number. This proves our assertion. Due to Theorem \[Abel\], $$G(X_1,\ldots, X_n):=\sum_{k=1}^\infty \sum_{|\alpha|=k} \overline{a}_\alpha X_\alpha^* +a_0 I+ \sum_{k=1}^\infty \sum_{|\alpha|=k} {a_\alpha} X_\alpha$$ represents a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ if and only if $$\limsup_{k\to\infty}\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2k}\leq 1.$$ If $H_1$ and $H_2$ are self-adjoint free pluriharmonic functions on $[B({{\mathcal H}})^n]_1$, we say that $H:=H_1+iH_2$ is a free pluriharmonic function on $[B({{\mathcal H}})^n]_1$. Notice that any free holomorphic function on $[B({{\mathcal H}})^n]_1$ is a free pluriharmonic function. This is due to the fact that $f=\frac{f+f^*}{2}+i\frac{f-f^*}{2i}$. Let $g$ be a free pluriharmonic function on the open operatorial $n$-ball of radius $1+\epsilon$, $\epsilon>0$. Then $$g(X_1,\ldots, X_n)=P_X(g(S_1,\ldots, S_n)),\quad X:=[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1,$$ where $P_X$ is the noncommutative Poisson transform at $X$. Moreover, if ${{\mathcal H}}$ is an infinite dimensional Hilbert space, then $g(S_1,\ldots, S_n)\geq 0$ if and only if $g(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Without loss of generality, we can assume that $g$ is a self-adjoint free pluriharmonic function and $g(X_1,\ldots, X_n)=f(X_1,\ldots, X_n)+f(X_1,\ldots, X_n)^*$ for any $[X_1,\ldots, X_n)]\in [B({{\mathcal H}})^n]_{1+\epsilon}$, where the function $ f(X_1,\ldots, X_n)=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha X_\alpha$ is free holomorphic on $[B({{\mathcal H}})^n]_{1+\epsilon}$. According to Theorem \[caract-shifts\], the series $\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} a_\alpha S_\alpha$ converges in the operator norm for any $r\in [0,1+\epsilon )$. Due to relation \[pol\] and taking limits in the operator norm, we have $$\begin{split} f(X_1,\ldots, X_n)&=\sum_{k=0}^\infty \sum_{|\alpha|=k} a_\alpha X_\alpha= P_X[f(S_1,\ldots, S_n)] \ \text{ and}\\ f(X_1,\ldots, X_n)^*&=\sum_{k=0}^\infty \sum_{|\alpha|=k} \overline{a}_\alpha X_\alpha^*= P_X[f(S_1,\ldots, S_n)^*]. \end{split}$$ Consequently, $$g(X_1,\ldots, X_n)=P_X[g(S_1,\ldots,S_n)], \quad [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ We prove now the last part of the proposition. One implication is obvious due to the above relation. Conversely, assume that $g(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Then, since ${{\mathcal H}}$ is infinite dimensional, we deduce that $g(rS_1,\ldots, rS_n)\geq 0$ for any $r\in [0,1)$. On the other hand, due to Theorem \[continuous\], $\lim\limits_{r\to 1} g(rS_1,\ldots, rS_n)=g(S_1,\ldots, S_n)$ in the operator norm. Hence, $g(S_1,\ldots, S_n)\geq 0$, and the proof is complete. Now, we obtain a noncommutative multivariable version of Herglotz theorem (see [@H]). \[Herglotz\] Let $f\in (F_n^\infty)^*+ F_n^\infty$ and let $u=P[f]$ be its noncommutative Poisson transform. Then $u$ is a free pluriharmonic function on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is a Hilbert space. Moreover, $u\geq 0$ on $[B({{\mathcal H}})^n]_1$, where ${{\mathcal H}}$ is an infinite dimensional Hilbert space, if and only if $f\geq 0$. First, notice that, without loss of generality, we can assume that $f=f^*$. Then, one can prove that $f=g^*+g$ for some $g\in F_n^\infty$. Indeed, if $f=h^*+g$ for some $h,g\in F_n^\infty$, the we must have $(g-h)^*=g-h$. Hence, $(g-h)^*1=(g-h)1$ and one can easily deduce that $g-h$ is a constant, which proves our assertion. According to Theorem \[behave\], $P[g]$ is a free holomorphic function on the open operatorial unit $n$-ball. On the other hand, due to [@Po-varieties], we have $$\text{\rm SOT}-\lim_{r\to 1} g_r(S_1,\ldots, S_n)^*=L_g^*.$$ Hence, using the properties of the Poisson tranform and Theorem \[continuous\], we deduce that $$\begin{split} \left< P[g^*] x,y\right>&= \lim_{r\to 1}\left< K_X(g_r(S_1,\ldots, S_n)^*\otimes I_{{\mathcal H}})K_X x,y\right>\\ &= \lim_{r\to 1}\left< g_r(X_1,\ldots, X_n)^*x,y\right>\\ &= \left< g(X_1,\ldots, X_n)^* x,y\right>\\ &=\left< P[g]^*x,y\right>. \end{split}$$ Hence, we have $P[g]^*=P[g^*]$. Consequently, $$u=P[f]=P[g^*]+P[g]=P[g]^*+P[g],$$ which proves that $u$ is a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$. Now, it is clear that if $f\geq 0$ then $u=P[f]\geq 0$. Conversely, assume that $u(X_1,\ldots, X_n)\geq 0$ for any $[X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. Since ${{\mathcal H}}$ is an infinite dimensional Hilbert space ${{\mathcal H}}$, we deduce that $$u(rS_1,\ldots, rS_n)=g(rS_1,\ldots, rS_n)^*+ g(rS_1,\ldots, rS_n)\geq 0,\qquad r\in [0,1).$$ Due to Theorem \[behave\], we have $$\text{\rm WOT}-\lim_{r\to 1} [g(rS_1,\ldots, rS_n)^*+ g(rS_1,\ldots, rS_n)]=L_g^*+L_g\geq 0.$$ Under the identification of $g$ with $L_g$, we deduce $f=g^*+g\geq 0$, and complete the proof. Here again, we remark that $ f$ plays the role of the boundary function from the classical complex analysis. Our version of the classical Dirichlet extension problem for the unit disc (see [@Co], [@H]) is the following extension of Theorem \[A-infty\]. \[Dirichlet\] If $f\in {{\mathcal A}}_n^*+{{\mathcal A}}_n$, then $u:=P[f]$ is a free pluriharmonic function on the open operatorial unit $n$-ball such that 1. $u$ has a continuous extension $\tilde u$ to $[B({{\mathcal H}})^n]_1^-$ for any Hilbert space ${{\mathcal H}}$, in the operator norm; 2. $\tilde u(S_1,\ldots, S_n)=f$. Without loss of generality, we can assume that $f$ is self-adjoint. As in the proof of Theorem \[Herglotz\], one can prove that $f=g^*+g$ for some $g\in {{\mathcal A}}_n$ and $u:=P[f]=P[g]^*+P[g]$ is a self-adjoint pluriharmonic function on the open operatorial unit $n$-ball. Since $g\in {{\mathcal A}}_n$, we know that $g_r(S_1,\ldots, S_n)\to L_g$ in norm, as $r\to 1$. Consequently, $$f_r(S_1,\ldots, S_n):=g_r(S_1,\ldots, S_n)^*+g_r(S_1,\ldots, S_n)\to L_f^*+ L_f,\quad \text{ as } \ r\to1,$$ in norm. As in the proof of Theorem \[Herglotz\], we have $$u(X_1,\ldots, X_n)=f(X_1,\ldots, X_n),\quad \text{ for } \ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1.$$ Moreover, $v:=P[g]$ is a free holomorphic function such that $v(X_1,\ldots, X_n)=g(X_1,\ldots, X_n)$,  for any $ [X_1,\ldots, X_n]\in [B({{\mathcal H}})^n]_1$. For each $n$-tuple $[Y_1,\ldots, Y_n]\in [B({{\mathcal H}})^n]_1^-$, we define $$\tilde v(Y_1,\ldots, Y_n):=\lim_{r\to 1} P_{rY}[g],$$ where $rY:=[rY_1,\ldots, rY_n]$. Hence, we have $\tilde v(Y_1,\ldots, Y_n)=\lim_{r\to 1} g(rY_1,\ldots, rY_n)$. Now, as in the proof of Theorem \[A-infty\], we deduce that the map $\tilde v:[B({{\mathcal H}})^n]_1^-\to B({{\mathcal H}})$ is a continuous extension of $v$. Therefore, the map $\tilde u:={\tilde v}^*+\tilde v$ is a continuous extension of $u$ to $[B({{\mathcal H}})^n]_1^-$. To prove (ii), apply part (i) when ${{\mathcal H}}=F^2(H_n)$ and take into account Theorem \[A-infty\]. We obtain $$\tilde v(S_1,\ldots, S_n)=\lim_{r\to 1} g(rS_1,\ldots, rS_n)=g,$$ where we used the identification of $g$ with $L_g$, and the limit is in the operator norm. Therefore, $$\tilde u(S_1,\ldots, S_n)=\tilde v(S_1,\ldots, S_n)^*+\tilde v(S_1,\ldots, S_n)=g^*+g=f.$$ This completes the proof. Let $u$ and $v$ be two self-adjoint free pluriharmonic functions on $[B({{\mathcal H}})^n]_1$. We say that $v$ is the pluriharmonic conjugate of $u$ if $u+iv$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. The pluriharmonic conjugate of a self-adjoint free pluriharmonic function on $[B({{\mathcal H}})^n]_1$ is unique up to an additive real constant. Let $f$ be a free holomorphic function on $[B({{\mathcal H}})^n]_1$ and $u=\text{\rm Re}\, f$. Assume that $v$ is a selfadjoint free pluriharmonic function such that $u+iv=g$ is a free holomorphic function on $[B({{\mathcal H}})^n]_1$. Hence, we have $$\label{v} v=\frac{2g-f-f^*}{2i}.$$ Since $v=v^*$, we must have $(g-f=(g-f)^*$, i.e., $\text{\rm Re}\, (g-f)=0$. Based on the remarks following Theorem \[behave\], we have $g-f=w$, where $w$ is an imaginary complex number. Consequently, relation , implies $v=\frac{f-f^*}{2i}-iw$. This proves the assertion. We remark that if $u=\text{\rm Re}\, f$ and $f(0)$ is real then $v=\frac{f-f^*}{2i}$ is the unique pluriharmonic conjugate of $u$ such that $v(0)=0$. \[cauch-conj\] Let $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ be an $n$-tuple of operators with joint spectral radius $r(T_1,\ldots, T_n)<1$. If $f\in H^\infty (B({{\mathcal X}})^n_1) $, $u=\text{\rm Re}\,f$, and $f(0)$ is real, then $$\left<f(T_1,\ldots, T_n)x,y\right>=\left<(u(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>$$ for any $x,y\in {{\mathcal H}}$, where $u(S_1,\ldots, S_n)$ is the boundary function of $u$. Due to Theorem \[an=cauch\], we have $$\begin{split} \left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x)\right. &, \left. [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &= 2\left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), [C_T(R_1,\ldots, R_n)](1\otimes y)\right>\\ &\qquad-\left<(f(S_1,\ldots, S_n)\otimes I_{{\mathcal H}})(1\otimes x), 1\otimes y\right>\\ &= 2\left< f(T_1,\ldots, T_n)x,y\right>-f(0)\left< x,y\right>. \end{split}$$ On the other hand, it is easy to see that $$\begin{split} \left<(f(S_1,\ldots, S_n)^*\otimes I_{{\mathcal H}})(1\otimes x)\right.&,\left. [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &= \left<(\overline{(f(0)}\otimes I_{{\mathcal H}})(1\otimes x), [2C_T(R_1,\ldots, R_n)-I](1\otimes y)\right>\\ &=\overline{(f(0)}\left<x,y\right>. \end{split}$$ If $f(0)\in {{\mathbb R}}$, then adding up the above relations, we complete the proof. We remark that under the conditions of Theorem \[cauch-conj\] and using the noncommutative Cauchy transform, one can express the pluriharmonic conjugate of $u$ in terms of $u$. In a forthcoming paper [@Po-Bohr], we will consider operator-valued Bohr type inequalities for classes of free pluriharmonic functions on the open operatorial unit $n$-ball with operator-valued coefficients. Hardy spaces of free holomorphic functions {#Banach} ============================================ In this section, we define the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$, and the symmetrized Hardy space $H^\infty_{\text{\rm sym}}({{\mathbb B}}_n)$, and prove that they are Banach spaces with respect to some appropriate norms. In this setting, we obtain von Neumann type inequalities for $n$-tuples of operators. Let $F$ be a free holomorphic function on the open operatorial unit $n$-ball. The map $\varphi:[0,1)\to B(F^2(H_n))$ defined by $\varphi(r):= F(rS_1,\ldots, rS_n)$ is called the [*radial boundary function*]{} associated with $F$. Due to Theorem \[continuous\], $\varphi$ is continuous with respect to the operator norm topology of $B(F^2(H_n))$. When $\lim\limits_{r\to 1} \varphi(r)$ exists, in one of the classical topologies of $B(F^2(H_n))$, we call it the [*boundary function*]{} of $F$. Due to the maximum principle for free holomorphic functions (see Theorem \[max-mod1\]), we have $$\|\varphi(r)\|=\sup \|F(X_1,\ldots, X_n)\|,\quad 0\leq r<1,$$ where the supremum is taken over all $n$ tuples of operators $[X_1,\ldots, X_n]$ in either one of the following sets $[B({{\mathcal H}})^n]_r,\ [B({{\mathcal H}})^n]_r^-$, or $$\{[X_1,\ldots, X_n]\in B({{\mathcal H}})^n: \ \|[X_1,\ldots, X_n]\|=r\},$$ where ${{\mathcal H}}$ is an arbitrary infinite dimensional Hilbert space. The [*radial maximal function*]{} $M_F:[0,1)\to [0,\infty)$ associated with a free holomorphic function $F\in Hol(B({{\mathcal X}})^n_1)$ is defined by $$M_F(r):=\|\varphi(r)\|=\|F(rS_1,\ldots, rS_n)\|.$$ $M_F$ is an increasing continuous function (see the proof of Theorem \[f-infty\]). We define the [*radial maximal Hardy space*]{} $H^p(B({{\mathcal X}})^n_1)$,  $p\geq 1$, as the set of all free holomorphic functions $F\in Hol(B({{\mathcal X}})^n_1)$ such that $M_F$ is in the Lebesque space $ L^p[0,1]$. Setting $$\|F\|_p:=\|M_F\|_p:=\left(\int_0^1\|F(rS_1,\ldots, rS_n)\|^p dr \right)^{1/p},$$ it is easy to see that $\|\cdot \|_p$ is a norm on the linear space $H^p(B({{\mathcal X}})^n_1)$. \[radial-Banach\] If $p\geq 1$, then the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$ is a Banach space. First we prove the result for $p=1$. Let $\{F_k\}_{k=1}^\infty \subset H^1(B({{\mathcal X}})^n_1)$ be a sequence such that $$\label{ser-conv-1} \sum_{k=1}^\infty \|F_k\|_1\leq M<\infty.$$ We need to prove that $\sum_{k=1}^\infty F_k$ converges in $\|\cdot\|_1$. By , we have $$\sum_{k=1}^m \int_0^1 \|F_k(rS_1,\ldots, rS_n)\| dr\leq M,\quad \text{ for any } \ m\in {{\mathbb N}}.$$ Using Fatou’s lemma, we deduce that the function $\psi(r):=\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|$ is integrable on $[0,1]$. Notice that the series $\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|<\infty$ for any $r\in [0,1)$. Indeed, assume that there exists $r_0\in [0,1)$ such that $\sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty$. Since the radial maximal function is increasing, we have $$\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\geq \sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty$$ for any $r\in [r_0,1)$. Hence, we deduce that $$\int_0^1\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr\geq (1-r_0) \sum_{k=1}^\infty \|F_k(r_0S_1,\ldots, r_0S_n)\|=\infty,$$ which contradicts the fact that $\psi$ is integrable on $[0,1]$. Therefore, we deduce that $\sum\limits_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|$ is convergent for any $r\in [0,1)$. Hence, the series $\sum\limits_{k=1}^\infty F_k(rS_1,\ldots, rS_n)$ is convergent in the operator norm of $B(F^2(H_n))$ for each $r\in [0,1)$. For each $m\geq 1$, define $g_m:=\sum_{k=1}^m F_k$. Since $\{g_m\}_{m=1}^\infty$ is a sequence of free holomorphic functions such that $\{g_m(rS_1,\ldots, rS_n)\}_{m=1}^\infty$ is convergent in norm for each $r\in [0,1)$, we deduce that $\{g_m\}_{m=1}^\infty$ is uniformly convergent on any closed operatorial ball $[B({{\mathcal X}})^n]_r^-$, $r\in [0,1)$. According to our noncommutative Weierstrass type result, Theorem \[Weierstrass\], there is a free holomorphic function $g$ on the open operatorial unit $n$-ball such that $\|g_m(rS_1,\ldots, rS_n)-g(rS_1,\ldots, rS_n)\|\to 0$, as $m\to\infty,$ and therefore $$g(rS_1,\ldots, rS_n)=\sum_{k=1}^\infty F_k(rS_1,\ldots, rS_n)\quad \text{ for any } \ r\in [0,1).$$ Moreover, due to the fact that $\psi$ is integrable, we have $$\int_0^1 \|g(rS_1,\ldots, rS_n)\| dr \leq \int_0^1 \sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr <\infty,$$ which shows that $g\in H^1(B({{\mathcal X}})^n_1)$. Now, notice that $$\begin{split} \|g-g_m\|_1&=\int_0^1 \|g(rS_1,\ldots, rS_n)-g_m(rS_1,\ldots, rS_n)\| dr\\ &=\int_0^1\left\|\sum_{k=m+1}^\infty F_k(rS_1,\ldots, rS_n)\right\| dr\\ &\leq \int_0^1 \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr \end{split}$$ Since $\sum_{k=1}^\infty \|F_k(rS_1,\ldots, rS_n)\|<\infty$, we have $$\lim_{m\to\infty} \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|=0 \quad \text{ for any} \ r\in [0,1).$$ On the other hand, $\sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\leq \psi(r)$ for any $m\in {{\mathbb N}}$. Since $\psi$ is integrable on $[0,1]$, we can apply Lebesgue’s dominated convergence theorem and deduce that $$\lim_{m\to\infty} \int_0^1\sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\| dr=0.$$ Now, we deduce that $\|g-g_m\|_1\to 0$, as $m\to\infty$, which shows that the series $\sum_{k=1}^\infty F_k$ is convergent in $\|\cdot\|_1$. This completes the proof when $p=1$. Assume now that $p>1$ and let $\{F_k\}_{k=1}^\infty\subset H^p((B({{\mathcal X}})^n_1)$ be a sequence such that $\sum_{k=1}^\infty \|F\|_p\leq M<\infty$. Since $\|F_k\|_1\leq \|F_k\|_p$, we have $\sum_{k=1}^\infty \|F\|_1\leq M$. Applying the first part of the proof, we find $g\in H^1(B({{\mathcal X}})^n_1)$ such that, for each $r\in [0,1)$, $$g(rS_1,\ldots, rS_n)=\sum_{k=1}^\infty F_k(rS_1,\ldots, rS_n),$$ where the convergence is in the operator norm of $B(F^2(H_n))$. Moreover, we have $$\begin{split} \int_0^1\left\|\sum_{k=1}^m F_k(rS_1,\ldots, rS_n)\right\|^p dr &\leq \int_0^1\left( \sum_{k=1}^m \|F_k(rS_1,\ldots, rS_n)\|\right)^{p} dr\\ &\leq \left[\sum_{k=1}^m\left(\int_0^1 \|F_k(rS_1,\ldots, rS_n)\|^p\right)^{1/p}\right]^p\\ &= \left(\sum_{k=1}^m \|F_k\|_p\right)^p\leq M^p. \end{split}$$ Using Fatou’s lemma, we deduce that the function $r\mapsto \left\| \sum\limits_{k=1}^\infty F_k(rS_1,\ldots, rS_n)\right\|^p$ is integrable on $[0,1]$ and therefore $g\in H^p((B({{\mathcal X}})^n_1)$. Notice also that $$\label{norm-int} \|g-g_m\|_p\leq \left[\int_0^1 \left( \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\right)^p\right]^{1/p}.$$ Since $ \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\leq \psi$ for any $m\in {{\mathbb N}}$, and $$\lim\limits_{m\to\infty} \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|=0\quad \text{ for any }\ r\in [0,1),$$ we can apply again Lebesgue’s dominated convergence theorem and deduce that $$\lim_{m\to\infty} \left[\int_0^1 \left( \sum_{k=m+1}^\infty \|F_k(rS_1,\ldots, rS_n)\|\right)^p\right]^{1/p}=0.$$ Hence and using inequality , we deduce that $\|g-g_m\|_p\to0$ as $m\to\infty$. Consequently, the series $\sum_{k=1}^\infty F_k$ converges in the norm $\|\cdot\|_p$. This completes the proof. \[prop-Hp\] Let $p\geq 1$. 1. If $f\in H^\infty(B({{\mathcal X}})^n_1)$, then $\|f\|_1\leq \|f\|_p\leq \|f\|_\infty$. Moreover, $$H^\infty(B({{\mathcal X}})^n_1)\subset H^p(B({{\mathcal X}})^n_1)\subset H^1(B({{\mathcal X}})^n_1)\subset Hol(B({{\mathcal X}})^n_1).$$ 2. If $f\in H^\infty(B({{\mathcal X}})^n_1)$, then $$\|f\|_\infty=\lim_{p\to\infty}\left(\int_0^1\|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}.$$ 3. If $f=\sum\limits_{k=0}^\infty \sum\limits_{|\alpha|=k} a_\alpha Z_\alpha$ is in $H^p(B({{\mathcal X}})^n_1)$, then $$\left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\leq (pk+1)^{1/p} \|f\|_p.$$ Part (i) follows as in the classical theory of $L^p$ spaces. To prove (ii), define the function $G:[0,1]\to [0,\infty)$ by setting $G(r):=\|f(rS_1,\ldots, rS_n)\|$ if $r\in [0,1)$ and $G(1):=\lim\limits_{r\to 1}\|f(rS_1,\ldots, rS_n)\|$. Due to Theorem \[f-infty\], $G$ is an increasing continuous function and $G(1)=\|f\|_\infty$. Therefore, $$\begin{split} \lim_{p\to\infty}\left(\int_0^1\|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}&= \lim_{p\to\infty} \left( \int_0^1 G(r)^p\right)^{1/p}\\ &=\max_{r\in [0,1]} G(r)=G(1)=\|f\|_\infty. \end{split}$$ To prove (iii), notice that Theorem \[Cauchy-est\] implies $$r^k \left(\sum_{|\alpha|=k}|a_\alpha|^2\right)^{1/2}\leq \|f(rS_1,\ldots, rS_n)\|,\quad r\in [0,1).$$ Integrating over $[0,1]$, we complete the proof of (iii). The next result extends the noncommutative von Neumann inequality from $H^\infty(B({{\mathcal X}})^n_1)$ to the radial maximal Hardy space $H^p(B({{\mathcal X}})^n_1)$, $p\geq 1$. \[vN-Hp\] If $T:=[T_1,\ldots, T_n]\in [B({{\mathcal H}})^n]_1$ and $p\geq 1$, then the mapping $$\Psi_T:H^p(B({{\mathcal X}})^n_1)\to B({{\mathcal H}})\quad \text{ defined by } \ \Psi_T(f):=f(T_1,\ldots, T_n)$$ is continuous, where $f(T_1,\ldots, T_n)$ is defined by the free analytic functional calculus and $B({{\mathcal H}})$ is considered with the operator norm topology. Moreover, $$\|f(T_1,\ldots, T_n)\|\leq\frac{1}{(1-\|[T_1,\ldots, T_n]\|)^{1/p}} \|f\|_p$$ for any $f\in H^p(B({{\mathcal X}})^n_1)$. Assume that $\|[T_1,\ldots, T_n]\|=r_0<1$ and let $f\in H^p(B({{\mathcal X}})^n_1)$. Since the radial maximal function is increasing and and due to Corollary \[max-mod2\], we have $$\begin{split} \|f\|_p&\geq \left(\int_{r_0}^1 \|f(rS_1,\ldots, rS_n)\|^p dr\right)^{1/p}\\ &\geq (1-r_0)^{1/p} \|f(r_0S_1,\ldots, r_0S_n)\|\\ &\geq (1-r_0)^{1/p}\|f(T_1,\ldots, T_n)\|. \end{split}$$ Hence, we deduce the above von Neumann type inequality, which can be used to prove the continuity of $\Psi_T$. We remark that if $f\in H^\infty(B({{\mathcal X}})^n_1)$, then one can recover the noncommutative von Neumann inequality [@Po-von] for strict row contractions, i.e., $\|f(T_1,\ldots, T_n)\|\leq \|f\|_\infty$. Indeed, take $p\to\infty$ in the above inequality and use part (ii) of Proposition \[prop-Hp\]. In the last part of this paper, we introduce a Banach space of analytic functions on the open unit ball of ${{\mathbb C}}^n$ and obtain a von Neumann type inequality in this setting. We use the standard multi-index notation. Let ${\bf p}:=(p_1,\ldots, p_n)$ be a multi-index in ${{\mathbb Z}}_+^n$. We denote $|{\bf p}|:=p_1+\cdots + p_n$ and ${\bf p} !:={ p}_1 !\cdots { p}_n !$. If $\lambda:=(\lambda_1,\ldots,\lambda_n)$, then we set $\lambda^{\bf p}:=\lambda_1^{p_1}\cdots \lambda_n^{p_n}$ and define the symmetrized functional calculus $$(\lambda^{\bf p})_{\text{\rm sym}} (S_1,\ldots, S_n):=\frac {{\bf p}!} {|{\bf p}|! }\sum_{\alpha\in \Lambda_{\bf p}} S_\alpha,$$ where $$\Lambda_{\bf p}:=\{\alpha\in {{\mathbb F}}_n^+: \lambda_\alpha= \lambda^{\bf p} \text{ for any } \lambda\in {{\mathbb B}}_n\}$$ and $S_1,\ldots, S_n$ are the left creation operators on the Fock space $F^2(H_n)$. Notice that card$\Lambda_{\bf p}=\frac {|{\bf p}|!} {{\bf p}!}$. Denote by $H_{\text{\rm sym}}({{\mathbb B}}_n)$ the set of all analytic functions on ${{\mathbb B}}_n$ with scalar coefficients $$f(\lambda_1,\ldots,\lambda_n):=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} \lambda^{\bf p} a_{\bf p}, \quad a_{\bf p}\in {{\mathbb C}},$$ such that $$\label{sup-AA} \limsup_{k\to \infty}\left( \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} \frac {|{\bf p}|!}{{\bf p}!} |a_{\bf p}|^2\right)^{1/2k}\leq 1.$$ Then $$\begin{split} f_{\text{\rm sym}}(rS_1,\ldots, rS_n) &:=\sum_{k=0}^\infty \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} r^k a_{{\bf p}}[(\lambda^{\bf p})_{\text{\rm sym}} (S_1,\ldots, S_n)]\\ &=\sum_{k=0}^\infty \sum_{|\alpha|=k} r^{|\alpha|} c_{\alpha}S_\alpha, \end{split}$$ where $c_{0}:=a_{0}$ and $c_{\alpha}:= \frac {{\bf p}!}{|{\bf p}|!}a_{{\bf p}}$ for ${\bf p}\in {{\mathbb Z}}_+^n$, ${\bf p}\neq (0,\ldots, 0)$, and $\alpha\in \Lambda_{\bf p}$. It is clear that, for each $k=1,2,\ldots, $ we have $$\begin{split} \sum_{|\alpha|=k}|c_{\alpha}|^2&= \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k}\left(\sum_{\alpha\in \Lambda_{\bf p}} |c_{\alpha}|^2\right)\\ &=\sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k} \frac {{\bf p}!}{|{\bf p}|!} |a_{\alpha}|^2. \end{split}$$ Due to Theorem \[Abel\], condition implies that $f_{\text{\rm sym}}(rS_1,\ldots, rS_n)$ is norm convergent for each $r\in [0,1)$, and $f_{\text{\rm sym}}(Z_1,\ldots, Z_n)$ is a free holomorphic function on the open operatorial unit $n$-ball. We define $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n) $ as the set of all functions $f\in H_{\text{\rm sym}}({{\mathbb B}}_n)$ such that $$\|f\|_{\text{\rm sym}}:=\sup_{0\leq r<1}\left\| f_{\text{\rm sym}}(rS_1,\ldots, rS_n)\right\|<\infty.$$ \[sym\] $\left(H_{\text{\rm sym}}^\infty({{\mathbb B}}_n), \|\cdot\|_{\text{\rm sym}}\right)$ is a Banach space. First notice that if $f\in H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ then $f_{\text{\rm sym}}(rS_1,\ldots, rS_n)$ is norm convergent and $f_{\text{\rm sym}}(Z_1,\ldots, Z_n)$ is a free holomorphic function on the open operatorial unit $n$-ball. Using Theorem \[operations\], it is easy to see that $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ is a vector space and $\|\cdot \|_{\text{\rm sym}}$ is a norm. Let $\{f_m\}_{m=1}^\infty$ be a Cauchy sequence of functions in $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$. According to Theorem \[f-infty\], $(f_m)_{\text{\rm sym}}\in F_n^\infty$ and $\{(f_m)_{\text{\rm sym}}\}_{m=1}^\infty$ is a Cauchy sequence in $\|\cdot\|_\infty$, the norm of the Banach algebra $F_n^\infty$. Therefore, there exists $g\in F_n^\infty$ such that $\|(f_m)_{\text{\rm sym}}-L_g\|_\infty\to 0$, as $m\to\infty$. If $ f(\lambda_1,\ldots,\lambda_n)=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} a_{\bf p}^{(m)}\lambda^{\bf p}, \quad a_{\bf p}\in {{\mathbb C}}, $ then $(f_m)_{\text{\rm sym}}(S_1,\ldots, S_n)=\sum_{k=0}^\infty\sum_{|\alpha|=k} c_\alpha^{(m)} S_\alpha$, where $c_\alpha^{(m)}:=\frac {|{\bf p}|!}{{\bf p}!} a_{\bf p}^{(m)}$ for ${\bf p}\in {{\mathbb Z}}_+^n$, ${\bf p}\neq (0,\ldots,0)$ and $\alpha\in \Lambda_{\bf p}$. If $g=\sum_{\alpha\in {{\mathbb F}}_n^+} b_\alpha e_\alpha$ is the Fourier representation of $g$ as an element of $F^2(H_n)$, then we have $$\begin{split} |c_\alpha^{(m)}-b_\alpha|&= \left|\left<[(f_m)_{\text{\rm sym}}(S_1,\ldots, S_n)-L_g]1,1\right>\right|\\ &\leq \|(f_m)_{\text{\rm sym}}-L_g\|_\infty. \end{split}$$ Taking $m\to \infty$, we deduce that $c_\alpha^{(m)}\to b_\alpha$ for each $\alpha\in {{\mathbb F}}_n^+$. Since $c_\alpha^{(m)}=c_\beta^{(m)}$ for any $\alpha,\beta \in \Lambda_{\bf p}$, we get $b_\alpha=b_\beta$. Setting $h(\lambda_1,\ldots, \lambda_n):= \sum_{k=0}^\infty \sum_{|\alpha|=k} b_\alpha \lambda_\alpha$, one can see that $h$ is holomorphic in ${{\mathbb B}}_n$ and $h_{\text{\rm sym}}=L_g$. Moreover, $\|h\|_{\text{\rm sym}}=\|g\|_\infty<\infty$. This shows that $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$ is a Banach space. Now, using Theorem \[abel\] in the scalar case, we can deduce the following. If $T:=[T_1,\ldots, T_n]\in B({{\mathcal H}})^n$ is a commuting $n$-tuple of operators with the joint spectral radius $r(T_1,\ldots, T_n)<1$ and $f(\lambda_1,\ldots,\lambda_n):=\sum\limits_{\bf p\in {{\mathbb Z}}_+^n} a_{\bf p} \lambda^{\bf p} $ is in $ H_{\text{\rm sym}}({{\mathbb B}}_n)$, then $$f(T_1,\ldots, T_n):= \sum_{k=0}^\infty \sum\limits_{{\bf p}\in {{\mathbb Z}}_+^n,|{\bf p}|=k}a_{\bf p} T^{\bf p}$$ is a well-defined operator in $B({{\mathcal H}})$, where the series is convergent in the operator norm topology. Moreover, the map $$\Psi_T: H_{\text{\rm sym}}({{\mathbb B}}_n)\to B({{\mathcal H}})\qquad \Psi_T(f)=f(T_1,\ldots, T_n)$$ is continuous and $$\|f(T_1,\ldots, T_n) \|\leq M\|f\|_{\text{\rm sym}},$$ where $M=\sum_{k=0}^\infty \left\|\sum_{|\alpha|=k}T_\alpha T_\alpha^*\right\|^{1/2}$. In a forthcoming paper [@Po-Bohr], we obtain operator-valued Bohr type inequalities for the Banach space $H_{\text{\rm sym}}^\infty({{\mathbb B}}_n)$. [99]{} , Analytic functions of several Banach algebra elements, [*Ann. of Math.*]{} [**62**]{} (1955), 204–216. , [ Factorization and reflexivity on Fock spaces,]{} [*Integr. Equat. Oper. Th.*]{} [**23**]{} (1995), 268–286. , [Noncommutative interpolation and Poisson transforms,]{} [*Israel J. Math.*]{} [**115**]{} (2000), 205–234. , [Subalgebras of $C^*$-algebras III: Multivariable operator theory,]{} [*Acta Math.*]{} [**181**]{} (1998), 159–228. , The curvature invariant of a Hilbert module over ${{\mathbb C}}[z_1,\ldots, z_n]$, [*J. Reine Angew. Math.*]{} [**522**]{} (2000), 173–236. , Interpolation and commutant lifting for multipliers on reproducing kernels Hilbert spaces, [*Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume*]{}, pages 89–138, [**OT 122**]{}, Birkhauser-Verlag, Basel-Boston-Berlin, 2001. , Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting, [*Mem. Amer. Math. Soc.*]{} [**837**]{} (2005). , A model theory for $q$-commuting contractive tuples, [*J. Operator Theory*]{} [**47**]{} (2002), 1551–1568. , Standard noncommuting and commuting dilations of commuting tuples, [*Trans. Amer. Math. Soc.*]{} [**356**]{} (2003), 1551–1568. , Models for n-tuples of noncommuting operators, [*J. Funct. Anal.*]{} [**57**]{}(1984), 21–30. , [*Functions of one complex variable. I.*]{} Second Edition. Graduate Texts in Mathematics [**159**]{}. [ Springer-Verlag, New York]{}, 1995. , Automorphism invariance of the operator-valued Poisson transform [*Acta Sci. Math. (Szeged)*]{} [**57**]{} (1993), 65–78. , The structure of free semigroup algebras, [*J. Reine Angew. Math.*]{} [**533**]{} (2001), 99–125. , Isometric dilations of non-commuting finite rank $n$-tuples, [*Canad. J. Math.*]{} [**53**]{} (2001), 506–545. , Nevanlinna-Pick interpolation for noncommutative analytic Toeplitz algebras, [*Integr. Equat. Oper.Th.*]{} [**31**]{} (1998), 321–337. , [Invariant subspaces and hyper-reflexivity for free semigroup algebras,]{} [*Proc. London Math. Soc.*]{} [**78**]{} (1999), 401–430. , Automorphisms and representations of the noncommutative analytic Toeplitz algebras, [*Math. Ann.*]{} [**311**]{} (1998), 275–303. , [*Operator spaces*]{}, London Mathematical Society Monographs. New Series, [**23**]{}. The Clarendon Press, Oxford University Press, New York, 2000. , Models for noncommuting operators, [*J. Funct. Anal.*]{} [**48**]{} (1982), 1–11. , [*Banach Spaces of Analytic Functions*]{}, Englewood Cliffs: Prentice-Hall, 1962. , The curvature invariant of a non-commuting $N$-tuple, [*Integral Equations Operator Theory*]{} [**41**]{} (2001), no.4, 426–454. , [*Combinatorial group theory. Presentations of groups in terms of generators and relations.*]{} Reprint of the 1976 second edition. [ Dover Publications, Inc., Mineola, NY]{}, 2004. , On the joint spectral radius, [*Ann. Polon. Math.*]{} [**66**]{} (1997), 173–182. , On Bohr’s inequality, [*Proc. London Math. Soc.*]{} [**85**]{} (2002), 493–512. , Models for infinite sequences of noncommuting operators, [*Acta. Sci. Math. (Szeged)*]{} [**53**]{} (1989), 355–368. , Isometric dilations for infinite sequences of noncommuting operators, [*Trans. Amer. Math. Soc.*]{} [**316**]{} (1989), 523–536. , Characteristic functions for infinite sequences of noncommuting operators, [*J. Operator Theory*]{} [**22**]{} (1989), 51–71. , Multi-analytic operators and some factorization theorems, [*Indiana Univ. Math. J.*]{} [**38**]{} (1989), 693–710. (1991), 292–304. , [Functional calculus for noncommuting operators,]{} [*Michigan Math. J.*]{} [**42**]{} (1995), 345–356. , [Multi-analytic operators on Fock spaces,]{} [*Math. Ann.*]{} [**303**]{} (1995), 31–46. , [Noncommutative disc algebras and their representations,]{} [*Proc. Amer. Math. Soc.*]{} [**124**]{} (1996), 2137–2148. , Interpolation problems in several variables, [*J. Math Anal. Appl.*]{} [**227**]{} (1998), 227–250. , [Poisson transforms on some $C^*$-algebras generated by isometries,]{} [*J. Funct. Anal.*]{} [**161**]{} (1999), 27–61. , Curvature invariant for Hilbert modules over free semigroup algebras, [*Adv. Math.*]{} [**158**]{} (2001), 264–309. , Similarity and ergodic theory of positive linear maps, [*J. Reine Angew. Math.*]{} [**561**]{} (2003), 87–129. , Entropy and Multivariable Interpolation, [*Mem. Amer. Math. Soc.*]{} [**184**]{} (868) (2006). , Multivariable Bohr inequalities, [*Trans. Amer. Math. Soc.*]{}, to appear. , Operator theory on noncommutative varieties, [*Indiana Univ. Math. J.*]{} [**55**]{} (2) (2006), 389–442. , [Unitary invariants in multivariable operator theory]{}, preprint 2004. , [*Hardy classes and operator theory*]{}, Oxford University Press-New York, 1985. , [*Real and Complex Analysis*]{}, [ McGraw-Hill Book Co.]{} (1966) , [*Function theory in the unit ball of ${{\mathbb C}}^n$*]{}, [ Springer-verlag, New-York/Berlin]{}, 1980 , A cyclic derivative in noncommutative algebra, [*J. Algebra*]{} [**64**]{} (1980), 54-75. , On the decomposition of a normed ring into a direct sum of ideals, [*Amer. Math. Soc. Transl.*]{} [**1**]{} (1955), 37-48. , A joint spectrum for several commuting operators, [*J. Funct. Anal.*]{} [**6**]{} (1970), 172–191 , The analytic functional calculus for several commuting operators, [*Acta. Math.*]{} [**125**]{} (1970), 1–38. , A general framework for a multi-operator functional calculus, [*Adv. Math.*]{} [**9**]{} (1972), 183-252. , [An operator-valued Poisson kernel]{}, [*J. Funct. Anal.*]{} [**110**]{} (1992), 47–72. , [*Harmonic Analysis of Operators on Hilbert Space*]{}, North Holland, New York 1970. , [Eine Spectraltheorie für allgemeine Operatoren eines unitären Raumes,]{} [*Math. Nachr.*]{} [**4**]{} (1951), 258–281. [^1]: Research supported in part by an NSF grant

ArXiv

--- abstract: 'In this paper, we give a very simple proof of the main result of Dafni (Canad Math Bull 45:46–59, 2002) concerning with weak$^*$-convergence in the local Hardy space $h^1({{\mathbb R}}^d)$.' address: - 'High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam' - 'Department of Natural Science and Technology, Tay Nguyen University, Daklak, Vietnam.' - 'Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnam' author: - Ha Duy HUNG - Duong Quoc Huy - 'Luong Dang Ky $^*$' title: 'A note on weak$^*$-convergence in $h^1({{\mathbb R}}^d)$' --- [^1] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] Introduction ============ A famous and classical result of Fefferman [@Fef] states that the John-Nirenberg space $BMO(\mathbb R^d)$ is the dual of the Hardy space $H^1(\mathbb R^d)$. It is also well-known that $H^1(\mathbb R^d)$ is one of the few examples of separable, nonreflexive Banach space which is a dual space. In fact, let $C_c({{\mathbb R}}^d)$ be the space of all continuous functions with compact support and denote by $VMO(\mathbb R^d)$ the closure of $C_c({{\mathbb R}}^d)$ in $BMO(\mathbb R^d)$, Coifman and Weiss showed in [@CW] that $H^1(\mathbb R^d)$ is the dual space of $VMO(\mathbb R^d)$, which gives to $H^1(\mathbb R^d)$ a richer structure than $L^1(\mathbb R^d)$. For example, the classical Riesz transforms $\nabla (-\Delta)^{-1/2}$ are not bounded on $L^1(\mathbb R^d)$, but are bounded on $H^1(\mathbb R^d)$. In addition, the weak$^*$-convergence is true in $H^1(\mathbb R^d)$ (see [@JJ]), which is useful in the application of Hardy spaces to compensated compactness (see [@CLMS]) and in studying the endpoint estimates for commutators of singular integral operators (see [@Ky1; @Ky2; @Ky3]). Recently, Dafni showed in [@Da] that the local Hardy space $h^1({{\mathbb R}}^d)$ of Goldberg [@Go] is in fact the dual space of $vmo({{\mathbb R}}^d)$ the closure of $C_c({{\mathbb R}}^d)$ in $bmo(\mathbb R^d)$. Moreover, the weak$^*$-convergence is true in $h^1(\mathbb R^d)$. More precisely, in [@Da], the author proved: \[Dafni 1\] The space $h^1({{\mathbb R}}^d)$ is the dual of the space $vmo({{\mathbb R}}^d)$. \[Dafni 2\] Suppose that $\{f_n\}_{n=1}^\infty$ is a bounded sequence in $h^1(\mathbb R^d)$, and that $\lim_{n\to\infty} f_n(x) = f(x)$ for almost every $x\in\mathbb R^d$. Then, $f\in h^1(\mathbb R^d)$ and $\{f_n\}_{n= 1}^\infty$ weak$^*$-converges to $f$, that is, for every $\phi\in vmo(\mathbb R^d)$, we have $$\lim_{n\to\infty} \int_{\mathbb R^d} f_n(x) \phi(x)dx = \int_{\mathbb R^d} f(x) \phi(x) dx.$$ The aim of the present paper is to give very simple proofs of the two above theorems. It should be pointed out that our method is different from that of Dafni and it can be generalized to the setting of spaces of homogeneous type (see [@Ky4]). To this end, we first recall some definitions of the function spaces. As usual, $\mathcal S(\mathbb R^d)$ denotes the Schwartz class of test functions on $\mathbb R^d$. The subspace $\mathcal A$ of $\mathcal S(\mathbb R^d)$ is then defined by $$\mathcal A=\Big\{\phi\in \mathcal S(\mathbb R^d): |\phi(x)|+ |\nabla\phi(x)|\leq (1+ |x|^2)^{-(d+1)}\Big\},$$ where $\nabla= (\partial/\partial x_1,..., \partial/\partial x_d)$ denotes the gradient. We define $$\mathfrak M f(x):= \sup\limits_{\phi\in\mathcal A}\sup\limits_{|y-x|<t}|f*\phi_t(y)|\quad\mbox{and}\quad \mathfrak mf(x):= \sup\limits_{\phi\in\mathcal A}\sup\limits_{|y-x|<t<1}|f*\phi_t(y)|,$$ where $\phi_t(\cdot)= t^{-d}\phi(t^{-1}\cdot)$. The space $H^1(\mathbb R^d)$ is the space of all integrable functions $f$ such that $\mathfrak M f\in L^1(\mathbb R^d)$ equipped with the norm $\|f\|_{H^1}= \|\mathfrak M f\|_{L^1}$. The space $h^1(\mathbb R^d)$ denotes the space of all integrable functions $f$ such that $\mathfrak m f\in L^1(\mathbb R^d)$ equipped with the norm $\|f\|_{h^1}= \|\mathfrak m f\|_{L^1}$. We remark that the local real Hardy space $h^1(\mathbb R^d)$, first introduced by Goldberg [@Go], is larger than $H^1(\mathbb R^d)$ and allows more flexibility, since global cancellation conditions are not necessary. For example, the Schwartz class $\mathcal S({{\mathbb R}}^d)$ is contained in $h^1(\mathbb R^d)$ but not in $H^1(\mathbb R^d)$, and multiplication by cutoff functions preserves $h^1(\mathbb R^d)$ but not $H^1(\mathbb R^d)$. Thus it makes $h^1(\mathbb R^d)$ more suitable for working in domains and on manifolds. It is well-known (see [@Fef]) that the dual space of $H^1(\mathbb R^d)$ is $BMO(\mathbb R^d)$ the space of all locally integrable functions $f$ with $$\|f\|_{BMO}:=\sup\limits_{B}\frac{1}{|B|}\int_B \Big|f(x)-\frac{1}{|B|}\int_B f(y) dy\Big|dx<\infty,$$ where the supremum is taken over all balls $B\subset {{\mathbb R}}^d$. It was also shown in [@Go] that the dual space of $h^1(\mathbb R^d)$ can be identified with the space $bmo(\mathbb R^d)$, consisting of locally integrable functions $f$ with $$\|f\|_{bmo}:= \sup\limits_{|B|\leq 1}\frac{1}{|B|}\int_B \Big|f(x)-\frac{1}{|B|}\int_B f(y) dy\Big|dx+ \sup\limits_{|B|\geq 1}\frac{1}{|B|}\int_B |f(x)|dx<\infty,$$ where the supremums are taken over all balls $B\subset {{\mathbb R}}^d$. It is clear that, for any $f\in H^1({{\mathbb R}}^d)$ and $g\in bmo({{\mathbb R}}^d)$, $$\|f\|_{h^1} \leq \|f\|_{H^1}\quad\mbox{and}\quad \|g\|_{BMO} \leq \|g\|_{bmo}.$$ Recall that the space $VMO(\mathbb R^d)$ (resp., $vmo(\mathbb R^d)$) is the closure of $C_c(\mathbb R^d)$ in $(BMO(\mathbb R^d),\|\cdot\|_{BMO})$ (resp., $(bmo(\mathbb R^d),\|\cdot\|_{bmo})$). The following theorem is due to Coifman and Weiss [@CW]. \[Coifman-Weiss\] The space $H^1({{\mathbb R}}^d)$ is the dual of the space $VMO({{\mathbb R}}^d)$. Throughout the whole paper, $C$ denotes a positive geometric constant which is independent of the main parameters, but may change from line to line. Proof of Theorems \[Dafni 1\] and \[Dafni 2\] ============================================= In this section, we fix $\varphi\in C_c({{\mathbb R}}^d)$ with supp $\varphi\subset B(0,1)$ and $\int_{{{\mathbb R}}^d} \varphi(x) dx=1$. Let $\psi:= \varphi*\varphi$. The following lemma is due to Goldberg [@Go]. \[Golberg\] There exists a positive constant $C=C(d,\varphi)$ such that [i)]{} for any $f\in L^1({{\mathbb R}}^d)$, $$\|\varphi*f\|_{h^1}\leq C \|f\|_{L^1};$$ [ii)]{} for any $g\in h^1({{\mathbb R}}^d)$, $$\|g- \psi*g\|_{H^1}\leq C \|g\|_{h^1}.$$ As a consequence of Lemma \[Golberg\](ii), for any $\phi\in C_c({{\mathbb R}}^d)$, $$\label{from big bmo to small bmo} \|\phi - \overline{\psi}*\phi\|_{bmo} \leq C \|\phi\|_{BMO},$$ here and hereafter, $\overline{\psi}(x):= \psi(-x)$ for all $x\in {{\mathbb R}}^d$. Since $vmo(\mathbb R^d)$ is a subspace of $bmo(\mathbb R^d)$, which is the dual space of $h^1(\mathbb R^d)$, every function $f$ in $h^1(\mathbb R^d)$ determines a bounded linear functional on $vmo(\mathbb R^d)$ of norm bounded by $\|f\|_{h^1}$. Conversely, given a bounded linear functional $L$ on $vmo(\mathbb R^d)$. Then, $$|L(\phi)|\leq \|L\| \|\phi\|_{vmo}\leq \|L\| \|\phi\|_{L^\infty}$$ for all $\phi\in C_c(\mathbb R^d)$. This implies (see [@Ro]) that there exists a finite signed Radon measure $\mu$ on ${{\mathbb R}}^d$ such that, for any $\phi\in C_c(\mathbb R^d)$, $$L(\phi)= \int_{{{\mathbb R}}^d} \phi(x) d\mu(x),$$ moreover, the total variation of $\mu$, $|\mu|({{\mathbb R}}^d)$, is bounded by $\|L\|$. Therefore, $$\label{Dafni 1, 1} \|\psi*\mu\|_{h^1} = \|\varphi*(\varphi*\mu)\|_{h^1} \leq C \|\varphi*\mu\|_{L^1}\leq C |\mu|({{\mathbb R}}^d)\leq C \|L\|$$ by Lemma \[Golberg\]. On the other hand, by (\[from big bmo to small bmo\]), we have $$\begin{aligned} |(L-\psi*L)(\phi)|=|L(\phi - \overline{\psi}*\phi)|&\leq& \|L\| \|\phi - \overline{\psi}*\phi\|_{vmo}\\ &\leq& C \|L\| \|\phi\|_{BMO}\end{aligned}$$ for all $\phi\in C_c({{\mathbb R}}^d)$. Consequently, by Theorem \[Coifman-Weiss\], there exists a function $h$ belongs $H^1({{\mathbb R}}^d)$ such that $\|h\|_{H^1}\leq C \|L\|$ and $$(L-\psi*L)(\phi)= \int_{{{\mathbb R}}^d} h(x) \phi(x) dx$$ for all $\phi\in C_c({{\mathbb R}}^d)$. This, together with (\[Dafni 1, 1\]), allows us to conclude that $$L(\phi)= \int_{{{\mathbb R}}^d} f(x) \phi(x) dx$$ for all $\phi\in C_c({{\mathbb R}}^d)$, where $f:= h+ \psi*\mu\in h^1({{\mathbb R}}^d)$ satisfying $\|f\|_{h^1}\leq \|h\|_{H^1} + \|\psi*\mu\|_{h^1}\leq C \|L\|$. The proof of Theorem \[Dafni 1\] is thus completed. Let $\{f_{n_k}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{f_n\}_{n=1}^\infty$. As $\{f_{n_k}\}_{k=1}^\infty$ is a bounded sequence in $h^1({{\mathbb R}}^d)$, by Theorem \[Dafni 1\] and the Banach-Alaoglu theorem, there exists a subsequence $\{f_{n_{k_j}}\}_{j=1}^\infty$ of $\{f_{n_k}\}_{k=1}^\infty$ such that $\{f_{n_{k_j}}\}_{j=1}^\infty$ weak$^*$-converges to $g$ for some $g\in h^1({{\mathbb R}}^d)$. Therefore, for any $x\in {{\mathbb R}}^d$, $$\lim_{j\to\infty} \int_{{{\mathbb R}}^d} f_{n_{k_j}}(y) \psi(x-y) dy = \int_{{{\mathbb R}}^d} g(y) \psi(x-y) dy.$$ This implies that $\lim_{j\to\infty}[f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]= f(x) - (g*\psi)(x)$ for almost every $x\in\mathbb R^d$. Hence, by Lemma \[Golberg\](ii) and the Jones-Journé’s theorem (see [@JJ]), $$\|f- g*\psi\|_{H^1}\leq \sup_{j\geq 1}\|f_{n_{k_j}} - f_{n_{k_j}}*\psi\|_{H^1}\leq C \sup_{j\geq 1} \|f_{n_{k_j}}\|_{h^1}<\infty,$$ moreover, $$\lim_{j\to\infty} \int_{\mathbb R^d} [f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]\phi(x)dx = \int_{\mathbb R^d} [f(x) - (g*\psi)(x)]\phi(x) dx$$ for all $\phi\in C_c(\mathbb R^d)$. As a consequence, we obtain that $$\begin{aligned} \|f\|_{h^1} \leq \|f- g*\psi\|_{h^1} + \|g*\psi\|_{h^1}&\leq& \|f- g*\psi\|_{H^1} + C\|g\|_{h^1}\\ &\leq& C \sup_{j\geq 1} \|f_{n_{k_j}}\|_{h^1}<\infty,\end{aligned}$$ moreover, $$\begin{aligned} &&\lim_{j\to\infty} \int_{\mathbb R^d} f_{n_{k_j}}(x)\phi(x)dx \\ &=& \lim_{j\to\infty} \int_{\mathbb R^d} [f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]\phi(x)dx + \lim_{j\to\infty} \int_{\mathbb R^d} f_{n_{k_j}}(x) (\overline\psi*\phi)(x)dx\\ &=& \int_{\mathbb R^d} [f(x) - (g*\psi)(x)]\phi(x) dx + \int_{\mathbb R^d} g(x) (\overline\psi*\phi)(x)dx\\ &=& \int_{\mathbb R^d} f(x)\phi(x)dx\end{aligned}$$ since $\{f_{n_{k_j}}\}_{j=1}^\infty$ weak$^*$-converges to $g$ in $h^1(\mathbb R^d)$. This, by $\{f_{n_k}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{f_n\}_{n=1}^\infty$, allows us to complete the proof of Theorem \[Dafni 2\]. [MTW1]{} R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286. G. Dafni, Local VMO and weak convergence in $h^1$. Canad. Math. Bull. 45 (2002), no. 1, 46–59. C. Fefferman, Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77 (1971), no. 4, 587–588. D. Goldberg, A local version of real Hardy spaces. Duke J. Math. 46 (1979), 27–42. P. W. Jones and J-L. Journé, On weak convergence in $H^1({\bf R}^d)$. Proc. Amer. Math. Soc. 120 (1994), no. 1, 137–138. L. D. Ky, Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931–2958. L. D. Ky, On weak$^*$-convergence in $H^1_L(\mathbb R^d)$. Potential Anal. 39 (2013), no. 4, 355–368. L. D. Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators, Rev. Mat. Iberoam. (to appear) or arXiv:1203.6335. L. D. Ky, On weak$^*$-convergence in $H^1_\rho(\mathcal X)$, preprint. H. L. Royden, Real analysis. Third edition. Macmillan Publishing Company, New York, 1988. [^1]: The paper was completed when the third author was visiting to Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for financial support and hospitality.\ $^{*}$Corresponding author: Luong Dang Ky

ArXiv

--- abstract: 'A junction between two boundaries of a topological superconductor (TSC), mediated by localized edge modes of Majorana fermions, is investigated. The tunneling of fermions across the junction depends on the magnetic flux. It breaks the time-reversal symmetry at the boundary of the sample. The persistent current is determined by the emergence of Majorana edge modes. The structure of the edge modes depends on the magnitude of the tunneling amplitude across the junction. It is shown that there are two different regimes, which correspond to strong and weak tunneling of Majorana fermions, distinctive in the persistent current behavior. In a strong tunneling regime, the fermion parity of edge modes is not conserved and the persistent current is a $2\pi$-periodic function of the magnetic flux. When the tunneling is weak the chiral Majorana states, which are propagating along the edges have the same fermion parity. They form a $4\pi$-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of $2\pi$- and of $4\pi$-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied.' author: - 'Igor N. Karnaukhov' title: Persistent current in 2D topological superconductors --- Introduction {#introduction .unnumbered} ============ The phase coherent tunneling across a junction between two superconductors implies the presence of a $2 \pi-$ periodic persistent current, which is defined by the phase difference between the superconducting order parameters. The Josephson effect has been considered in Refs [@I1; @Er] in the framework of the well-known Kitaev chain model [@Kitaev]. The Kitaev’s proposal that, in the case of the fermion parity conservation, zero-energy Majorana fermion states, which are localized at the ends of the superconducting wire, trigger a $4 \pi$-periodic persistent current explains the so-called ’topological (or fractional) Josephson effect. The $2 \pi$- and $4 \pi$-harmonics of a persistent current correspond to the respective ground states of the system with different fermion parity when the magnetic flux is greater than $\pi$. In [@Kitaev] the author stimulates further research of new topological states that are realized at junctions between 1D TSCs, and Luttinger liquids [@I2; @I3]. In the absence of fermion parity conservation (that is, in those superconductors, in which the total number of particles is not conserved), the system under consideration is relaxing to the phase state with the lowest energy, which leads to the emergence of a $2\pi$-periodic persistent current. Below we discuss the persistent current in a 2D $(p+ip)$ TSC that has the spatial form of a hollow cylinder and is penetrated by a magnetic flux Q. We expect a nontrivial behavior of the persistent current depending on the magnitude of the applied magnetic flux. Due to their nontrivial topology [@TSC5; @TSC6; @TSC7; @TSC8], the superconductors with $(d +id)$ and $(p+ip)$ order parameters exhibit exotic phenomena such as Majorana vortex bound states and gapless chiral edge modes. The 2D TSCs with the ($p+ip)$-pairing of spinless fermions, which have chiral Majorana fermion states propagating along the edges, have been considered in [@TSC5a]. The behavior of topological states in the presence of disorder has been studied in Refs [@A1; @D1; @D3; @D4; @D5; @D6]. A nontraditional approach for description of TSCs has been proposed in [@K1] (see also [@K2]). It was shown that spontaneous breaking of time reversal symmetry is realized due to nontrivial stable phases of the superconducting order parameter (new order parameter). At that, the models of the TSC with the $p-$ and $(p+ip)$-wave superconducting pairing of spinless fermions are the simplest and the most straightforward examples of relevant model systems. In a finite system, the gapless chiral edge modes are localized at the boundaries. The tunneling of fermions across a junction leads to gapped edge modes due to the hybridization (through the weak link) of chiral edge modes localized at the different boundaries of the junction. In the case of a 1D superconductor the fermion parity is associated with zero energy Majorana edge states [@Kitaev; @I5; @I6; @I7; @D7], for a 2D TSC a persistent current is determined by the presence of the Majorana gapless edge modes localized at the boundaries of the junction. The ground state fermion parity changes whenever the energy of a pair of Majorana fermions crosses the zero energy. In the superconductor-topological insulator system the fermion parity of the ground state was associated with the Hopf index [@I4]. The fermion parity conservation, as a rule, is the result of the conservation of the total number of particles in the system, while the total number of particles is not conserved in those superconductors, which were studied in the framework of the Bogoliubov-de Gennes formalism. Nevertheless, we show that the fermion parity conservation is realized due to the conservation of the Chern number that determines the chiral current at the ends of the cylinder. The key point of the paper is that the unconventional behavior of the persistent current is determined by a chiral current along the boundaries of the TSC, while the behavior of the persistent current depends on the value of the tunneling amplitude of Majorana fermions across the junction. We should expect that behavior of the persistent current differs in the cases of the strong and weak tunneling of Majorana fermions. Model Hamiltonian, edge modes {#model-hamiltonian-edge-modes .unnumbered} ============================= We consider a junction between two boundaries of the TSC. The lattice Hamiltonian for a $(p+ip)$-wave superconductor of spinless fermions consists of two terms: ${\cal H} = {\cal H}_{TSC} + {\cal H}_{tun}$. At that, the first term describes the TSC per se: $${\cal H}_{TSC}= - \sum_{<ij>}a^\dagger_{i}a_j - 2\mu \sum_{j} n_j+ (i\Delta \sum_{<ij> x-links} a^\dagger_{i}a^\dagger_{j}+\Delta\sum_{<ij> y-links} a^\dagger_{i}a^\dagger_{j}+h.c.) , \label{eq-H}$$ and the second term describes the tunneling of fermions between two boundaries of a TSC with a junction along the x-direction $${\cal H}_{tun}= - 2\tau e^{i\frac{Q}{2}} \sum_{x-links} a^\dagger_ {x,1} a_{x,L} +h.c., \label{eq-Htun}$$ where $a^\dagger_{j} $ and $a_{j}$ are the spinless fermion operators on a site $j = {x,y}$ obeying usual anticommutation relations, and $n_j$ denotes the density operator. The first term in (1) describes hoppings of spinless fermions between nearest-neighbor lattice sites with equal to the unity magnitude, $\mu$ is the chemical potential (by choosing $ 0 < \mu < 1$ we do not restrict the generality of the study). Remaining terms describe pairing with superconducting order parameter $\Delta > 0$, which is defined along the link. Links are divided into two types depending on their direction: real $\Delta$ along y-links and complex $i\Delta$ along x-links. In practice, values of $\Delta,|\mu| << 1$. Therefore, we consider low energy excitations for $\Delta,|\mu| < 1$. The term ${\cal H}_{tun}$ contains the tunneling amplitude $0 < \tau < 1$ and takes into account the applied flux $Q$. The value of $ Q$ is measured in units of the quantum of flux $hc/(2e)$. Energies of spinless fermions E in the TSC that is described by the Hamiltonian (\[eq-H\]) are arranged symmetrically with respect to the zero energy and are given by the following dispersion relation $$E=\pm[(\mu+\cos k_x + \cos k_y)^2 +\Delta^2 (\sin^2 k_x +\sin^2 k_y)]^{1/2}, \label{eq-3}$$ where the wave vector $\textbf{k}=\{k_x,k_y\}$. In a finite system, the one-particle spectrum of the Hamiltonian $ {\cal H}$ (\[eq-H\]), (\[eq-Htun\]), is also symmetric edge states including. The corresponding edge states are determined by the particle-hole states of Majorana fermions. ![(Color online) Low energy spectra with edge modes of the one-dimensional strip along the *x*-direction as a function of the momentum directed along the edge. The energies are calculated at the Kitaev point $\Delta =1$ for $\mu=\frac{1}{5}$, $Q=\pi$ left), $Q=1\frac{1}{4}\pi$ right) and for different $\tau$. []{data-label="fig:3"}](3.eps){width="1\linewidth"} We analyze the formation of Majorana modes at the edges of the TSC. The gapped spectrum of excitations (\[eq-3\]) is realized in the topological nontrivial phase at $0<| \mu |<2$ (see the excitation spectra in Figs \[fig:1\]a),c)). The topological properties of a system are manifested in the existence of a nontrivial Chern number $C$ and chiral gapless edge modes (see in Figs \[fig:1\]), which are robust to effects of disorder and interactions. The excitation spectrum of the TSC includes chiral edge modes that connect the lower and upper fermion subbands. They are localized near the boundaries of the sample, and, therefore, amplitudes of the corresponding wave functions decrease exponentially with receding from the boundaries. The chiral gapless edge modes do exist in the gap if the Chern number of isolated bands located below the gap is nonzero. The gap of the superconductor collapses at $\mu=0$ and $\mu=\pm 2$. The TSC state with $C=\texttt{sgn} (\mu) $ is realized at $|\mu|<2$ , whereas the Chern number is equal to zero in a trivial topological state at $|\mu| > 2$ [@D7]. ### The Kitaev point $\Delta=1$ {#the-kitaev-point-delta1 .unnumbered} In order to describe in detail edge states of TSCs, we consider a superconductor in the form of a right square prism. Its base is taken to be LxL in size, while its height is assumed to be smaller than the superconducting coherence length. The superconductor can effectively be described by the 2D model (see Fig.\[fig:2\]a)). In the Appendix, we rigorously prove that chiral edge modes exist in the TSC for open boundary conditions. At that, their energy is determined by the wave vector component $k$ that is parallel to the boundary. These edge modes intersect each other at the Dirac point according to the dispersion relation $E_{edge}=\pm \Delta \sin k $. This dispersion relation is valid up to the points, at which edge modes are entering the domain of bulk states (see Figs \[fig:1\]). In the topological state with the Chern number equal to 1, gapless edge modes with wavevectors directed along the x-,y-boundaries have the corresponding Dirac points at $k_x = \pi , k_y =0$ and $k_x=0 , k_y=\pi$, respectively. In the topological state with $C=-1$ gapless edge modes have a different chirality, and their Dirac points are shifted to $\pi$ at $k_x=0$, $k_y =\pi$ and $k_x=\pi , k_y=0$, the TSC state is characterized by a chiral current along the boundary of the 2D system (see in Fig \[fig:2\]a)). This current can be of different chirality depending on the sign of the Chern number. The behavior of chiral edge modes at the junction (see in Fig.\[fig:2\]b)) is examined for the sample in the form of a hollow cylinder with varying the applied flux $Q$. In the case of a contact interaction between fermions at the boundaries, the tunneling Hamiltonian can be expressed in Majorana operators $f_{x,1},g_{x,1}$ and $ f_{x,L},g_{x,L}$ as follows: ${\cal H}_{tun}=- i \tau \cos(Q/2)\sum_{x-links} \delta_{1,L} (f_{x,1} g_{x,L} - g_{x,1} f_{x,L})$. Gapless edge modes are associated with Majorana operators $g_1(k_x)$, $f_1(k_x)$ and $g_L(k_x)$, $f_L(k_x)$ that belong to the boundaries. Cases of $Q=\pm \pi$ are particular because the contact interaction between particles vanishes at the boundaries for arbitrary $\tau$. Thus, the system is reduced to the TSC with open boundary conditions, in which (as noted above) the chiral gapless modes are realized in the topological phase (see Figs \[fig:1\]a),b)).At $\tau \neq 0$ and $Q\neq \pm \pi$ the edge modes at the junction are gapped, as a result of their hybridization at the Dirac point. In addition, we will demonstrate that their behavior depends on the magnitude of $\tau$. Majorana edge states are gapless at the points $Q = \pm \pi$ with the linear dispersion in $k$: $E_{edge}(\delta k)\sim \delta k$, $k=\pi+\delta k$ which is given by Eq (\[A7\]) (see in Figs \[fig:1\],\[fig:3\], the appendix contains some calculation details). Numerical calculations show that in a weak tunneling regime $\tau<\tau_c$ two chiral gapless edge modes are realized in the spectrum of the TSC for $Q=\pm \pi$ (see in Figs \[fig:3\]). These edge modes merge with the bulk states for any other $\tau$ and $Q$ in a weak tunneling regime. The edge modes with different chirality are localized at the different boundaries (at y = 1 and y = L) of the junction. The chiral edge modes yield chiral currents along the boundaries of the junction and form a chiral boundary current. The numerical calculation of $\tau_c$ at $Q=\pi$ as a function of $\mu$ is shown in Fig.\[fig:4\]a) at the Kitaev point. The calculations of $\tau_c$ for arbitrary $Q$ demonstrate that $\tau_c$ has the maximum value at $Q=\pi$. At the point $Q=\pi$ for $\tau >\tau_c$ gapless modes are localized at the junction, but they are non-chiral and do not touch fermion subbands at the arbitrary $Q$ (see Figs \[fig:3\]b),c),d)). For $\tau =1$ the linear dispersion of edge modes vanishes at $k=\pi$. We see that the behavior of the edge modes changes radically at $\tau >1$. In the strong tunneling regime $\tau>\tau_c$ the edge modes are localized at both ($1$ and $L$) boundaries of the junction. They do not connect lower and upper subbands of the superconductor and form localized standing waves. Arbitrary $\Delta$ {#arbitrary-delta .unnumbered} ------------------ The critical value $\tau_c$ depends on $\Delta$ and $\mu$. The minimal value of $\tau_c =\frac{1}{2}$ is reached in the $\Delta\rightarrow 0$ limit. The value of $\tau_c$ calculated for $Q=\pi$ as a function of $\mu$ and $\Delta$ is shown in Fig.\[fig:4\] b). We have plotted the width of the gap in the spectrum of Majorana bound states as a function of $Q$ for different values of $\tau$ (see in Fig\[fig:5\]). It follows from numerical calculations that this gap width is an even functions of $Q$, which can be approximated by $ \pm \tau^* \cos (Q/2)$, where, in the case of a weak tunneling the amplitude $\tau^*\sim \tau$ at $\tau <0.3$. Persistent current {#persistent-current .unnumbered} ================== The current along the boundaries is divided into chiral currents at the ends of the cylinder (red lines in Fig.\[fig:2\]b)) and chiral currents along the junction (blue lines). Chiral currents at the ends of the cylinder are described by the Hamiltonian (\[eq-H\]) with open boundary conditions and do not depend on the tunneling term (\[eq-Htun\]), whereas currents along the junction are described by the total Hamiltonian $\cal H$. The energy of the system $E^P(\tau,Q)=E_{cyl}^{p'} + E_{bulk}^p(\tau,Q)$ is determined by two terms: the energy of chiral edge modes at the ends of the cylinder with fermion parity $p'$ $E_{cyl}^{p'}$ (which does not dependent on $\tau$) and the energy of the superconductor, which takes into account the tunneling of fermions across the junction $E_{bulk}^p(\tau,Q)$, where $p,p'=f,h$ denote the fermion parity of the edge states: fermion (f) or hole (h), the symbol $P$ denotes the fermion parity of the ground state. In the strong tunneling regime, the edge modes, which occur at the junction, are represented by localized standing waves at all Q’s including the points $Q = \pm \pi$. Chiral currents at the ends of the cylinder and the current flowing along the junction, which is equal to zero, are not connected. Their fermion parities $p'$ and $p$ are not conserved. The fermion parities of the edge states are independent. In a contrast, in the weak tunneling regime, chiral currents flowing at the ends of the cylinder and along the junction are connected with each other due to the chiral current along the boundaries. Therefore, the fermion parities of Majorana-bound states located at the ends of the cylinder and at the junction are the same $p=p'$. Let us consider the behavior of the persistent current in the TSC in detail. In the limit $T\to 0$, the magnitude of the persistent current $I(\tau,Q)$ is determined by the ground-state energy of the system $I(\tau, Q)=\partial E^P(\tau, Q)/\partial Q$ (in unities of $2e/\hbar$), where $E^P(\tau,Q)=\int dk \sum_{\epsilon_n (k) <0} \epsilon_n(k)$ is determined by the quasi-particle excitations $\epsilon_n (k)$, and the Fermi energy is equal to zero at half-filling. In the strong coupling regime of tunneling $\tau>\tau_c$, the magnitude $I(\tau, Q)$ is a generic periodic function of the magnetic flux with the period of $2\pi$, so that $I(\tau,Q)=\partial E^f(\tau, Q)/\partial Q$ (see in Fig\[fig:6\] a)). In this case, the persistent current is determined by the energy of the superconductor, which takes into account both bulk excitations renormalized via the tunneling across the junction and the energies of edge modes at the junction. The fermion parity of edge states of Majorana fermions at the ends of the cylinder $p' = f$ and the Chern number, associated with these edge modes, are conserved. The fermion parity of edge Majorana fermions at the junction is not conserved. The system relaxes to the phase state with the minimum energy. In strong coupling tunneling regime $\tau>\tau_c$, $I(\tau,Q)$ is a typically periodic function of a magnetic flux with the period $2\pi$ (see in Fig\[fig:6\] a)). The persistent current is determined by the energy of the superconductor which takes into account the bulk excitations renormalized via the tunneling across the junction and the energies of edge modes at the junction. The fermion parity of edge states of Majorana fermions at the ends of the cylinder $p`=f$ and the Chern number, associated with these edge modes, are conserved. The fermion parity of edge Majorana fermions at the junction is not conserved, the system relaxes to the phase state with the minimum energy. In the weak tunneling regime $\tau<\tau_c$ all edge modes have the same chirality and fermion parity. This leads to a periodic persistent current having the period of $4\pi$.. The fermion parities of edge modes, which form the current along the boundaries, are identical. At $Q<\pi$ and $Q>\pi$ the ground state energy is determined by $E_{cyl}^{p'}+E_{bulk}^p(\tau,Q)$ with $p'=p=f$. At $Q>\pi$ the energy of the edge modes at the ends of the cylinder is negative, while the energy of the edge modes at the junction is positive. The balance of these energies determines the total energy of the system for given values of $\tau$ and $Q>\pi$. According to numerical calculations, a critical value of $Q_c$, at which energy difference of edge modes with different fermion parity changes its sign, is greater than $\pi$. The Chern number of the TSC is conserved, while the phase state of the system may not have the minimum energy at $Q>Q_c$. The resistive current is a periodic function of $Q$ with period $4\pi$, and $I(\tau,Q)$ is a continuous function of $Q$ within the whole interval $[-2\pi,2\pi]$ (see in Fig. \[fig:6\] b)). Conclusions {#conclusions .unnumbered} =========== This work is a step in our understanding of the behavior of a persistent current in topological systems. We have discussed the emergence of a persistent current in 2D TSC, pierced by a magnetic flux. It is proved that, the behavior of a persistent current is different in the case of strong and weak tunneling of Majorana fermions across a junction. The fermion parity of edge modes, forming a current along the boundaries of the sample, is the same, therefore the Chern number conserves a fermion parity of edge modes in the case of a weak interaction. Bulk edge correspondence leads to $4\pi$-periodic tunnel current. In a strong tunneling regime the currents at the ends of the cylinder and along the junction are not connected, therefore the fermion parities of the edge modes at the ends of the cylinder and at the junction are not conserved. At $Q=\pi$ in a strong tunneling regime spontaneous breaking of a fermion parity is realized. In the absence of fermion parity conservation the system relaxes to the minimum energy state, thus triggering a $2\pi$ periodic persistent current in TSC at strong tunneling of Majorana fermions across the junction. The results can be generalized to other topological phases, in particular, to topological insulators. Methods {#methods .unnumbered} ======= Edge modes in the 2D topological superconductor {#edge-modes-in-the-2d-topological-superconductor .unnumbered} ----------------------------------------------- Below we discuss the solution of the Schr$\ddot{o}$dinger equation for the chosen Hamiltonian ${\cal H}$ at the special point $\Delta=\pm 1$ using the formalism proposed for the calculation of Kitaev’s chain in Refs [@A1; @A2]. We focus on a 2D superconductor in the form of a square with the LxL size. Its sketch is shown in Figs \[fig:2\]. The wave function $\psi =\sum_{j=1}^L\sum_{s=1}^L[a^\dagger_{j,s}u_ {j, s}+a_{j,s}v_ {j,s}]$ is determined by amplitudes $u_ {j,s}$ and $v_ {j,s}$,that are solutions of the following equations: for $-L<j,s<L$ $$\begin{aligned} &&(E +\mu) u_ {j, s} = -\frac{1}{2}( u_ {j + 1, s} + u_ {j - 1, s} + u_ {j, s + 1} + u_ {j, s - 1}) + \frac{i}{2} (v_ {j + 1, s} - v_ {j - 1, s}) + \frac{1}{2} (v_ {j, s + 1} - v_ {j, s - 1}),\nonumber\\ && (E -\mu) v_ {j, s} = \frac{1}{2}( v_ {j + 1, s} + v_ {j - 1, s} + v_ {j, s + 1} + v_ {j, s - 1}) + \frac{i}{2} (u_ {j + 1, s} - u_ {j - 1, s}) - \frac{1}{2} (u_ {j, s + 1} - u_ {j, s - 1}), \label{A1}\end{aligned}$$ for $s=1,L$, where $1<j<L$ $$\begin{aligned} &&(E + \mu) u_ {j, 1} = -\frac{1}{2} (u_ {j,2} + u_ {j+1,1}+u_ {j-1,1} - v_ {j,2}) + \frac{i}{2} (v_ {j+1, 1} - v_ {j-1, 1}) - \tau e^{-i\frac{Q}{2}} u_ {j,L},\nonumber\\ &&(E - \mu) v_ {j, 1} =\frac{1}{2}( v_ {j,2} + v_ {j+1,1} +v_ {j-1,1} -u_ {j,2}) + \frac{i}{2} (u_ {j+1,1} - u_ {j-1,1}) + \tau e^{i\frac{Q}{2}} v_ {j,L},\nonumber\\ && (E +\mu) u_ {j,L} = -\frac{1}{2}(u_ {j,L - 1} +u_ {j+1,L}+u_ {j-1,L}+ v_ {j,L - 1}) + \frac{i}{2} (v_ {j+1,L} - v_ {j-1,L})- \tau e^{i\frac{Q}{2}}u_{j,1}, \nonumber\\ &&(E - \mu) v_ {j,L} = \frac{1}{2} (v_ {j,L - 1} + v_ {j+1,L} +v_ {j-1,L}+ u_ {j,L - 1}) +\frac{i}{2} (u_ {j+1,L} - u_ {j-1,L})+ \tau e^{-i\frac{Q}{2}}v_{j,1}, \label{A2}\end{aligned}$$ for $j=1,L$, where $1<s<L$ $$\begin{aligned} (E + \mu) u_ {1, s} = -\frac{1}{2} (u_ {2,s} + u_ {1,s+1}+u_ {1,s-1}- v_ {1,s+1}+v_{1,s-1}) + \frac{i}{2} v_ {2,s},\nonumber\\ (E - \mu) v_ {1,s} =\frac{1}{2}( v_ {2,s} + v_ {1,s+1} +v_ {1,s-1} -u_ {1,s+1}+u_{1,s-1})+\frac{i}{2} u_ {2,s},\nonumber\\ (E +\mu) u_ {L,s} = -\frac{1}{2}(u_ {L-1,s} +u_ {L,s+1}+u_ {L,s-1} -v_ {L,s+1}+v_ {L,s-1})- \frac{i}{2}v_ {L-1,s}\nonumber, \\ (E - \mu) v_ {L,s} = \frac{1}{2} (v_ {L - 1,s} + v_ {L,s+1} +v_ {L,s-1}- u_ {L,s+1}+ u_ {L,s-1}) - \frac{i}{2} u_ {L-1,s}, \label{A3}\end{aligned}$$ and the similar equations for the vertices of the square $\{1,1\};\{1,L\};\{L,1\};\{L,L\}$. The solutions of Eqs (\[A1\]) also satisfy Eqs (\[A2\])-(\[A3\]) at $\tau =0$ and the following boundary conditions $v_ {j,0} + u_ {j,0} = 0$, $v_ {j,L + 1} - u_ {j,L + 1} = 0$ and $u_ {0,s} + i v_ {0,s} = 0$, $u_ {L + 1,s} - iv_ {L + 1,s} = 0$. We determine the amplitudes of the wave function in accordance with the following Ansatz $$\begin{gathered} u_{j,s}=A_u(k_x,k_y)e^{ik_x j+i k_y s}+B_u(k_x,k_y)e^{i k_x j-i k_y s}+C_u(k_x,k_y)e^{-i k_x j +i k_y s}+D_u(k_x,k_y) e^{-i k_x j -i k_y s},\\ v_{j,s}=A_v(k_x,k_y)e^{i k_x j+i k_y s}+B_v(k_x,k_y)e^{i k_x j-i k_y s}+C_v(k_x,k_y)e^{-i k_x j +i k_y s}+D_v(k_x,k_y) e^{-i k_x j -i k_y s}. \label{A5}\end{gathered}$$ Unknown amplitudes in (\[A5\]) are defined as $A_u(k_x,k_y)=G_u(k_x,k_y)$, $B_u(k_x,k_y)=G_u(k_x,k_y)e^{i(-\chi+ \alpha)}$, $C_u=G_u(k_x,k_y)e^{i(- \chi+ \beta)}$, $D_u(k_x,k_y)=G_u(k_x,k_y)e^{i\gamma}$, $A_v(k_x,k_y)=G_v(k_x,k_y)$, $B_v(k_x,k_y)=-G_v(k_x,k_y)e^{i( \chi+ \alpha)}$, $D_v=G_v(k_x,k_y)e^{i(\chi + \beta)}$, $D_v(k_x,k_y)=-G_v(k_x,k_y)e^{i \gamma}$, where $e^{2 i \chi} =\frac{i\sin k_y -\sin k_x}{i\sin k_y +\sin k_x}$, the energies of the eigenstates are determined by Eq (\[eq-3\]) at $\Delta=1$ and the constants $\alpha,\beta,\gamma$ are determined by the boundary conditions. We redefine the unknown $G_u(k_x,k_y)$ and $G_v (k_x,k_y)$ as $G_u(k_x,k_y) =G \cos\varphi/2$ and $G_v(k_x,k_y) = iG \sin\varphi/2$, where $\tan \varphi =\frac{ \sin k_y - i \sin k_x}{\mu +\cos k_y + \cos k_x}$, $G$ is a normalization constant. Let us consider the points $k_y=0$ and $k_y=\pi$ at $\tau =0$ that correspond to zero energy of Majorana modes localized at the boundaries (see Figs \[fig:1\] for the illustration). The solutions for particle-hole excitations localized at the boundary are determined by complex $k_y$-wave vectors $k_y=\pm i \varepsilon $ or $k_y=\pi \pm i \epsilon $ with $$\begin{aligned} \varepsilon =2\sinh^{-1}\left(\frac{1}{2}\sqrt{\frac{ \mu^2+4(1+\mu)\cos^2(k_x/2)-E^2}{-\cos k_x- \mu}}\right),\nonumber\\ - \cos k_x >\mu \nonumber\\ \varepsilon =2\sinh^{-1}\left(\frac{1}{2}\sqrt{\frac{ \mu^2+4(1-\mu)\sin^2(k_x/2)-E^2}{\cos k_x + \mu}}\right),\nonumber\\ \cos k_x >-\mu \label{A6}\end{aligned}$$ $\mu$ defines the bulk gap. Solution (\[A6\]) determines the momentum of an excitation at a given energy, we can invert Eq (\[eq-3\]) yielding the momentum with energy E. At $k_y=0,\pi$ or $k_x=0,\pi$ $\chi=\frac{\pi}{2}$ or $\chi=0$, therefore the boundary conditions are reduced to the following equations $\sin[k_y(L+1)- \varphi] = 0$, $ \sin[k_x(L+1)]=0$. Similar to the 1D model [@A1], the energy of level localized at a boundary is equal to zero at $k_y= i \varepsilon , k_x=\pi$, in the $L\to \infty$ limit $E\sim (-1+\cosh\varepsilon -\mu)\exp(-2\varepsilon L)$. Complex solution for $k_y$ describes the edge modes localized at the $x$-boundary with $k_x$-dispersion. The boundary conditions describe free fermion states with the wave vector directed along the boundary. The solution of Eqs (\[A1\]) are a x-y symmetric. Let us consider the edge modes localized at the $x$-boundary with $k_y=i\varepsilon$ which have zero energy $E\to 0$ at the Dirac point $k_x=\pi$. The solution $E=0$ corresponds to the degenerate solution of Eqs (\[A1\]) for the amplitudes of the wave function $u(k_x,i\varepsilon)=v(k_x,i\varepsilon)$. This solution is valid for arbitrary $k_x$ at $E^\ast\rightarrow E-\sin k_x=0$. We do not use the boundary conditions for calculation of the wave function, as a result, the dispersion of edge modes is determined for an arbitrary value of $\Delta$. We find that the dispersion relation for the energy of the edge modes reads: $E_{edge}(k_x)=\pm \Delta \sin k_x$. The numerical calculations of the spectrum of the edge modes, obtained for arbitrary $\mu$ and $\Delta$, confirm the dispersion (see in Fig.\[fig:1\]b) for example). The energy of the edge modes at the $x-$ and $y-$boundaries have the intebtical dispersion for the wave vector directed along the boundary, that triggers a chiral current along the boundaries of the sample. As we already noted above, points $Q=\pm \pi$ are the special since the gapless edge modes are realized at them for $\tau \neq 0$. The zero-energy solutions for the edge modes at the Dirac point follow from the solutions of Eqs (\[A1\])-(\[A3\]) in the $L\to \infty$ limit at $\tau \neq 0$. Using appropriate boundary conditions we calculate a low energy dispersion of gapless edge modes at $Q=\pm \pi$. The energies of the edge modes propagating along the junction have the following form $$E_{edge}(\delta k_x)=\pm \frac{1}{3}\sin \delta k_x \mp\frac{2}{3} \sqrt{\sin^2 \delta k_x +3 w(0)} \cos (\zeta - 2 \pi/3) \label{A7}$$ where $k_x=\pi+\delta k_x$, $w(Q)=(\mu+1-\cos \delta k_x)^2+\sin^2 \delta k_x + \tau^2 \cos Q$, $\zeta =\frac{1}{3}\arccos\left(\frac{27}{54}\frac{-2 \sin^3 \delta k_x - 9 w(0) \sin \delta k_x + 27 w(Q)}{ (\sin^2 \delta k_x +3 w(0))^{3/2}}\right)$. Equation (\[A7\]) is derived from the low energy solution of the following equation $ E^3 - E^2 \sin \delta k_x- w(0) E +\sin \delta k_x w(Q)=0$. We consider the zero $\delta k_x$ limit of Eq (\[A7\]) and obtain the linear dispersion of edge modes at the Dirac point $E_{edge}(\delta k_x)=\pm v_{edge} \delta k_x$, where $v_{edge}=1-\frac{2\tau^2 }{\mu^2+\tau^2 }$. The linear dispersion of the edge modes vanishes at $\tau=1$. The solutions (\[A5\]) do not satisfy the boundary conditions at $\tau=1$ and, as a result, solution (\[A7\]) does not hold. According to numerical calculations the edge modes have a parabolic dispersion (see in Fig. \[fig:3\]c)). [31]{} Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M.P.A. [Non-Abelian Statistics and Topological Quantum Information Processing in 1D Wire Networks.]{} *Nature Physics*, **7**, 412-417 (2011). Nogueira, F.S. & Eremin, I. [Strong-coupling topological Josephson effect in quantum wires.]{} *J.Phys.:Condens. Matter*, **24**, 325701 (2012). Kitaev, A.Yu. [Unpaired Majorana fermions in quantum wires]{} *Phys. Usp.* **44,** 131-136 (2001). Fidkowski, L., Alicea, J.,, Lindner, N.H., Lutchyn, R.M. & Fisher, M.P.A. [Universal transport signatures of Majorana fermions in superconductor-Luttinger liquid junctions.]{} *Phys. Rev. B*, **85**, 245121 (2012). Affleck, I. & Giuliano, D., [Topological superconductor-Luttinger liquid junctions.]{} *Journal of Statistical Mechanics: Theory and Experiment*, P06011 (2013). Fischer, M.H. *et al.* [Chiral d-wave superconductivity in SrPtAs.]{} *Phys.Rev.B*, **89**, 020509(R) (2014). Wang, Y., Cho, G.Y., Hughes, T.L. & Fradkin, E. [Topological superconducting phases from inversion symmetry breaking order in spin-orbit-coupled systems.]{} *Phys.Rev.B*, **93**, 134512 (2016). Asahi, D. & Nagaosa, N. [Topological indices, defects, and Majorana fermions in chiral superconductors.]{} *Phys.Rev.B*, **86**, 100504R (2012). Diez, M., Fulga, I.C., Pikulin, D.I., Tworzyd, J. & Beenakker, C.W.J. [Bimodal conductance distribution of Kitaev edge modes in topological superconductors.]{} *New J.Phys.*, **16**, 63049 (2014). Read, N. & Green, D. [Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect.]{} *Phys. Rev. B* **61**, 10267 (2000). Nava, A., Giuliano, R., Campagnano,G. & Giuliano, D. [Persistent current and zero-energy Majorana modes in a p-wave disordered superconducting ring.]{} arXiv:1612.03740v1. Beenakker, C.W.J. [Random-matrix theory of Majorana fermions and topological superconductors.]{} *Rev. Mod. Phys.* **87**, 1037 (2015). Sau J.D. & Das Sarma, S. [Density of states of disordered topological superconductor-semiconductor hybrid nanowires.]{} *Phys. Rev. B* **88**, 064506 (2013). Brouwer, P.W., Duckheim,M., Romito, A., & von Oppen, F. [Probability Distribution of Majorana End-State Energies in Disordered Wires.]{} *Phys. Rev. Lett.* **107**, 196804 (2011). Pientka, F., Kells, G., Romito, A., Brouwer, P.W. & von Oppen, F. [Enhanced Zero-Bias Majorana Peak in the Differential Tunneling Conductance of Disordered Multisubband Quantum-Wire/Superconductor Junctions.]{} *Phys. Rev. Lett.* **109**, 227006 (2012). Gergs, M., Fritz, L. & Schuricht, D. [Topological order in the Kitaev/Majorana chain in the presence of disorder and interactions.]{} *Phys. Rev. B* **93**, 075129 (2016). Karnaukhov, I.N. *Spontaneous breaking of time-reversal symmetry in topological superconductors*, *Scientific Reports* **7**, 7008 (2017). Karnaukhov, I.N. *Spontaneous breaking of time-reversal symmetry in topological insulators*, *Phys. Lett. A* **381**, 1967 (2017). Hasan, M.Z. & Kane, C.L. [Colloquium: Topological insulators.]{} *Rev. Mod. Phys.* **82**, 3045 (2010). Qi, X.-L. & Zhang, S.-C. [Topological insulators and superconductors.]{} *Rev. Mod. Phys.* **83**, 1057 (2011). Ando, Y. & Fu, L. [Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials.]{} *Ann. Rev. Condens. Matter Phys.* **6**, 361 (2015). Alicea, J. [New directions in the pursuit of Majorana fermions in solid state systems.]{} *Rep. Prog. Phys.* **75**, 076501 (2012). Ran, Y., Hosur, P. & Vishwanath, A. [Fermionic Hopf solitons and Berry phase in topological surface superconductors]{} *Phys. Rev. B* **84**, 184501 (2011). Nava, A., Giuliano, R., Campagnano,G. & Giuliano, D. [Transfer matrix approach to the persistent current in quantum rings: Application to hybrid normal-superconducting rings.]{} *Phys. Rev. B* **94**, 205125 (2016). Author contributions statement {#author-contributions-statement .unnumbered} ============================== I.K. is an author of the manuscript Additional information {#additional-information .unnumbered} ====================== The author declares no competing financial interests.

ArXiv

--- author: - bibliography: - 'biblio.bib' title: | [**Pulsatile lipid vesicles under osmotic stress**]{}\ \ --- [ABSTRACT ]{} The response of lipid bilayers to osmotic stress is an important part of cellular function. Recent experimental studies showed that when cell-sized giant unilamellar vesicles (GUVs) are exposed to hypotonic media, they respond to the osmotic assault by undergoing a cyclical sequence of swelling and bursting events, coupled to the membrane’s compositional degrees of freedom. Here, we establish a fundamental and quantitative understanding of the essential pulsatile behavior of GUVs under hypotonic conditions by advancing a comprehensive theoretical model of vesicle dynamics. The model quantitatively captures the experimentally measured swell-burst parameters for single-component GUVs, and reveals that thermal fluctuations enable rate-dependent pore nucleation, driving the dynamics of the swell-burst cycles. We further extract constitutional scaling relationships between the pulsatile dynamics and GUV properties over multiple time scales. Our findings provide a fundamental framework that has the potential to guide future investigations on the non-equilibrium dynamics of vesicles under osmotic stress. INTRODUCTION {#introduction .unnumbered} ============ In their constant struggle with the environment, living cells of contemporary organisms employ a variety of highly sophisticated molecular mechanisms to deal with sudden changes in their surroundings. One often encountered environmental assault on cells is osmotic stress, where the amount of dissolved molecules in the extracellular environment drops suddenly [@christensen1987; @hoffmann2009]. If left unchecked, this perturbation will result in a rapid flow of water into the cell through osmosis, causing it to swell, rupture, and die. To avoid this catastrophic outcome, even bacteria have evolved complex molecular machineries, such as mechanosensitive channel proteins, which allow them to release excess water from their interior [@berrier1996; @blount1997; @levina1999; @wood1999]. This then raises an intriguing question of how might primitive cells, or cell-like artificial constructs, that lack the sophisticated protein machinery for osmosensing and osmoregulation, respond to such environmental insults and preserve their structural integrity. Using rudimentary cell-sized giant unilamellar vesicles (GUVs) devoid of proteins and consisting of amphiphilic lipids and cholesterol as models for simple protocells, we showed previously that vesicular compartments respond to osmotic assault created by the exposure to hypotonic media by undergoing a cyclical sequence of swelling and poration [@oglecka2014]. In each cycle, osmotic influx of water through the semi-permeable boundary swells the vesicles and renders the bounding membrane tense, which in turn, opens a microscopic transient pore, releasing some of the internal solutes before resealing. This swell-burst process, [depicted in Fig. \[fig1\](A), ]{}repeats multiple times producing a pulsating pattern in the size of the vesicle undergoing osmotic relaxation. From a dynamical point of view, this autonomous osmotic response results from an initial, far-from-equilibrium, thermodynamically unstable state generated by the sudden application of osmotic stress. The subsequent evolution of the system, characterized by the swell-burst sequences described above, occurs in the presence of a global constraint, namely constant membrane area, during a dissipation-dominated process [@peterlin2008; @ho2016]. The study of osmotic response of lipid vesicles has a rich history in theoretical biophysics, beginning with the pioneering work by Koslov & Markin [@koslov1984], who provided some of the early theoretical foundations of osmotic swelling of lipid vesicles. In this work, they predicted that the response of a sub-micrometer sized vesicles to osmotic stress is likely pulsatile and due to the formation of successive transient pores (see Fig. 9, in [@koslov1984], for a schematic for the volume change of the vesicle over time). They further approximated the characteristic quantities of swell-burst cycles (e.g. swelling time, critical volumes), based on the probability of the membrane overcoming the nucleation energy barrier to form a pore. Independently, the dynamics of a single transient pore in a tense membrane were first theorized by Litster [@litster1975], and later investigated theoretically and experimentally by Brochard-Wyart and coworkers [@sandre1999; @brochard-wyart2000]. Idiart & Levin [@idiart2004] combined the osmotic swelling theory and pore dynamics, and calculated the dynamics of a pulsatile behavior assuming a constant lytic tension. These modeling efforts made great strides in our understanding of some of the essential physics underlying vesicle responses to osmotic stress. Previously, we used these ideas to provide a qualitative interpretation of pulsatile behavior of GUVs (see schematics in [@oglecka2014] Fig. 7h,i). However a general framework that quantitatively describes the response of pulsatile vesicles to osmotic stress at all relevant time scales is still missing. The success of such a model must rely on (a) the integration of vesicle dynamics, pore dynamics with nucleation, and long-time solute concentration dynamics within a unified framework, and (b) the assessment of the model predictions with respect to experimental measurements, in order to establish the physical relevance of the essential parameters that govern the system dynamics. Here, we build on the findings and theories reported previously [@litster1975; @koslov1984; @brochard-wyart2000; @idiart2004; @evans2003; @ryham2011] to develop such a quantitative model for the dynamics of swell-burst cycles in giant lipid vesicles subject to osmotic stress. In analyzing the pulsatile dynamics of GUVs, a number of general questions naturally arise: (i) Is the observed condition for membrane poration deterministic or stochastic? (ii) Is poration controlled by a unique value of membrane tension (*i.e.* lytic tension) introduced by the area-volume changes, which occur during osmotic influx, or does it involve coupling of the membrane response to thermal fluctuations? (iii) Does the critical lytic tension depend on the strain rate, and thus the strength of the osmotic gradient? Such questions arise beyond the present context of vesicle osmoregulation in other important scenarios where the coupling between the dissipation of osmotic energy and cellular compartmentalization has important biological ramifications [@rand2004; @diz-munoz2013; @stroka2014; @porta2015]. Motivated by these considerations, we carried out a combined theoretical-experimental study integrating membrane elasticity, continuum transport, and statistical thermodynamics. We gathered quantitative experimental data to address the questions above, and developed a general model that recapitulates the essential qualitative features of the experimental observations, emphasizes the importance of dynamics, and places the heretofore neglected contribution of thermal fluctuations in driving osmotic response of stressed vesicular compartments. MATERIALS AND METHODS {#materials-and-methods .unnumbered} ===================== The detailed materials and methods used in this work are available in Supporting Materials and Methods in the Supporting Material. The experimental configuration is similar to that already described [@angelova1992; @oglecka2014]. Briefly, we prepared GUVs consisting essentially of a single amphiphile, namely 1-palmitoyl-2-oleoyl-sn-1-glycero-3-phosphocholine (POPC), doped with a small concentration (1 mol$\%$) of a fluorescently labeled phospholipid (1,2-dipalmitoyl -sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl)) or Rho-DPPE using standard electroformation technique [@angelova1992]. The GUVs thus obtained were typically between 7 and 20 $\mu$m in radius, encapsulated 200 mM sucrose, and were suspended in the isotonic glucose solution of identical osmolarity. Diluting the extra-vesicular dispersion medium with deionized water produces a hypotonic bath depleted in osmolytes, subjecting the GUVs to osmotic stress. Shortly ($\sim$1 min delay) after subjecting the GUVs to the osmotic differential, GUVs were monitored using time-lapse epifluorescence microscopy at a rate of 1 image per 150 ms, and images were analyzed using a customized MATLAB code to extract the evolution of the GUV radii with time, with a precision of about 0.1 $\mu$m. We developed a mathematical model predicting the pulsatile behavior of GUVs in hypotonic environment. Essentially, the model couples pore nucleation by thermal fluctuations, osmotic swelling, and solute transport. These aspects are represented by Eqs. \[eq:sde\_r\], \[eq:ode\_R\] and \[eq:ode\_c\] respectively, and discussed below. Details regarding the theory and its numerical implementation are reported in Model Development and Simulations in the Supporting Materials. RESULTS {#results .unnumbered} ======= Homogeneous GUVs display swell-burst cycles in hypotonic conditions {#homogeneous-guvs-display-swell-burst-cycles-in-hypotonic-conditions .unnumbered} -------------------------------------------------------------------  A selection of snapshots, revealing different morphological states, and a detailed trace showing [the time-dependence of the vesicle radius $R$ and corresponding area strain ($\epsilon_\text{exp}=(R^2-R_0^2)/R_0^2$, where and $R_0$ is the resting initial vesicle radius) are shown]{} in Fig. \[fig1\](B, C, and D), for a representative GUV. Swelling phases are characterized by a quasi-linear increase of the GUV radius, while pore openings cause a sudden decrease of the vesicle radius. We outline here three key observations about the dynamics of swell-burst cycles from these experiments. 1. The period between two consecutive bursting events increases with each cycle, starting from a few tenths of a second for the early cycles, to several hundreds of seconds after the tenth cycle. 2. The maximum radius and therefore the maximum strain at which a pore opens decreases with cycle number, suggesting that lytic tension is a dynamic property of the membrane. 3. The observed transient pores are short lived, stay open for about a hundred milliseconds, and reach a maximum radius of up to 60 $\%$ of the GUV radius. We seek to explain these observations through a quantitative understanding of the pulsatile GUVs in hypotonic conditions. To do so, we first investigate the mechanics of pore nucleation and its relationship to the GUV swell-burst dynamics. Thermal fluctuations drive the dynamics of pore nucleation {#thermal-fluctuations-drive-the-dynamics-of-pore-nucleation .unnumbered} ----------------------------------------------------------  [ In the framework of classical nucleation theory [@litster1975], the energy potential $V(r,\epsilon)$ of a pore of radius $r$ in a lipid membrane under surface tension $\sigma$, is the balance of two competitive terms: $V_s(\epsilon)$, the strain energy, and $V_p(r)$, the pore energy. The strain energy tends to favor the opening and enlargement of the pore while the pore closure is driven by the pore line tension $\gamma$. Accordingly, the energy potential reads]{} $$\begin{aligned} V(r,\epsilon)=&V_s(\epsilon) + V_p(r) \nonumber\\ =&\dfrac{1}{2}\kappa_\text{eff} A_0 \epsilon^2 + 2\pi r \gamma \;. \end{aligned}$$ [The area strain is defined as $\epsilon=(A-A_0)/A_0$, where $A=4\pi R^2-\pi r^2$ is the surface of the membrane, and $A_0=4\pi R_0^2$ is the resting vesicle area. Here $V_s(\epsilon)$ is assumed to have a Hookean form, where $\kappa_\text{eff}$ is the effective stretching modulus, which relates the surface tension to the strain as $\sigma=\kappa_\text{eff}\epsilon$ (see next section for a discussion on $\kappa_\text{eff}$). These two energetic terms oppose each other, resulting in an energy barrier that the system has to overcome in order for a pore to nucleate. The competition between the strain and pore energy is expressed by the ratio $r_b=\gamma/\sigma$, which is the critical radius associated with the crossing of the energy barrier. That is, if a pore in a tensed membrane has a radius $r<r_b$, the pore energy $V_p(r)$ dominates and the pore closes. On the contrary, for $r>r_b$, the strain energy $V_s(\epsilon)$ prevails and the pore grows. The energy required to open a pore of radius $r$ in a tensed GUV is given by $\Delta V(r,\epsilon) = V(r,\epsilon)-V(0,\epsilon)$ and is represented in Fig. \[fig2\](A). The corresponding critical radius of the the energy barrier $r_b$ is shown as a function of the strain $\epsilon$ in Fig. \[fig2\](B). The height of the energy barrier and its critical radius are dependent on the membrane strain; the more the membrane is stretched, the lower the energy barrier is, and the smaller the amount of energy required to nucleate a pore. ]{} The amplitude of this energy barrier is strictly positive for finite strain values, making pore nucleation impossible without the addition of external energy. This issue has been often resolved by assuming a predetermined and *constant* lytic [strain ($\epsilon^*$)]{} corresponding to a critical energy barrier under which the pore opens (Fig. \[fig2\](C and E)). However, this approach is in contradiction with our experimental observations that the lytic strain in the membrane varies with each swell-burst cycle (Fig. \[fig1\](D)), due to a dependence on the strain rate [@evans2003]. In order to account for this variation, we included thermal fluctuations associated with the pore nucleation barrier in our analysis [@ting2011; @bicout2012]. In this scenario, increasing the membrane tension of the vesicle reduces the minimum pore radius $r_b$ at which a pore opens (Fig. \[fig2\](A and B)), lowering the energy barrier down to the range of thermal fluctuations, eventually letting the free energy of the system to overcome the nucleation barrier (Fig. \[fig2\](D and F)). The stochastic nature of the fluctuations can then explain a distribution of pore opening tensions, eliminating the need to assume constant lytic tension. A direct consequence of the fluctuation-mediated pore nucleation is that the membrane rupture properties become dynamic. Indeed, fluctuations naturally cause the strain at which the membrane ruptures to be dependent on the *strain rate*, as illustrated in Fig. \[fig2\](D). In order to understand this dynamic nucleation process, consider stretching the membrane at different strain rates $\dot{\epsilon}$. Doing so decreases the radius of the nucleation barrier at corresponding speeds, as shown in Fig. \[fig2\](F). For slow strain rates, as $r_b$ tends to zero, it spends more time in the accessible range of the thermal pore fluctuations, increasing the probability that a fluctuation will overcome the energy barrier. On the other hand, at faster strain rates, $r_b$ decreases quickly, reaching small values in less time, lowering the probability for above average fluctuations to occur during this shorter time. We use a Langevin equation to capture the stochastic nature of pore nucleation and the subsequent pore dynamics. This equation includes membrane viscous dissipation, a conservative force arising from the membrane potential, friction with water, and thermal fluctuations for pore nucleation (see Model Development and Simulations in the Supporting Material for detailed derivation). This yields the stochastic differential equation for the pore radius $r$ $$\label{eq:sde_r} \overbrace{\left(h \eta_m + C \eta_s r\right)}^{\mathclap{\text{viscous drag}}} \underbrace{\frac{d}{dt}(2\pi r)}_{\mathclap{\text{change of pore radius}}} = \overbrace{2\pi \left(\sigma r-\gamma\right)}^{\mathclap{\substack{\text{surface and}\\\text{line tension}}}} + \underbrace{\xi(t)}_{\mathclap{\substack{\text{thermal}\\\text{pore fluctuations}}}} \;,$$ where the noise source $\xi(t)$ has zero mean and satisfies, $\langle \xi(t)\xi(t^\prime) \rangle = 2\left(h \eta_m + C \eta_s r\right) k_B T \delta(t-t^\prime)$ according to the fluctuation dissipation theorem [@kubo1966]. Here, $\eta_m$ and $\eta_s$ are the membrane and solute viscosities respectively, $h$ is the membrane thickness, $C$ is a geometric coefficient [@ryham2011; @aubin2016], $k_B$ is the Boltzmann constant and $T$ is the temperature. [We assume here that the pore nucleation probability is independent on the total membrane surface area.]{} The values of the different parameters used in the model are given in Table S\[tab\_parameters\] in the Supporting Material. Model captures experimentally observed pulsatile GUV behavior {#model-captures-experimentally-observed-pulsatile-guv-behavior .unnumbered} ------------------------------------------------------------- ![Dynamics of swell-burst cycles from the model for a GUV of radius 14 $\mu$m in 200 mM hypotonic stress. (A and C) GUV radius and (B and D) pore radius as a function of time. The model captures the dynamics of multiple swell-burst cycles, in particular the decrease of maximum GUV radius and increase of cycle period with cycle number (A). Looking closely at a single pore opening event corresponding to the grey region, the model predicts three stage pore dynamics (C and D), namely opening, closing, and resealing, with a characteristic time of a few hundred milliseconds. Numerical reconstruction of the GUV is shown in Movies S3 and S4 in the Supporting Material. Results for $R_0=8$ and 20 $\mu$m are shown in Fig. S\[figS\_R\] in the Supporting Material.[]{data-label="fig3"}](fig_cycles_num) In addition to pore dynamics (Eq. \[eq:sde\_r\]), we need to consider mass conservation of the solute and the solvent. We assume that the GUV remains spherical at all times and neglect spatial effects. The GUV volume changes because of osmotic influx through the semi-permeable membrane and the leak-out of the solvent through the pore. The osmotic influx is the result of two competitive pressures, the osmotic pressure driven by the solute differential ($\Delta p_{osm} = k_B T N_A \Delta c$), and the Laplace pressure, arising from the membrane tension ($\Delta p_L = 2\sigma / R$), resulting in the following equation for the GUV radius $R$: $$\label{eq:ode_R} \underbrace{\frac{d}{dt}\left( \frac{4}{3}\pi R^3 \right)}_{\mathclap{\text{change of GUV volume}}} = \overbrace{\frac{P \nu_s }{k_B T N_A} \left( \Delta p_{osm} - \Delta p_L \right) A}^{\substack{\text{influx of solvent}\\\text{ through the membrane} }} - \underbrace{ v_L \pi r^2}_{\mathclap{\substack{\text{leak-out}\\\text{through the pore}}}} \;.$$ Here $A=4\pi R^2$ is the membrane area, $P$ is the membrane permeability to the solvent, $\nu_s$ is the solvent molar volume, and $N_A$ is the Avogadro number. Assuming low Reynolds number regime, the leak-out velocity is given by $ v_L = \Delta p_L r / (3\pi\eta_s)$ [@happel1983; @aubin2016]. Mass conservation of solute in the GUV is governed by the diffusion of sucrose and convection of the solution through the pore, which gives the governing equation for the solute concentration differential $\Delta c$: $$\label{eq:ode_c} \underbrace{\frac{d}{dt}\left(\frac{4}{3}\pi R^3 \Delta c \right)}_{\mathclap{\substack{\text{molar differential}\\\text{of solute}}}} = -\pi r^2 \bigg( \overbrace{D \frac{\Delta c}{R}}^{\mathclap{\substack{\text{diffusion through}\\\text{the pore}}}} + \underbrace{v_L\Delta c}_{\mathclap{\substack{\text{convection}\\\text{through the pore}}}} \bigg) \;,$$ where $D$ is the solute diffusion coefficient. These three coupled equations (Eqs. \[eq:sde\_r\] to \[eq:ode\_c\]) constitute the mathematical model. In order to completely define the system, we need to specify the relationship between the membrane surface tension $\sigma$ and the area strain of the GUV. We note that the GUV has irregular contours during the pore opening event and for a short time afterwards, when “nodules" are observed at the opposite end from the pore, indicating accumulation of excess membrane generated by pore formation (Fig. \[fig1\](B) middle and right panels). In the low tension regime, GUVs swell by unfolding these membrane nodules, and the stretching is controlled by the membrane bending modulus $\kappa_b$ and thermal energy, yielding an effective “unfolding modulus" $\kappa_u = 48\pi\kappa_b^2 / (R_0^2k_BT)$ of the order of 10$^{-5}$ N/m [@brochard1976]. In contrast, in the high tension regime, elastic stretching is dominant, and the elastic area expansion modulus $\kappa_e$ is roughly equal to 0.2 N/m [@evans1990]. Since the maximum area strain plotted in Fig. \[fig1\](D), is about 15 $\%$, significantly larger than the expected 4 $\%$ for a purely elastic membrane deformation, the experimental data suggests the occurrence of two stretching regimes: an unfolding driven stretching, and an elasticity driven stretching [@ertel1993; @hallett1993; @karatekin2003a]. [Therefore, for simplicity, we assume an effective stretching modulus $\kappa_\text{eff}$, which takes into account both unfoldoing and elastic regimes [@evans1990; @bloom1991] through a linear dependence between the membrane tension and the strain ($\sigma = \kappa_\text{eff} \epsilon$). Note that $\kappa_\text{eff}$ is the only adjustable parameter of the model.]{} We solved the three coupled equations (Eqs. \[eq:sde\_r\] to \[eq:ode\_c\]) for an initial inner solute concentration of $c_0=200$ mM, and different GUV radii of $R_0=8$, 14 and 20 $\mu$m. All the results presented here are obtained for $\kappa_\text{eff}=2 \times10^{-3}$N/m, the value that best fits the experimental observations (see Supplemental Fig. \[figS\_kappa\] for the effect of this parameter on the GUV dynamics). Dynamics of the GUV radius and the pore radius are shown in Fig. \[fig3\] for a typical simulation with $R_0=14$ $\mu$m (see Supplemental Fig. \[figS\_R\] for simulations with different values of $R_0$). Our model qualitatively reproduces the dynamics of the GUV radius during the swell burst cycle (compare Figs. \[fig1\](C) and \[fig3\](A)). Importantly, we recover the key features of the swell-burst cycle – namely an increase of the cycle period with each bursting event (point 1), and a decrease of the maximum radius with time (point 2). The stochastic nature of the thermodynamic fluctuations leads to variations and irregularities in the pore opening events, and therefore, the cycle period and maximum strain. The dynamics of a single cycle is shown in Fig. \[fig3\](C and D). Our numerical results show an abrupt drop in the GUV radius, followed by a slower decrease, suggesting a sequence of two leak-out regimes: a fast-burst releasing most of the membrane tension, and a low tension leak-out. This two-step tension release is confirmed by the pore radius dynamics, which after suddenly opening (release of membrane tension), reseals quasi-linearly due to dominance of line tension compared to membrane tension in Eq. \[eq:sde\_r\]. Furthermore, the computed pore amplitude and lifetime are in agreement with experimental observations (point 3). Overall, our model is able to reproduce the quantitative features of GUV response to hypotonic stress over multiple time scales. If thermal fluctuations are ignored, the strain to rupture needs to be adjusted to roughly 15$\%$ in order to match the range of maximum GUV radius observed experimentally (Fig. S\[figS\_det\] in the Supporting Material). However such a deterministic model does not capture the pulsatile dynamics as well as the stochastic model in terms of cycle period and strain rate (Fig. S\[figS\_comparison\] in the Supporting Material), and fails to reproduce a strain rate dependent maximum stress (Fig. S\[figS\_det\]). Solute diffusion is dominant during the low tension regime of pore resealing {#solute-diffusion-is-dominant-during-the-low-tension-regime-of-pore-resealing .unnumbered} ---------------------------------------------------------------------------- ![[Diffusion of sucrose through the transient pore produces a step-wise decrease of the inner solute concentration. (A) In hypotonic conditions, the model predicts a step-wise decrease of solute concentration differential with time (blue line), which is solely due to diffusion of solute through the transient pore. In comparison, when diffusion is neglected in the model (grey line), the solute concentration differential decreases smoothly (also see Fig. S\[figS\_D\] in the Supporting Material for further analysis on the effect of diffusion). In isotonic conditions (dashed line), the solute concentration differential is constant with time. (Here $t_0=$ 40 s).]{} (B) Time evolution of the normalized fluorescence intensity of a GUV in hypotonic condition, encapsulating fluorescent glucose analog. $\Delta I$ is the difference in mean intensity between the inside of the GUV and the background. In hypotonic conditions (solid lines) the normalized intensity decreases with time due to the constant influx of water through the membrane, and shows sudden drops in intensity at each pore opening (indicated by arrows), due to diffusion of sucrose through the pore (see Movie S5 in the Supporting Material). In comparison, GUVs in an isotonic environment (dashed lines) exhibit a rather constant fluorescence intensity (see Movie S6 in the Supporting Material). (C) Micrographs of a GUV in hypotonic condition, encapsulating fluorescent glucose analog, just prior to bursting (left panel), with an open pore (middle panel), and just after pore resealing (right panel). The leak-out of fluorescent dye is observed in the middle frame, coinciding with a drop of the GUV radius. Frames extracted from Movie S7 in the Supporting Material. (D) Same as panel (C), with the images processed to increase contrast and attenuate noise. The blue, red, and white lines are the isocontours of the 90, 75, and 60 grey scale values respectively, highlighting the leak-out of fluorescent dye.[]{data-label="fig4"}](fig_efflux) The concentration differential of sucrose decreases exponentially and drops from 200 mM to about 10 mM in about 1000 seconds (Fig. S\[figS\_R\] in the Supporting Material). Even after 2000 s when the concentration differential is as low as 10 mM, the osmotic influx is still large enough to maintain the dynamics of swell-burst cycles (Fig. \[fig1\](C), Fig. S\[figS\_R\]). We further observe that every pore opening event produces a sudden drop in inner solute concentration (Fig. \[fig4\](A), blue line). This suggests that diffusion of sucrose plays an important role in governing the dynamics of solute. In the absence of diffusive effects, the model does not show the abrupt drops in concentration but a rather smooth exponential decay (Fig. \[fig4\](A), grey line). To experimentally verify the model predictions of sucrose dynamics, we quantified the evolution of fluorescence intensity in GUVs encapsulating 200 mM sucrose plus 58.4 $\mu$M 2-NBDG, a fluorescent glucose analog (see Supporting Material and Methods). Fig. \[fig4\](B) presents the evolution of fluorescent intensity of sucrose in time. GUVs in isotonic conditions (dashed lines) do not show a significant change in fluorescence intensity. GUVs in hypotonic conditions (solid lines) exhibit an overall decrease of intensity due to permeation of water through the membrane. Strikingly, consecutive drops of fluorescence intensity are observed coinciding with the pore opening events (Fig. \[fig4\](C and D) middle panels), and point out the importance of sucrose diffusion through the pore. While the quantitative dynamics of sucrose depends on the value of the diffusion constant (Fig. S\[figS\_D\]), the qualitative effect of diffusion on the dynamics remains unchanged. On the other hand, leak-out induced convection does not influence the inner concentration of sucrose, as both solvent and solute are convected, conserving their relative amounts. These observations are in agreement with the existence of the low tension pore closure regime discussed above, where Laplace pressure produces negligible convective transport compared to solute diffusion though the pore. Cycle period and strain rate are explicit functions of the cycle number and GUV properties {#cycle-period-and-strain-rate-are-explicit-functions-of-the-cycle-number-and-guv-properties .unnumbered} ------------------------------------------------------------------------------------------   Given that lytic tension is a dynamic quantity, we asked how cycle period and strain rate evolve along with the cycles. We analyzed the simulated dynamics of GUVs with resting radii of 8, 14 and 20$\mu m$, each data point representing the mean and the standard deviation of 10 simulations with identical parameters (the variations being due to the stochastic nature of the model). [The details of this burst cycle analysis is reported in the Supporting Material]{}. Cycle periods and strain rates show a dependence on the GUV radius, as depicted in Fig. \[fig5\_1\] where larger GUVs have slower dynamics, resulting in smaller strain rates and longer cycle periods (Fig. S\[figS\_R\]). To verify this experimentally, a total of eight GUVs were similarly analyzed with resting radii ranging from 7.02 to 18.76 $\mu$m (Fig. S\[figS\_exp\] in the Supporting Material). The measured cycle period and strain rate as a function of the cycle number (corrected for the lag between the application of the hypotonic stress and the beginning of the observations) are shown in Fig. \[fig5\_1\](A) and (B), respectively. Experimental and model results quantitatively agree, and show a exponential dependence of the cycle period and strain rate on cycle number (Insets Fig. \[fig5\_1\](A and B)). Two further questions arise: How can we relate the cycle number to the driving force of the process, namely the osmotic differential? And, is there a scaling law that governs the GUV swell-burst dynamics? To answer these questions we computed the cycle solute concentration (defined as the solute concentration at the beginning of each cycle) as a function of the cycle number (Fig. S\[figS\_cn\] in the Supporting Material). We found that the solute concentration follows an exponential decay function of the cycle number, and is independent of the GUV radius. Additionally, plotting the cycle period and strain rate against the cycle solute concentration (Fig. \[fig5\_1\](C and D)), we observe that the cycle period increases as $\Delta c$ decreases, while the strain rate is a linear function of $\Delta c$. The data presented in Fig. \[fig5\_1\] suggest that the dynamics of GUVs swell-burst cycle can be scaled to their size. From the non-dimensional form of Eq. \[eq:ode\_R\], we extracted a characteristic time associated with swelling, defined by $\tau = R_0 / (P \nu_s c_0)$, and scaled the cycle period and strain rates with this quantity. As shown in Fig. \[fig5\_2\], all the scaled experimental and model data collapse onto the same curve, within the range of the standard deviations. The scaled relationships can be justified analytically, by estimating the cycle period and strain rates as [ $$\label{eq:cycle_period} \frac{ T_n}{\tau} \simeq \frac{\epsilon^*}{\left( 2\sqrt{\epsilon^*+1} \Delta c/c_0 \right)} \qquad \text{and} \qquad \tau \dot{\epsilon} \simeq \frac{2\sqrt{\epsilon^*+1} \Delta c}{c_0}$$ ]{} respectively (see Supporting Material for full derivation). These analytical expressions are plotted in Figs. \[fig5\_1\](C, D) and \[fig5\_2\](C, D) for a characteristic lytic strain of $\varepsilon^*=0.15$, showing good agreement with the numerical data. Taken together, these results suggest that the GUV pulsatile dynamics is governed by the radius, the membrane permeability, the solute concentration, and importantly the stochastic pore nucleation mechanism which determines the strain to rupture. DISCUSSION {#discussion .unnumbered} ========== Explaining how membrane-enclosed compartments regulate osmotic stress is a first step towards understanding how cells control volume homeostasis in response to environmental stressors. In this work, we have used a combination of theory, computation, and experiments in a simple model system to study how swell-burst cycles control the dynamics of GUV response to osmotic stress. Using this system, we show that the pulsatile dynamics of GUVs under osmotic stress is controlled through thermal fluctuations that govern pore nucleation and lytic tension. The central feature of a GUV’s osmotic response is the nucleation of a pore. Even though Evans and coworkers [@evans2003; @evans2011] identified that rupture tension was not governed by an intrinsic critical stress, but rather by the load rate, the idea of a constant lytic tension has persisted in the literature [@idiart2004; @popescu2008; @peterlin2008]. By coupling fluctuations to pore energy, we have now reconciled the dynamics of the GUV over several swell-burst cycles with pore nucleation and dependence on strain rate. Our model is not only able to capture the experimentally observed pulsatile dynamics of GUV radius and solute concentration (Figs. \[fig3\] and \[fig4\]), but also predicts pore formation events and pore dynamics (Fig. \[fig3\](B and D)). We also found that during the pore opening event, a low-tension regime enables a diffusion dominated transport of solute through the pore (Fig. \[fig4\]), a feature that has been until now neglected in the literature. Specifically, we have identified a scaling relationship between (a) the cycle period and cycle number and (b) the strain rate and the cycle number, highlighting that swell-burst cycles of the GUVs in response to hypotonic stress is a dynamic response (Fig. \[fig5\_2\]). One of the key features of the model is that we relate the cycle number, an experimentally observable quantity, to the concentration difference of the solute, a quantity that is hard to measure in experiments (Fig. S\[figS\_cn\]). This allows to interpret the scaling relationships described above in terms of solute concentration differential. The cycle period increases as the solute concentration difference decreases, while the strain rate is a linear function of the concentration difference. Both relationships are derived theoretically in the Supplemental material. These features indicate long time scale relationships of pulsatile vesicles in osmotic stress. Thermal fluctuations and stochasticity are known to play diverse roles in cell biology. Well-recognized examples include Brownian motors and pumps [@julicher1997; @oster2002], noisy gene expression [@elowitz2002], and red blood cell flickering [@turlier2016]. The pulsatile vesicles presented here provide yet another example of how fluctuations can be utilized by simple systems to produce dynamical adaptive behavior. Given the universality of fluctuations in biological processes, it appears entirely reasonable that simple mechanisms similar to these pulsatile vesicles may have been exploited by early cells, conferring them with a thermodynamic advantage against environmental osmotic assaults. On the other hand, if such swell-burst mechanisms were at play, the chronic leak-out of inner content could have led protocells to evolve active transport mechanisms to compensate for volume loss, and endure osmotic stress without a high energetic cost. In this study, we experimentally measure the dynamics of swell-burst cycles in GUVs, and provide for the first time a model that captures quantitatively the pulsatile behavior of GUVs under hypotonic conditions for long time scales. In order to do so, we developed a general framework which integrated parts of existing models [@koslov1984; @brochard-wyart2000; @ryham2011], with novel key elements: (a) the explicit inclusion of thermal pore fluctuations, which enables dynamic pore nucleation; (b) the definition of an effective stretching modulus, which combines membrane unfolding and elastic stretching; (c) the incorporation of solute diffusion through the pore, which results in a non-trivial contribution to the evolution of the osmotic differential. The coupling of these key features results in a unified model that is valid in all regimes of the vesicle, pore, and solute dynamics. While we have been able to explain many fundamental features of the pulsatile GUVs in response to osmotic stress, our approach has some limitations and there is a need for further experiments. We have assumed a linear relationship between stress and strain. Although this assumption is reasonable and appears to work well for the present experimental conditions, a more general expression should be considered to include both membrane (un)folding and elastic deformation [@helfrich1984]. Another important aspect of biological relevance is membrane composition, where the abundance of proteins and heterogeneous composition leading to in-plane order and asymmetry across leaflets influence the membrane mechanics [@alberts2014; @rangamani2014]. We have previously found experimentally that the dynamics of swell-burst cycles is related to the compositional degrees of freedom of the membrane [@oglecka2014]. Future efforts will be oriented toward the development of theoretical framework and quantitative experimental measures that provide insight into how the membrane’s compositional degrees of freedom influence the pulsatile dynamics of cell-size vesicles. [In addition to osmotic response and membrane composition, we will focus on how membrane components such as aquaporins and ion channels may couple thermal fluctuations with membrane tension to regulate their functions. Additionally, we are also investigating how the properties of the encapsulated bulk fluid phase may affect the response of the GUV in response to osmotic shock. The current work is a first and critical step in these directions. ]{} AUTHOR CONTRIBUTIONS {#author-contributions .unnumbered} ==================== J.C.S.H. and A.N.P. designed the experiments; J.C.S.H. performed the experiments; M.C. and P.R. derived the model; M.C. performed the simulations; M.C. and J.C.S.H. analyzed the data; all authors discussed and interpreted results; all authors wrote and agreed on the manuscript. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We are grateful to Prof. Wouter-Jan Rappel and Prof. Alex Mogilner for insightful comments on the manuscript. We also thank Prof. Daniel Tartakovsky for enriching discussions. This work was supported in part by the FISP 3030 for the year 2015-2016 to M.C., NTU provost office to J.C.S.H., AFOSR FA9550-15-1-0124 award to P.R., and NSF PHY-1505017 award to P.R. and A.N.P. SUPPORTING CITATIONS {#supporting-citations .unnumbered} ==================== References (42-51) appear in the Supporting Material. Table of Content {#table-of-content .unnumbered} ================ **** ---------------------------------------------------------------------------------------------------------- --- Supporting Materials and Methods 1 Model Development and Simulations 2 Derivation of the analytical relations between cycle period, strain rate, and concentration differential 4 Supporting References 5 Supporting Figures and Movies 6 ---------------------------------------------------------------------------------------------------------- --- Supporting Materials and Methods {#supporting-materials-and-methods .unnumbered} ================================ **Swell-burst cycle experiments.** The experimental methods for the GUVs preparation has been described in [@oglecka2014; @angelova1992]. Briefly, GUVs (100$\%$ POPC + 1mol$\%$ Rho-DPPE) containing 200 mM of sucrose were prepared by electroformation, yielding vesicles with radii ranging from 7 to 20 $\mu$m. GUVs were then placed in a bath of deionized water at room temperature, inducing hypotonic stress proportional to the inner sucrose concentration. The kinetics of eight GUVs were recorded by time-lapse microscopy at 1/150 images/ms. In order to allow for the sedimentation of GUVs to the bottom of the well, observations were started about one minute after the GUVs were subject to hypotonic conditions. For each frame, the GUV radius was measured using a customized MATLAB (Mathworks, Natick, MA) code to streamline the image analysis. This code uses a Circular Hough Transform method based on a phase-coding algorithm to detect circles , and measure their radii and centers. For our data, this custom code gives the evolution of the GUV radius in time with a precision of about 0.1 $\mu$m. Due to slow movement of the GUVs, in some cases the observation fields had to be adjusted to follow the GUVs, and the recording was paused. These are indicated by black dashed lines in Fig. S\[figS\_exp\]. In order to define a systematic experimental initial GUV radius, $R_0$ was determined for each GUV as 0.995 times the first measured local minimum GUV radius, in accordance with our numerical results. Furthermore, burst events were identified by drops of GUV radius larger than 0.2 $\mu$m within a 1.5 s interval, and are plotted as solid red triangle. Bursting events that were likely to happen during the video gaps were indicated by plain red triangle (these “likely" bursting events were not taken into account in the data processing for Figs. \[fig5\_1\] and \[fig5\_2\]). **Leak-out quantification.** To quantify the leak-out amount when a membrane pore is formed, giant unilamellar vesicles (GUVs) were electroformed in 200 mM sucrose, supplemented with 58.4 $\mu$M 2-NBDG (2-(N- (7-Nitrobenz-2-oxa-1,3-diazol-4-yl) Amino) -2-Deoxyglucose), a fluorescent glucose analog that has an almost identical molecular weight as sucrose. Fluorescence imaging was performed on a deconvolution microscope, equipped with a FITC filter. Time-lapse imaging of the vesicles was performed approximately one minute after exposing the vesicles to either deionized water (hypotonic conditions $n=3$) or glucose (isotonic conditions $n=3$) environment to ensure sedimentation of GUVs to the bottom of the well. All acquisitions were performed using identical settings to facilitate comparison of vesicles submerged in water or equi-osmotic glucose environment. For Fig. \[fig4\](C), the GUVs were detected with a MATLAB (Mathworks, Natick, MA) code adapted from the one described above, where the mean gray intensity inside and outside of the GUV are measured. For every time frame, the difference between the inner and outer mean intensity $\Delta I(t)$ was computed, and normalized by the intensity difference of the first frame $\Delta I(t_0)$. Bursting events were identified by visual inspection of the videos, and reported by arrows on Fig. \[fig4\](B). In order to highlight the efflux of fluorescent dyes during a GUV bursting event, three frames (before, during and after the event) were extracted form the video of a GUV containing 200 mM sucrose + 58.4 $\mu$M 2-NBDG in hypotonic conditions (Fig. \[fig4\](C)). These images were further processed with ImageJ software to plot Fig. \[fig4\](D). Briefly, the noise was attenuated by successively applying ImageJ built-in routines (background suppression, contrast enhancing, median filter), and ploting the isovalues of gray at 90, 75 and 60 with the pluggin Contour Plotter (http://rsb.info.nih.gov/ij/plugins/contour-plotter.html). **Burst cycle analysis.** [ The following analysis has been applied for both experimental and numerical data in order to produce Figs. \[fig5\_1\], \[fig5\_2\] and S\[figS\_comparison\]. For a given GUV radius dynamics, a swell-burst cycle was defined between two successive minimum GUV radii that immediately followed a bursting event. Cycle periods were computed as the time between two consecutive minima in vesicular radii. For experimental data, if there was a video gap between two consecutive radius minima, the cycle was not taken into account for the analysis, that is to say, only cycles between two successive solid triangles in Figs. \[fig1\](C) and S\[figS\_exp\] were taken into account. The strain rate was computed as the difference between the maximum and minimum radii within these cycles, divided by the time between these two events. For experimental data, because of the lag between the beginning of the experiments and the beginning of the video recordings, the initial observed cycle number was adjusted between $n=1$ and $n=4$, depending on $R_0$.]{} Model Development and Simulations {#model-development-and-simulations .unnumbered} ================================= Here we derive a theoretical model to describe the swell-burst cycle of a GUV under hypotonic conditions. In line with previous work [@koslov1984; @idiart2004; @popescu2008], the model has three conservation equations, governing the dynamics of the solvent, solute, and membrane pore. **Mass conservation of solvent.** Mass conservation of the solvent (water) within the vesicle is governed by the flux through the membrane ($j_w$), and the leak-out through the pore. For a spherical GUV, the general form of the mass conservation equation for the solvent is $$\label{eq:sol_mass_cons1} \frac{d}{dt}\left( \frac{4}{3}\pi R^3 \rho_s \right) = j_w - \pi r^2 \rho_s v_L \;,$$ where $R$ and $r$ are the radius of the vesicle and the pore respectively, $\rho_s$ is the mass density of the solvent, and $v_L$ is the leak-out velocity of the solvent. The osmotic flux is influenced by the permeability of the membrane to the solvent ($P$), the osmotic pressure ($\Delta p_{osm}$), and the Laplace pressure ($\Delta p_L$). A phenomenological expression for the osmotic flux is [@koslov1984; @popescu2008] $$\label{eq:jw} j_w = \frac{P \nu_s \rho_s}{k_B T N_A} A \left( \Delta p_{osm} - \Delta p_L \right) \;,$$ where $\nu_s$ is the solvent molar volume, and the membrane area is defined as $A = 4\pi R^2 - \pi r^2$. The two pressures involved in Eq. \[eq:jw\] are defined as $$\left\{ \begin{array}{l} \Delta p_{osm} = k_B T N_A \Delta c \\ \Delta p_L = \dfrac{2\sigma}{R} \end{array}\right. \;.$$ The Laplace pressure originates from the surface tension in the membrane $\sigma$, which we assume to be proportional to the membrane strain $$\label{eq:sigma_linear} \sigma = \kappa_\text{eff} \frac{A-A_0}{A_0} = \kappa_\text{eff} \epsilon \;.$$ Here $\kappa_\text{eff}$ is the effective area extension modulus (combining the effects of membrane unfolding and elastic deformation), and $A_0 = 4\pi R_0^2$ is the surface of the vesicle in its unstretched state. The leak-out velocity $v_L$ can be analytically approximated at low Reynolds number in order to relate it to the Laplace pressure [@happel1983] $$\label{eq:vl} v_L = \frac{\Delta p_L r}{3\pi\eta_s} \;.$$ Substituting these definitions into Eq. \[eq:sol\_mass\_cons1\], the mass conservation equation for the solvent takes the form of an ordinary differential equation (ODE) for the GUV radius $$\label{eq:sol_mass_cons2} 4\pi R^2\frac{dR}{dt}= \frac{P \nu_s }{k_B T N_A} A \left( k_B T N_A \Delta c - \frac{2\sigma}{R} \right) - \frac{2\sigma }{3\eta_s R} r^3 \;.$$ **Mass conservation of solute.** The permeability of lipid membranes to water is several orders of magnitude larger than for most solutes . Consequently the lipid bilayer is supposed to be semi-permeable, neglecting sucrose transport through the membrane. Thus, variation of solute in the vesicle is exclusively limited to diffusive and convective transport through the pore, such that $$\label{eq:solute_mass_cons1} \frac{d}{dt}\left(\frac{4}{3}\pi R^3 \Delta c \right) = \pi r^2 \left( - D \frac{\Delta c}{R} - v_L\Delta c \right) \;.$$ While the diffusive flux through the pore is usually neglected over the convective efflux of solute, theoretical analysis of long lived pores indicates that the Laplace pressure decreases rapidly after the pore opening, and stays low for most of the pore life time [@brochard-wyart2000]. This suggests that the convective efflux directed by the leak-out velocity may not always be the dominant solute transport mechanism, as confirmed by our numerical and experimental results (see main text Fig. \[fig4\]). Expanding Eq. \[eq:solute\_mass\_cons1\] we obtain an ODE for the concentration difference in solute $$\label{eq:solute_mass_cons2} \frac{4}{3}\pi R^3 \frac{d \Delta c }{dt} = - D\pi r^2 \frac{\Delta c}{R} - \frac{2\sigma }{3\eta_s R} r^3\Delta c - 4\pi R^2\Delta c \frac{d R}{dt} \;.$$ **Pore force balance.** The pore in the lipid bilayer is modeled as an overdamped system, where the pore radius is governed by the following Langevin equation $$\label{eq:pore_energy1} \zeta \frac{d}{dt}\left( 2\pi r \right) = F(r,t) + \xi(t) \;,$$ where $\zeta$ is the membrane drag coefficient (inverse of the mobility), $F(r,t)$ is a conservative force, and $\xi$ is a noise term accounting for independent thermally-induced pore fluctuations. The drag coefficient includes two in-plane contributions $\zeta = \zeta_m + \zeta_s$: one from membrane dissipation, proportional to the membrane viscosity and thickness $\zeta_m=\eta_m h$ [@brochard-wyart2000], and a second from the friction of the solvent with the moving pore – proportional to the solvent viscosity $\zeta_s = C \eta_s r $, where $C=2\pi$ is a geometric coefficient [@ryham2011; @aubin2016]. The conservative force $F(r,t)=-\partial V(r,t)/\partial r$ arises from the membrane potential $V(r,t)$, which is equal to the sum of the strain energy $V_s$, and the pore energy $V_p$. We assume the membrane strain energy to take a Hookean form $V_s = \kappa_\text{eff} \left(A-A_0\right)^2 /(2 A_0)$, where $\kappa_\text{eff}$ is an effective stretching modulus approximating the combined contributions of membrane unfolding and elastic stretching. The pore energy depends on the edge energy and length as $V_p = 2\pi r \gamma$, where $\gamma$ is the pore line tension, here assumed independent of the pore radius. Using the definition $\sigma = \partial V_s/\partial A$, we can therefore express the force as $$F(r,t) =2\pi\sigma r - 2\pi\gamma \;.$$ The fluctuation term has a zero mean, and a correlation function given by $$\langle \xi(t)\xi(t^\prime) \rangle = 2\zeta k_B T \delta(t-t^\prime) \;,$$ following the dissipation-fluctuation theorem, where $\delta$ is the Dirac delta function. Rearranging Eq. \[eq:pore\_energy1\] with these definitions, we obtain a stochastic differential equation for the pore radius $$\label{eq:pore_energy2} \left(\eta_m h + C \eta_s r\right)\frac{d}{dt}(2\pi r) = 2\pi(\sigma r - \gamma) + \xi(t) \;,$$ with $r\ge0$. The last term in Eq. \[eq:pore\_energy2\] is responsible for thermally driven pore nucleation. **Deterministic model.** In the absence of thermal fluctuations, a critical value for the membrane tension (or strain) has to be defined, and an initial pore has to be set artificially in order for a large pore to open. In that case, Eq. \[eq:pore\_energy2\] is simply $$\label{eq:pore_energy_det} \left(\eta_m h + C \eta_s r\right)\frac{dr}{dt} = \sigma r - \gamma \;.$$ When the pore is closed ($r=0$) and the strain overcomes the predetermined critical value ($\epsilon\ge\epsilon^*$), an initial pore large enough to overcome the nucleation barrier ($r=\gamma/\sigma$) is artificially created. **Numerical implementation.** All numerical computations have been carried out using a custom code in MATLAB (Mathworks, Natick, MA). The stochastic model, composed of Eqs. \[eq:sde\_r\], \[eq:ode\_R\] and \[eq:ode\_c\] was solved using an order-1 Runge-Kutta scheme. Because a pore nucleation event occurs due to a single fluctuation overcoming the energy barrier, the numerical implementation of the noise requires the definition of a fluctuation frequency $f_T$ [(number of fluctuation “kicks” per seconds)]{} that is independent of the time step. For comparison a deterministic model (Eqs. \[eq:sol\_mass\_cons2\], \[eq:solute\_mass\_cons2\], and \[eq:pore\_energy\_det\]) was solved using Euler method. All parameters are shown in Table S\[tab\_parameters\]. All time steps were taken as 0.1 ms, (smaller time steps did not improve the accuracy of the results significantly). For the cycle analysis of the stochastic model, Figs. \[fig5\_1\] and \[fig5\_2\], shows the average and standard deviations of 10 runs with same parameters. Parameter Typical value References --------------------- ----------------------------------------- -------------- $R_0$ 8-20 $\mu$m this work $c_0$ 200 mM this work $d$ 3.5 nm $\rho_s$ 1000 kg m$^{-3}$ $\nu_s$ 18.04$\times$10$^{-6}$ m$^3$ mol$^{-1}$ $P$ 20 $\mu$m/s $T$ 294 K $\gamma$ 5 pN $\kappa_\text{eff}$ 2$\times$10$^{-3}$ N/m this work $\eta_m$ 5 Pa s $\eta_s$ 0.001 Pa s $D$ 5$\times$10$^{-10}$ m$^2$/s $C$ $2\pi$ [@aubin2016] $f_T$ 150 Hz this work Derivation of the analytical relations between cycle period, strain rate, and concentration differential {#derivation-of-the-analytical-relations-between-cycle-period-strain-rate-and-concentration-differential .unnumbered} ======================================================================================================== First, we derive the linear dependence of the strain rate on the concentration difference shown in Fig. \[fig5\_2\](D). For a closed vesicle ($r=0$), the membrane area is $A=4\pi R^2$, and the strain rate is $$\dot{\epsilon} = \frac{d}{dt} \left( \frac{A - A_0}{A_0} \right) = \frac{2R}{R_0}\frac{dR}{dt} \;.$$ This allows us to write Eq. \[eq:sol\_mass\_cons2\] in terms of the strain rate as $$\dot{\epsilon} = \frac{P \nu_s }{k_B T N_A} \frac{A}{2\pi R R_0^2} \left( k_B T N_A \Delta c - \frac{2\sigma}{R} \right)\;.$$ When the osmotic pressure is the dominant process influencing GUV swelling, we can neglect the Laplace pressure and obtain $$\label{eq:eps_dot} \tau \dot{\epsilon} \simeq \frac{2R}{R_0}\frac{\Delta c}{c_0} \;,$$ where $\tau=R_0/(P \nu_s c_0)$. At maximum GUV radius amplitude, $R/R_0$ can be expressed in term of the lytic strain as $R_\text{max}/R_0 = \sqrt{\epsilon^*+1}$, allowing to write Eq. \[eq:eps\_dot\] as $$\label{eq:eps_dot_s} \tau \dot{\epsilon} \simeq 2\sqrt{\epsilon^*+1}\frac{\Delta c}{c_0} \;.$$ Plotting this relationship in Figs. \[fig5\_1\](D) and \[fig5\_2\](D) for a typical lytic strain $\epsilon^*=0.15$, we get a good agreement with the numerical results from the stochastic model. We now derive an approximate relation between the cycle period and the strain rate. During a cycle of period $T_n$, the lytic strain can be written $$\epsilon^* \simeq T_n \dot{\epsilon} \;.$$ Introducing Eq. \[eq:eps\_dot\_s\], we get $$\frac{T_n}{\tau} \simeq \frac{\epsilon^*}{2\sqrt{\epsilon^*+1}} \left( \frac{\Delta c}{c_0}\right)^{-1} \;.$$ Taking $\epsilon^*=0.15$, this relationship fits well the simulation results, as shown in Figs. \[fig5\_1\](C) and \[fig5\_2\](C). It should be noted that, because the Laplace pressure is neglected in the derivation of Eq. \[eq:eps\_dot\_s\], the analytical expression slightly overestimates the strain rate as shown in Figs. \[fig5\_1\](D) and \[fig5\_2\](D). Moreover the cycle period is also overestimated for low solute concentrations due to the constant lytic strain assumed in the analytical expression (Figs. \[fig5\_1\](C) and \[fig5\_2\](C)). Supporting Figures and Movies {#supporting-figures-and-movies .unnumbered} ============================= #### Supporting Movie 1 Membrane nodules appearance after membrane pore reseals. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 2 fps; total duration, 17 s; image size, 82.43 $\mu$m $\times$ 82.43 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM). #### Supporting Movie 2 Multiple swell-burst cycles of GUVs subject to hypotonic stress. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 77 s; image size, 82.43 $\mu$m $\times$ 82.43 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM). #### Supporting Movie 3 Model results showing multiple swell-burst cycles of a GUV subject to hypotonic stress. GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. Right panel is a representation of the numerical GUV in time, where the grey intensity is proportional to the inner sucrose concentration. GUV initial radius is $R_0=14\;\mu$m, initial solute concentration is $c_0=200$ mM. All parameters are shown in Supporting Table S\[tab\_parameters\]. #### Supporting Movie 4 Model results showing a single pore opening dynamics of a GUV subject to hypotonic stress. GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. Right panel is a representation of the numerical GUV in time, where the grey intensity is proportional to the inner sucrose concentration. GUV initial radius is $R_0=14\;\mu$m, initial solute concentration is $c_0=200$ mM. All parameters are shown in Supporting Table \[tab\_parameters\]. #### Supporting Movie 5 Solute leakage of a GUV in multiple swell-burst cycles under hypotonic condition. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 11 s; image size, 119.14 $\mu$m $\times$ 125.58 $\mu$m; scale bar, 20 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM). #### Supporting Movie 6 GUV under isotonic condition. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 8 s; image size, 101.11 $\mu$m $\times$ 101.11 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a isotonic solution (no osmotic differential). #### Supporting Movie 7 Solute efflux from GUV during one swell-burst cycle. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 12 fps; total duration, 8 s; image size, 164.86 $\mu$m $\times$ 164.86 $\mu$m; scale bar, 20 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM). {width="100.00000%"} {width="50.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="100.00000%"} {width="60.00000%"}

ArXiv

--- abstract: 'This paper studies how to verify the conformity of a program with its specification and proposes a novel constraint-programming framework for bounded program verification (CPBPV). The CPBPV framework uses constraint stores to represent the specification and the program and explores execution paths nondeterministically. The input program is partially correct if each constraint store so produced implies the post-condition. CPBPV does not explore spurious execution paths as it incrementally prunes execution paths early by detecting that the constraint store is not consistent. CPBPV uses the rich language of constraint programming to express the constraint store. Finally, CPBPV is parametrized with a list of solvers which are tried in sequence, starting with the least expensive and less general. Experimental results often produce orders of magnitude improvements over earlier approaches, running times being often independent of the variable domains. Moreover, CPBPV was able to detect subtle errors in some programs while other frameworks based on model checking have failed.' author: - 'Hélène Collavizza, Michel Rueher, Pascal Van Hentenryck' title: | CPBPV: A Constraint-Programming Framework\ For Bounded Program Verification --- Introduction ============ This paper is concerned with software correctness, a critical issue in software engineering. It proposes a novel constraint-programming framework for bounded program verification (CPBPV), i.e., when the program inputs (e.g., the array lengths and the variable values) are bounded. The goal is to verify the conformity of a program with its specification, that is to demonstrate that the specification is a consequence of the program. The key idea of CPBPV is to use constraint stores to represent the specification and the program, and to non-deterministically explore execution paths over these constraint stores. This non-deterministic constraint-based symbolic execution incrementally refines the constraint store, which initially consists of the precondition. Non-determinism occurs when executing conditional or iterative instructions and the non-deterministic execution refines the constraint store by adding constraints coming from conditions and from assignments. The input program is partially correct if each constraint store produced by the symbolic execution implies the post-condition. It is important to emphasize that CPBPV considers programs with complete specifications and that verifying the conformity between a program and its specification requires to check (explicitly or implicitly) all executables paths. This is not the case in model-checking tools designed to detect violations of some specific property, e.g., safety or liveness properties. The CPBPV framework has a number of fundamental benefits. First, contrary to earlier work using constraint programming or SMT [@ABM07; @CoR06; @CoR07], CPBPV does not use predicate abstraction or explore spurious execution paths, i.e., paths that do not correspond to actual executions over inputs satisfying the pre-condition. CPBPV incrementally prunes execution paths early by detecting that the constraint store is not consistent. Second, CPBPV uses the rich language of constraint programming to express the constraint store, including arbitrary logical and threshold combination of constraints, the [*element*]{} constraint, and global/combinatorial constraints that express complex relationships on a set of variables. Finally, CPBPV is parametrized with a list of solvers which are tried in sequence, starting with the least expensive and less general. The CPBPV framework was evaluated experimentally on a series of benchmarks from program verification. Experimental results of our (slow) prototype often produce orders of magnitude improvements over earlier approaches, and indicate that the running times are often independent of the variable domains. Moreover, CPBPV was able to found subtle errors in some programs that some other verification frameworks based on model-checking could not detect. The rest of the paper is organized as follows. Section \[motivation\] illustrates how CPBPV handles constraints store on a motivating example. Section \[formalization\] formalizes the CPBPV framework for a small programming language and Section \[implementation\] discusses the implementation issues. Section \[experimental\] presents experimental results on a number of verification problems, comparing our approach with state of the art model-checking based verification frameworks. Section \[related\] discusses related work in test generation, bounded program verification and software model checking. Section \[conclusion\] summarizes the contributions and presents future research directions. The Constraint-Programming Framework at Work {#motivation} ============================================ This section illustrates the CPBPV verifier on a motivating example, the binary search program. CPBPV uses Java programs and JML specifications for the pre- and post-conditions, appropriately enhanced to support the expressivity of constraint programming. Figure \[BsearchFig\] depicts a binary search program to determine if a value $v$ is present in a sorted array $t$. (Note that $\backslash$[result]{} in JML corresponds to the value returned by the program). To verify this program, our prototype implementation requires a bound on the length of array $t$, on its elements, and on $v$. We will verify its correctness for specific lengths and simply assume that the values are signed integers on a number of bits. /*@ requires (\forall int i; i>=0 && i<t.length-1;t[i]<=t[i+1]) @ ensures @ (\result != -1 ==> t[\result] == v) && @ (\result == -1 ==> \forall int k; 0 <= k < t.length ; t[k] != v) @*/ 1 static int binary_search(int[] t, int v) { 2 int l = 0; 3 int u = t.length-1; 4 while (l <= u) { 5 int m = (l + u) / 2; 6 if (t[m]==v) 7 return m; 8 if (t[m] > v) 9 u = m - 1; 10 else 11 l = m + 1; } // ERROR else u = m - 1; 12 return -1; } The initial constraint store of the CPBPV verifier, assuming an input array of length 8, is the precondition[^1] $c_{pre} \equiv \forall 0 \leq i < 7: t^0[i] \leq t^0[i+1]$ where $t^0$ is an array of constraint variables capturing the input. The constraint variables are annotated with a version number as CPBPV performs a SSA-like renaming [@CFR91] on the fly since each assignment generates constraints possibly linking the old and the new values of the assigned variable. The assignments in lines 2–3 add the constraints $l^0 = 0 \wedge u^0 = 7$. CPBPV then considers the loop instruction. Since $l^0 \leq u^0$, it enters the loop body, adds the constraint $m^0 = (l^0 + u^0)/2$, which simplifies to $m^0 = 3$, and considers the conditional statement on line 6. The execution of the statement is nondeterministic: Indeed, both $t^0[3] = v^0$ and $t^0[3] \neq v^0$ are consistent with the constraint store, so that the two alternatives, which give rise to two execution paths, must be explored. Note that these two alternatives correspond to actual execution paths in which $t[3]$ in the input is equal to, or different from, input $v$. The first alternative adds the constraint $t^0[3] = v^0$ to the store and executes line 7 which adds the constraint $result = m^0$. CPBPV has thus obtained an execution path $p$ whose final constraint store $c_p$ is: $ c_{pre} \; \wedge \; l^0 = 0 \wedge u^0 = 7 \; \wedge \; m^0 = (l^0 + u^0)/2 \; \wedge \; t^0[m^0] = v^0 \; \wedge \; result = m^0 $\ CPBPV then checks whether this store $c_p$ implies the post-condition $c_{post}$ by searching for a solution to $c_p \; \wedge \; \neg c_{post}$. This test fails, indicating that the computation path $p$, which captures the set of actual executions in which $t[3] = v$, satisfies the specification. CPBPV then explores the other alternatives to the conditional statement in line 6. It adds the constraint $t^0[m^0] \neq v^0$ and executes the conditional statement in line 8. Once again, this statement is nondeterministic. Its first alternative assumes that the test holds, generating the constraint $t^0[m^0] > v^0$ and executing the instruction in line 9. Since $u$ is (re-)assigned, CPBPV creates a new variable $u^1$ and posts the constraint $u^1 = m^0 - 1 =2$. The execution returns to line 4, where the test now reads $l^0 \leq u^1$, since CPBPV always uses the most recent version for each variable. Since the constraint stores entails $l^0 \leq u^1$, the only extension to the current path consists of executing line 5, adding the constraint $m^1 = (l^0 + u^1)/2$, which actually simplifies to $m^1 = 1$. Another complete execution path is then obtained by executing lines 6 and 7. Consider now a version of the program in which line 11 is replaced by [u = m-1]{}. To illustrate the CPBPV verifier, we specify partial execution paths by indicating which alternative is selected for each nondeterministic instruction. For instance, $\langle T_4,F_6,T_8,T_5,T_6\rangle$ denotes the last execution path discussed above in which the true alternative is selected for the first execution of the instruction in line 4, the false alternative for the first execution of instruction 6, the true alternative for the first instruction of instruction 8, the true alternative of the second execution of instruction 5, and the true alternative of the second execution of instruction 6. Consider the partial path $\langle T_4,F_6,F_8 \rangle$ and let us study how it can be extended. The partial path $\langle T_4,F_6,F_8,T_4,T_6 \rangle$ is not explored, since it produces a constraint store containing\ $ c_{pre} \; \wedge \; t^0[3] \neq v^0 \; \wedge \; t^0[3] \leq v^0 \; \wedge \; t^0[1] = v^0 $\ which is clearly inconsistent. Similarly, the path $\langle T_4,F_6,F_8,T_4,F_6,T_8\rangle$ cannot be extended. The output of CPBPV on this incorrect program when executed on an array of length 8 (with integers coded on 8-bits to make it readable) produces, in 0.025 seconds, the counterexample:\ $ v^0 = -126 \ \wedge \ t^0 = [-128,-127,-126,-125,-124,-123,-122,-121] \ \wedge \ result = -1. $\ This example highlights a few interesting benefits of CPBPV. 1. The verifier only considers paths that correspond to collections of actual inputs (abstracted by constraint stores). The resulting execution paths must all be explored since our goal is to prove the partial correctness of the program. 2. The performance of the verifier is independent of the integer representation on this application: it only requires a bound on the length of the array. 3. The verifier returns a counter-example for debugging the program. Note that $CBMC$ and $ESC/Java 2$, two state-of-the-art model checkers fail to verify this example as discussed in Section \[experimental\]. Formalization of the Framework {#formalization} ============================== \[semantics\] This section formalizes the CPBPV verifier on a small abstract language using a small-step SOS semantics. The semantics primarily specifies the execution paths over constraint stores explored by the verifier. It features `assert` and `enforce` constructs which are necessary for modular composition. #### **Syntax** Figure \[syntax-c\] depicts the syntax of the programs and the constraints generated by the verifier. In the following, we use $s$, possibly subscripted, to denote elements of a syntactic entity $S$. $$\begin{aligned} \begin{array}{l} L: \mbox{\it list of instructions}; I: {\it instructions}; B: \mbox{\it Boolean expressions} \\ E: \mbox{\it integer expressions}; A: {\it arrays}; V: \mbox{\it variables} \\ \\ L ::= I ; L \; | \; \epsilon \\ I ::= A[E] \leftarrow E \; | \; V \leftarrow E \; | \; {\bf\it if} \ B \ I \; | \; {\bf\it while} \ B \ I \; | \; {\bf\it assert(B)} \; | \; {\bf\it enforce(B)} \; | \; {\bf\it return} \ E \; | \; \{ L \} \\ B ::= true \;|\; false \;|\; E > E \;| \; E \geq E \;| \; E = E \;| \; E \neq E \;| \; E \leq E \;| \; E < E \\ B ::= \neg B \;| \; B \wedge B \; | \; B \vee B \;| \; B \Rightarrow B \\ E ::= V \;|\; A[E] \;|\; E + E \;|\; E - E \;|\; E \times E \;|\; E / E \;|\; \\ \\ C: \mbox{\it constraints} \hspace*{2cm} E^+: \mbox{\it solver expressions} \\ V^+ = \{ v^i \ | \ v \in V \ \& \ i \in {\cal N}\}: \mbox{\it solver variables} \\ A^+ = \{ a^i \ | \ a \in A \ \& \ i \in {\cal N}\}: \mbox{\it solver arrays} \\ \\ C ::= true \;|\; false \;|\; E^+ > E^+ \;| \; E^+ \geq E^+ \;| \; E^+ = E^+ \;| \; E^+ \neq E^+ \;| \; E^+ \leq E^+ \;| \; E^+ < E^+ \\ C ::= \neg C \;| \; C \wedge C \; | \; C \vee C \;| \; C \Rightarrow C \\ E^+ ::= V \;|\; A[E^+] \;|\; E^+ + E^+ \;|\; E^+ - E^+ \;|\; E^+ \times E^+ \;|\; E^+ / E^+ \;|\; \end{array}\end{aligned}$$ #### **Renamings** CPBPV creates variables and arrays of variables “on-the-fly” when they are needed. This process resembles an SSA normalization but does not introduce the join nodes, since the results of different execution paths are not merged. Similar renamings are used in model checking. The renaming uses mappings of type $V \cup A \rightarrow {\cal N}$ which maps variables and arrays into a natural numbers denoting their current “version numbers”. In the semantics, the version number is incremented each time a variable or an array element is assigned. We use $\sigma_{\bot}$ to denote the uniform mapping to zero (i.e., $\forall x \in V \cup A: \sigma_{\bot}(x) = 0$) and $\sigma[x/i]$ the mapping $\sigma$ where $x$ now maps to $i$, i.e., $\sigma[x/i](y) = {\it if} x = y \mbox{ {\it then} } i \mbox{ {\it else} } \sigma(y).$ These mappings are used by a polymorphic renaming function $\rho$ to transform program expressions into constraints. For example, $\rho \ \sigma \ b_1 \oplus b_2 = (\rho \ \sigma \ b_1) \oplus (\rho \ \sigma \ b_2) (\mbox{where } \oplus \in \{\wedge,\vee,\Rightarrow\})$ is the rule used to transform a logical expression. #### **Configurations** The CPBCV semantics mostly uses configurations of the type $\langle l, \sigma, c \rangle$, where $l$ is the list of instructions to execute, $\sigma$ is a version mapping, and $c$ is the set of constraints generated so far. It also uses configurations of the form $\langle \top, \sigma, c \rangle$ to denote final states and configurations of the form $\langle \bot, \sigma, c \rangle$ to denote the violation of an assertion. The semantics is specified by rules of the form $ \frac{\mbox{conditions}} {\gamma_1 \longmapsto \gamma_2} $ stating that configuration $\gamma_1$ can be rewritten into $\gamma_2$ when the conditions hold. #### **Conditional Instructions** The conditional instruction ${\bf\it if} \ b \ i$ considers two cases. If the constraint $c_b$ associated with $b$ is consistent with the constraint store, then the store is augmented with $c_b$ and the body is executed. If the negation $\neg c_b$ is consistent with the store, then the constraint store is augmented with $\neg c_b$. Both rules may apply, since the store may represent some memory states satisfying the condition and some violating it. $$\frac{c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it if} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle i \; ; \; l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ $\;\;\; \;\;\;$ $$\frac{c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it if} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge \neg (\rho \ \sigma \ b) \rangle}$$ #### **Iterative Instructions** The while instruction ${\bf\it while} \ b \ i$ also considers two cases. If the constraint $c_b$ associated with $b$ is consistent with the constraint store, then the constraint store is augmented with $c_b$, the body is executed, and the while instruction is reconsidered. If the negation $\neg c_b$ is consistent with the constraint store, then the constraint store is augmented with $\neg c_b$. $$\frac{c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it while} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle i;while \ b \ i \; ; \; l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ $$\frac{c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable}} {\langle {\bf\it while} \ b \ i \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge \neg (\rho \ \sigma \ b) \rangle}$$ #### **Scalar Assignments** Scalar assignments create a new constraint variable for the program variable to be assigned and add a constraint specifying that the variable is equal to the right-hand side. A new renaming mapping is produced. $$\frac{\sigma_2 = \sigma_1[v/\sigma_1(v)+1] \;\; \& \;\; c_2 \equiv (\rho \ \sigma_2 \ v) = (\rho \ \sigma_1 \ e)} {\langle v \leftarrow e \; ; \; l, \sigma_1, c_1 \rangle \longmapsto \langle l, \sigma_2, c_1 \wedge c_2 \rangle}$$ #### **Assignments of Array Elements** The assignment of an array element creates a new constraint array, add a constraint for the index being indexed and posts constraints specifying that all the new constraint variables in the array are equal to their earlier version, except for the element being indexed. Note that the index is an expression which may contain variables as well, giving rise to the well-known [*element*]{} constraint in constraint programming [@VanHentenryck89]. $$\frac{\begin{array}{l} \sigma_2 = \sigma_1[a/\sigma_1(a)+1] \\ c_2 \equiv (\rho \ \sigma_2 \ a) [\rho \ \sigma_1 \ e_1] = (\rho \ \sigma_1 \ e_2) \\ c_3 \equiv \forall i \in 0..{\it a.length}: (\rho \ \sigma_1 \ e_1) \neq i \; \Rightarrow \; (\rho \ \sigma_2 \ a) [i] = (\rho \ \sigma_1 \ a) [i] \end{array}} {\langle a[e_1] \leftarrow e_2, \sigma_1\; ; \; l, c_1 \rangle \longmapsto \langle l, \sigma_2, c_1 \wedge c_2 \wedge c_3 \rangle}$$ #### **Assert Statements** An assert statement checks whether the assertion is implied by the control store in which case it proceeds normally. Otherwise, it terminates the execution with an error. $$\frac{ c \Rightarrow (\rho \ \sigma \ b) } {\langle {\bf\it assert} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \rangle}$$ $\;\;\;$ $$\frac{ c \wedge \neg (\rho \ \sigma \ b) \mbox{ is satisfiable} } {\langle {\bf\it assert} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle \bot, \sigma, c \rangle}$$ #### **Enforce Statements** An enforce statement adds a constraint to the constraint store if it is satisfiable. $$\frac{ c \wedge (\rho \ \sigma \ b) \mbox{ is satisfiable} } {\langle {\bf\it enforce} \; b \; ; \; l, \sigma, c \rangle \longmapsto \langle l, \sigma, c \wedge (\rho \ \sigma \ b) \rangle}$$ #### **Block Statements** Block statements simply remove the braces. $${\langle \{ l_1 \}\; ; \; l_2, \sigma, c \rangle \longmapsto \langle l_1:l_2, \sigma, c \rangle}$$ #### **Return Statements** A return statement simply constrains the [*result*]{} variable. $$\frac{ c_2 \equiv (\rho \ \sigma_1 \ result) = (\rho \ \sigma_1 \ e) } {\langle {\bf\it return} \ e \; ; \; l, \sigma_1, c_1 \rangle \longmapsto \langle \sigma_1, c_1 \wedge c_2 \rangle}$$ #### **Termination** Termination also occurs when no instruction remains. $${\langle \epsilon, \sigma, c \rangle \longmapsto \langle \top, \sigma, c \rangle}$$ #### **The CPBPV Semantics** Let ${\cal P}$ be program $b_{pre} \; l \; b_{post}$ in which $b_{pre}$ denotes the precondition, $l$ is a list of instructions, and $b_{post}$ the post-condition. Let $\stackrel{*}{\longmapsto}$ be the transitive closure of $\longmapsto$. The final states are specified by the set $${\it SFN}(b_{pre},{\cal P}) = \{ \ \langle f, \sigma, c \rangle | \langle i, \sigma_{\bot}, \rho \ \sigma_{\bot} \ b_{pre} \rangle \stackrel{*}{\longmapsto}{*} \langle f , \sigma, c \rangle \; \wedge \; f \in \{\bot,\top\} \ \}$$ The program violates an assertion if the set $${\it SFE}(b_{pre},{\cal P},b_{post}) = \{ \langle \bot, \sigma, c \rangle \in {\it SFN}(b_{pre},{\cal P}) \}$$ is not empty. It violates its specification if the set $${\it SFE}(b_{pre},{\cal P},b_{post}) = \{ \top, \sigma, c \rangle \in {\it SFN}(b_{pre},{\cal P}) \ | \ c \ \wedge \ (\rho \ \sigma \ \neg b_{post}) \mbox{ satisfiable} \}$$ is not empty. It is partially correct otherwise. Implementation issues {#implementation} ===================== The CPBPV framework is parametrized by a list of solvers $(S_1,\ldots,S_k)$ which are tried in sequence, starting with the least expensive and less general. When checking satisfiability, the verifier never tries solver $S_{i+1},\ldots,S_{k}$ if solver $S_i$ is a decision procedure for the constraint store. If solver $S_i$ is not a decision procedure, it uses an abstraction $\alpha$ of the constraint store $c$ satisfying $c \Rightarrow \alpha$ and can still detect failed execution paths quickly. The last solver in the sequence is a constraint-programming solver (CP solver) over finite domains which iterates pruning and searching to find solutions or prove infeasibility. When the CP solver makes a choice, the earlier solvers in the sequence are called once again to prune the search space or find solutions if they have become decision procedures. Our prototype implementation uses a sequence $(MIP,CP)$, where MIP is the mixed integer-programming tool ILOG CPLEX[^2] and CP is the constraint-programming tool Ilog JSOLVER. Our Java implementation also performs some trivial simplifications such as constant propagation but is otherwise not optimized in its use of the solvers and in its renaming process whose speed and memory usage could be improved substantially. Practically, simplifications are done on the fly and the MIP solver is called at each node of the executable paths. The CP solver is only called at the end of the executable paths when the complete post condition is considered. Currently, the implementation use a depth-first strategy for the CP solver, but modern CP languages now offer high-level abstractions to implement other exploration strategies. In practice, when CPBPV is used for model checking as discussed below, it is probably advisable to use a depth-first iterative deepening implementation. Experimental results {#experimental} ==================== In this section, we report experimental results for a set of traditional benchmarks for program verification. We compare CPBVP with the following frameworks: - ESC/Java is an Extended Static Checker for Java to find common run-time errors in JML-annotated Java programs by static analysis of the code and its annotations. See http://kind.ucd.ie/products/opensource/ESCJava2/. - CBMC is a Bounded Model Checker for ANSI-C and C++ programs. It allows for the verification of array bounds (buffer overflows), pointer safety, exceptions, and user-specified assertions. See http://www.cprover.org/cbmc/. - BLAST, the Berkeley Lazy Abstraction Software Verification Tool, is a software model checker for C programs. See http://mtc.epfl.ch/software-tools/blast/. - EUREKA is a C bounded model checker which uses an SMT solver instead of an SAT solver. See http://www.ai-lab.it/eureka/. - Why is a software verification platform which integrates many existing provers (proof assistants such as Coq, PVS, HOL 4,...) and decision procedures such as Simplify, Yices, ...). See http://why.lri.fr/. Of course, neither the expressiveness nor the objectives of all these systems are the same as the one of CPBPV. For instance, some of them can handle CTL/LTL constraints whereas CPBPV dos not yet support this kind of constraints. Nevertheless, this comparison is useful to illustrate the capabilities of CPBPV. All experiments were performed on the same machine, an Intel(R) Pentium(R) M processor 1.86GHz with 1.5G of memory, using the version of the verifiers that can be downloaded from their web sites (except for EUREKA for which the execution times given in [@ABM07; @AMP06] are reported.) For each benchmark program, we describe the data entries and the verification parameters. In the tables, “UNABLE” means that the corresponding framework is unable to validate the program either because a lack of expressiveness or because of time or memory limitations, “NOT\_FOUND” that it does not detect an error, and “FALSE\_ERROR” that it reports an error in a correct program. Complete details of the experiments, including input files and error traces, can be found in [@CRV08]. #### **Binary search** We start with the binary search program presented in figure \[BsearchFig\]. ESC/Java is applied on the program described in Figure \[BsearchFig\]. ESC/Java requires a limit on the number of loop unfoldings, which we set to $log(n)+1$ which is the worst case complexity of binary search algorithm for an array of length $n$. Similarly, CBMC requires an overestimate of the number of loop unfoldings. Since CBMC does not support first-order expressions such as JML $\setminus forall$ statement, we generated a C program for each instance of the problem (i.e., each array length). For example, the postcondition for an array of length $8$ is given by (result!=-1 && a[result]==x)|| (result==-1 && (a[0]!=x&&a[1]!=x&&a[2]!=x&&a[3]!=x&&a[4]!=x&&a[5]!=x&&a[6]!=x&&a[7]!=x) For the Why framework, we used the binary search version given in their distribution. This program uses an assert statement to give a loop invariant. Note that CPBPV does not require any additional information: no invariant and no limits on loop unfoldings. During execution, it selects a path by nondeterministically applying the semantic rules for conditional and loop expressions. Table \[tabsearch\] reports the experimental results. Execution times for CPBPV are reported as a function of the array length for integers coded on 31 bits.[^3] Our implementation is neither optimized for time or space at this stage and times are only given to demonstrate the feasibility of the CPBPV verifier. The “Why” framework [@FiM07] was unable to verify the correctness without the loop invariant; 60% of the proof obligations remained unknown. The CBMC framework was not able to do the verification for an instance of length 32 (it was interrupted after 6691,87s). ESC/Java was unable to verify the correctness of this program unless complete loop invariants are provided [^4]. [[|c||l|c|c|c|c|c|c|]{}]{} \*[CPBPV]{} & array length & 8 & 16 & 32 & 64 & 128 & 256\ & time & 1.081s & 1.69s & 4.043s & 17.009s & 136.80s& 1731.696s\ \*[CBMC]{} & array length & 8 & 16 & 32 & 64 & 128 & 256\ & time & 1.37s & 1.43s & UNABLE & UNABLE &UNABLE &UNABLE\ \*[Why]{} & with invariant &\ & without invariant &\ ESC/Java &\ BLAST &\ #### **An Incorrect Binary search** Table \[tabsearchKO\] reports experimental results for an incorrect [*binary search*]{} program (see Figure \[BsearchFig\], line 11) for CPBPV, ESC/Java, CBMC, and Why using an invariant. The error trace found with CPBPV has been described in Section \[motivation\]. The error traces provided by CBMC and ESC/Java only show the decisions taken along the faulty path can be found in [@CRV08]. In contrast to CPBPV, they do not provide any value for the array nor the searched data. Observe that CPBPV provides orders of magnitude improvements in efficiency over CBMC and also outperforms ESC/Java by almost a factor 8 on the largest instance. CPBPV ESC/Java CBMC WHY with invariant BLAST ------------ -------- ---------- --------- -------------------- -------- -- length 8 0.027s 1.21 s 1.38s NOT\_FOUND UNABLE length 16 0.037s 1.347 s 1.69s NOT\_FOUND UNABLE length 32 0.064s 1.792 s 7.62s NOT\_FOUND UNABLE length 64 0.115s 1.886 s 27.05s NOT\_FOUND UNABLE length 128 0.241s 1.964 s 189.20s NOT\_FOUND UNABLE : Experimental Results for an Incorrect Binary Search[]{data-label="tabsearchKO"} #### **The Tritype Program** The tritype program is a standard benchmark in test case generation and program verification since it contains numerous non-feasible paths: only 10 paths correspond to actual inputs because of complex conditional statements in the program. The program takes three positive integers as inputs (the triangle sides) and returns 2 if the inputs correspond to an isosceles triangle, 3 if they correspond to an equilateral triangle, 1 if they correspond to some other triangle, and 4 otherwise. The [ tritype]{} program in Java with its specification in JML can be found in[@CRV08]. Table \[tabTritype\] depicts the experimental results for CPBPV, ESC/Java, CBMC, BLAST and Why. BLAST was unable to validate this example because the current version does not handle linear arithmetic. Observe the excellent performance of CPBPV and note that our previous approach using constraint programming and Boolean abstraction to abstract the conditions, validated this benchmark in $8.52$ seconds when integers were coded on 16 bits [@CoR07]. It also explored 92 spurious paths. CPBPV ESC/Java CBMC Why BLAST ------ -------- ---------- ------- ------- -------- time 0.287s 1.828s 0.82s 8.85s UNABLE : Experimental Results on the Tritype Program[]{data-label="tabTritype"} #### **An Incorrect Tritype Program** Consider now an incorrect version of [*Tritype*]{} program in which the test [*“if ((trityp==2)&&(i+k$>$j))”*]{} in line 22 (see [@CRV08]) is replaced by [*“if ((trityp==1)&&(i+k$>$j))”*]{}. Since the local variable [ *trityp*]{} is equal to [*2*]{} when [*i==k*]{}, the condition [ *(i+k)$>$j*]{} implies that [*(i,j,k)*]{} are the sides of an isosceles triangle (the two other triangular inequalities are trivial because j$>$0). But, when [*trityp=1*]{}, [*i==j*]{} holds and this incorrect version may answer that the triangle is isosceles while it may not be a triangle at all. For example, it will return [*2*]{} when [ *(i,j,k)=(1,1,2)*]{}. Table \[tabTritypeKO\] depicts the experimental results. Execution times correspond to the time required to find the first error. The error found with CPBPV corresponds to input values $(i,j,k)=(1,1,2)$ mentioned earlier. Once again, observe the excellent behavior of CPBPV compared to the remaining tools. [^5] CPBPV ESC/Java CBMC WHY ------ ---------- ---------- ------------ ------------ time 0.056s s 1.853s NOT\_FOUND NOT\_FOUND : Experimental Results for the Incorrect Tritype Program[]{data-label="tabTritypeKO"} #### **Bubble Sort with initial condition** This benchmark (see [@CRV08]) is taken from [@ABM07] and performs a bubble sort of an array $t$ which contains integers from $0$ to $t.length$ given in decreasing order. Table \[tabbuble\] shows the comparative results for this benchmark. CPBPV was limited on this benchmark because its recursive implementation uses up all the JAVA stack space. This problem should be remedied by removing recursion in CPBPV. CPBPV ESC/Java CBMC EUREKA ----------- -------- ---------- -------- -------- length 8 1.45s 3.778 s 1.11s 91s length 16 2.97s UNABLE 2.01s UNABLE length 32 UNABLE UNABLE 6.10s UNABLE length 64 UNABLE UNABLE 37.65s UNABLE : Experimental Results for Bubble Sort[]{data-label="tabbuble"} #### **Selection Sort** We now present a benchmark to highlight both modular verification and the [element]{} constraint of constraint programming to index arrays with arbitrary expressions. The benchmark described in [@CRV08]. Assume that function `findMin` has been verified for arbitrary integers. When encountering a call to `findMin`, CPBPV first checks if its precondition is entailed by the constraint store, which requires a consistency check of the constraint store with respect to the negation of the precondition. Then CPBPV replaces the call by the post-condition where the formal parameters are replaced by the actual variables. In particular, for the first iteration of the loop and an array length of 40, CPBPV generates the conjunction $ 0 \leq k^0 < 40 \; \wedge \; t^0[k^0] \leq t^0[0] \; \wedge \; \ldots \; \wedge \; t^0[k^0] \leq t^0[39] $ which features [element]{} constraint [@VanHentenryck89]. Indeed, $k^0$ is a variable and a constraint like $t^0[k^0] \leq t^0[0]$ indexes the array $t^0$ of variables using $k^0$. The modular verification of the selection sort explores only a single path, is independent of the integer representation, and takes less than $0.01s$ for arrays of size 40. The bottleneck in verifying selection sort is the validation of function `findMin`, which requires the exploration of many paths. However the complete validation of selection sort takes less than 4 seconds for an array of length 6. Once again, this should be contrasted with the model-checking approach of Eureka [@ABM07]. On a version of selection sort where all variables are assigned specific values (contrary to our verification which makes no assumptions on the inputs), Eureka takes 104 seconds on a faster machine. Reference [@ABM07] also reports that CBMC takes 432.6 seconds, that BLAST cannot solve this problem, and that SATABS [@CKS05] only verifies the program for an array with 2 elements. #### **Sum of Squares** Our last benchmark is described in [@CRV08] and computes the sum of the square of the $n$ first integers stored in an array. The precondition states that $n$ is the size of the array and that $t$ must contain any possible permutation of the $n$ first integers. The postcondition states that the result is $n\times(n+1)\times(2\times n+1)/6$. The benchmark illustrates two functionalities of constraint programming: the ability of specifying combinatorial constraints and of solving nonlinear problems. The `alldifferent` constraint[@Reg94] in the pre-condition specifies that all the elements of the array are different, while the program constraints and postcondition involves quadratic and cubic constraints. The maximum instance that we were able to solve with CPBPV was an array of size 10 in 66.179s. CPLEX, the MIP solver, plays a key role in all these benchmarks. For instance, the CP solver is never called in the Tritype benchmark. For the Binary search benchmark, there are length calls to the CP solver but almost 75% of the CPU time is spent in the CP solver. Since there is only path in the Buble sort benchmark, the CP solver is only called once. In the Sum of squares example, 80% of the CPU time is spent in the CP solver. Discussion and Related Work {#related} =========================== We briefly review recent work in constraint programming and model checking for software testing, validation, and verification. We outline the main differences between our CPBPV framework and existing approaches. #### **Constraint Logic Programming** Constraint logic programming (CLP) was used for test generation of programs (e.g., [@GBR98; @JaV00; @SyD01; @GLM08]) and provides a nice implementation tool extending symbolic execution techniques [@BGM06]. Gotlieb et al. showed how to represent imperative programs as constraint logic programs and used predicate abstraction (from model checking) and conditional constraints within a CLP framework. Flanagan [@Fla04] formalized the translation of imperative programs into CLP, argued that it could be used for bounded model checking, but did not provide an implementation. The test-generation methodology was generalized and applied to bounded program verification in [@CoR06; @CoR07]. The implementation used dedicated predicate abstractions to reduce the exploration of spurious execution paths. However, as shown in the paper, the CPBPV verifier is significantly more efficient and often avoids the generation of spurious execution paths completely. #### **Model Checking** It is also useful to contrast the CPBPV verifier with model-checking of software systems. SAT-based bounded model checking for software[@CBR01] consists in building a propositional formula whose models correspond to execution paths of bounded length violating some properties and in using SAT solvers to check whether the resulting formula is satisfiable. SAT-based model-checking platforms [@CBR01] have been widely popular thanks to significant progress in SAT solvers. A fundamental issue faced by model checkers is the state space explosion of the resulting model. Various techniques have been proposed to address this challenge, including generalized symbolic execution (e.g., [@KPV03]), SMT-based model checking, and abstraction/refinement techniques. SMT-based model checking is the idea of representing and checking quantifier-free formulas in a more general decidable theory (e.g. [@GHN04; @DuM06; @NOR07]). These SMT solvers integrate dedicated solvers and share some of the motivations of constraint programming. Predicate abstraction is another popular technique to address the state space explosion. The idea consists in abstracting the program to obtain an abstract program on which model checking is performed. The model checker may then generate an abstract counterexample which must be checked to determine if it corresponds to a concrete execution path. If the counterexample is spurious, the abstract program is refined and the process is iterated. A successful predicate abstraction consists of abstracting the concrete program into a Boolean program (e.g., [@BPR01; @CKL04; @CKS04]). In recent work [@AMP06; @ABM07], Armando & al proposed to abstract concrete programs into linear programs and used an abstraction of sets of variables and array indices. They showed that their tool compares favourably and, on some of the programs considered in this paper, outperforms model checkers based on predicate abstraction.\ Our CPBPV verifier contrasts with SAT-based model checkers, SMT-based model checkers and predicate abstraction based approaches: It does not abstract the program and does not generate spurious execution paths. Instead it uses a constraint-solver and nondeterministic exploration to incrementally construct abstractions of execution paths. The abstraction uses constraint stores to represent sets of concrete stores. On many bounded verification benchmarks, our preliminary experimental results show significant improvements over the state-of-the-art results in [@ABM07]. Model checking is well adapted to check low-level C program and hardware applications with numerous Boolean constraints and bitwise operations: It was successfully used to compare an ANSI C program with a circuit given as design in Verilog [@CKL04]. However, it is important to observe that in model checking, one is typically interested in checking some specific properties such as buffer overflows, pointer safety, or user-specified assertions. These properties are typically much less detailed than our post-conditions and abstracting the program may speed up the process significantly. In our CPBPV verifier, it is critical to explore all execution paths and the main issue is how to effectively abstract memory stores by constraints and how to check satisfiability incrementally. It is an intriguing issue to determine whether an hybridization of the two approaches would be beneficial for model checking, an issue briefly discussed in the next section. Observe also that this research provides convincing evidence of the benefits of Nieuwenhuis’ challenge [@NOR07] aiming at extending SMT[^6] with CP techniques. Perspectives and Future Work {#conclusion} ============================ This paper introduced the CPBPV framework for bounded program verification. Its novelty is to use constraints to represent sets of memory stores and to explore execution paths over these constraint stores nondeterministically and incrementally. The CPBPV verifier exploits the fact that, when variables and arrays are bounded, the constraint store can always be checked for feasibility. As a result, it never explores spurious execution path contrary to earlier approaches combining constraint programming and predicate abstraction [@CoR06; @CoR07] or integrating SMT solvers and the abstraction/refinement approach from model checking [@ABM07]. We demonstrated the CPBPV verifier on a number of standard benchmarks from model checking and program checking as well as on nonlinear programs and functions using complex array indexings, and showed how to perform modular verification. The experimental results demonstrate the potential of the approach: The CPBPV verifier provides significant gain in performance and functionalities compared to other tools. Our current work aims at improving and generalizing the framework and implementation. In particular, we would like to include tailored, light-weight solvers for a variety of constraint classes, the optimization of the array implementation, and the integration of Java objects and references. There are also many research avenues opened by this research, two of which are reviewed now. Currently, the CPBPV verifier does not check for variable overflows: the constraint store enforces that variables take values inside their domains and execution paths violating these constraints are thus not considered. It is possible to generalize the CPBPV verifier to check overflows as the verification proceeds. The key idea is to check before each assignment if the constraint store entails that the value produced fits in the selected integer representation and generate an error otherwise. (Similar assertions must in fact be checked for each subexpression in the right hand-side in the language evaluation order. Interval techniques on floats [@BGM06] may be used to obtain conservative checking of such assertions. An intriguing direction is to use the CPBPV approach for properties checking. Given an assertion to be verified, one may perform a backward execution from the assertion to the function entry point. The negation of the assertion is now the pre-condition and the pre-condition becomes the post-condition. This requires to specify inverse renaming and executions of conditional and iterative statements but these have already been studied in the context of test generation. #### **Acknowledgements** Many thanks to Jean-François Couchot for many helps on the use of the [*[Why]{}*]{} framework. [10]{} Aït-Kaci H., Berstel B., Junker U., Leconte M., Podelski A. : Satisfiability Modulo Structures as Constraint Satisfaction : An Introduction. Procs of JFLA 2007. Armando A., Benerecetti M., and Montovani J. Abstraction Refinement of Linear Programs with Arrays. Proceedings of TACAS 2007, LNCS 4424: 373–388. Armando A., Mantovani J., and Platania L. Bounded Model Checking of C Programs using a SMT solver instead of a SAT solver. Proc. SPIN’06. LNCS 3925, Pages 146-162. Botella B., Gotlieb A., Michel C. Symbolic execution of floating-point computations. Software Testing, Verification and Reliability. 16:2:97–121.2006. Thomas Ball, Andreas Podelski, Sriam K. Rajamani Boolean and Cartesian Abstraction for Model Checking C Programs. Proc. of TACAS 2001. E. Clarke, A. Biere, R. Raimi, and Y. Zhu. Bounded Model Checking using Satisfiability Solving. FMSD, 19(1):7–34, 2001. Clarke E., Kroening D., Lerda F. : A Tool for Checking ANSI-C programs. Tacas 2004, LNCS 2988, pp 168-176, 2004 Clarke E., Kroening D., Sharygina N., Yorav K. : Predicate abstraction of ANSI-C Programs using SAT. FMSD, 25:105–127, 2004 Clarke E., Kroening D., Sharygina N., Yorav K. : SATABS: SAT-Based Predicate Abstraction for ANSI-C. TACAS’05, 570–574, 2005. Cytron R., Ferrante J., Rosen B., Wegman M., and Zadeck K. : fficently [C]{}omputing [S]{}tatic [S]{}ingle [A]{}ssignment [F]{}orm and the [C]{}ontrol [D]{}ependence [G]{}raph. , 13(4):451–490, October 1991. Collavizza H. and Rueher M. : Software Verification using Constraint Programming Techniques. Procs of TACAS 2006, LNCS 3920: 182-196. Collavizza H. and Rueher M. : Exploring different constraint-based modelings for program verification Procs of CP 2007, LNCS 3920: 182-196 Collavizza H. Rueher M., Van Hentenryck P. : Comparison between CPBPV with ESC/Java, CBMC, Blast, EUREKA and Why. http://www.i3s.unice.fr/\~rueher/verificationBench.pdf Bruno Dutertre and Leonardo Mendon¸ca de Moura. A fast linear-arithmetic solver for DPLL(T). CAV 2006, pages 81–94. LNCS 4144. Cormac Flanagan, “Automatic software model checking via constraint logic” (2004). Science of Computer Programming. 50 (1-3), pp. 253-270. Filliâtre J.C., Claude Marché.The Why/Krakatoa/Caduceus Platform for Deductive Program Verification Proc. CAV’2007, LNCS 4590. pp 173-177. Gotlieb A., Botella B. and Rueher M : Automatic Test Data Generation using Constraint Solving Techniques. Proc. ISSTA 98, ACM SIGSOFT (2), 1998. Ganzinger H., Hagen G., Nieuwenhuis R.,Oliveras A., and Tinelli C.: DPLL(T): Fast Decision Procedures. Proc. of CAV 2004, 175-188, 2004. P. Godefroid, M. Y. Levin, D. Molnar: Automated Whitebox Fuzz Testing, NDSS 2008, Network and Distributed System Security Symposium. Daniel Jackson and Mandana Vaziri, Finding Bugs with a Constraint Solver, ACM SIGSOFT Symposium on Software Testing and Analysis, 14–15, 2000. Khurshid, S., Pasareanu, C.S., and Vissser, W. “Generalized Symbolic Execution for Model Checking and Testing”, in TACAS 2003, Warsaw, Poland. R. Nieuwenhuis, A. Oliveras, E. Rodríguez-Carbonell and A. Rubio: Challenges in Satisfiability Modulo Theories. Invited Talk. RTA 2007, LNCS 4533, pp 2-18. J-C. Régin. A filtering algorithm for constraints of difference in CSPs. AAAI-94, Seattle, WA, USA, pp 362–367, 1994. Sy N.T. and Deville Y.: Automatic Test Data Generation for Programs with Integer and Float Variables. Proc of. 16th IEEE ASE01, 2001. VanHentenryck P. (1989) Constraint Satisfaction in Logic Programming, MIT Press. Numerica: A Modeling Language for Global Optimization Pascal Van Hentenryck, Laurent Michel, Yves Deville. MIT Press, 1997. [^1]: We omit the domain constraints on the variables for simplicity. [^2]: See http://www.ilog.com/products. [^3]: The commercial MIP solver fails with 32-bit domains because of scaling issues. [^4]: a version with loop invariants that allows to show the correctness of this program has been written by David Cok, a developper of ESC/Java, after we contacted him. [^5]: For CBMC, we have contacted D. Kroening who has recommended to use the option CPROVER\_assert. If we do so, CBMC is able to find the error, but we must add some assumptions to mean that there is no overflow into the sums, in order to prove the correct version of tritype with this same option. [^6]: See also [@ABJ07] for a study of the relations between constraint programming and Satisfiability Modulo Theories (SMT)

ArXiv

--- abstract: 'Frenkel and Reshetikhin [@Fre] introduced $q$-characters to study finite dimensional representations of the quantum affine algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. In the simply laced case Nakajima [@Naa][@Nab] defined deformations of $q$-characters called $q,t$-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non necessarily simply laced) new approach to $q,t$-characters motivated by the deformed screening operators [@Her01]. The $t$-deformations are naturally deduced from the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$: the parameter $t$ is analog to the central charge $c\in{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. The $q,t$-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.' address: 'David Hernandez: École Normale Supérieure - DMA, 45, Rue d’Ulm F-75230 PARIS, Cedex 05 FRANCE' author: - David Hernandez title: 'Algebraic Approach to $q,t$-Characters' --- Introduction ============ We suppose $q\in{\ensuremath{\mathbb{C}}}^*$ is not a root of unity. In the case of a semi-simple Lie algebra ${\mathfrak{g}}$, the structure of the Grothendieck ring $\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))$ of finite dimensional representations of the quantum algebra ${\mathcal{U}}_q({\mathfrak{g}})$ is well understood. It is analogous to the classical case $q=1$. In particular we have ring isomorphisms: $$\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))\simeq \text{Rep}({\mathfrak{g}})\simeq {\mathbb{Z}}[\Lambda]^W\simeq {\mathbb{Z}}[T_1,...,T_n]$$ deduced from the injective homomorphism of characters $\chi$: $$\chi(V)=\underset{\lambda\in\Lambda}{\sum}\text{dim}(V_{\lambda})\lambda$$ where $V_{\lambda}$ are weight spaces of a representation $V$ and $\Lambda$ is the weight lattice. For the general case of Kac-Moody algebras the picture is less clear. In the affine case ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$, Frenkel and Reshetikhin [@Fre] introduced an injective ring homomorphism of $q$-characters: $$\chi_q:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\rightarrow {\mathbb{Z}}[Y_{i,a}^{\pm}]_{1\leq i\leq n,a\in{\ensuremath{\mathbb{C}}}^*}={\mathcal{Y}}$$ The homomorphism $\chi_q$ allows to describe the ring $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\simeq{\mathbb{Z}}[X_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$, where the $X_{i,a}$ are fundamental representations. It particular $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is commutative. The morphism of $q$-characters has a symmetry property analogous to the classical action of the Weyl group $\text{Im}(\chi)={\mathbb{Z}}[\Lambda]^W$: Frenkel and Reshetikhin defined $n$ screening operators $S_i$ such that $\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$ (the result was proved by Frenkel and Mukhin for the general case in [@Fre2]). In the simply laced case Nakajima introduced $t$-analogues of $q$-characters ([@Naa], [@Nab]): it is a ${\mathbb{Z}}[t^{\pm}]$-linear map $$\chi_{q,t}:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]\rightarrow{\mathcal{Y}}_t={\mathbb{Z}}[Y_{i,a}^{\pm},t^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$$ which is a deformation of $\chi_q$ and multiplicative in a certain sense. A combinatorial axiomatic definition of $q,t$-characters is given. But the existence is non-trivial and is proved with the geometric theory of quiver varieties which holds only in the simply laced case. In [@Her01] we introduced $t$-analogues of screening operators $S_{i,t}$ such that in the simply laced case: $$\underset{i\in I}{\bigcap}\text{Ker}(S_{i,t})=\text{Im}(\chi_{q,t})$$ It is a first step in the algebraic approach to $q,t$-characters proposed in this article: we define and construct $q,t$-characters in the general (non necessarily simply laced) case. The motivation of the construction appears in the non-commutative structure of the Cartan subalgebra ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$, the study of screening currents and of deformed screening operators. As an application we construct a deformed algebra structure and an involution of the Grothendieck ring, and analogues of Kazhdan-Lusztig polynomials in the general case in the same spirit as Nakajima did for the simply laced case. In particular this article proves a conjecture that Nakajima made for the simply laced case (remark 3.10 in [@Nab]): there exists a purely combinatorial proof of the existence of $q,t$-characters. This article is organized as follows: after some backgrounds in section \[back\], we define a deformed non-commutative algebra structure on ${\mathcal{Y}}_t={\mathbb{Z}}[Y_{i,a}^{\pm},t^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$ (section \[defoal\]): it is naturally deduced from the relations of ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ (theorem \[dessus\]) by using the quantization in the direction of the central element $c$. In particular in the simply laced case it can be used to construct the deformed multiplication of Nakajima [@Nab] (proposition \[form\]) and of Varagnolo-Vasserot [@Vas] (section \[varva\]). This picture allows us to introduce the deformed screening operators of [@Her01] as commutators of Frenkel-Reshetikhin’s screening currents of [@Freb] (section \[scr\]). In [@Her01] we gave explicitly the kernel of each deformed screening operator (theorem \[her\]). In analogy to the classic case where $\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$, we have to describe the intersection of the kernels of deformed screening operators. We introduce a completion of this intersection (section \[complesection\]) and give its structure in proposition \[thth\]. It is easy to see that it is not too big (lemma \[leasto\]); but the point is to prove that it contains enough elements: it is the main result of our construction in theorem \[con\] which is crucial for us. It is proved by induction on the rank $n$ of ${\mathfrak{g}}$. We define a $t$-deformed algorithm (section \[defialgo\]) analog to the Frenkel-Mukhin’s algorithm [@Fre2] to construct $q,t$-characters in the completion of ${\mathcal{Y}}_t$. An algorithm was also used by Nakajima in the simply laced case in order to compute the $q,t$-characters for some examples ([@Naa]) assuming they exist (which was geometrically proved). Our aim is different : we do not know [*a priori*]{} the existence in the general case. That is why we have to show the algorithm is well defined, never fails (lemma \[nfail\]) and gives a convenient element (lemma \[conv\]). This construction gives $q,t$-characters for fundamental representations; we deduce from them the injective morphism of $q,t$-characters $\chi_{q,t}$ (definition \[mqt\]). We study the properties of $\chi_{q,t}$ (theorem \[axiomes\]). Some of them are generalization of the axioms that Nakajima defined in the simply laced case ([@Nab]); in particular we have constructed the morphism of [@Nab]. We have some applications: the morphism gives a deformation of the Grothendieck ring because the image of $\chi_{q,t}$ is a subalgebra for the deformed multiplication (section \[quanta\]). Moreover we define an antimultiplicative involution of the deformed Grothendieck ring (section \[invo\]); the construction of this involution is motivated by the new point view adopted in this paper : it is just replacing $c$ by $-c$ in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. In particular we define constructively analogues of Kazhdan-Lusztig polynomials and a canonical basis (theorem \[expol\]) motivated by the introduction of [@Nab]. We compute explicitly the polynomials for some examples. In section \[quest\] we raise some questions : we conjecture that the coefficients of $q,t$-characters are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]\subset{\mathbb{Z}}[t^{\pm}]$. In the $ADE$-case it a result of Nakajima; we give an alternative elementary proof for the $A$-cases in section \[acase\]. The cases $G_2, B_2, C_2$ are also checked in section \[fin\]. The cases $F_4, B_n, C_n$ ($n\leq 10$) have been checked on a computer. We also conjecture that the generalized analogues to Kazhdan-Lusztig polynomials give at $t=1$ the multiplicity of simple modules in standard modules. We propose some generalizations and further applications which will be studied elsewhere. In the appendix (section \[fin\]) we give explicit computations of $q,t$-characters for semi-simple Lie algebras of rank 2. They are used in the proof of theorem \[con\]. For convenience of the reader we give at the end of this article an index of notations defined in the main body of the text. [**Acknowledgments.**]{} The author would like to thank M. Rosso for encouragements and precious comments on a previous version of this paper, I. B. Frenkel for having encouraged him in this direction, E. Frenkel for encouragements, useful discussions and references, E. Vasserot for very interesting explanations about [@Vas], O. Schiffmann for valuable comments and his kind hospitality in Yale university, and T. Schedler for help on programming. Background {#back} ========== Cartan matrix {#recalu} ------------- A generalized Cartan matrix of rank $n$ is a matrix $C=(C_{i,j})_{1\leq i,j\leq n}$\[carmat\] such that $C_{i,j}\in{\mathbb{Z}}$ and: $$C_{i,i}=2$$ $$i\neq j\Rightarrow C_{i,j}\leq 0$$ $$C_{i,j}=0\Leftrightarrow C_{j,i}=0$$ Let $I = \{1,...,n\}$. We say that $C$ is symmetrizable if there is a matrix $D=\text{diag}(r_1,...,r_n)$ ($r_i\in{\ensuremath{\mathbb{N}}}^*$) such that $B=DC$\[symcar\] is symmetric. Let $q\in{\ensuremath{\mathbb{C}}}^*$ be the parameter of quantization. In the following we suppose it is not a root of unity. $z$ is an indeterminate.\[qz\] If $C$ is symmetrizable, let $q_i=q^{r_i}$, $z_i=z^{r_i}$ and $C(z)=(C(z)_{i,j})_{1\leq i,j\leq n}$ the matrix with coefficients in ${\mathbb{Z}}[z^{\pm}]$ such that: $$C(z)_{i,j}=[C_{i,j}]_z\text{ if $i\neq j$}$$ $$C(z)_{i,i}=[C_{i,i}]_{z_i}=z_i+z_i^{-1}$$ where for $l\in{\mathbb{Z}}$ we use the notation: $$[l]_z=\frac{z^l-z^{-l}}{z-z^{-1}}\text{ ($=z^{-l+1}+z^{-l+3}+...+z^{l-1}$ for $l\geq 1$)}$$ In particular, the coefficients of $C(z)$ are symmetric Laurent polynomials (invariant under $z\mapsto z^{-1}$). We define the diagonal matrix $D_{i,j}(z)=\delta_{i,j}[r_i]_z$ and the matrix $B(z)=D(z)C(z)$. In the following we suppose that $C$ is of finite type, in particular $\text{det}(C)\neq 0$. In this case $C$ is symmetrizable; if $C$ is indecomposable there is a unique choice of $r_i\in{\ensuremath{\mathbb{N}}}^*$ such that $r_1\wedge...\wedge r_n=1$. We have $B_{i,j}(z)=[B_{i,j}]_z$ and $B(z)$ is symmetric. See [@bou] or [@Kac] for a classification of those finite Cartan matrices. We say that $C$ is simply-laced if $r_1=...=r_n=1$. In this case $C$ is symmetric, $C(z)=B(z)$ is symmetric. In the classification those matrices are of type $ADE$. Denote by $\mathfrak{U}\subset{\ensuremath{\mathbb{Q}}}(z)$\[mathu\] the subgroup ${\mathbb{Z}}$-linearly spanned by the $\frac{P(z)}{Q(z^{-1})}$ such that $P(z)\in{\mathbb{Z}}[z^{\pm}]$, $Q(z)\in{\mathbb{Z}}[z]$, the zeros of $Q(z)$ are roots of unity and $Q(0)=1$. It is a subring of ${\ensuremath{\mathbb{Q}}}(z)$, and for $R(z)\in\mathfrak{U},m\in{\mathbb{Z}}$ we have $R(q^m)\in\mathfrak{U}$ and $R(q^m)\in{\ensuremath{\mathbb{C}}}$ makes sense. It follows from lemma 1.1 of [@Fre2] that $C(z)$ has inverse $\tilde{C}(z)$\[invcar\] with coefficients of the form $R(z)\in\mathfrak{U}$. Finite quantum algebras ----------------------- We refer to [@Ro] for the definition of the finite quantum algebra ${\mathcal{U}}_q({\mathfrak{g}})$ associated to a finite Cartan matrix, the definition and properties of the type $1$-representations of ${\mathcal{U}}_q({\mathfrak{g}})$, the Grothendieck ring $\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))$ and the injective ring morphism of characters $\chi:\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))\rightarrow {\mathbb{Z}}[y_i^{\pm}]$. Quantum affine algebras ----------------------- The quantum affine algebra associated to a finite Cartan matrix $C$ is the ${\ensuremath{\mathbb{C}}}$-algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$\[qadefi\] defined (Drinfeld new realization) by generators $x_{i,m}^{\pm}$ ($i\in I$, $m\in{\mathbb{Z}}$), $k_i^{\pm}$ ($i\in I$), $h_{i,m}$ ($i\in I$, $m\in{\mathbb{Z}}^*$), central elements $c^{\pm\frac{1}{2}}$, and relations: $$k_ik_j=k_jk_i$$ $$k_ih_{j,m}=h_{j,m}k_i$$ $$k_ix_{j,m}^{\pm}k_i^{-1}=q^{\pm B_{ij}}x_{j,m}^{\pm}$$ $$[h_{i,m},x_{j,m'}^{\pm}]=\pm \frac{1}{m}[mB_{ij}]_qc^{\mp\frac{\mid m\mid}{2}} x_{j,m+m'}^{\pm}$$ $$x_{i,m+1}^{\pm}x_{j,m'}^{\pm}-q^{\pm B_{ij}}x_{j,m'}^{\pm}x_{i,m+1}^{\pm}=q^{\pm B_{ij}}x_{i,m}^{\pm}x_{j,m'+1}^{\pm}-x_{j,m'+1}^{\pm}x_{i,m}^{\pm}$$ $$[h_{i,m},h_{j,m'}]=\delta_{m,-m'}\frac{1}{m}[mB_{ij}]_q\frac{c^m-c^{-m}}{q-q^{-1}}$$ $$[x_{i,m}^+,x_{j,m'}^-]= \delta_{ij}\frac{c^{\frac{m-m'}{2}}\phi^+_{i,m+m'}-c^{-\frac{m-m'}{2}}\phi^-_{i,m+m'}}{q_i-q_i^{-1}}$$ $$\underset{\pi\in \Sigma_s}{\sum}\underset{k=0..s}{\sum}(-1)^k\begin{bmatrix}s\\k\end{bmatrix}_{q_i}x_{i,m_{\pi(1)}}^{\pm}...x_{i,m_{\pi(k)}}^{\pm}x_{j,m'}^{\pm}x_{i,m_{\pi(k+1)}}^{\pm}...x_{i,m_{\pi(s)}}^{\pm}=0$$ where the last relation holds for all $i\neq j$, $s=1-C_{ij}$, all sequences of integers $m_1,...,m_s$. $\Sigma_s$ is the symmetric group on $s$ letters. For $i\in I$ and $m\in{\mathbb{Z}}$, $\phi_{i,m}^{\pm}\in {\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ is determined by the formal power series in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})[[u]]$ (resp. in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})[[u^{-1}]]$): $$\underset{m=0..\infty}{\sum}\phi_{i,\pm m}^{\pm}u^{\pm m}=k_i^{\pm}\text{exp}(\pm(q-q^{-1})\underset{m'=1..\infty}{\sum}h_{i,\pm m'}u^{\pm m'})$$ and $\phi_{i,m}^+=0$ for $m<0$, $\phi_{i,m}^-=0$ for $m>0$. One has an embedding ${\mathcal{U}}_q({\mathfrak{g}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ and a Hopf algebra structure on ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ (see [@Fre] for example). The Cartan algebra ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$\[qhdefi\] is the ${\ensuremath{\mathbb{C}}}$-subalgebra of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ generated by the $h_{i,m},c^{\pm}$ ($i\in I, m\in{\mathbb{Z}}-\{0\}$). Finite dimensional representations of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ ----------------------------------------------------------------------------- A finite dimensional representation $V$ of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ is called of type $1$ if $c$ acts as $\text{Id}$ and $V$ is of type $1$ as a representation of ${\mathcal{U}}_q({\mathfrak{g}})$. Denote by $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the Grothendieck ring of finite dimensional representations of type $1$. The operators $\{\phi^{\pm}_{i,\pm m},i\in I, m\in{\mathbb{Z}}\}$ commute on $V$. So we have a pseudo-weight space decomposition: $$V=\underset{\gamma\in {\ensuremath{\mathbb{C}}}^{I\times {\mathbb{Z}}}\times{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}}{\bigoplus} V_{\gamma}$$ where for $\gamma=(\gamma^+,\gamma^-)$, $V_{\gamma}$ is a simultaneous generalized eigenspace: $$V_{\gamma}=\{x\in V/\exists p\in{\ensuremath{\mathbb{N}}},\forall i\in\{1,...,n\},\forall m\in{\mathbb{Z}},(\phi_{i,m}^{\pm}-\gamma_{i,m}^{\pm})^p.x=0\}$$ The $\gamma_{i,m}^{\pm}$ are called pseudo-eigen values of $V$. ([**Chari, Pressley**]{} [@Cha],[@Cha2]) Every simple representation $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is a highest weight representation $V$, that is to say there is $v_0\in V$ (highest weight vector) $\gamma_{i,m}^{\pm}\in{\ensuremath{\mathbb{C}}}$ (highest weight) such that: $$V={\mathcal{U}}_q(\hat{{\mathfrak{g}}}).v_0\text{ , }c^{\frac{1}{2}}.v_0=v_0$$ $$\forall i\in I,m\in{\mathbb{Z}},x_{i,m}^+.v_0=0\text{ , }\text{ , }\phi_{i,m}^{\pm}.v_0=\gamma_{i,m}^{\pm}v_0$$ Moreover we have an $I$-uplet $(P_i(u))_{i\in I}$ of (Drinfeld-)polynomials such that $P_i(0)=1$ and: $$\gamma_i^{\pm}(u)=\underset{m\in{\ensuremath{\mathbb{N}}}}{\sum}\gamma_{i,\pm m}^{\pm}u^{\pm}=q_i^{\deg(P_i)}\frac{P_i(uq_i^{-1})}{P_i(uq_i)}\in{\ensuremath{\mathbb{C}}}[[u^{\pm}]]$$ and $(P_i)_{i\in I}$ parameterizes simple modules in $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$. ([**Frenkel, Reshetikhin**]{} [@Fre]) The eigenvalues $\gamma_i(u)^{\pm}\in{\ensuremath{\mathbb{C}}}[[u]]$ of a representation $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ have the form: $$\gamma_i^{\pm}(u)=q_i^{deg(Q_i)-deg(R_i)}\frac{Q_i(uq_i^{-1})R_i(uq_i)}{Q_i(uq_i)R_i(uq_i^{-1})}$$ where $Q_i(u),R_i(u)\in{\ensuremath{\mathbb{C}}}[u]$ and $Q_i(0)=R_i(0)=1$. Note that the polynomials $Q_i,R_i$ are uniquely defined by $\gamma$. Denote by $Q_{\gamma,i}$, $R_{\gamma,i}$ the polynomials associated to $\gamma$. q-characters {#qcar} ------------ Let ${\mathcal{Y}}$ be the commutative ring ${\mathcal{Y}}={\mathbb{Z}}[Y_{i,a}^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$. For $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ a representation, the $q$-character $\chi_q(V)$\[chiqdefi\] of $V$ is: $$\chi_q(V)=\underset{\gamma}{\sum}\text{dim}(V_{\gamma})\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}Y_{i,a}^{\lambda_{\gamma,i,a}-\mu_{\gamma,i,a}}\in {\mathcal{Y}}$$ where for $\gamma\in{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}\times{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}$, $i\in I$, $a\in{\ensuremath{\mathbb{C}}}^*$ the $\lambda_{\gamma,i,a},\mu_{\gamma,i,a}\in{\mathbb{Z}}$ are defined by: $$Q_{\gamma, i}(z)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-za)^{\lambda_{\gamma,i,a}}\text{ , }R_{\gamma, i}(z)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-za)^{\mu_{\gamma,i,a}}$$ ([**Frenkel, Reshetikhin**]{} [@Fre]) The map $$\chi_q:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\rightarrow {\mathcal{Y}}$$ is an injective ring homomorphism and the following diagram is commutative: $$\begin{array}{rcccl} \text{Rep} ({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))&\stackrel{\chi_q}{\longrightarrow}&{\mathbb{Z}}[Y^{\pm}_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}\\ \downarrow res&&\downarrow\beta\\ \text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))&\stackrel{\chi}{\longrightarrow}&{\mathbb{Z}}[y^{\pm}_i]_{i\in I}\\\end{array}$$ where $\beta$ is the ring homomorphism such that $\beta(Y_{i,a})=y_i$ ($i\in I,a\in{\ensuremath{\mathbb{C}}}^*$). For $m\in{\mathcal{Y}}$ of the form $m=\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}Y_{i,a}^{u_{i,a}(m)}$ ($u_{i,a}(m)\geq 0$), denote $V_m\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the simple module with Drinfeld polynomials $P_i(u)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-ua)^{u_{i,a}(m)}$. In particular for $i\in I,a\in{\ensuremath{\mathbb{C}}}^*$ denote $V_{i,a}=V_{Y_{i,a}}$ and $X_{i,a}=\chi_q(V_{i,a})$. The simple modules $V_{i,a}$ are called fundamental representations. Denote by $M_m\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the module $M_m=\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\bigotimes}V_{i,a}^{\otimes u_{i,a}(m)}$. It is called a standard module and his $q$-character is $\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}X_{i,a}^{u_{i,a}(m)}$. ([**Frenkel, Reshetikhin**]{} [@Fre]) The ring $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is commutative and isomorphic to ${\mathbb{Z}}[X_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$. ([**Frenkel, Mukhin**]{} [@Fre2])\[aidafm\] For $i\in I,a\in{\ensuremath{\mathbb{C}}}^*$, we have $X_{i,a}\in{\mathbb{Z}}[Y_{j,aq^l}^{\pm}]_{j\in I,l\geq 0}$. In particular for $a\in{\ensuremath{\mathbb{C}}}^*$ we have an injective ring homomorphism: $$\chi_q^a:\text{Rep}_a={\mathbb{Z}}[X_{i,aq^l}]_{i\in I,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}_a={\mathbb{Z}}[Y_{i,aq^l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$$ For $a,b\in{\ensuremath{\mathbb{C}}}^*$ denote $\alpha_{b,a}:\text{Rep}_a\rightarrow\text{Rep}_b$ and $\beta_{b,a}:{\mathcal{Y}}_a\rightarrow{\mathcal{Y}}_b$ the canonical ring homomorphism. We have a commutative diagram: $$\begin{array}{rcccl} \text{Rep}_a&\stackrel{\chi_q^a}{\longrightarrow}&{\mathcal{Y}}_a\\ \alpha_{b,a}\downarrow &&\downarrow\beta_{b,a}\\ \text{Rep}_b&\stackrel{\chi_q^b}{\longrightarrow}&{\mathcal{Y}}_b\\\end{array}$$ This result is a consequence of theorem \[simme\] (or see [@Fre], [@Fre2]). In particular it suffices to study $\chi_q^1$. In the following denote $\text{Rep}=\text{Rep}_1$, $X_{i,l}=X_{i,q^l}$\[xil\], ${\mathcal{Y}}={\mathcal{Y}}_1$ and $\chi_q=\chi_q^1:\text{Rep}\rightarrow {\mathcal{Y}}$.\[rep\] Twisted polynomial algebras related to quantum affine algebras {#defoal} ============================================================== The aim of this section is to define the $t$-deformed algebra ${\mathcal{Y}}_t$ and to describe its structure (theorem \[dessus\]). We define the Heisenberg algebra $\mathcal{H}$, the subalgebra ${\mathcal{Y}}_u\subset \mathcal{H}[[h]]$ and eventually ${\mathcal{Y}}_t$ as a quotient of ${\mathcal{Y}}_u$. Heisenberg algebras related to quantum affine algebras ------------------------------------------------------ ### The Heisenberg algebra $\mathcal{H}$ $\mathcal{H}$\[zq\] is the ${\ensuremath{\mathbb{C}}}$-algebra defined by generators $a_i[m]$\[aim\] ($i\in I, m\in{\mathbb{Z}}-\{0\}$), central elements $c_r$\[cr\] ($r>0$) and relations ($i,j\in I,m,r\in{\mathbb{Z}}-\{0\}$): $$[a_i[m],a_j[r]]=\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}$$ This definition is motivated by the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$: in $\mathcal{H}$ the $c_r$ are algebraically independent, but we have a surjective homomorphism from $\mathcal{H}$ to ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})$ such that $a_i[m]\mapsto (q-q^{-1})h_{i,m}$ and $c_r\mapsto\frac{c^{r}-c^{-r}}{r}$. ### Properties of $\mathcal{H}$ For $j\in I,m\in{\mathbb{Z}}$ we set\[yim\]: $$y_j[m]=\underset{i\in I}{\sum} \tilde{C}_{i,j}(q^m)a_i[m]\in \mathcal{H}$$ \[calczq\] We have the Lie brackets in $\mathcal{H}$ ($i,j\in I,m,r\in{\mathbb{Z}}$): $$[a_i[m],y_j[r]]=(q^{mr_i}-q^{-r_im})\delta_{m,-r}\delta_{i,j}c_{|m|}$$ $$[y_i[m],y_j[r]]=\delta_{m,-r}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_{|m|}$$ [[*Proof:*]{}]{}We compute in $\mathcal{H}$: $$[a_i[m],y_j[r]] =[a_i[m],\underset{k\in I}{\sum} \tilde{C}_{k,j}(q^r)a_k[r]] =\delta_{m,-r}c_{|m|}\underset{k\in I}{\sum}\tilde{C}_{k,j}(q^{-m})[r_i]_{q^m}C_{i,k}(q^m)(q^m-q^{-m})$$ $$=\delta_{i,j}\delta_{m,-r}(q^{mr_i}-q^{-mr_i})c_{|m|}$$ $$[y_i[m],y_j[r]] =[\underset{k\in I}{\sum} \tilde{C}_{k,i}(q^m)a_k[m],y_j[r]] =\delta_{m,-r}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_{|m|}$$ Let $\pi_+$\[piplus\] and $\pi_-$ be the ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms of $\mathcal{H}$ such that ($i\in I$, $m>0$, $r<0$): $$\pi_+(a_i[m])=a_i[m]\text{ , }\pi_+(a_i[r])=0\text{ , }\pi_+(c_m)=0$$ $$\pi_-(a_i[m])=0\text{ , }\pi_-(a_i[r])=a_i[r]\text{ , }\pi_-(c_m)=0$$ They are well-defined because the relations are preserved. We set $\mathcal{H}^+=\text{Im}(\pi_+)\subset \mathcal{H}$\[zqplus\] and $\mathcal{H}^-=\text{Im}(\pi_-)\subset \mathcal{H}$. Note that $\mathcal{H}^+$ (resp. $\mathcal{H}^-$) is the subalgebra of $\mathcal{H}$ generated by the $a_i[m]$, $i\in I,m>0$ (resp. $m<0$). So $\mathcal{H}^+$ and $\mathcal{H}^-$ are commutative algebras, and: $$\mathcal{H}^+\simeq \mathcal{H}^-\simeq {\ensuremath{\mathbb{C}}}[a_i[m]]_{i\in I,m>0}$$ We say that $m\in \mathcal{H}$ is a $\mathcal{H}$-monomial if it is a product of the generators $a_i[m],c_r$. There is a unique ${\ensuremath{\mathbb{C}}}$-linear endomorphism $::$ of $\mathcal{H}$ such that for all $\mathcal{H}$-monomials $m$ we have: $$:m:=\pi_+(m)\pi_-(m)$$ In particular there is a vector space triangular decomposition $\mathcal{H}\simeq \mathcal{H}^+\otimes {\ensuremath{\mathbb{C}}}[c_r]_{r>0}\otimes \mathcal{H}^-$. [[*Proof:*]{}]{}The $\mathcal{H}$-monomials span the ${\ensuremath{\mathbb{C}}}$-vector space $\mathcal{H}$, so the map is unique. But there are non trivial linear combinations between them because of the relations of $\mathcal{H}$: it suffices to show that for $m_1$, $m_2$ $\mathcal{H}$-monomials the definition of $::$ is compatible with the relations ($i,j\in I$, $l,k\in{\mathbb{Z}}-\{0\}$): $$m_1a_i[k]a_j[l]m_2-m_1a_j[l]a_i[k]m_2=\delta_{k,-l}(q^k-q^{-k})B_{i,j}(q^k)m_1c_{|k|}m_2$$ As $\mathcal{H}^+$ and $\mathcal{H}^-$ are commutative, we have: $$\pi_+(m_1a_i[k]a_j[l]m_2)\pi_-(m_1a_i[k]a_j[l]m_2)=\pi_+(m_1a_j[l]a_i[k]m_2)\pi_-(m_1a_j[l]a_i[k]m_2)$$ and we can conclude because $\pi_+(m_1c_{|k|}m_2)=\pi_-(m_1c_{|k|}m_2)=0$. The deformed algebra ${\mathcal{Y}}_u$ -------------------------------------- ### Construction of ${\mathcal{Y}}_u$ Consider the ${\ensuremath{\mathbb{C}}}$-algebra $\mathcal{H}_h=\mathcal{H}[[h]]$\[zqh\]. The application $\text{exp}$ is well-defined on the subalgebra $h\mathcal{H}_h$: $$\text{exp}:h\mathcal{H}_h\rightarrow \mathcal{H}_h$$ For $l\in{\mathbb{Z}}$, $i\in I$, introduce $\tilde{A}_{i,l},\tilde{Y}_{i,l}\in \mathcal{H}_h$\[tail\] such that: $$\tilde{A}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})$$ $$\tilde{Y}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})$$ Note that $\tilde{A}_{i,l}$ and $\tilde{Y}_{i,l}$ are invertible in $\mathcal{H}_h$ and that: $$\tilde{A}_{i,l}^{-1}=\text{exp}(-\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})\text{exp}(-\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$ $$\tilde{Y}_{i,l}^{-1}=\text{exp}(-\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})\text{exp}(-\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})$$ Recall the definition $\mathfrak{U}\subset{\ensuremath{\mathbb{Q}}}(z)$ of section \[recalu\]. For $R\in \mathfrak{U}$, introduce $t_{R}\in \mathcal{H}_h$\[tr\]: $$t_R=\text{exp}(\underset{m>0}{\sum}h^{2m}R(q^m)c_m)$$ \[yu\] ${\mathcal{Y}}_u$ is the ${\mathbb{Z}}$-subalgebra of $\mathcal{H}_h$ generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t_R$ ($i\in I,l\in{\mathbb{Z}},R\in\mathfrak{U}$). In this section we give properties of ${\mathcal{Y}}_u$ and subalgebras of ${\mathcal{Y}}_u$ which will be useful in section \[studyyt\]. ### Relations in ${\mathcal{Y}}_u$ {#defij} \[relu\] We have the following relations in ${\mathcal{Y}}_u$ ($i,j\in I$ $l,k\in{\mathbb{Z}}$): $$\label{ay}\tilde{A}_{i,l}\tilde{Y}_{j,k}\tilde{A}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1} =t_{\delta_{i,j}(z^{-r_i}-z^{r_i})(-z^{(l-k)}+z^{(k-l)})}$$ $$\label{yy}\tilde{Y}_{i,l}\tilde{Y}_{j,k}\tilde{Y}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1} =t_{\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)})}$$ $$\label{aa}\tilde{A}_{i,l}\tilde{A}_{j,k}\tilde{A}_{i,l}^{-1}\tilde{A}_{j,k}^{-1}=t_{B_{i,j}(z)(z^{-1}-z)(-z^{(l-k)}+z^{(k-l)})}$$ [[*Proof:*]{}]{}For $A,B\in h\mathcal{H}_h$ such that $[A,B]\in h{\ensuremath{\mathbb{C}}}[c_r]_{r>0}$, we have: $$\text{exp}(A)\text{exp}(B)=\text{exp}(B)\text{exp}(A)\text{exp}([A,B])$$ So we can compute (see lemma \[calczq\]): $\tilde{A}_{i,l}\tilde{A}_{j,k} \\=\text{exp}(\underset{m>0}{\sum}h^m a_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km}))\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}B_{i,j}(q^m)(q^{-m}-q^{m})q^{m(k-l)}c_m) \\\text{exp}(\underset{m>0}{\sum}h^m a_{i}[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m a_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}B_{i,j}(q^m)(q^{-m}-q^{m})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{A}_{j,k}\tilde{A}_{i,l}$ $\tilde{A}_{i,l}\tilde{Y}_{j,k} \\=\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{km}))\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}(q^{-mr_i}-q^{mr_i})q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{ml}) \\\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}(q^{-mr_i}-q^{mr_i})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{Y}_{j,k}\tilde{A}_{i,l}$ $\tilde{Y}_{i,l}\tilde{Y}_{j,k} \\=\text{exp}(\underset{m>0}{\sum}h^my_i[m]q^{ml})(\text{exp}(\underset{m>0}{\sum}h^my_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk}))\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}q^{m(k-l)}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_m)\text{exp}(\underset{m>0}{\sum}h^my_{i}[m]q^{ml}) \\\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{Y}_{j,k}\tilde{Y}_{i,l}$ ### Commutative subalgebras of $\mathcal{H}_h$ {#defipip} The ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms $\pi_+,\pi_-$ of $\mathcal{H}$ are naturally extended to ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms of $\mathcal{H}_h$. As ${\mathcal{Y}}_u\subset \mathcal{H}_h$, we have by restriction the ${\mathbb{Z}}$-algebra morphisms $\pi_{\pm}:{\mathcal{Y}}_u\rightarrow \mathcal{H}_h$. Introduce ${\mathcal{Y}}=\pi_+({\mathcal{Y}}_u)\subset \mathcal{H}^+[[h]]$\[y\]. In this section \[defipip\] we study ${\mathcal{Y}}$. In particular we will see in proposition \[circ\] that the notation ${\mathcal{Y}}$ is consistent with the notation of section \[qcar\]. For $i\in I,l\in{\mathbb{Z}}$, denote\[ail\]: $$Y_{i,l}^{\pm}=\pi_+(\tilde{Y}_{i,l}^{\pm})=\text{exp}(\pm\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})$$ $$A_{i,l}^{\pm}=\pi_+(\tilde{A}_{i,l}^{\pm})=\text{exp}(\pm\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$ \[gen\] For $i\in I,l\in{\mathbb{Z}}$, we have: $$A_{i,l}=Y_{i,l-r_i}Y_{i,l+r_i}(\underset{j/C_{j,i}=-1}{\prod}Y_{j,l}^{-1})(\underset{j/C_{j,i}=-2}{\prod}Y_{j,l+1}^{-1}Y_{j,l-1}^{-1})(\underset{j/C_{j,i}=-3}{\prod}Y_{j,l+2}^{-1}Y_{j,l}^{-1}Y_{j,l-2}^{-1})$$ In particular ${\mathcal{Y}}$ is generated by the $Y_{i,l}^{\pm}$ ($i\in I,l\in{\mathbb{Z}}$). [[*Proof:*]{}]{} We have $a_i[m]=\underset{j\in I}{\sum}C_{j,i}(q^m)y_j[m]$, and: $$\pi_+(\tilde{A}_{i,l})=\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})=\underset{j\in I}{\prod}\text{exp}(\underset{m>0}{\sum}h^m C_{j,i}(q^m)y_j[m]q^{lm})$$ As $C_{i,i}(q)=q^{r_i}+q^{-r_i}$, we have: $$\text{exp}(\underset{m>0}{\sum}h^m C_{i,i}(q^m)y_i[m]q^{lm})=\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{(l-r_i)m})\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{(l+r_i)m})=Y_{i,l-r_i}Y_{i,l+r_i}$$ If $C_{j,i}<0$, we have $C_{j,i}(q)=-\underset{k=C_{j,i}+1, C_{j,i}+3...-C_{j,i}-1}{\sum}q^{k}$ and: $$\text{exp}(-\underset{m>0}{\sum}h^m C_{j,i}(q^m)y_j[m]q^{lm})=\underset{k=C_{j,i}+1, C_{j,i}+3...-C_{j,i}-1}{\prod}\text{exp}(-\underset{m>0}{\sum}h^m y_j[m]q^{(l+k)m})$$ As ${\mathcal{Y}}_u$ is generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t_R$ we get the last point. Note that the formula of lemma \[gen\] already appeared in [@Fre]. We need a general technical lemma to describe ${\mathcal{Y}}$: \[indgene\] Let $J=\{1,...,r\}$ and let $\Lambda$ be the polynomial commutative algebra\ $\Lambda={\ensuremath{\mathbb{C}}}[\lambda_{j,m}]_{j\in J,m\geq 0}$. For $R=(R_1,...,R_r)\in\mathfrak{U}^{r}$, consider: $$\Lambda_R=\text{exp}(\underset{j\in J,m>0}{\sum}h^mR_j(q^m)\lambda_{j,m})\in\Lambda[[h]]$$ Then the $(\Lambda_R)_{R\in\mathfrak{U}^r}$ are ${\ensuremath{\mathbb{C}}}$-linearly independent. In particular the $\Lambda_{j,l}=\Lambda_{(0,...,0,z^l,0,...,0)}$ ($j\in J$, $l\in{\mathbb{Z}}$) are ${\ensuremath{\mathbb{C}}}$-algebraically independent. [[*Proof:*]{}]{}Suppose we have a linear combination ($\mu_R\in{\ensuremath{\mathbb{C}}}$, only a finite number of $\mu_R\neq 0$): $$\underset{R\in\mathfrak{U}^r}{\sum}\mu_R\Lambda_R=0$$ The coefficients of $h^L$ in $\Lambda_R$ are of the form $R_{j_1}(q^{l_1})^{L_1}R_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}\lambda_{j_1,l_1}^{L_1}\lambda_{j_2,l_2}^{L_2}...\lambda_{j_N,l_N}^{L_N}$ where $l_1L_1+...+l_NL_N=L$. So for $N\geq 0$, $j_1,...,j_N\in J$, $l_1,...,l_N>0$, $L_1,...,L_N\geq 0$ we have: $$\underset{R\in\mathfrak{U}^r}{\sum}\mu_RR_{j_1}(q^{l_1})^{L_1}R_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}=0$$ If we fix $L_2,...,L_{N}$, we have for all $L_1=l\geq 0$: $$\underset{\alpha_1\in{\ensuremath{\mathbb{C}}}}{\sum}\alpha_1^l\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1}{\sum}\mu_RR_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}=0$$ We get a Van der Monde system which is invertible, so for all $\alpha_1\in{\ensuremath{\mathbb{C}}}$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1}{\sum}\mu_RR_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{l_N}=0$$ By induction we get for $r'\leq N$ and all $\alpha_1,...,\alpha_{r'}\in{\ensuremath{\mathbb{C}}}$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1,...,R_{j_{r'}}(q^{l_{r'}})=\alpha_{r'}}{\sum}\mu_{R}R_{j_{r'+1}}(q^{l_{r'+1}})^{L_{r'+1}}...R_{j_N}(q^{l_N})^{L_N}=0$$ And so for $r'=N$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1,...,R_{j_{N}}(q^{l_{N}})=\alpha_{N}}{\sum}\mu_{R}=0$$ Let be $S\geq 0$ such that for all $\mu_R,\mu_{R'}\neq 0$, $j\in J$ we have $R_j-R_j'=0$ or $R_j-R_j'$ has at most $S-1$ roots. We set $N=Sr$ and $((j_1,l_1),...,(j_S,l_S))=((1,1),(1,2),...,(1,S),(2,1),...,(2,S),(3,1),...,(r,S))$. We get for all $\alpha_{j,l}\in{\ensuremath{\mathbb{C}}}$ ($j\in J,1\leq l\leq S$): $$\underset{R\in\mathfrak{U}^r/\forall j\in J,1\leq l\leq S, R_{j}(q^{l})=\alpha_{j,l}}{\sum}\mu_{R}=0$$ It suffices to show that there is at most one term is this sum. But consider $P,Q\in\mathfrak{U}$ such that for all $1\leq l\leq S$, $P(q^l)=P'(q^l)$. As $q$ is not a root of unity the $q^l$ are different and $P-P'$ has $S$ roots, so is $0$. For the last assertion, we can write a monomial $\underset{j\in J,l\in{\mathbb{Z}}}{\prod}\Lambda_{j,l}^{u_{j,l}}=\Lambda_{\underset{l\in{\mathbb{Z}}}{\sum}u_{1,l}z^l,...,\underset{l\in{\mathbb{Z}}}{\sum}u_{r,l}z^l}$. In particular there is no trivial linear combination between those monomials. It follows from lemma \[gen\] and lemma \[indgene\]: \[circ\] The $Y_{i,l}\in{\mathcal{Y}}$ are ${\mathbb{Z}}$-algebraically independent and generate the ${\mathbb{Z}}$-algebra ${\mathcal{Y}}$. In particular, ${\mathcal{Y}}$ is the commutative polynomial algebra ${\mathbb{Z}}[Y_{i,l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$. The $A_{i,l}^{-1}\in{\mathcal{Y}}$ are ${\mathbb{Z}}$-algebraically independent. In particular the subalgebra of ${\mathcal{Y}}$ generated by the $A_{i,l}^{-1}$ is the commutative polynomial algebra ${\mathbb{Z}}[A_{i,l}^{-1}]_{i\in I,l\in{\mathbb{Z}}}$. ### Generators of ${\mathcal{Y}}_u$ {#ptpt} The ${\ensuremath{\mathbb{C}}}$-linear endomorphism $::$ of $\mathcal{H}$ is naturally extended to a ${\ensuremath{\mathbb{C}}}$-linear endomorphism of $\mathcal{H}_h$. As ${\mathcal{Y}}_u\subset \mathcal{H}_h$, we have by restriction a ${\mathbb{Z}}$-linear morphism $::$ from ${\mathcal{Y}}_u$ to $\mathcal{H}_h$. We say that $m\in{\mathcal{Y}}_u$ is a ${\mathcal{Y}}_u$-monomial if it is a product of generators $\tilde{A}_{i,l}^{\pm},\tilde{Y}_{i,l}^{\pm},t_R$. In the following, for a product of non commuting terms, denote $\overset{\rightarrow}{\underset{s=1..S}{\prod}}U_s=U_1U_2...U_S$. \[rel\] The algebra ${\mathcal{Y}}_u$ is generated by the $\tilde{Y}_{i,l}^{\pm},t_R$ ($i\in I,l\in{\mathbb{Z}},R\in\mathfrak{U}$). [[*Proof:*]{}]{}Let be $i\in I$, $l\in{\mathbb{Z}}$. It follows from proposition \[circ\] that $\pi_+(\tilde{A}_{i,l})$ is of the form $\pi_+(\tilde{A}_{i,l})=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}}$ and that $:m:=:\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}}:$. So it suffices to show that for $m$ a ${\mathcal{Y}}_u$-monomial, there is a unique $R_m\in \mathfrak{U}$ such that $m=t_{R_m}:m:$. Let us write $m=t_R\overset{\rightarrow}{\underset{s=1.. S}{\prod}}U_s$ where $U_s\in\{\tilde{A}_{i,l}^{\pm},\tilde{Y}_{i,l}^{\pm}\}_{i\in I,l\in{\mathbb{Z}}}$ are generators. Then: $$:m:=(\underset{s=1..S}{\prod}\pi_+(U_s))(\underset{s=1..S}{\prod}\pi_-(U_s))$$ And we can conclude because it follows from the proof of lemma \[relu\] that for $1\leq s,s'\leq S$, there is $R_{s,s'}\in\mathfrak{U}$ such that $\pi_+(U_s)\pi_-(U_{s'})=t_{R_{s,s'}}\pi_-(U_{s'})\pi_+(U_s)$. In particular it follows from this proof that $:{\mathcal{Y}}_u:\subset {\mathcal{Y}}_u$. The deformed algebra ${\mathcal{Y}}_t$ {#studyyt} -------------------------------------- ### Construction of ${\mathcal{Y}}_t$ {#defipire} Denote by ${\mathbb{Z}}((z^{-1}))$ the ring of series of the form $P=\underset{r\leq R_P}{\sum}P_rz^r$ where $R_P\in{\mathbb{Z}}$ and the coefficients $P_r\in{\mathbb{Z}}$. Recall the definition $\mathfrak{U}$ of section \[recalu\]. We have an embedding $\mathfrak{U}\subset{\mathbb{Z}}((z^{-1}))$ by expanding $\frac{1}{Q(z^{-1})}$ in ${\mathbb{Z}}[[z^{-1}]]$ for $Q(z)\in{\mathbb{Z}}[z]$ such that $Q(0)=1$. So we can introduce maps:\[pir\] $$\pi_r:\mathfrak{U}\rightarrow {\mathbb{Z}}\text{ , }P=\underset{k\leq R_P}{\sum}P_k z^k\mapsto P_r$$ Note that we could have consider the expansion in ${\mathbb{Z}}((z))$ and that the maps $\pi_r$ are not independent of our choice. We define ${\mathcal{Y}}_t$\[tyt\] (resp. $\mathcal{H}_t$)\[zqt\] as the algebra quotient of ${\mathcal{Y}}_u$ (resp. $\mathcal{H}_h$) by relations: $$t_R=t_{R'}\text{ if $\pi_0(R)=\pi_0(R')$}$$ We keep the notations $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm}$ for their image in ${\mathcal{Y}}_t$. Denote by $t$ the image of $t_1=\text{exp}(\underset{m>0}{\sum}h^{2m}c_m)$ in ${\mathcal{Y}}_t$. As $\pi_0$ is additive, the image of $t_R$ in ${\mathcal{Y}}_t$ is $t^{\pi_o(R)}$\[t\]. In particular ${\mathcal{Y}}_t$ is generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t^{\pm}$. As the defining relations of $\mathcal{H}_t$ involve only the $c_l$ and $\pi_+(c_l)=\pi_-(c_l)=0$, the algebra endomorphisms $\pi_+,\pi_-$ of $\mathcal{H}_t$ are well-defined. So we can define\[zqtplus\] $\mathcal{H}_t^+,\mathcal{H}_t^-,{\mathcal{Y}}_t^+,{\mathcal{Y}}_t^-$\[tytplus\] in the same way as in section \[defipip\] and $::$ a ${\ensuremath{\mathbb{C}}}$-linear endomorphism of $\mathcal{H}_t$ as in section \[ptpt\]. The ${\mathbb{Z}}[t^{\pm}]$-subalgebra ${\mathcal{Y}}_t\subset \mathcal{H}_t$ verifies $:{\mathcal{Y}}_t:\subset{\mathcal{Y}}_t$ (proof of lemma \[rel\]). We have ${\mathcal{Y}}_t^+\simeq{\mathcal{Y}}$. We say that $m\in{\mathcal{Y}}_t$ (resp. $m\in{\mathcal{Y}}$) is a ${\mathcal{Y}}_t$-monomial (resp. a ${\mathcal{Y}}$-monomial) if it is a product of the generators $\tilde{Y}_{i,m}^{\pm},t^{\pm}$ (resp. $Y_{i,m}^{\pm}$). ### Structure of ${\mathcal{Y}}_t$ {#dessusdeux} The following theorem gives the structure of ${\mathcal{Y}}_t$: \[dessus\] The algebra ${\mathcal{Y}}_t$ is defined by generators $\tilde{Y}_{i,l}^{\pm}$ $(i\in I,l\in{\mathbb{Z}})$, central elements $t^{\pm}$ and relations ($i,j\in I, k,l\in{\mathbb{Z}}$): $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}=t^{\gamma(i,l,j,k)}\tilde{Y}_{j,k}\tilde{Y}_{i,l}$$ where $\gamma: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by (recall the maps $\pi_r$ of section \[defipire\])\[gamma\]: $$\gamma(i,l,j,k)=\underset{r\in{\mathbb{Z}}}{\sum}\pi_r(\tilde{C}_{j,i}(z))(-\delta_{l-k,-r_j-r}-\delta_{l-k,r-r_j}+\delta_{l-k,r_j-r}+\delta_{l-k,r_j+r})$$ [[*Proof:*]{}]{}As the image of $t_R$ in ${\mathcal{Y}}_t$ is $t^{\pi_o(R)}$, we can deduce the relations from lemma \[relu\]. For example formula \[yy\] (p. ) gives: $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}\tilde{Y}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1}=t^{\pi_0((\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)}))}$$ where: $$\pi_0(\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)}))$$ $$=\underset{r\in{\mathbb{Z}}}{\sum}\pi_r(\tilde{C}_{j,i}(z))(\delta_{r_j+r+k-l,0}+\delta_{-r_j+r+l-k,0}-\delta_{r_j+r+l-k,0}-\delta_{-r_j+r+k-l,0})=\gamma(i,l,j,k)$$ It follows from lemma \[gen\] that ${\mathcal{Y}}_t$ is generated by the $\tilde{Y}_{i,l}^{\pm},t^{\pm}$. It follows from lemma \[indgene\] that the $t_R\in{\mathcal{Y}}_u$ ($R\in\mathfrak{U}$) are ${\mathbb{Z}}$-linearly independent. So the ${\mathbb{Z}}$-algebra ${\mathbb{Z}}[t_R]_{R\in\mathfrak{U}}$ is defined by generators $(t_R)_{R\in\mathfrak{U}}$ and relations $t_{R+R'}=t_Rt_{R'}$ for $R,R'\in\mathfrak{U}$. In particular the image of ${\mathbb{Z}}[t_R]_{R\in\mathfrak{U}}$ in ${\mathcal{Y}}_t$ is ${\mathbb{Z}}[t^{\pm}]$. Let $A$ be the classes of ${\mathcal{Y}}_t$-monomials modulo $t^{{\mathbb{Z}}}$. So we have: $$\underset{m\in A}{\sum}{\mathbb{Z}}[t^{\pm}].m={\mathcal{Y}}_t$$ We prove the sum is direct: suppose we have a linear combination $\underset{m\in A}{\sum}\lambda_m(t)m=0$ where $\lambda_m(t)\in{\mathbb{Z}}[t^{\pm}]$. We saw in proposition \[circ\] that ${\mathcal{Y}}\simeq {\mathbb{Z}}[{Y}_{i,l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$. So $\lambda_m(1)=0$ and $\lambda_m(t)=(t-1)\lambda_m^{(1)}(t)$ where $\lambda_m^{(1)}(t)\in{\mathbb{Z}}[t^{\pm}]$. In particular $\underset{m\in A}{\sum}\lambda_m(t)^{(1)}(t)m=0$ and we get by induction $\lambda_m(t)\in (t-1)^r{\mathbb{Z}}[t^{\pm}]$ for all $r\geq 0$. This is possible if and only if all $\lambda_m(t)=0$. In the same way using the last assertion of proposition \[circ\], we have: \[yenga\] The sub ${\mathbb{Z}}[t^{\pm}]$-algebra of ${\mathcal{Y}}_t$ generated by the $\tilde{A}_{i,l}^{-1}$ is defined by generators $\tilde{A}_{i,l}^{-1},t^{\pm}$ $(i\in I,l\in{\mathbb{Z}})$ and relations\[alpha\]: $$\tilde{A}_{i,l}^{-1}\tilde{A}_{j,k}^{-1}=t^{\alpha(i,l,j,k)}\tilde{A}_{j,k}^{-1}\tilde{A}_{i,l}^{-1}$$ where $\alpha: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by: $$\alpha(i,l,i,k)=2(-\delta_{l-k,2r_i}+\delta_{l-k,-2r_i})$$ $$\alpha(i,l,j,k)=2\underset{r=C_{i,j}+1,C_{i,j}+3,...,-C_{i,j}-1}{\sum}(-\delta_{l-k,-r_i+r}+\delta_{l-k,r_i+r})\text{ (if $i\neq j$)}$$ Moreover we have the following relations in ${\mathcal{Y}}_t$: $$\tilde{A}_{i,l}\tilde{Y}_{j,k}=t^{\beta(i,l,j,k)}\tilde{Y}_{j,k}\tilde{A}_{i,l}$$ where $\beta: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by\[beta\]: $$\beta(i,l,j,k)=2\delta_{i,j}(-\delta_{l-k,r_i}+\delta_{l-k,-r_i})$$ Notations and properties related to monomials --------------------------------------------- In this section we study some technical properties of the ${\mathcal{Y}}$-monomials and the ${\mathcal{Y}}_t$-monomials which will be used in the following. ### Basis Denote by $A$\[a\] the set of ${\mathcal{Y}}$-monomials. It is a ${\mathbb{Z}}$-basis of ${\mathcal{Y}}$ (proposition \[circ\]). Let us define an analog ${\mathbb{Z}}[t^{\pm}]$-basis of ${\mathcal{Y}}_t$: denote $A'$\[ap\] the set of ${\mathcal{Y}}_t$-monomials of the form $m=:m:$. It follows from theorem \[dessus\] that: $${\mathcal{Y}}_t=\underset{m\in A'}{\bigoplus}{\mathbb{Z}}[t^{\pm}]m$$ The map $\pi:A'\rightarrow A$ defined by $\pi(m)=\pi_+(m)$\[pi\] is a bijection. In the following we identify $A$ and $A'$. In particular we have an embedding ${\mathcal{Y}}\subset{\mathcal{Y}}_t$ and an isomorphism of ${\mathbb{Z}}[t^{\pm}]$-modules ${\mathcal{Y}}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]\simeq{\mathcal{Y}}_t$. Note that it depends on the choice of the ${\mathbb{Z}}[t^{\pm}]$-basis of ${\mathcal{Y}}_t$. We say that $\chi_1\in{\mathcal{Y}}_t$ has the same monomials as $\chi_2\in {\mathcal{Y}}$ if in the decompositions $\chi_1=\underset{m\in A}{\sum}\lambda_m(t)m$, $\chi_2=\underset{m\in A}{\sum}\mu_mm$ we have $\lambda_m(t)=0\Leftrightarrow\mu_m=0$. ### The notation $u_{i,l}$ {#notuil} For $m$ a ${\mathcal{Y}}$-monomial we set $u_{i,l}(m)\in{\mathbb{Z}}$\[uil\] such that $m=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}{Y}_{i,l}^{u_{i,l}(m)}$ and $u_i(m)=\underset{l\in{\mathbb{Z}}}{\sum}u_{i,l}(m)$\[ui\]. For $m$ a ${\mathcal{Y}}_t$-monomial, we set $u_{i,l}(m)=u_{i,l}(\pi_+(m))$ and $u_i(m)=u_i(\pi_+(m))$. Note that $u_{i,l}$ is invariant by multiplication by $t$ and compatible with the identification of $A$ and $A'$. Note that section \[dessusdeux\] implies that for $i\in I, l\in {\mathbb{Z}}$ and $m$ a ${\mathcal{Y}}_t$-monomial we have: $$\tilde{A}_{i,l}m=t^{-2u_{i,l-r_i}(m)+2u_{i,l+r_i}(m)}m\tilde{A}_{i,l}$$ Denote by $B_i\subset A$ the set of $i$-dominant ${\mathcal{Y}}$-monomials, that is to say $m\in B_i$\[bi\] if $\forall l\in{\mathbb{Z}}$, $u_{i,l}(m)\geq 0$. For $J\subset I$ denote $B_J=\underset{i\in J}{\bigcap}B_i$\[bj\] the set of $J$-dominant ${\mathcal{Y}}$-monomials. In particular, $B=B_I$ is the set of dominant ${\mathcal{Y}}$-monomials.\[b\] We recall we can define a partial ordering on $A$ by putting $m\leq m'$ if there is a ${\mathcal{Y}}$-monomial $M$ which is a product of $A_{i,l}^{\pm}$ ($i\in I,l\in{\mathbb{Z}}$) such that $m=Mm'$ (see for example [@Her01]). A maximal (resp. lowest, higher...) weight ${\mathcal{Y}}$-monomial is a maximal (resp. minimal, higher...) element of $A$ for this ordering. We deduce from $\pi_+$ a partial ordering on the ${\mathcal{Y}}_t$-monomials. Following [@Fre2], a ${\mathcal{Y}}$-monomial $m$ is said to be right negative if the factors $Y_{j,l}$ appearing in $m$, for which $l$ is maximal, have negative powers. A product of right negative ${\mathcal{Y}}$-monomials is right negative. It follows from lemma \[gen\] that the $A_{i,l}^{-1}$ are right negative. A ${\mathcal{Y}}_t$-monomial is said to be right negative if $\pi_+(m)$ is right negative. ### Some technical properties \[genea\] Let $(i_1,l_1),...,(i_K,l_K)$ be in $(I\times{\mathbb{Z}})^K$. For $U\geq 0$, the set of the $m=\underset{k=1...K}{\prod}A_{i_k,l_k}^{-v_{i_k,l_k}(m)}$ ($v_{i_k,l_k}(m)\geq 0$) such that $\underset{i\in I,k\in{\mathbb{Z}}}{\text{min}}u_{i,k}(m)\geq -U$ is finite. [[*Proof:*]{}]{}Suppose it is not the case: let be $(m_p)_{p\geq 0}$ such that $\underset{i\in I,k\in{\mathbb{Z}}}{\text{min}}u_{i,k}(m_p)\geq -U$ but\ $\underset{k=1...K}{\sum}v_{i_k,l_k}(m_p)\underset{p\rightarrow\infty}{\rightarrow}+\infty$. So there is at least one $k$ such that $v_{i_k,l_k}(m_p)\underset{p\rightarrow\infty}{\rightarrow}+\infty$. Denote by $\mathfrak{R}$ the set of such $k$. Among those $k\in\mathfrak{R}$, such that $l_k$ is maximal suppose that $r_{i_k}$ is maximal (recall the definition of $r_i$ in section \[recalu\]). In particular, we have $u_{i_k,l_k+r_{i_k}}(m_p)=-v_{i_k,l_k}(m_p)+f(p)$ where $f(p)$ depends only of the $v_{i_{k'},l_{k'}}(m_p)$, $k'\notin\mathfrak{R}$. In particular, $f(p)$ is bounded and $u_{i_k,l_k+r_{i_k}}(m_p)\underset{p\rightarrow\infty}{\rightarrow}-\infty$. \[fini\] For $M\in B$, $K\geq 0$ the set of ${\mathcal{Y}}$-monomials $\{MA_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}/R\geq 0,l_1,...,l_R\geq K\}\cap B$ is finite. [[*Proof:*]{}]{}Let us write $M=Y_{i_1,l_1}...Y_{i_R,l_R}$ such that $l_1=\underset{r=1...R}{\text{min}}l_r$, $l_R=\underset{r=1...R}{\text{max}}l_r$ and consider $m$ in the set. It is of the form $m=MM'$ where $M'=\underset{i\in I,l\geq K}{\prod}A_{i,l}^{-v_{i,l}}$ ($v_{i,l}\geq 0$). Let $L=\text{max}\{l\in{\mathbb{Z}}/\exists i\in I, u_{i,l}(M')<0\}$. $M'$ is right negative so for all $i\in I$, $l>L\Rightarrow v_{i,l}=0$. But $m$ is dominant, so $L\leq l_R$. In particular $M'=\underset{i\in I,K\leq l\leq l_R}{\prod}A_{i,l}^{-v_{i,l}}$. It suffices to prove that the $v_{i,l}(m_r)$ are bounded under the condition $m$ dominant. This follows from lemma \[genea\]. Presentations of deformed algebras ---------------------------------- Our construction of ${\mathcal{Y}}_t$ using $\mathcal{H}_h$ (section \[studyyt\]) is a “concrete” presentation of the deformed structure. Let us look at another approach: in this section we define two bicharacters $\mathcal{N},\mathcal{N}_t$ related to basis of ${\mathcal{Y}}_t$. All the information of the multiplication of ${\mathcal{Y}}_t$ is contained is those bicharacters because we can construct a deformed $*$ multiplication on the “abstract” ${\mathbb{Z}}[t^{\pm}]$-module ${\mathcal{Y}}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]$ by putting for $m_1,m_2\in A$ ${\mathcal{Y}}$-monomials: $$m_1*m_2=t^{\mathcal{N}(m_1,m_2)-\mathcal{N}(m_2,m_1)}m_2*m_1$$ or $$m_1*m_2=t^{\mathcal{N}_t(m_1,m_2)-\mathcal{N}_t(m_2,m_1)}m_2*m_1$$ Those presentations appeared earlier in the literature [@Nab], [@Vas] for the simply laced case. In particular this section identifies our approach with those articles and gives an algebraic motivation of the deformed structures of [@Nab], [@Vas] related to the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. ### The bicharacter $\mathcal{N}$ It follows from the proof of lemma \[rel\] that for $m$ a ${\mathcal{Y}}_t$-monomial, there is $N(m)\in{\mathbb{Z}}$\[n\] such that $m=t^{N(m)}:m:$. For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we define $\mathcal{N}(m_1,m_2)=N(m_1m_2)-N(m_1)-N(m_2)$. We have $N(Y_{i,l})=N(A_{i,l})=0$. Note that for $\alpha,\beta\in{\mathbb{Z}}$ we have: $$N(t^{\alpha}m)=\alpha+N(m)\text{ , }\mathcal{N}(t^{\alpha}m_1,t^{\beta}m_2)=\mathcal{N}(m_1,m_2)$$ In particular the map $\mathcal{N}:A\times A\rightarrow{\mathbb{Z}}$ is well-defined and independent of the choice of a representant in $\pi_+^{-1}(A)$. \[aaa\] For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials, we have in $\mathcal{H}_t$: $$\pi_-(m_1)\pi_+(m_2)=t^{\mathcal{N}(m_1,m_2)}\pi_+(m_2)\pi_-(m_1)$$ [[*Proof:*]{}]{}We have: $$m_1=t^{N(m_1)}\pi_+(m_1)\pi_-(m_1)\text{ , }m_2=t^{N(m_2)}\pi_+(m_2)\pi_-(m_2)$$ and so: $$m_1m_2=t^{N(m_1m_2)}\pi_+(m_1)\pi_+(m_2)\pi_-(m_1)\pi_-(m_2)=t^{N(m_1)+N(m_2)}\pi_+(m_1)\pi_-(m_1)\pi_+(m_2)\pi_-(m_2)$$ \[bibi\] The map $\mathcal{N}:A\times A\rightarrow {\mathbb{Z}}$ is a bicharacter, that is to say for $m_1,m_2,m_3\in A$, we have: $$\mathcal{N}(m_1m_2,m_3)=\mathcal{N}(m_1,m_3)+\mathcal{N}(m_2,m_3)\text{ and }\mathcal{N}(m_1,m_2m_3)=\mathcal{N}(m_1,m_2)+\mathcal{N}(m_1,m_3)$$ Moreover for $m_1,...,m_k$ ${\mathcal{Y}}_t$-monomials, we have: $$N(m_1m_2...m_k)=N(m_1)+N(m_2)+...+N(m_k)+\underset{1\leq i<j\leq k}{\sum}\mathcal{N}(m_i,m_j)$$ [[*Proof:*]{}]{}For the first point it follows from lemma \[aaa\]: $$\pi_-(m_1m_2)\pi_+(m_3)=t^{\mathcal{N}(m_1m_2,m_3)}\pi_+(m_3)\pi_-(m_1m_2)=t^{\mathcal{N}(m_2,m_3)}\pi_-(m_1)\pi_+(m_3)\pi_-(m_2)$$ $$=t^{\mathcal{N}(m_1,m_3)+\mathcal{N}(m_2,m_3)}\pi_+(m_3)\pi_-(m_1m_2)$$ For the second point we have first: $$N(m_1m_2)=N(m_1)+N(m_2)+\mathcal{N}(m_1,m_2)$$ and by induction: $$N(m_1m_2...m_k)=N(m_1)+N(m_2...m_k)+\mathcal{N}(m_1,m_2...m_k)$$ $$=N(m_1)+N(m_2)+...+N(m_k)+\underset{1<i<j\leq k}{\sum}\mathcal{N}(m_i,m_j)+\mathcal{N}(m_1,m_2)+...+\mathcal{N}(m_1,m_k)$$ ### The bicharacter $\mathcal{N}_t$ {#tildeat} For $m$ a ${\mathcal{Y}}_t$-monomial and $l\in{\mathbb{Z}}$, denote $\pi_l(m)=\underset{j\in I}{\prod}\tilde{Y}_{j,l}^{u_{j,l}(m)}$. It is well defined because for $i,j\in I$ and $l\in{\mathbb{Z}}$ we have $\tilde{Y}_{i,l}\tilde{Y}_{j,l}=\tilde{Y}_{j,l}\tilde{Y}_{i,l}$ (theorem \[dessus\]). Moreover for $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we have $\pi_l(m_1m_2)=\pi_l(m_1)\pi_l(m_2)=\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}(m_1)+u_{i,l}(m_2)}$. For $m$ a ${\mathcal{Y}}_t$-monomial denote $\tilde{m}={\underset{l\in{\mathbb{Z}}}{\overset{\rightarrow}{\prod}}}\pi_l(m)$\[tm\], and $A_t$\[tat\] the set of ${\mathcal{Y}}_t$-monomials of the form $\tilde{m}$. From theorem \[dessus\] there is a unique $N_t(m)\in{\mathbb{Z}}$\[nt\] such that $m=t^{N_t(m)}\tilde{m}$, and: $${\mathcal{Y}}_t=\underset{m\in A_t}{\bigoplus}{\mathbb{Z}}[t^{\pm}]m$$ For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we define $\mathcal{N}_t(m_1,m_2)=N_t(m_1m_2)-N_t(m_1)-N_t(m_2)$. We have $N_t(Y_{i,l})=0$. Note that for $\alpha,\beta\in{\mathbb{Z}}$ we have: $$N_t(t^{\alpha}m)=\alpha+N_t(m)\text{ , }\mathcal{N}_t(t^{\alpha}m_1,t^{\beta}m_2)=\mathcal{N}_t(m_1,m_2)$$ In particular the map $\mathcal{N}_t:A\times A\rightarrow{\mathbb{Z}}$ is well-defined and independent of the choice of $A$. For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials, we have: $$\mathcal{N}_t(m_1,m_2)=\underset{l>l'}{\sum}(\mathcal{N}(\pi_l(m_1),\pi_{l'}(m_2))-\mathcal{N}(\pi_{l'}(m_2),\pi_{l}(m_1)))$$ In particular, $\mathcal{N}_t$ is a bicharacter and for $m_1,...,m_k$ ${\mathcal{Y}}_t$-monomials, we have: $$N_t(m_1m_2...m_k)=N_t(m_1)+N_t(m_2)+...+N_t(m_k)+\underset{1\leq i<j\leq k}{\sum}\mathcal{N}_t(m_i,m_j)$$ [[*Proof:*]{}]{}For the first point, it follows from the definition that $\tilde{(m_1m_2)}=t^{\mathcal{N}_t(m_1,m_2)}\tilde{m_1}\tilde{m_2}$. But: $$\tilde{(m_1m_2)}=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_1)\pi_l(m_2)\text{ , }\tilde{m_1}\tilde{m_2}=(\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_1))(\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_2))$$ So we have to commute $\pi_l(m_1)$ and $\pi_{l'}(m_2)$ for $l>l'$. The last assertion is proved as in lemma \[bibi\]. ### Presentation related to the basis $A_t$ and identification with [@Nab] We suppose we are in the $ADE$-case. Let be $m_1=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}}\tilde{A}_{i,l}^{-v_{i,l}}:,m_2=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}'}\tilde{A}_{i,l}^{-v_{i,l}'}:\in{\mathcal{Y}}_t$. We set $m_1^y=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}}:$ and $m_2^y=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}'}:$. \[form\] We have $\mathcal{N}_t(m_1,m_2)=\mathcal{N}_t(m_1^y,m_2^y)+2d(m_1,m_2)$, where:\[d\] $$d(m_1,m_2)=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}u_{i,l}'+y_{i,l+1}v_{i,l}'=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}u_{i,l+1}v_{i,l}'+v_{i,l+1}y_{i,l}'$$ where $u_{i,l}=y_{i,l}-v_{i,l-1}-v_{i,l+1}+\underset{j/C_{i,j}=-1}{\sum}v_{j,l}$ and $u_{i,l}'=y_{i,l}'-v_{i,l-1}'-v_{i,l+1}'+\underset{j/C_{i,j}=-1}{\sum}v_{j,l}'$. [[*Proof:*]{}]{} First notice that we have ($i\in I,l\in{\mathbb{Z}}$): $$\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},A_{i,l-1}^{-1})=-2$$ $$\mathcal{N}_t(Y_{i,l+1}^{-1},Y_{i,l-1}^{-1})=-2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2$$ For example $\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=\mathcal{N}(Y_{i,l},A_{i,l-1}^{-1})-\mathcal{N}(A_{i,l-1}^{-1},Y_{i,l})=2$ because $\tilde{Y}_{i,l}\tilde{A}_{i,l-1}^{-1}=t^2\tilde{A}_{i,l-1}^{-1}\tilde{Y}_{i,l}$. We have $\mathcal{N}_t(m_1,m_2)=A+B+C+D$ where: $A=\mathcal{N}_t(m_1^y,m_2^y)$ $B=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}y_{i,l}v_{j,k}'\mathcal{N}_t(Y_{i,l},A_{j,k}^{-1})=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}y_{i,l}v_{i,l-1}'\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}y_{i,l}v_{i,l-1}'$ $C=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}v_{i,l}y_{j,k}'\mathcal{N}_t(A_{i,l}^{-1},Y_{j,k})=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}y_{i,l}'\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}y_{i,l}'$ $D=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}v_{i,l}v_{j,k}'\mathcal{N}_t(A_{i,l}^{-1},A_{j,k}^{-1}) \\=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{i,l-1}'\mathcal{N}_t(A_{i,l+1}^{-1},A_{i,l-1}^{-1})+v_{i,l}v_{i,l}'\mathcal{N}_t(Y_{i,l+1}^{-1},Y_{i,l-1}^{-1})+\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l}) \\=-2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}(v_{i,l+1}v_{i,l-1}'+v_{i,l}v_{i,l'})+2\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'$ In particular, we have: $$B+C+D=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}(y_{i,l}v_{i,l-1}'+v_{i,l+1}y_{i,l}'-v_{i,l+1}v_{i,l-1}'-v_{i,l}v_{i,l}')+2\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'$$ The bicharacter $d$ was introduced for the $ADE$-case by Nakajima in [@Nab] motivated by geometry. It particular this proposition \[form\] gives a new motivation for this deformed structure. ### Presentation related to the basis $A$ and identification with [@Vas] {#varva} For $m_1,m_2\in A$, we have: $$\mathcal{N}(m_1,m_2)=\underset{i,j\in I, l,k\in{\mathbb{Z}}}{\sum}u_{i,l}(m_1)u_{j,k}(m_2)((\tilde{C}_{j,i}(z))_{r_j+l-k}-(\tilde{C}_{j,i}(z))_{-r_j+l-k})$$ [[*Proof:*]{}]{}First we can compute in ${\mathcal{Y}}_u$: $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}=\text{exp}(\underset{m>0}{\sum}h^{2m}[y_i[-m],y_j[m]]q^{m(k-l)}):\tilde{Y}_{i,l}\tilde{Y}_{j,k}:=t_{\tilde{C}_{j,i}(z)z^{k-l}(z^{-r_j}-z^{r_j})}:\tilde{Y}_{i,l}\tilde{Y}_{j,k}:$$ and as $N(\tilde{Y}_{i,l})=N(\tilde{Y}_{j,k})=0$ we have $\mathcal{N}(\tilde{Y}_{i,l},\tilde{Y}_{j,k})=(\tilde{C}_{j,i}(z))_{r_j+l-k}-(\tilde{C}_{j,i}(z))_{-r_j+l-k}$. In $sl_2$-case we have $C(z)=z+z^{-1}$ and $\tilde{C}(z)=\frac{1}{z+z^{-1}}=\underset{r\geq 0}{\sum}(-1)^rz^{-2r-1}$. So: $$\tilde{Y}_l\tilde{Y}_k=t^s:\tilde{Y}_l\tilde{Y}_k:$$ where: $s=0$ if $l-k=1+2r$, $r\in{\mathbb{Z}}$ $s=0$ if $l-k=2r$, $r>0$ $s=2(-1)^{r+1}$ if $l-k=2r$, $r<0$ $s=-1$ if $l=k$ It is analogous to the multiplication introduced for the $ADE$-case by Varagnolo-Vasserot in [@Vas]: we suppose we are in the $ADE$-case, denote $P=\underset{i\in I}{\bigoplus}{\mathbb{Z}}\omega_i$ (resp. $Q=\underset{i\in I}{\bigoplus}{\mathbb{Z}}\alpha_i$) the weight-lattice (resp. root-lattice) and: $\bar{}:P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow P\otimes{\mathbb{Z}}[z^{\pm}]$ is defined by $\overline{\lambda \otimes P(z)}=\lambda\otimes P(z^{-1})$. $(,):Q\otimes {\mathbb{Z}}((z^{-1}))\times P\otimes{\mathbb{Z}}((z^{-1}))\rightarrow {\mathbb{Z}}((z^{-1}))$ is the ${\mathbb{Z}}((z^{-1}))$-bilinear form defined by $(\alpha_i,\omega_j)=\delta_{i,j}$. $\Omega^{-1}:P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow Q\otimes{\mathbb{Z}}((z^{-1}))$ is defined by $\Omega^{-1}(\omega_i)=\underset{k\in I}{\sum}\tilde{C}_{i,k}(z)\alpha_k$. The map $\epsilon:P\otimes{\mathbb{Z}}[z^{\pm}]\times P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow {\mathbb{Z}}$ is defined by: $$\epsilon_{\lambda,\mu}=\pi_0((z^{-1}\Omega^{-1}(\bar{\lambda})|\mu))$$ The multiplication of [@Vas] is defined by: $$Y_{i,l}Y_{j,m}=t^{2\epsilon_{z^l\omega_i,z^m\omega_j}-2\epsilon_{z^m\omega_j,z^l\omega_i}}Y_{j,m}Y_{i,l}$$ So we can compute: $$\epsilon_{z^l\omega_i,z^m\omega_j}=\pi_0((z^{-1}\Omega^{-1}(z^{-l}\omega_i)|z^m\omega_j))=\pi_0(\underset{k\in I}{\sum}(z^{-1-l}\tilde{C}_{i,k}(z)\alpha_k|z^m\omega_j))$$ $$=\pi_0(z^{m-l-1}\tilde{C}_{i,j}(z))=(\tilde{C}_{i,j}(z))_{l+1-m}$$ If we set $\epsilon'_{\lambda,\mu}=\pi_0((z\Omega^{-1}(\bar{\lambda})|\mu))$ then we have $\epsilon'_{z^l\omega_i,z^m\omega_j}=(\tilde{C}_{i,j}(z))_{l-1-m}$ and: $$\epsilon_{z^l\omega_i,z^m\omega_j}-\epsilon'_{z^l\omega_i,z^m\omega_j}=\mathcal{N}({Y}_{i,l},{Y}_{j,m})$$ Deformed screening operators {#scr} ============================ Motivated by the screening currents of [@Freb] we give in this section a “concrete” approach to deformations of screening operators. In particular the $t$-analogues of screening operators defined in [@Her01] will appear as commutators in $\mathcal{H}_h$. Let us begin with some background about classic screening operators. Reminder: classic screening operators ([@Fre],[@Fre2]) {#screclas} ------------------------------------------------------ ### Classic screening operators and symmetry property of $q$-characters Recall the definition of\ $\pi_+(\tilde{A}_{i,l}^{\pm})=A_{i,l}^{\pm}\in {\mathcal{Y}}$ and of $u_{i,l}:A\rightarrow{\mathbb{Z}}$ in section \[defoal\]. The $i^{\text{th}}$-screening operator is the ${\mathbb{Z}}$-linear map defined by:\[si\] $$S_i:{\mathcal{Y}}\rightarrow{\mathcal{Y}}_i=\frac{\underset{l\in{\mathbb{Z}}}{\bigoplus}{\mathcal{Y}}.S_{i,l}}{\underset{l\in{\mathbb{Z}}}{\sum}{\mathcal{Y}}.(S_{i,l+2r_i}-A_{i,l+r_i}.S_{i,l})}$$\[yi\] $$\forall m\in A, S_i(m)=\underset{l\in{\mathbb{Z}}}{\sum}u_{i,l}(m)S_{i,l}$$ Note that the $i^{\text{th}}$-screening operator can also be defined as the derivation such that: $$S_i(1)=0\text{ , }\forall j\in I,l\in{\mathbb{Z}},S_i(Y_{j,l})=\delta_{i,j}Y_{i,l}.S_{i,l}$$ ([**Frenkel, Reshetikhin, Mukhin**]{} [@Fre],[@Fre2])\[simme\] The image of $\chi_q:{\mathbb{Z}}[X_{i,l}]_{i\in I,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}$ is: $$\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$$ It is analogous to the classical symmetry property of $\chi$: $\text{Im}(\chi)={\mathbb{Z}}[y_i^{\pm}]_{i\in I}^W$. ### Structure of the kernel of $S_i$ {#mi} Let $\mathfrak{K}_i=\text{Ker}(S_i)$\[ki\]. It is a ${\mathbb{Z}}$-subalgebra of ${\mathcal{Y}}$. \[deux\]([**Frenkel, Reshetikhin, Mukhin**]{} [@Fre],[@Fre2]) The ${\mathbb{Z}}$-subalgebra $\mathfrak{K}_i$ of ${\mathcal{Y}}$ is generated by the $Y_{i,l}(1+A_{i,l+r_i}^{-1}),Y_{j,l}^{\pm}$ ($j\neq i,l\in {\mathbb{Z}}$). For $m\in B_i$, we denote: $$E_i(m)=m\underset{l\in{\mathbb{Z}}}{\prod}(1+{{A_{i,l+r_i}}}^{-1})^{u_{i,l}(m)}\in\mathfrak{K}_i$$\[eim\] In particular: \[el\] The ${\mathbb{Z}}$-module $\mathfrak{K}_i$ is freely generated by the $E_i(m)$ ($m\in B_i$): $$\mathfrak{K}_i=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}E_i(m)\simeq {\mathbb{Z}}^{(B_i)}$$ ### Examples in the $sl_2$-case {#exlsl} We suppose in this section that we are in the $sl_2$-case. For $m\in B$, let $L(m)=\chi_q(V_m)$ be the $q$-character of the ${\mathcal{U}}_q(\hat{sl_2})$-irreducible representation of highest weight $m$. In particular $L(m)\in\mathfrak{K}$ and $\mathfrak{K}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}L(m)$. In [@Fre] an explicit formula for $L(m)$ is given: a $\sigma\subset{\mathbb{Z}}$ is called a $2$-segment if $\sigma$ is of the form $\sigma=\{l,l+2,...,l+2k\}$. Two $2$-segment are said to be in special position if their union is a $2$-segment that properly contains each of them. All finite subset of ${\mathbb{Z}}$ with multiplicity $(l,u_l)_{l\in{\mathbb{Z}}}$ ($u_l\geq 0$) can be broken in a unique way into a union of $2$-segments which are not in pairwise special position. For $m\in B$ we decompose $m=\underset{j}{\prod}\underset{l\in \sigma_j}{\prod}Y_l\in B$ where the $(\sigma_j)_j$ is the decomposition of the $(l,u_l(m))_{l\in{\mathbb{Z}}}$. We have: $$L(m)=\underset{j}{\prod}L(\underset{l\in \sigma_j}{\prod}Y_l)$$ So it suffices to give the formula for a $2$-segments: $$L(Y_lY_{l+2}Y_{l+4}...Y_{l+2k})=Y_lY_{l+2}Y_{l+4}...Y_{l+2k}+Y_lY_{l+2}...Y_{l+2(k-1)}Y_{l+2(k+1)}^{-1}$$ $$+Y_lY_{l+2}...Y_{l+2(k-2)}Y_{l+2k}^{-1}Y_{l+2(k+1)}^{-1}+...+Y_{l+2}^{-1}Y_{l+2}^{-1}...Y_{l+2(k+1)}^{-1}$$ We say that $m$ is irregular if there are $j_1\neq j_2$ such that $$\sigma_{j_1}\subset \sigma_{j_2}\text{ and }\sigma_{j_1}+2\subset\sigma_{j_2}$$ ([**Frenkel, Reshetikhin**]{} [@Fre])\[dominl\] There is a dominant ${\mathcal{Y}}$-monomial other than $m$ in $L(m)$ if and only if $m$ is irregular. ### Complements: another basis of $\mathfrak{K}_i$ {#compl} Let us go back to the general case. Let ${\mathcal{Y}}_{sl_2}={\mathbb{Z}}[Y_l^{\pm}]_{l\in{\mathbb{Z}}}$ the ring ${\mathcal{Y}}$ for the $sl_2$-case. Let $i$ be in $I$ and for $0\leq k\leq r_i-1$, let $\omega_k: A\rightarrow {\mathcal{Y}}_{sl_2}$ be the map defined by: $$\omega_k(m)=\underset{l\in{\mathbb{Z}}}{\prod}Y_l^{u_{i,k+lr_i}(m)}$$ and $\nu_k:{\mathbb{Z}}[(Y_{l-1}Y_{l+1})^{-1}]_{l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}$ be the ring homomorphism such that $\nu_k((Y_{l-1}Y_{l+1})^{-1})=A_{i,k+lr_i}^{-1}$. For $m\in B_i$, $\omega_k(m)$ is dominant in ${\mathcal{Y}}_{sl_2}$ and so we can define $L(\omega_k(m))$ (see section \[exlsl\]). We have $L(\omega_k(m))\omega_k(m)^{-1}\in{\mathbb{Z}}[(Y_{l-1}Y_{l+1})^{-1}]_{l\in{\mathbb{Z}}}$. We introduce: $$L_i(m)=m\underset{0\leq k\leq r_i-1}{\prod}\nu_k(L(\omega_k(m))\omega_k(m)^{-1})\in \mathfrak{K}_i$$\[lim\] In analogy with the corollary \[el\] the ${\mathbb{Z}}$-module $\mathfrak{K}_i$ is freely generated by the $L_i(m)$ ($m\in B_i$): $$\mathfrak{K}_i=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}L_i(m)\simeq {\mathbb{Z}}^{(B_i)}$$ Screening currents ------------------ Following [@Freb], for $i\in I,l\in{\mathbb{Z}}$, introduce $\tilde{S}_{i,l}\in \mathcal{H}_h$:\[tsil\] $$\tilde{S}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q_i^{m}-q_i^{-m}}q^{lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q_i^{-m}-q_i^{m}}q^{-lm})$$ \[currents\] We have the following relations in $\mathcal{H}_h$: $$\tilde{A}_{i,l}\tilde{S}_{i,l-r_i}=t_{-z^{-2r_i}-1}\tilde{S}_{i,l+r_i}$$ $$\tilde{S}_{i,l}\tilde{A}_{j,k}=t_{C_{i,j}(z)(z^{(k-l)}+z^{(l-k)})}\tilde{A}_{j,k}\tilde{S}_{i,l}$$ $$\tilde{S}_{i,l}\tilde{Y}_{j,k}=t_{\delta_{i,j}(z^{(k-l)}+z^{(l-k)})}\tilde{Y}_{j,k}\tilde{S}_{i,l}$$ [[*Proof:*]{}]{} As for lemma \[relu\] we compute in $\mathcal{H}_h$: $\tilde{A}_{i,l}\tilde{S}_{i,l-r_i} \\=\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{m(l-r_i)}) \\\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{m(r_i-l)}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{-q^{2mr_i}+q^{-2mr_i}}{q^{mr_i}-q^{-mr_i}}q^{-mr_i}c_m)\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm}+h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{m(l-r_i)}) \\\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{m(r_i-l)}+h^ma_i[-m]q^{-lm}) \\=t_{-z^{-2r_i}-1}\text{exp}(\underset{m>0}{\sum}h^m a_i[m](1+\frac{q^{-mr_i}}{q^{mr_i}-q^{-mr_i}})q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[m](1+\frac{q^{mr_i}}{q^{-mr_i}-q^{mr_i}})q^{-lm}) \\=t_{-z^{-2r_i}-1}\tilde{S}_{i,l+r_i}$ $\tilde{S}_{i,l}\tilde{A}_{j,k} \\=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})(\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km})) \\\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{(q^{-mr_i}-q^{mr_i})C_{i,j}(q^m)}{q^{-mr_i}-q^{mr_i}}q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm}) \\\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}C_{i,j}(q^m)(q^{m(k-l)}+q^{(l-k)m})c_m)\tilde{A}_{j,k}\tilde{S}_{i,l}$ Finally: $\tilde{S}_{i,l}\tilde{Y}_{j,k} \\=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})(\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_{j}[m]q^{km})) \\\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}\frac{(q^{-mr_i}-q^{mr_i})}{q^{-mr_i}-q^{mr_i}}q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm}) \\\text{exp}(\underset{m>0}{\sum}h^my_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km}) \\=\text{exp}(\underset{m>0}{\sum}\delta_{i,j}h^{2m}(q^{m(k-l)}+q^{(l-k)m})c_m)\tilde{Y}_{j,k}\tilde{S}_{i,l}$ Deformed bimodules ------------------ In this section we define and study a $t$-analogue ${\mathcal{Y}}_{i,t}$ of the module ${\mathcal{Y}}_i$. For $i\in I$, let ${\mathcal{Y}}_{i,u}$\[yiu\] be the ${\mathcal{Y}}_u$ sub left-module of $\mathcal{H}_h$ generated by the $\tilde{S}_{i,l}$ ($l\in {\mathbb{Z}}$). It follows from lemma \[currents\] that $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$ generate ${\mathcal{Y}}_{i,u}$ and that it is also a subbimodule of $\mathcal{H}_h$. Denote by $\tilde{S}_{i,l}\in \mathcal{H}_t$\[tsit\] the image of $\tilde{S}_{i,l}\in \mathcal{H}_h$ in $\mathcal{H}_t$. ${\mathcal{Y}}_{i,t}$\[yit\] is the sub left-module of $\mathcal{H}_t$ generated by the $\tilde{S}_{i,l}$ ($l\in{\mathbb{Z}}$). In particular it is to say the image of ${\mathcal{Y}}_{i,u}$ in $\mathcal{H}_t$. It follows from lemma \[currents\] that for $l\in{\mathbb{Z}}$, we have in ${\mathcal{Y}}_{i,t}$: $$\tilde{A}_{i,l}\tilde{S}_{i,l-r_i}=t^{-1}\tilde{S}_{i,l+r_i}$$ It particular ${\mathcal{Y}}_{i,t}$ is generated by the $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$. It follows from lemma \[currents\] that for $l\in{\mathbb{Z}}$, we have: $$\tilde{S}_{i,l}.\tilde{Y}_{j,k}=t^{2\delta_{i,j}\delta_{l,k}}\tilde{Y}_{j,k}.\tilde{S}_{i,l}\text{ , }\tilde{S}_{i,l}.t=t.\tilde{S}_{i,l}$$ In particular ${\mathcal{Y}}_{i,t}$ a subbimodule of $\mathcal{H}_t$. Moreover: $$\tilde{S}_{i,l}.\tilde{A}_{i,k}=t^{2\delta_{l-k,r_i}+2\delta_{l-k,-r_i}}\tilde{A}_{i,k}.\tilde{S}_{i,l}$$ $$\tilde{S}_{i,l}.\tilde{A}_{j,k}=t^{-2\underset{r=C_{i,j}+1,C_{i,j}+3,...,-C_{i,j}-1}{\sum}\delta_{l-k,r}}\tilde{A}_{j,k}.\tilde{S}_{i,l}\text{ (if $i\neq j$)}$$ \[tcur\] The ${\mathcal{Y}}_t$ left module ${\mathcal{Y}}_{i,t}$ is freely generated by $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$: $${\mathcal{Y}}_{i,t}=\underset{-r_i\leq l <r_i}{\bigoplus}{\mathcal{Y}}_t\tilde{S}_{i,l}\simeq {\mathcal{Y}}_t^{2r_i}$$ [[*Proof:*]{}]{}We saw that $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$ generate ${\mathcal{Y}}_{i,t}$. We prove they are ${\mathcal{Y}}_t$-linearly independent:\ for $(R_1,...,R_n)\in\mathfrak{U}^n$, introduce: $$Y_{R_1,...,R_n}=\text{exp}(\underset{m>0, j\in I}{\sum}h^my_j[m]R_j(q^m))\in \mathcal{H}_t^+$$ It follows from lemma \[indgene\] that the $(Y_{R})_{R\in\mathfrak{U}^n}$ are ${\mathbb{Z}}$-linearly independent. Note that we have $\pi_+({\mathcal{Y}}_{i,t})\subset \underset{R\in \mathfrak{U}^n}{\bigoplus}{\mathbb{Z}}Y_R$ and that ${\mathcal{Y}}=\underset{R\in{\mathbb{Z}}[z^{\pm}]^n}{\bigoplus}{\mathbb{Z}}Y_R$. Suppose we have a linear combination ($\lambda_r\in{\mathcal{Y}}_t$): $$\lambda_{-r_i}\tilde{S}_{i,-r_i}+...+\lambda_{r_i-1}\tilde{S}_{i,r_i-1}=0$$ Introduce $\mu_{k,R}\in{\mathbb{Z}}$ such that: $$\pi_+(\lambda_k)=\underset{R\in{\mathbb{Z}}[z^{\pm}]^n}{\sum}\mu_{k,R}Y_R$$ and $R_{i,k}=(R_{i,k}^{1}(z),...,R_{i,k}^{n}(z))\in\mathfrak{U}^n$ such that $\pi_+(\tilde{S}_{i,k})=Y_{R_{i,k}}$. If we apply $\pi_+$ to the linear combination, we get: $$\underset{R\in{\mathbb{Z}}[z^{\pm}]^n,-r_i\leq k\leq r_i-1}{\sum}\mu_{k,R}Y_RY_{R_{i,k}}=0$$ and we have for all $R'\in \mathfrak{U}$: $$\underset{-r_i\leq k\leq r_i-1/R'-R_{i,k}\in{\mathbb{Z}}[z^{\pm}]^n}{\sum}\mu_{k,R'-R_{i,k}}=0$$ Suppose we have $-r_i\leq k_1\neq k_2\leq r_i -1$ such that $R'-R_{i,k_1},R'-R_{i,k_2}\in {\mathbb{Z}}[z^{\pm}]^n$. So $R_{i,k_1}-R_{i,k_2}\in{\mathbb{Z}}[z^{\pm}]^n$. But $a_i[m]=\underset{j\in I}{\sum}C_{j,i}(q^m)y_j[m]$, so for $j\in I$: $$C_{j,i}(z)\frac{z^{k_1}-z^{k_2}}{z^{r_i}-z^{-r_i}}=(R_{i,k_1}^j(z)-R_{i,k_2}^j(z))\in {\mathbb{Z}}[z^{\pm}]$$ In particular for $j=i$ we have $C_{i,i}(z)\frac{z^{k_1}-z^{k_2}}{z^{r_i}-z^{-r_i}}=\frac{(z^{r_i}+z^{-r_i})(z^{k_1}-z^{k_2})}{z^{r_i}-z^{-r_i}}\in {\mathbb{Z}}[z^{\pm}]$. This is impossible because $|k_1-k_2|<2r_i$. So we have only one term in the sum and all $\mu_{k,R}=0$. So $\pi_+(\lambda_k)=0$, and $\lambda_k\in (t-1){\mathcal{Y}}_t$. We have by induction for all $m>0$, $\lambda_k\in (t-1)^m{\mathcal{Y}}_t$. It is possible if and only if $\lambda_k=0$. Denote by ${\mathcal{Y}}_i$ the ${\mathcal{Y}}$-bimodule $\pi_+({\mathcal{Y}}_{i,t})$. It is consistent with the notations of section \[screclas\]. $t$-analogues of screening operators ------------------------------------ We introduced $t$-analogues of screening operators in [@Her01]. The picture of the last section enables us to define them from a new point of view. For $m$ a ${\mathcal{Y}}_t$-monomial, we have: $$[\tilde{S}_{i,l},m]=\tilde{S}_{i,l}m-m\tilde{S}_{i,l}=(t^{2u_{i,l}(m)}-1)m\tilde{S}_{i,l}=t^{u_{i,l}(m)}(t-t^{-1})[u_{i,l}(m)]_tm\tilde{S}_{i,l}$$ So for $\lambda\in {\mathcal{Y}}_t$ we have $[\tilde{S}_{i,l},\lambda]\in (t^2-1){\mathcal{Y}}_{i,t}$, and $[\tilde{S}_{i,l},\lambda]\neq 0$ only for a finite number of $l\in{\mathbb{Z}}$. So we can define: The $i^{th}$ $t$-screening operator is the map $S_{i,t}:{\mathcal{Y}}_t\rightarrow {\mathcal{Y}}_{i,t}$ such that ($\lambda\in{\mathcal{Y}}_t$): $$S_{i,t}(\lambda)=\frac{1}{t^2-1}\underset{l\in{\mathbb{Z}}}{\sum}[\tilde{S}_{i,l},\lambda]\in {\mathcal{Y}}_{i,t}$$ In particular, $S_{i,t}$ is ${\mathbb{Z}}[t^{\pm}]$-linear and a derivation. It is our map of [@Her01]. For $m$ a ${\mathcal{Y}}_t$-monomial, we have $\pi_+(S_{i,t}(m))=\pi_+(t^{u_{i,l}(m)-1}[u_{i,l}(m)]_t)\pi_+(m\tilde{S}_{i,l})=u_{i,l}(m)\pi_+(m\tilde{S}_{i,l})$ and the following commutative diagram: $$\begin{array}{rcccl} {\mathcal{Y}}_t&\stackrel{S_{i,t}}{\longrightarrow}&{\mathcal{Y}}_{i,t}\\ \pi_+\downarrow &&\downarrow&\pi_+\\ {\mathcal{Y}}&\stackrel{S_i}{\longrightarrow}&{\mathcal{Y}}_i\end{array}$$ Kernel of deformed screening operators {#kernelun} -------------------------------------- ### Structure of the kernel {#defitl} We proved in [@Her01] a $t$-analogue of theorem \[deux\]: \[her\] ([@Her01]) The kernel of the $i^{th}$ $t$-screening operator $S_{i,t}$ is the ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$ generated by the $\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}),\tilde{Y}_{j,l}^{\pm}$ ($j\neq i,l\in {\mathbb{Z}}$). [[*Proof:*]{}]{}For the first inclusion we compute: $$S_{i,t}(\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))=\tilde{Y}_{i,l}\tilde{S}_{i,l}+t\tilde{Y}_{i,l}\tilde{A}_{i,l+r_i}^{-1}(-t^{-2})\tilde{S}_{i,l+2r_i}=\tilde{Y}_{i,l}(\tilde{S}_{i,l}-t^{-1}\tilde{A}_{i,l+r_i}^{-1}\tilde{S}_{i,l+2r_i})=0$$ For the other inclusion we refer to [@Her01]. Let $\mathfrak{K}_{i,t}=\text{Ker}(S_{i,t})$\[kit\]. It is a ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$. In particular we have $\pi_+(\mathfrak{K}_{i,t})=\mathfrak{K}_i$ (consequence of theorem \[deux\] and \[her\]). For $m\in B_i$ introduce: (recall that $\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}U_l$ means $...U_{-1}U_0U_1U_2...$): $$E_{i,t}(m)=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}((\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))^{u_{i,l}(m)}\underset{j\neq i}{\prod}\tilde{Y}_{j,l}^{u_{j,l}(m)})$$\[teitm\] It is well defined because it follows from theorem \[dessus\] that for $j\neq i, l\in{\mathbb{Z}}$, $(\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))$ and $\tilde{Y}_{j,l}$ commute. For $m\in B_i$, the formula shows that the ${\mathcal{Y}}_t$-monomials of $E_{i,t}(m)$ are the ${\mathcal{Y}}$-monomials of $E_i(m)$ (with identification by $\pi_+$). Such elements were used in [@Nab] for the $ADE$ case. The theorem \[her\] allows us to describe $\mathfrak{K}_{i,t}$: \[hers\] For all $m\in B_i$, we have $E_{i,t}(m)\in\mathfrak{K}_{i,t}$. Moreover: $$\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_{i,t}(m)\simeq {\mathbb{Z}}[t^{\pm}]^{(B_i)}$$ [[*Proof:*]{}]{}First $E_{i,t}(m)\in\mathfrak{K}_{i,t}$ as product of elements of $\mathfrak{K}_{i,t}$. We show easily that the $E_{i,t}(m)$ are ${\mathbb{Z}}[t^{\pm}]$-linearly independent by looking at a maximal ${\mathcal{Y}}_t$-monomial in a linear combination. Let us prove that the $E_{i,t}(m)$ are ${\mathbb{Z}}[t^{\pm}]$-generators of $\mathfrak{K}_{i,t}$: for a product $\chi$ of the algebra-generators of theorem \[her\], let us look at the highest weight ${\mathcal{Y}}_t$-monomial $m$. Then $E_{i,t}(m)$ is this product up to the order in the multiplication. But for $p=1$ or $p\geq 3$, $Y_{i,l}Y_{i,l+pr_i}$ is the unique dominant ${\mathcal{Y}}$-monomial of $E_i(Y_{i,l})E_i(Y_{i,l+pr_i})$, so: $$\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+pr_i}(1+t\tilde{A}_{i,l+pr_i+r_i}^{-1})\in t^{{\mathbb{Z}}}\tilde{Y}_{i,l+pr_i}(1+t\tilde{A}_{i,l+pr_i+r_i}^{-1})\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})$$ And for $p=2$: $$\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})-\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})$$ $$\in {\mathbb{Z}}[t^{\pm}]+t^{{\mathbb{Z}}}\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})$$ ### Elements of $\mathfrak{K}_{i,t}$ with a unique $i$-dominant ${\mathcal{Y}}_t$-monomial \[defifprem\] For $m\in B_i$, there is a unique $F_{i,t}(m)\in\mathfrak{K}_{i,t}$\[tfitm\] such that $m$ is the unique $i$-dominant ${\mathcal{Y}}_t$-monomial of $F_{i,t}(m)$. Moreover : $$\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}F_{i,t}(m)$$ [[*Proof:*]{}]{}It follows from corollary \[hers\] that an element of $\mathfrak{K}_{i,t}$ has at least one $i$-dominant ${\mathcal{Y}}_t$-monomial. In particular we have the uniqueness of $F_{i,t}(m)$. For the existence, let us look at the $sl_2$-case. Let $m$ be in $B$. It follows from the lemma \[fini\] that $\{MA_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}/R\geq 0,l_1,...,l_R\geq l(M)\}\cap B$ is finite (where $l(M)=\text{min}\{l\in{\mathbb{Z}}/\exists i\in I,u_{i,l}(M)\neq 0\}$). We define on this set a total ordering compatible with the partial ordering: $m_L=m>m_{L-1}>...>m_1$. Let us prove by induction on $l$ the existence of $F_t(m_l)$. The unique dominant ${\mathcal{Y}}_t$-monomial of $E_t(m_1)$ is $m_1$ so $F_t(m_1)=E_t(m_1)$. In general let $\lambda_1(t),...,\lambda_{l-1}(t)\in{\mathbb{Z}}[t^{\pm}]$ be the coefficient of the dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_{l-1}$ in $E_t(m_l)$. We put: $$F_t(m_l)=E_t(m_l)-\underset{r=1...l-1}{\sum}\lambda_r(t)F_t(m_r)$$ Notice that this construction gives $F_t(m)\in m{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_{l\in{\mathbb{Z}}}$. For the general case, let $i$ be in $I$ and $m$ be in $B_i$. Consider $\omega_k(m)$ as in section \[compl\]. The study of the $sl_2$-case allows us to set $\chi_k=\omega_k(m)^{-1}F_t(\omega_k(m))\in{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_l$. And using the ${\mathbb{Z}}[t^{\pm}]$-algebra homomorphism $\nu_{k,t}:{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_{l\in{\mathbb{Z}}}\rightarrow {\mathbb{Z}}[\tilde{A}_{i,l}^{-1},t^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$ defined by $\nu_{k,t}(\tilde{A}_l^{-1})=\tilde{A}_{i,k+lr_i}^{-1}$, we set (the terms of the product commute): $$F_{i,t}(m)=m\underset{0\leq k\leq r_i-1}{\prod}\nu_{k,t}(\chi_k)\in\mathfrak{K}_{i,t}$$ For the last assertion, we have $E_{i,t}(m)=\underset{l=1...L}{\sum}\lambda_l(t)F_{i,t}(m_l)$ where $m_1,...,m_L$ are the $i$-dominant ${\mathcal{Y}}_t$-monomials of $E_{i,t}(m)$ with coefficients $\lambda_1(t),...,\lambda_L(t)\in{\mathbb{Z}}[t^{\pm}]$. In the same way there is a unique $F_i(m)\in\mathfrak{K}_i$\[fim\] such that $m$ is the unique $i$-dominant ${\mathcal{Y}}$-monomial of $F_i(m)$. Moreover $F_i(m)=\pi_+(F_{i,t}(m))$. ### Examples in the $sl_2$-case {#examples-in-the-sl_2-case} In this section we suppose that ${\mathfrak{g}}=sl_2$ and we compute $F_t(m)=F_{1,t}(m)$ in some examples with the help of section \[exlsl\]. \[fexp\] Let $\sigma=\{l,l+2,...,l+2k\}$ be a $2$-segment and $m_{\sigma}=\tilde{Y}_l\tilde{Y}_{l+2}...\tilde{Y}_{l+2k}\in B$. Then we have the formula: $$F_t(m_{\sigma})=m_{\sigma}(1+t\tilde{A}_{l+2k+1}^{-1}+t^2\tilde{A}_{l+(2k+1)}^{-1}\tilde{A}_{l+(2k-1)}^{-1}+...+t^k\tilde{A}_{l+(2k+1)}^{-1}\tilde{A}_{l+(2k-1)}^{-1}...\tilde{A}_{l+1}^{-1})$$ If $\sigma_1,\sigma_2$ are $2$-segments not in special position, we have: $$F_t(m_{\sigma_1})F_t(m_{\sigma_2})=t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}F_t(m_{\sigma_2})F_t(m_{\sigma_1})$$ If $\sigma_1,...,\sigma_R$ are $2$-segments such that $m_{\sigma_1}...m_{\sigma_r}$ is regular, we have: $$F_t(m_{\sigma_1}...m_{\sigma_R})=F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$$ In particular if $m\in B$ verifies $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$ then it is of the form $m=m_{\sigma_1}...m_{\sigma_R}$ where the $\sigma_r$ are $2$-segments such that $\text{max}(\sigma_r)+2<\text{min}(\sigma_{r+1})$. So the lemma \[fexp\] gives an explicit formula $F_t(m)=F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$. [[*Proof:*]{}]{}First we need some relations in ${\mathcal{Y}}_{1,t}$ : we know that for $l\in{\mathbb{Z}}$ we have $t\tilde{S}_{l-1}=\tilde{A}_l^{-1}\tilde{S}_{l+1}=t^2\tilde{S}_{l+1}\tilde{A}_l^{-1}$, so $t^{-1}\tilde{S}_{l-1}=\tilde{S}_{l+1}\tilde{A}_l^{-1}$. So we get by induction that for $r\geq 0$: $$t^{-r}\tilde{S}_{l+1-2r}=\tilde{S}_{l+1}\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}$$ As $u_{i,l+1}(\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1})=u_{i,l+1}(\tilde{A}_l^{-1})=-1$, we get: $$t^{-r}\tilde{S}_{l+1-2r}=t^{-2}\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}\tilde{S}_{l+1}$$ For $r'\geq 0$, by multiplying on the left by $\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l+2}^{-1}$, we get: $$t^{-r}\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l+2}^{-1}\tilde{S}_{l+1-2r}=t^{-2}\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}\tilde{S}_{l+1}$$ If we put $r'=1+R',r=R-R',l=L-1-2R'$, we get for $0\leq R'\leq R$: $$t^{R'}\tilde{A}_{L+1}^{-1}\tilde{A}_{L-1}^{-1}...\tilde{A}_{L+1-2R'}^{-1}\tilde{S}_{L-2R}=t^{R-2}\tilde{A}_{L+1}^{-1}\tilde{A}_{L-1}^{-1}...\tilde{A}_{L+1-2R}^{-1}\tilde{S}_{L-2R'}$$ Now let be $m=\tilde{Y}_0\tilde{Y}_2...\tilde{Y}_l$ and $\chi\in{\mathcal{Y}}_t$ given by the formula in the lemma. Let us compute $\tilde{S}_t(\chi)$: $$\tilde{S}_t(\chi)=m(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_l)$$ $$+tm\tilde{A}^{-1}_{l+1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-2}-t^{-2}\tilde{S}_{l+2})$$ $$+t^2m\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-4}-t^{-2}\tilde{S}_{l}-t^{-2}\tilde{S}_{l+2})$$ $$+...$$ $$+t^lm\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}...\tilde{A}^{-1}_{1}(-t^{-2}\tilde{S}_2+...-t^{-2}\tilde{S}_{l}-t^{-2}\tilde{S}_{l+2})$$ $$=m(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_l)$$ $$+tm\tilde{A}^{-1}_{l+1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-2})-m\tilde{S}_l$$ $$+t^2m\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-4})-tm\tilde{A}^{-1}_{l+1}\tilde{S}_{l-2}-m\tilde{S}_{l-2}$$ $$+...$$ $$-mt^{l-1}\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}...\tilde{A}^{-1}_{3}-...-t^{-2}\tilde{S}_{l}-m\tilde{S}_0$$ $$=0$$ So $\chi\in\mathfrak{K}_t$. But we see on the formula that $m$ is the unique dominant monomial of $\chi$. So $\chi=F_t(m)$. For the second point, we have two cases: if $m_{\sigma_1}m_{\sigma_2}$ is regular, it follows from lemma \[dominl\] that $L(m_{\sigma_1})L(m_{\sigma_2})=L(m_{\sigma_2})L(m_{\sigma_1})$ has no dominant monomial other than $m_{\sigma_1}m_{\sigma_2}$. But our formula shows that $F_t(m_{\sigma_1})$ (resp. $F_t(m_{\sigma_2}$)) has the same monomials than $L(m_{\sigma_1})$ (resp. $L(m_{\sigma_2})$). So $$F_t(m_{\sigma_1})F_t(m_{\sigma_2})-t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}F_t(m_{\sigma_2})F_t(m_{\sigma_1})$$ has no dominant ${\mathcal{Y}}_t$-monomial because $m_{\sigma_1}m_{\sigma_2}-t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}m_{\sigma_2}m_{\sigma_1}=0$. if $m_{\sigma_1}m_{\sigma_2}$ is irregular, we have for example $\sigma_{j_1}\subset \sigma_{j_2}$ and $\sigma_{j_1}+2\subset\sigma_{j_2}$. Let us write $\sigma_{j_1}=\{l_1,l_1+2,...,,p_1\}$ and $\sigma_2=\{l_2,l_2+2,...,,p_2\}$. So we have $l_2\leq l_1$ and $p_1\leq p_2-2$. Let $m=m_1m_2$ be a dominant ${\mathcal{Y}}$-monomial of $L(m_{\sigma_1}m_{\sigma_2})=L(m_{\sigma_1})L(m_{\sigma_2})$ where $m_1$ (resp. $m_2$) is a ${\mathcal{Y}}$-monomial of $L(m_{\sigma_1})$ (resp. $L(m_{\sigma_2})$). If $m_2$ is not $m_{\sigma_2}$, we have $Y_{p_2}^{-1}$ in $m_2$ which can not be canceled by $m_1$. So $m=m_1m_{\sigma_2}$. Let us write $m_1=m_{\sigma_1}A_{p_1+1}^{-1}...A_{p_1+1-2r}^{-1}$. So we just have to prove: $$\tilde{A}_{p_1+1}^{-1}...\tilde{A}_{p_1+1-2r}^{-1}m_{\sigma_2}=m_{\sigma_2}\tilde{A}_{p_1+1}^{-1}...\tilde{A}_{p_1+1-2r}^{-1}$$ This follows from ($l\in{\mathbb{Z}}$): $$\tilde{A}_l^{-1}\tilde{Y}_{l-1}\tilde{Y}_{l+1}=\tilde{Y}_{l-1}\tilde{Y}_{l+1}\tilde{A}_l^{-1}$$ For the last assertion it suffices to show that $F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$ has no other dominant ${\mathcal{Y}}_t$-monomial than $m_{\sigma_1}...m_{\sigma_R}$. But $F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$ has the same monomials than $L(m_{\sigma_1})...L(m_{\sigma_R})=L(m_{\sigma_1}...m_{\sigma_R})$. As $m_{\sigma_1}...m_{\sigma_R}$ is regular we get the result. ### Technical complements Let us go back to the general case. We give some technical results which will be used in the following to compute $F_{i,t}(m)$ in some cases (see proposition \[cpfacile\] and section \[fin\]). \[calcn\] Let $i$ be in $I$, $l\in{\mathbb{Z}}$, $M\in A$ such that $u_{i,l}(M)=1$ and $u_{i,l+2r_i}=0$. Then we have $\mathcal{N}(M,\tilde{A}_{i,l+r_i}^{-1})=-1$. In particular $\pi^{-1}(MA_{i,l+r_i}^{-1})=tM\tilde{A}_{i,l+r_i}^{-1}$. [[*Proof:*]{}]{}We can suppose $M=:M:$ and we compute in ${\mathcal{Y}}_u$: $$M\tilde{A}_{i,l+r_i}^{-1}=\pi_+(m)\text{exp}(\underset{m>0,r\in{\mathbb{Z}},j\in I}{\sum}u_{j,r}(M)h^mq^{-rm}y_j[-m])$$ $$\text{exp}(\underset{m>0}{\sum}-h^mq^{-(l+r_i)m}a_i[-m])\text{exp}(\underset{m>0}{\sum}-h^mq^{(l+r_i)m}a_i[m])$$ $$=:M\tilde{A}_{i,l+r_i}^{-1}:\text{exp}(\underset{m>0}{\sum}h^{2m}([a_i[-m],a_i[m]]-\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(m)[y_i[-m],a_i[m]]q^{(l+r_i-r)m}c_m)=t_R:\tilde{Y}_{i,l}\tilde{A}_{i,l+r_i}^{-1}:$$ where: $$R(z)=-(z^{2r_i}-z^{-2r_i})+\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(M)z^{(l+r_i-r)}(z^{r_i}-z^{-r_i})$$ So: $$\mathcal{N}(\tilde{Y}_{i,l},M\tilde{A}_{i,l+r_i}^{-1})=\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(M)(z^{2r_i+l-r}-z^{l-r})_0=-u_{i,l}(M)+u_{i,l+2r_i}(M)=-1$$ \[pidonne\] Let $m$ be in $B_i$ such that $\forall l\in{\mathbb{Z}}, u_{i,l}(m)\leq 1$ and for $1\leq r\leq 2r_i$ the set\ $\{l\in{\mathbb{Z}}/u_{i,r+2lr_i}(m)=1\}$ is a $1$-segment. Then we have $F_{i,t}(m)=\pi^{-1}(F_i(m))$. [[*Proof:*]{}]{}Let us look at the $sl_2$-case : $m=m_1m_2=m_{\sigma_1}m_{\sigma_2}$ where $\sigma_1,\sigma_2$ are $2$-segment. So the lemma \[fexp\] gives an explicit formula for $F_t(m)$ and it follows from lemma \[calcn\] that $F_t(m)=\pi^{-1}(F(m))$. We go back to the general case : let us write $m=m'm_1...m_{2r_i}$ where $m'=\underset{j\neq i,l\in{\mathbb{Z}}}{\prod}Y_{j,l}^{u_{j,l}(m)}$ and $m_r=\underset{l\in{\mathbb{Z}}}{\prod}Y_{i,r+2lr_i}^{u_{i,r+2lr_i}(m)}$. We have $m_r$ of the form $m_r=Y_{i,l_r}Y_{i,l_r+2r_i}...Y_{i,l_r+2n_ir_i}$. We have $F_{i,t}(m)=t^{-N(m'm_1...m_r)}m'F_{i,t}(m_1)...F_{i,t}(m_{2r_i})$. The study of the $sl_2$-case gives $F_{i,t}(m_r)=\pi^{-1}(F_i(m_r))$. It follows from lemma \[calcn\] that: $$t^{-N(m'm_1...m_r)}m'\pi^{-1}(F_i(m_1))...\pi^{-1}(F_i(m_r))=\pi^{-1}(m'F_i(m_1)...F_i(m_r))=\pi^{-1}(F_i(m))$$ Intersection of kernels of deformed screening operators {#rests} ======================================================= Motivated by theorem \[simme\] we study the structure of a completion of $\mathfrak{K}_t=\underset{i\in I}{\bigcap}\text{Ker}(S_{i,t})$ in order to construct $\chi_{q,t}$ in section \[conschi\]. Note that in the $sl_2$-case we have $\mathfrak{K}_t=\text{Ker}(S_{1,t})$ that was studied in section \[scr\]. Reminder: classic case ([@Fre], [@Fre2]) ---------------------------------------- ### The elements $E(m)$ and $q$-characters For $J\subset I$, denote the ${\mathbb{Z}}$-subalgebra $\mathfrak{K}_J=\underset{i\in J}{\bigcap}\mathfrak{K}_i\subset{\mathcal{Y}}$ and $\mathfrak{K}=\mathfrak{K}_I$. \[least\] ([@Fre], [@Fre2]) A non zero element of $\mathfrak{K}_J$ has at least one $J$-dominant ${\mathcal{Y}}$-monomial. [[*Proof:*]{}]{}It suffices to look at a maximal weight ${\mathcal{Y}}$-monomial $m$ of $\chi\in\mathfrak{K}_J$: for $i\in J$ we have $m\in B_i$ because $\chi\in\mathfrak{K}_i$. ([@Fre], [@Fre2]) For $i\in I$ there is a unique $E(Y_{i,0})\in \mathfrak{K}$\[ei\] such that ${Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}$-monomial in $E(Y_{i,0})$. The uniqueness follows from lemma \[least\]. For the existence we have $E(Y_{i,0})=\chi_q(V_{\omega_i}(1))$ (theorem \[simme\]). Note that the existence of $E(Y_{i,0})\in\mathfrak{K}$ suffices to characterize $\chi_q: \text{Rep}\rightarrow \mathfrak{K}$. It is the ring homomorphism such that $\chi_q(X_{i,l})=s_l(E(Y_{i,0}))$ where $s_l:{\mathcal{Y}}\rightarrow {\mathcal{Y}}$ is given by $s_l({Y}_{j,k})={Y}_{j,k+l}$. For $m\in B$, we defined the standard module $M_m$ in section \[back\]. We set: $$E(m)=\underset{m\in B}{\prod}s_l(E(Y_{i,0}))^{u_{i,l}(m)}=\chi_q(M_m)\in\mathfrak{K}$$\[em\] We defined the simple module $V_m$ in section \[back\]. We set $L(m)=\chi_q(V_m)\in\mathfrak{K}$\[lm\]. We have: $$\mathfrak{K}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}E(m)=\underset{m\in B}{\bigoplus}{\mathbb{Z}}L(m)\simeq {\mathbb{Z}}^{(B)}$$ For $m\in B$, we can also define a unique $F(m)\in\mathfrak{K}$\[fm\] such that $m$ is the unique dominant ${\mathcal{Y}}$-monomial which appears in $F(m)$ (see for example the proof of proposition \[defifprem\]). ### Technical complements For $J\subset I$, let ${\mathfrak{g}}_J$ be the semi-simple Lie algebra of Cartan Matrix $(C_{i,j})_{i,j\in J}$ and ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$ the associated quantum affine algebra with coefficient $(r_i)_{i\in J}$. In analogy with the definition of $E_i(m),L_i(m)$ using the $sl_2$-case (section \[compl\]), we define for $m\in B_J$: $E_J(m)$, $L_J(m)$, $F_J(m)\in \mathfrak{K}_J$ using ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$. We have: $$\mathfrak{K}_J=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}E_J(m)=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}L_J(m)=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}F_J(m)\simeq {\mathbb{Z}}^{(B_J)}$$ As a direct consequence of proposition \[aidafm\] we have : \[aidahfm\] For $m\in B$, we have $E(m)\in{\mathbb{Z}}[Y_{i,l}]_{i\in I,l\geq l(m)}$ where $l(m)=\text{min}\{l\in{\mathbb{Z}}/\exists i\in I,u_{i,l}(m)\neq 0\}$. Completion of the deformed algebras {#complesection} ----------------------------------- In this section we introduce completions of ${\mathcal{Y}}_t$ and of $\mathfrak{K}_{J,t}=\underset{i\in J}{\bigcap}\mathfrak{K}_{i,t}\subset {\mathcal{Y}}_t$ ($J\subset I$). We have the following motivation: we have seen $\pi_+(\mathfrak{K}_{J,t})\subset \mathfrak{K}_J$ (section \[scr\]). In order to prove an analogue of the other inclusion (theorem \[con\]) we have to introduce completions where infinite sums are allowed. ### The completion ${\mathcal{Y}}_t^{\infty}$ of ${\mathcal{Y}}_t$ Let $\overset{\infty}{A}_t$\[infat\] be the ${\mathbb{Z}}[t^{\pm}]$-module $\overset{\infty}{A}_t=\underset{m\in A}{\prod}{\mathbb{Z}}[t^{\pm}].m\simeq {\mathbb{Z}}[t^{\pm}]^A$. An element $(\lambda_m(t)m)_{m\in A}\in \overset{\infty}{A}_t$ is noted $\underset{m\in A}{\sum}\lambda_m(t)m$. We have $\underset{m\in A}{\bigoplus}{\mathbb{Z}}[t^{\pm}].m={\mathcal{Y}}_t\subset \overset{\infty}{A}_t$. The algebra structure of ${\mathcal{Y}}_t$ gives a ${\mathbb{Z}}[t^{\pm}]$-bilinear morphisms ${\mathcal{Y}}_t\otimes\overset{\infty}{A}_t\rightarrow\overset{\infty}{A}_t$ and $\overset{\infty}{A}_t\otimes{\mathcal{Y}}_t\rightarrow\overset{\infty}{A}_t$ such that $\overset{\infty}{A}_t$ is a ${\mathcal{Y}}_t$-bimodule. But the ${\mathbb{Z}}[t^{\pm}]$-algebra structure of ${\mathcal{Y}}_t$ can not be naturally extended to $\overset{\infty}{A}_t$. We define a ${\mathbb{Z}}[t^{\pm}]$-submodule ${\mathcal{Y}}_t^{\infty}$\[tytinf\] with ${\mathcal{Y}}_t\subset{\mathcal{Y}}_t^{\infty}\subset\overset{\infty}{A}_t$, for which it is the case: Let ${\mathcal{Y}}_t^A$\[tyta\] be the ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$ generated by the $(\tilde{A}_{i,l}^{-1})_{i\in I,l\in{\mathbb{Z}}}$. We gave in proposition \[yenga\] the structure of ${\mathcal{Y}}_t^A$. In particular we have ${\mathcal{Y}}_t^A=\underset{K\geq 0}{\bigoplus}{\mathcal{Y}}_t^{A,K}$ where for $K\geq 0$: $${\mathcal{Y}}_t^{A,K}=\underset{m=:\tilde{A}_{i_1,l_1}^{-1}...\tilde{A}_{i_K,l_K}^{-1}:}{\bigoplus}{\mathbb{Z}}[t^{\pm}].m\subset {\mathcal{Y}}_t^A$$ Note that for $K_1,K_2\geq 0$, ${\mathcal{Y}}_t^{A,K_1}{\mathcal{Y}}_t^{A,K_2}\subset{\mathcal{Y}}_t^{A,K_1+K_2}$ for the multiplication of ${\mathcal{Y}}_t$. So ${\mathcal{Y}}_t^A$ is a graded algebra if we set $\text{deg}(x)=K$ for $x\in{\mathcal{Y}}_t^{A,K}$. Denote by ${\mathcal{Y}}_t^{A,\infty}$ the completion of ${\mathcal{Y}}_t^A$ for this gradation. It is a sub-${\mathbb{Z}}[t^{\pm}]$-module of $\overset{\infty}{A}_t$. We define ${\mathcal{Y}}_t^{\infty}$ as the sub ${\mathcal{Y}}_t$-leftmodule of $\overset{\infty}{A}_t$ generated by ${\mathcal{Y}}_t^{A,\infty}$. In particular, we have: ${\mathcal{Y}}_t^{\infty}=\underset{M\in A}{\sum}M.{\mathcal{Y}}_t^{A,\infty}\subset \overset{\infty}{A}_t$. There is a unique algebra structure on ${\mathcal{Y}}_t^{\infty}$ compatible with the structure of ${\mathcal{Y}}_t\subset {\mathcal{Y}}_t^{\infty}$. [[*Proof:*]{}]{}The structure is unique because the elements of ${\mathcal{Y}}_t^{\infty}$ are infinite sums of elements of ${\mathcal{Y}}_t$. For $M\in A$, we have ${\mathcal{Y}}_t^{A,\infty}.M\subset M.{\mathcal{Y}}_t^{A,\infty}$, so ${\mathcal{Y}}_t^{\infty}$ is a sub ${\mathcal{Y}}_t$-bimodule of $\overset{\infty}{A}_t$. For $M\in A$ and $\lambda\in{\mathcal{Y}}_t^{A, \infty}$ denote $\lambda^M\in{\mathcal{Y}}_t^{A, \infty}$ such that $\lambda.M=M.\lambda^M$. We define the ${\mathbb{Z}}[t^{\pm}]$-algebra structure on ${\mathcal{Y}}_t^{\infty}$ by ($M,M'\in A,\lambda,\lambda'\in{\mathcal{Y}}_t^{A,\infty}$): $$(M.\lambda)(M'.\lambda')=MM'.(\lambda^{M'}\lambda')$$ It is well defined because for $M_1,M_2,M\in A,\lambda,\lambda_2\in{\mathcal{Y}}_t^A$ we have $M_1\lambda_1=M_2\lambda_2\Rightarrow M_1M\lambda_1^M=M_2M\lambda_2^M$. ### The completion $\mathfrak{K}_{i,t}^{\infty}$ of $\mathfrak{K}_{i,t}$ We define a completion of $\mathfrak{K}_{i,t}$ analog to the completed algebra ${\mathcal{Y}}_t^{\infty}$. For $M\in A$, we define a ${\mathbb{Z}}[t^{\pm}]$-linear endomorphism $E_{i,t}^M:M{\mathcal{Y}}_t^{A,\infty}\rightarrow M{\mathcal{Y}}_t^{A,\infty}$\[teitmap\] such that ($m$ ${\mathcal{Y}}_t^A$-monomial): $$E_{i,t}^M(Mm)=0\text{ if $:Mm:\notin B_i$}$$ $$E_{i,t}^M(Mm)=E_{i,t}(Mm)\text{ if $:Mm:\in B_i$}$$ It is well-defined because if $m\in{\mathcal{Y}}_t^{A,K}$ and $:Mm:\in B_i$ we have $E_{i,t}(Mm)\in M\underset{K'\geq K}{\bigoplus}{\mathcal{Y}}^{A,K'}$. We define $\mathfrak{K}_{i,t}^{\infty}=\underset{M\in A}{\sum}\text{Im}(E_{i,t}^M)\subset{\mathcal{Y}}_t^{\infty}$\[kitinf\]. For $J\subset I$, we set $\mathfrak{K}_{J,t}^{\infty}=\underset{i\in J}{\bigcap}\mathfrak{K}_{i,t}^{\infty}$ and $\mathfrak{K}_t^{\infty}=\mathfrak{K}_{I,t}^{\infty}$. \[leasto\] A non zero element of $\mathfrak{K}_{J,t}^{\infty}$ has at least one $J$-dominant ${\mathcal{Y}}_t$-monomial. [[*Proof:*]{}]{}Analog to the proof of lemma \[least\]. \[alginf\] For $J\subset I$, we have $\mathfrak{K}_{J,t}^{\infty}\cap {\mathcal{Y}}_t=\mathfrak{K}_{J,t}$. Moreover $\mathfrak{K}_{J,t}^{\infty}$ is a ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t^{\infty}$. [[*Proof:*]{}]{}It suffices to prove the results for $J=\{i\}$. First for $m\in B_i$ we have $E_{i,t}(m)=E_{i,t}^m(m)\in\mathfrak{K}_{i,t}^{\infty}$ and so $\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_{i,t}(m)\subset \mathfrak{K}_{i,t}^{\infty}\cap{\mathcal{Y}}_t$. Now let $\chi$ be in $\mathfrak{K}_{i,t}^{\infty}$ such that $\chi$ has only a finite number of ${\mathcal{Y}}_t$-monomials. In particular it has only a finite number of $i$-dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_r$ with coefficients $\lambda_1(t),...,\lambda_r(t)$. In particular it follows from lemma \[leasto\] that $\chi=\lambda_1(t)F_{i,t}(m_1)+...+\lambda_r(t)F_{i,t}(m_r)\in\mathfrak{K}_{i,t}$ (see proposition \[defifprem\] for the definition of $F_{i,t}(m)$). For the last assertion, consider $M_1$, $M_2\in A$ and $m_1$, $m_2$ ${\mathcal{Y}}_t^A$-monomials such that $:M_1m_1:,:M_2m_2:\in B_i$. Then $E_{i,t}(M_1m_1)E_{i,t}(M_2m_2)$ is in the the sub-algebra $\mathfrak{K}_{i,t}\subset{\mathcal{Y}}_t$ and in $\text{Im}(E_{i,t}^{M_1M_2})$. In the same way for $t=1$ we define the ${\mathbb{Z}}$-algebra $\mathfrak{{\mathcal{Y}}}^{\infty}$ and the ${\mathbb{Z}}$-subalgebras $\mathfrak{K}_J^{\infty}\subset{\mathcal{Y}}^{\infty}$. The surjective map $\pi_+:{\mathcal{Y}}_t\rightarrow{\mathcal{Y}}$ is naturally extended to a surjective map $\pi_+:{\mathcal{Y}}_t^{\infty}\rightarrow{\mathcal{Y}}^{\infty}$. For $i\in I$, we have $\pi_+(\mathfrak{K}_{i,t}^{\infty})=\mathfrak{K}_i^{\infty}$ and for $J\subset I$, $\pi_+(\mathfrak{K}_{J,t}^{\infty})\subset\mathfrak{K}_J^{\infty}$. The other inclusion is equivalent to theorem \[con\]. ### Special submodules of ${\mathcal{Y}}_t^{\infty}$ {#infsum} For $m\in A$, $K\geq 0$ we construct a subset $D_{m,K}\subset m\{\tilde{A}_{i_1,l_1}^{-1}...\tilde{A}_{i_K,l_K}^{-1}\}$\[dmk\] stable by the maps $E_{i,t}^{m}$ such that $\underset{K\geq 0}{\bigcup}D_{m,K}$ is countable: we say that $m'\in D_{m,K}$ if and only if there is a finite sequence $(m_0=m,m_1,...,m_R=m')$ of length $R\leq K$, such that for all $1\leq r\leq R$, there is $r'<r$, $J\subset I$ such that $m_{r'}\in B_J$ and for $r'<r''\leq r$, $m_{r''}$ is a ${\mathcal{Y}}$-monomial of $E_J(m_{r'})$ and $m_{r''}m_{r''-1}^{-1}\in\{A_{j,l}^{-1}/l\in{\mathbb{Z}},j\in J\}$. The definition means that “there is chain of monomials of some $E_J(m'')$ from $m$ to $m'$”. The set $D_{m,K}$ is finite. In particular, the set $D_m$ is countable. [[*Proof:*]{}]{}Let us prove by induction on $K\geq 0$ that $D_{m,K}$ is finite: we have $D_{m,0}=\{m\}$ and: $$D_{m,K+1}\subset \underset{J\subset I, m'\in D_{m,K}\cap B_J}{\bigcup}\{\text{${\mathcal{Y}}$-monomials of }E_J(m')\}$$ \[ordrep\] For $m,m'\in A$ such that $m'\in D_{m}$ we have $D_{m'}\subseteq D_{m}$. For $M\in A$, the set $B\cap D_M$ is finite. [[*Proof:*]{}]{}Consider $(m_0=m,m_1,...,m_R=m')$ a sequence adapted to the definition of $D_m$. Let $m''$ be in $D_{m'}$ and $(m_R=m',m_{R+1},...,m_{R'}=m'')$ a sequence adapted to the definition of $D_{m'}$. So $(m_0,m_1,...,m_{R'})$ is adapted to the definition of $D_m$, and $m''\in D_m$. Let us look at $m\in B\bigcap D_M$: we can see by induction on the length of a sequence $(m_0=M,m_1,...,m_R=m)$ adapted to the definition of $D_M$ that $m$ is of the form $m=MM'$ where $M'=\underset{i\in I,l\geq l_1}{\prod}A_{i,l}^{-v_{i,l}}$ ($v_{i,l}\geq 0$). So the last assertion follows from lemma \[fini\]. $\tilde{D}_m$\[tdm\] is the ${\mathbb{Z}}[t^{\pm}]$-submodule of ${\mathcal{Y}}_t^{\infty}$ whose elements are of the form $(\lambda_m(t)m)_{m\in D_m}$. For $m\in A$ introduce $m_0=m>m_1>m_2>...$ the countable set $D_m$ with a total ordering compatible with the partial ordering. For $k\geq 0$ consider an element $F_k\in \tilde{D}_{m_k}$. Note that some infinite sums make sense in $\tilde{D}_m$: for $k\geq 0$, we have $D_{m_k}\subset \{m_k,m_{k+1},...\}$. So $m_k$ appears only in the $F_{k'}$ with $k'\leq k$ and the infinite sum $\underset{k\geq 0}{\sum}F_k$ makes sense in $\tilde{D}_m$. Crucial result for our construction ----------------------------------- Our construction of $q,t$-characters is based on theorem \[con\] proved in this section. ### Statement \[tftm\] For $n\geq 1$ denote $P(n)$\[pn\] the property “for all semi-simple Lie-algebras ${\mathfrak{g}}$ of rank $\text{rk}({\mathfrak{g}})=n$, for all $m\in B$ there is a unique $F_t(m)\in\mathfrak{K}_t^{\infty}\cap \tilde{D}_m$ such that $m$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(m)$.”. \[con\] For all $n\geq 1$, the property $P(n)$ is true. Note that for $n=1$, that is to say ${\mathfrak{g}}=sl_2$, the result follows from section \[scr\]. The uniqueness follows from lemma \[leasto\] : if $\chi_1,\chi_2\in \mathfrak{K}_t^{\infty}$ are solutions, then $\chi_1-\chi_2$ has no dominant ${\mathcal{Y}}_t$-monomial, so $\chi_1=\chi_2$. Remark: in the simply-laced case the existence is a consequence of the geometric theory of quivers [@Naa], [@Nab], and in $A_n,D_n$-cases of algebraic explicit constructions [@Nac]. In the rest of this section \[rests\] we give an algebraic proof of this theorem in the general case. ### Outline of the proof First we give some preliminary technical results (section \[conste\]) in which we construct $t$-analogues of the $E(m)$. Next we prove $P(n)$ by induction on $n$. Our proof has 3 steps: Step 1 (section \[petitdeux\]): we prove $P(1)$ and $P(2)$ using a more precise property $Q(n)$ such that $Q(n)\Rightarrow P(n)$. The property $Q(n)$ has the following advantage: it can be verified by computation in elementary cases $n=1,2$. Step 2 (section \[consp\]): we give some consequences of $P(n)$ which will be used in the proof of $P(r)$ ($r>n$): we give the structure of $\mathfrak{K}_t^{\infty}$ (proposition \[thth\]) for $\text{rk}({\mathfrak{g}})=n$ and the structure of $\mathfrak{K}_{J,t}^{\infty}$ where $J\subset I$, $|J|=n$ and $|I|>n$ (corollary \[recufond\]). Step 3 (section \[proof\]): we prove $P(n)$ ($n \geq 3$) assuming $P(r)$, $r\leq n$ are true. We give an algorithm (section \[defialgo\]) to construct explicitly $F_t(m)$. It is called $t$-algorithm and is a $t$-analogue of Frenkel-Mukhin algorithm [@Fre2] (a deformed algorithm was also used by Nakajima in the $ADE$-case [@Naa]). As we do not know [*a priori*]{} the algorithm is well defined the general case, we have to show that it never fails (lemma \[nfail\]) and gives a convenient element (lemma \[conv\]). Preliminary: Construction of the $E_t(m)$ {#conste} ----------------------------------------- \[copieun\] We suppose that for $i\in I$, there is $F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t^{\infty}\cap \tilde{D}_{\tilde{Y}_{i,0}}$ such that $\tilde{Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$. Then: i\) All ${\mathcal{Y}}_t$-monomials of $F_t(\tilde{Y}_{i,0})$, except the highest weight ${\mathcal{Y}}_t$-monomial, are right negative. ii\) All ${\mathcal{Y}}_t$-monomials of $F_t(\tilde{Y}_{i,0})$ are products of $\tilde{Y}_{j,l}^{\pm}$ with $l\geq 0$. iii\) The only ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$ which contains a $\tilde{Y}_{j,0}^{\pm}$ ($j\in I$) is the highest weight monomial $\tilde{Y}_{i,0}$. iv\) The $F_t(\tilde{Y}_{i,0})$ ($i\in I$) commute. Note that (i),(ii) and (iii) appeared in [@Fre2]. [[*Proof:*]{}]{} i\) It suffices to prove that all ${\mathcal{Y}}_t$-monomials $m_0=Y_{i,0},m_1,...$ of $D_{Y_{i,0}}$ except $Y_{i,0}$ are right negative. But $m_1$ is the monomial $Y_{i,0}A_{i,1}^{-1}$ of $E_i(Y_{i,0})$ and it is right negative. We can now prove the statement by induction: suppose that $m_r$ is a monomial of $E_J(m_{r'})$, where $m_{r'}$ is right negative. So $m_r$ is a product of $m_{r'}$ by some $A_{j,l}^{-1}$ ($l\in{\mathbb{Z}}$).Those monomials are right negative because a product of right negative monomial is right negative. ii\) Suppose that $m\in A$ is product of $Y_{k,l}^{\pm}$ with $l\geq 0$. It follows from lemma \[aidahfm\] that all monomials of $D_{m}$ are product of $Y_{k,l}^{\pm}$ with $l\geq 0$. iii\) All ${\mathcal{Y}}$-monomials of $D_{Y_{i,0}}$ except $\tilde{Y}_{i,0}$ are in $D_{Y_{i,0}A_{i,r_i}^{-1}}$. But $l(Y_{i,0}A_{i,r_i}^{-1})\geq 1$ and we can conclude with the help of lemma \[aidahfm\]. iv\) Let $i\neq j$ be in $I$ and look at $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$. Suppose we have a dominant ${\mathcal{Y}}_t$-monomial $m_0=m_1m_2$ in $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$ different from the highest weight ${\mathcal{Y}}_t$-monomial $\tilde{Y}_{i,0}\tilde{Y}_{j,0}$. We have for example $m_1\neq \tilde{Y}_{i,0}$, so $m_1$ is right negative. Let $l_1$ be the maximal $l$ such that a $\tilde{Y}_{k,l}$ appears in $m_1$. We have $u_{k,l}(m_1)<0$ and $l>0$. As $u_{k,l}(m_0)\geq 0$ we have $u_{k,l}(m_2)>0$ and $m_2\neq {Y}_{j,0}$. So $m_2$ is right negative and there is $k'\in I$ and $l'>l$ such that $u_{k',l'}(m_2)<0$. So $u_{k',l'}(m_1)>0$, contradiction. So the highest weight ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$ is the unique dominant ${\mathcal{Y}}_t$-monomial. In the same way the highest weight ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{j,0})F_t(\tilde{Y}_{i,0})$ is the unique dominant ${\mathcal{Y}}_t$-monomial. But we have $\tilde{Y}_{i,0}\tilde{Y}_{j,0}=\tilde{Y}_{j,0}\tilde{Y}_{i,0}$, so $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})-F_t(\tilde{Y}_{j,0})F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t^{\infty}$ has no dominant ${\mathcal{Y}}_t$-monomial, so is equal to $0$. Denote, for $l\in{\mathbb{Z}}$, by $s_l:{\mathcal{Y}}_t^{\infty}\rightarrow {\mathcal{Y}}_t^{\infty}$ the endomorphism of ${\mathbb{Z}}[t^{\pm}]$-algebra such that $s_l(\tilde{Y}_{j,k})=\tilde{Y}_{j,k+l}$ (it is well-defined because the defining relations of ${\mathcal{Y}}_t$ are invariant for $k\mapsto k+l$). If the hypothesis of the lemma \[copieun\] are verified, we can define for $m\in t^{{\mathbb{Z}}}B$ : $$E_t(m)=m (\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}(m)})^{-1}\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}\in\mathfrak{K}_t^{\infty}$$\[tetm\] because for $l\in{\mathbb{Z}}$ the product $\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}$ is commutative (lemma \[copieun\]). Step 1: Proof of $P(1)$ and $P(2)$ {#petitdeux} ---------------------------------- The aim of this section is to prove $P(1)$ and $P(2)$. First we define a more precise property $Q(n)$ such that $Q(n)\Rightarrow P(n)$. ### The property $Q(n)$ {#qn} For $n\geq 1$ denote $Q(n)$ the property “for all semi-simple Lie-algebras ${\mathfrak{g}}$ of rank $\text{rk}({\mathfrak{g}})=n$, for all $i\in I$ there is a unique $F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t\cap \tilde{D}_{\tilde{Y}_{i,0}}$ such that $\tilde{Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$. Moreover $F_t(\tilde{Y}_{i,0})$ has the same monomials as $E(Y_{i,0})$”. The property $Q(n)$ is more precise than $P(n)$ because it asks that $F_t(\tilde{Y}_{i,0})$ has only a finite number of monomials. For $n\geq 1$, the property $Q(n)$ implies the property $P(n)$. [[*Proof:*]{}]{}We suppose $Q(n)$ is true. In particular the section \[conste\] enables us to construct $E_t(m)\in\mathfrak{K}_t^{\infty}$ for $m\in B$. The defining formula of $E_t(m)$ shows that it has the same monomials as $E(m)$. So $E_t(m)\in\tilde{D}_m$ and $E_t(m)\in\mathfrak{K}_t$. Let us prove $P(n)$: let $m$ be in $B$. The uniqueness of $F_t(m)$ follows from lemma \[leasto\]. Let $m_L=m>m_{L-1}>...>m_1$ be the dominant monomials of $D_m$ with a total ordering compatible with the partial ordering (it follows from lemma \[fini\] that $D_m\cap B$ is finite). Let us prove by induction on $l$ the existence of $F_t(m_l)$. The unique dominant of $D_{m_1}$ is $m_1$ so $F_t(m_1)=E_t(m_1)\in\tilde{D}_{m_1}$. In general let $\lambda_1(t),...,\lambda_{l-1}(t)\in{\mathbb{Z}}[t^{\pm}]$ be the coefficient of the dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_{l-1}$ in $E_t(m_l)$. We put: $$F_t(m_l)=E_t(m_l)-\underset{r=1...l-1}{\sum}\lambda_r(t)F_t(m_r)$$ We see in the construction that $F_t(m)\in\tilde{D}_m$ because for $m'\in D_m$ we have $E_t(m')\in \tilde{D}_{m'}\subseteq\tilde{D}_m$ (lemma \[ordrep\]). ### Cases $n=1,n=2$ {#petit} We need the following general technical result: \[cpfacile\] Let $m$ be in $B$ such that all monomial $m'$ of $F(m)$ verifies : $\forall i\in I, m'\in B_i$ implies $\forall l\in{\mathbb{Z}}, u_{i,l}(m')\leq 1$ and for $1\leq r\leq 2r_i$ the set $\{l\in{\mathbb{Z}}/u_{i,r+2lr_i}(m')=1\}$ is a $1$-segment. Then $\pi^{-1}(F(m))\in{\mathcal{Y}}_t$ is in $\mathfrak{K}_t$ and has a unique dominant monomial $m$. [[*Proof:*]{}]{}Let us write $F(m)=\underset{m'\in A}{\sum}\mu(m')m'$ ($\mu(m')\in{\mathbb{Z}}$). Let $i$ be in $I$ and consider the decomposition of $F(m)$ in $\mathfrak{K}_i$: $$F(m)=\underset{m'\in B_i}{\sum}\mu(m')F_i(m')$$ But $\mu(m')\neq 0$ implies the hypothesis of lemma \[pidonne\] is verified for $m'\in B_i$. So $\pi^{-1}(F_i(m'))=F_{i,t}(m')$. And: $$\pi^{-1}(F(m))=\underset{m'\in B_i}{\sum}\mu(m')F_{i,t}(m')\in \mathfrak{K}_{i,t}$$ For $n=1$ (section \[kernelun\]), $n=2$ (section \[fin\]), we can give explicit formula for the $E(Y_{i,0})=F(Y_{i,0})$. In particular we see that the hypothesis of proposition \[cpfacile\] are verified, so: The properties $Q(1)$, $Q(2)$ and so $P(1)$, $P(2)$ are true. This allow us to start our induction in the proof of theorem \[con\]. In section \[acase\] we will see other applications of proposition \[cpfacile\]. Note that the hypothesis of proposition \[cpfacile\] are not verified for fundamental monomials $m=Y_{i,0}$ in general: for example for the $D_5$-case we have in $F(Y_{2,0})$ the monomial $Y_{3,3}^2Y_{5,4}^{-1}Y_{2,4}^{-1}Y_{4,4}^{-1}$. Step 2: consequences of the property $P(n)$ {#consp} ------------------------------------------- Let be $n\geq 1$. We suppose in this section that $P(n)$ is proved. We give some consequences of $P(n)$ which will be used in the proof of $P(r)$ ($r>n$). Let $\mathfrak{K}_t^{\infty,f}$\[ktinff\] be the ${\mathbb{Z}}[t^{\pm}]$-submodule of $\mathfrak{K}_t^{\infty}$ generated by elements with a finite number of dominant ${\mathcal{Y}}_t$-monomials. \[thth\] We suppose $\text{rk}({\mathfrak{g}})=n$. We have: $$\mathfrak{K}_t^{\infty, f}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{(B)}$$ Moreover for $M\in A$, we have: $$\mathfrak{K}_t^\infty\cap\tilde{D}_M=\underset{m\in B\cap D_M}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{B\cap D_M}$$ [[*Proof:*]{}]{}Let $\chi$ be in $\mathfrak{K}_t^{\infty, f}$ and $m_1,...,m_L\in B$ the dominant ${\mathcal{Y}}_t$-monomials of $\chi$ and $\lambda_1(t),...,\lambda_L(t)\in{\mathbb{Z}}[t^{\pm}]$ their coefficients. It follows from lemma \[leasto\] that $\chi=\underset{l=1...L}{\sum}\lambda_l(t)F_t(m_l)$. Let us look at the second point: lemma \[ordrep\] shows that $m\in B\cap D_M\Rightarrow F_t(m)\in\tilde{D}_M$. In particular the inclusion $\supseteq$ is clear. For the other inclusion we prove as in the first point that $\mathfrak{K}_t^\infty\cap\tilde{D}_M=\underset{m\in B\cap D_M}{\sum}{\mathbb{Z}}[t^{\pm}]F_t(m)$. We can conclude because it follows from lemma \[fini\] that $D_M\cap B$ is finite. We recall that have seen in section \[infsum\] that some infinite sum make sense in $\tilde{D}_M$. \[recufond\] We suppose $\text{rk}({\mathfrak{g}})>n$ and let $J$ be a subset of $I$ such that $|J|=n$. For $m\in B_J$, there is a unique $F_{J,t}(m)\in\mathfrak{K}_{J,t}^{\infty}$ such that $m$ is the unique $J$-dominant ${\mathcal{Y}}_t$-monomial of $F_{J,t}(m)$. Moreover $F_{J,t}(m)\in\tilde{D}_{m}$. For $M\in A$, the elements of $\mathfrak{K}_{J,t}^\infty\cap\tilde{D}_M$ are infinite sums $\underset{m\in B_J\cap D_M}{\sum}\lambda_m(t)F_{J,t}(m)$. In particular: $$\mathfrak{K}_{J,t}^\infty\cap\tilde{D}_M\simeq{\mathbb{Z}}[t^{\pm}]^{B_J\cap D_M}$$ [[*Proof:*]{}]{}The uniqueness of $F_{J,t}(m)$ follows from lemma \[leasto\]. Let us write $m=m_Jm'$ where\ $m_J=\underset{i\in J,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}(m)}$. So $m_J$ is a dominant ${\mathcal{Y}}_t$-monomial of ${\mathbb{Z}}[Y_{i,l}^{\pm}]_{i\in J,l\in{\mathbb{Z}}}$. In particular the proposition \[thth\] with the algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$ of rank $n$ gives $m_J\chi$ where $\chi\in{\mathbb{Z}}[\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1},t^{\pm}]_{i\in J,l\in{\mathbb{Z}}}$ (where for $i\in I,l\in{\mathbb{Z}}$, $\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,\pm}=\beta_{I,J}(\tilde{A}_{i,l}^{\pm})$ where $\beta_{I,J}(\tilde{Y}_{i,l}^{\pm})=\delta_{i\in J}\tilde{Y}_{i,l}^{\pm}$). So we can put $F_t(m)=m\nu_{J,t}(\chi)$ where $\nu_{J,t}:{\mathbb{Z}}[\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1},t^{\pm}]_{i\in J,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}_t$ is the ring homomorphism such that $\nu_{J,t}(\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1})=\tilde{A}_{i,l}^{-1}$. The last assertion is proved as in proposition \[thth\]. Step 3: $t$-algorithm and end of the proof of theorem \[con\] {#proof} ------------------------------------------------------------- In this section we explain why the $P(r)$ ($r<n$) imply $P(n)$. In particular we define the $t$-algorithm which constructs explicitly the $F_t(m)$. ### The induction We prove the property $P(n)$ by induction on $n\geq 1$. It follows from section \[petitdeux\] that $P(1)$ and $P(2)$ are true. Let be $n\geq 3$ and suppose that $P(r)$ is proved for $r<n$. Let $m_+$ be in $B$ and $m_0=m_+>m_1>m_2>...$ the countable set $D_{m_+}$ with a total ordering compatible with the partial ordering. For $J\varsubsetneq I$ and $m\in B_J$, it follows from $P(r)$ and corollary \[recufond\] that there is a unique $F_{J,t}(m)\in\tilde{D}_m\cap\mathfrak{K}_{J,t}^{\infty}$ such that $m$ is the unique $J$-dominant monomial of $F_{J,t}(m)$ and that the elements of $\tilde{D}_{m_+}\cap \mathfrak{K}_{J,t}^{\infty}$ are the infinite sums of ${\mathcal{Y}}_t^{\infty}$: $\underset{m\in D_{m_+}\cap B_J}{\sum}\lambda_m(t)F_{J,t}(m)$ where $\lambda_m(t)\in{\mathbb{Z}}[t^{\pm}]$. If $m\in A-B_J$, denote $F_{J,t}(m)=0$. ### Definition of the $t$-algorithm {#defialgo} For $r,r'\geq 0$ and $J\subsetneq I$ denote $[F_{J,t}(m_{r'})]_{m_r}\in{\mathbb{Z}}[t^{\pm}]$ the coefficient of $m_r$ in $F_{J,t}(m_{r'})$. We call $t$-algorithm the following inductive definition of the sequences $(s(m_r)(t))_{r\geq 0}\in{\mathbb{Z}}[t^{\pm}]^{{\ensuremath{\mathbb{N}}}}$, $(s_J(m_r)(t))_{r\geq 0}\in{\mathbb{Z}}[t^{\pm}]^{{\ensuremath{\mathbb{N}}}}$ ($J\varsubsetneq I$)\[smrt\]: $$s(m_0)(t)=1\text{ , }s_J(m_0)(t)=0$$ and for $r\geq 1, J\subsetneq I$: $$s_J(m_r)(t)=\underset{r'<r}{\sum}(s(m_{r'})(t)-s_J(m_{r'})(t))[F_{J,t}(m_{r'})]_{m_r}$$ $$\text{ if }m_r\notin B_J, s(m_r)(t)=s_J(m_r)(t)$$ $$\text{ if }m_r\in B, s(m_r)(t)=0$$ We have to prove that the $t$-algorithm defines the sequences in a unique way. We see that if $s(m_r),s_J(m_r)$ are defined for $r\leq R$ so are $s_J(m_{R+1})$ for $J\subsetneq I$. The $s_J(m_R)$ impose the value of $s(m_{R+1})$ and by induction the uniqueness is clear. We say that the $t$-algorithm is well defined to step $R$ if there exist $s(m_{r}), s_J(m_r)$ such that the formulas of the $t$-algorithm are verified for $r\leq R$. The $t$-algorithm is well defined to step $r$ if and only if: $$\forall J_1,J_2\varsubsetneq I, \forall r'\leq r, m_{r'}\notin B_{J_1} \text{ and }m_{r'}\notin B_{J_2}\Rightarrow s_{J_1}(m_{r'})(t)=s_{J_2}(m_{r'})(t)$$ [[*Proof:*]{}]{}If for $r'<r$ the $s(m_{r'})(t),s_J(m_{r'})(t)$ are well defined, so is $s_J(m_r)(t)$. If $m_r\in B$, $s(m_r)(t)=0$ is well defined. If $m_r\notin B$, it is well defined if and only if $\{s_J(m_r)(t)/m_r\notin B_J\}$ has one unique element. ### The $t$-algorithm never fails If the $t$-algorithm is well defined to all steps, we say that the $t$-algorithm never fails. In this section we show that the $t$-algorithm never fails. If the $t$-algorithm is well defined to step $r$, for $J\varsubsetneq I$ we set: $$\mu_J(m_r)(t)=s(m_r)(t)-s_J(m_r)(t)$$ $$\chi_J^r=\underset{r'\leq r}{\sum}\mu_{J}(m_{r'}(t))F_{J,t}(m_{r'})\in\mathfrak{K}_{J,t}^{\infty}$$ If the $t$-algorithm is well defined to step $r$, for $J\subset I$ we have: $$\chi_J^r\in (\underset{r'\leq r}{\sum} s(m_{r'})(t)m_{r'})+s_J(m_{r+1})(t)m_{r+1}+\underset{r'>r+1}{\sum}{\mathbb{Z}}[t^{\pm}]m_{r'}$$ For $J_1\subset J_2\subsetneq I$, we have: $$\chi_{J_2}^r=\chi_{J_1}^r+\underset{r'>r}{\sum}\lambda_{r'}(t)F_{J_1,t}(m_{r'})$$ where $\lambda_{r'}(t)\in{\mathbb{Z}}[t^{\pm}]$. In particular, if $m_{r+1}\notin B_{J_1}$, we have $s_{J_1,t}(m_{r+1})=s_{J_2,t}(m_{r+1})$. [[*Proof:*]{}]{}For $r'\leq r$ let us compute the coefficient $(\chi_J^r)_{m_{r'}}\in{\mathbb{Z}}[t^{\pm}]$ of $m_{r'}$ in $\chi_J^r$: $$(\chi_J^r)_{m_{r'}}=\underset{r''\leq r'}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r'}}$$ $$=(s(m_{r'})(t)-s_J(m_{r'})(t))[F_{J,t}(m_{r'})]_{m_{r'}}+\underset{r''<r'}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r'}}$$ $$=(s(m_{r'})(t)-s_J(m_{r'})(t))+s_J(m_{r'})(t)=s(m_{r'})(t)$$ Let us compute the coefficient $(\chi_J^r)_{m_{r+1}}\in{\mathbb{Z}}[t^{\pm}]$ of $m_{r+1}$ in $\chi_J^r$: $$(\chi_J^r)_{m_{r+1}}=\underset{r''<r+1}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r+1}}=s_J(m_{r+1})$$ For the second point let $J_1\subset J_2\subsetneq I$. We have $\chi_{J_2}^r\in \mathfrak{K}_{J_1,t}^{\infty}\cap\tilde{D}_{m+}$ and it follows from $P(|J_1|)$ and corollary \[recufond\] (or section \[petit\] if $|J_1|\leq 2$) that we can introduce $\lambda_{m_{r'}}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that : $$\chi_{J_2}^r=\underset{r'\geq 0}{\sum}\lambda_{m_{r'}}(t)F_{J_1,t}(m_{r'})$$ We show by induction on $r'$ that for $r'\leq r$, $m_{r'}\in B_{J_1}\Rightarrow \lambda_{m_{r'}}(t)=\mu_{J_1}(m_{r'})(t)$. First we have $\lambda_{m_0}(t)=(\chi_{J_2}^r)_{m_0}=s(m_0)(t)=1=\mu_{J_1}(m_0)$. For $r'\leq r$: $$s(m_{r'})(t)=\lambda_{m_{r'}}(t)+\underset{r''<r'}{\sum}\lambda_{m_{r''}}(t)[F_{J_1,t}(m_{r''})]_{m_{r'}}$$ $$\lambda_{m_{r'}}(t)=s(m_{r'})(t)-\underset{r''<r'}{\sum}\mu_{J_1}(m_{r'})(t)[F_{J_1,t}(m_{r''})]_{m_{r'}}=s(m_{r'})(t)-s_{J_1}(m_{r'})(t)=\mu_{J_1}(m_{r'})(t)$$ For the last assertion if $m_{r+1}\notin B_{J_1}$, the coefficient of $m_{r+1}$ in $\underset{r'>r}{\sum}{\mathbb{Z}}[t^{\pm}]F_{J_1,t}(m_{r'})$ is 0, and $(\chi_{J_2}^r)_{m_{r+1}}=(\chi_{J_1}^r)_{m_{r+1}}$. It follows from the first point that $s_{J_1,t}(m_{r+1})=s_{J_2,t}(m_{r+1})$. \[nfail\] The $t$-algorithm never fails. [[*Proof:*]{}]{}Suppose the sequence is well defined until the step $r-1$ and let $J_1,J_2\varsubsetneq I$ such that $m_r\notin B_{J_1}$ and $m_r\notin B_{J_2}$. Let $i$ be in $J_1$, $j$ in $J_2$ such that $m_r\notin B_i$ and $m_r\notin B_j$. Consider $J=\{i,j\}\varsubsetneq I$. The $\chi_J^{r-1},\chi_i^{r-1},\chi_j^{r-1}\in{\mathcal{Y}}_t$ have the same coefficient $s(m_{r'})(t)$ on $m_{r'}$ for $r'\leq r-1$. Moreover: $$s_i(m_r)(t)=(\chi_i^{r-1})_{m_r}\text{ , }s_j(m_r)(t)=(\chi_j^{r-1})_{m_r}\text{ , }s_J(m_r)(t)=(\chi_J^{r-1})_{m_r}$$ But $m_r\notin B_J$, so: $$\chi_J^{r-1}=\underset{r'\leq r-1}{\sum}\mu_i(m_{r'})(t)F_{i,t}(m_{r'})+\underset{r'\geq r+1}{\sum}\lambda_{m_{r'}}(t)F_{i,t}(m_{r'})$$ So $(\chi_J^{r-1})_{m_r}=(\chi_i^{r-1})_{m_r}$ and we have $s_i(m_r)(t)=s_J(m_r)(t)$. In the same way we have $s_i(m_r)(t)=s_{J_1}(m_r)(t)$, $s_j(m_r)(t)=s_J(m_r)(t)$ and $s_j(m_r)(t)=s_{J_2}(m_r)(t)$. So we can conclude $s_{J_1}(m_r)(t)=s_{J_2}(m_r)(t)$. ### Proof of $P(n)$ It follows from lemma \[nfail\] that $\chi=\underset{r\geq 0}{\sum}s(m_r)(t)m_r\in{\mathcal{Y}}_t^{\infty}$ is well defined. \[conv\] We have $\chi\in \mathfrak{K}_t^{\infty}\bigcap\tilde{D}_{m_+}$. Moreover the only dominant ${\mathcal{Y}}_t$-monomial in $\chi$ is $m_0=m_+$. [[*Proof:*]{}]{}The defining formula of $\chi$ gives $\chi\in\tilde{D}_{m_+}$. Let $i$ be in $I$ and: $$\chi_i=\underset{r\geq 0}{\sum}\mu_i(m_r)(t)F_{i,t}(m_r)\in\mathfrak{K}_{i,t}^{\infty}$$ Let us compute for $r\geq 0$ the coefficient of $m_r$ in $\chi-\chi_i$: $$(\chi-\chi_i)_{m_r}=s(m_r)(t)-\underset{r'\leq r}{\sum}\mu_i(m_{r'})(t)[F_{i,t}(m_{r'})]_{m_r}$$ $$=s(m_r)(t)-s_i(m_r)(t)-\mu_i(m_r)(t)[F_{i,t}(m_{r})]_{m_r}=(s(m_r)(t)-s_i(m_r)(t))(1-[F_{i,t}(m_{r})]_{m_r})$$ We have two cases: if $m_r\in B_i$, we have $1-[F_{i,t}(m_{r})]_{m_r}=0$. if $m_r\notin B_i$, we have $s(m_r)(t)-s_i(m_r)(t)=0$. So $\chi=\chi_i\in\mathfrak{K}_{i,t}^{\infty}$, and $\chi\in \mathfrak{K}_t^{\infty}$. The last assertion follows from the definition of the algorithm: for $r>0$, $m_r\in B\Rightarrow s(m_r)(t)=0$. This lemma implies: For $n\geq 3$, if the $P(r)$ ($r<n$) are true, then $P(n)$ is true. In particular the theorem \[con\] is proved by induction on $n$. Morphism of $q,t$-characters and applications {#conschi} ============================================= Morphism of $q,t$-characters {#aida} ---------------------------- ### Definition of the morphism We set $\text{Rep}_t=\text{Rep}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]={\mathbb{Z}}[X_{i,l},t^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$\[rept\]. We say that $M\in\text{Rep}_t$ is a $\text{Rep}_t$-monomial if it is of the form $M=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}X_{i,l}^{x_{i,l}}$ ($x_{i,l}\geq 0$). In this case denote $x_{i,l}(M)=x_{i,l}$. Recall the definition of the $E_t(m)$ (section \[conste\]). \[mqt\] The morphism of $q,t$-characters is the ${\mathbb{Z}}[t^{\pm}]$-linear map $\chi_{q,t}:\text{Rep}_t\rightarrow {\mathcal{Y}}_t^{\infty}$\[chiqt\] such that ($u_{i,l}\geq 0$): $$\chi_{q,t}(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}X_{i,l}^{u_{i,l}})=E_t(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}})$$ ### Properties of $\chi_{q,t}$ \[axiomes\] We have $\pi_+(\text{Im}(\chi_{q,t}))\subset{\mathcal{Y}}$ and the following diagram is commutative: $$\begin{array}{rcccl} \text{Rep}&\stackrel{\chi_{q,t}}{\longrightarrow}&\text{Im}(\chi_{q,t})\\ \text{id}\downarrow &&\downarrow&\pi_+\\ \text{Rep}&\stackrel{\chi_q}{\longrightarrow}&{\mathcal{Y}}\end{array}$$ In particular the map $\chi_{q,t}$ is injective. The ${\mathbb{Z}}[t^{\pm}]$-linear map $\chi_{q,t}:\text{Rep}_t\rightarrow{\mathcal{Y}}_t^{\infty}$ is characterized by the three following properties: 1\) For a $\text{Rep}_t$-monomial $M$ define $m=\pi^{-1}(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M)})\in A$ and $\tilde{m}\in A_t$ as in section \[tildeat\]. Then we have : $$\chi_{q,t}(M)=\tilde{m}+\underset{m'<m}{\sum}a_{m'}(t)m'\text{ (where $a_{m'}(t)\in{\mathbb{Z}}[t^{\pm}]$)}$$ 2\) The image of $\text{Im}(\chi_{q,t})$ is contained in $\mathfrak{K}_t^{\infty}$. 3\) Let $M_1,M_2$ be $\text{Rep}_t$-monomials such that $\text{max}\{l/\underset{i\in I}{\sum}x_{i,l}(M_1)>0\}\leq \text{min}\{l/\underset{i\in I}{\sum}x_{i,l}(M_2)>0\}$. We have : $$\chi_{q,t}(M_1M_2)=\chi_{q,t}(M_1)\chi_{q,t}(M_2)$$ Note that the properties $1,2,3$ are generalizations of the defining axioms introduced by Nakajima in [@Nab] for the $ADE$-case; in particular in the $ADE$-case $\chi_{q,t}$ is the morphism of $q,t$-characters constructed in [@Nab]. [[*Proof:*]{}]{}$\pi_+(\text{Im}(\chi_{q,t}))\subset{\mathcal{Y}}$ means that only a finite number of ${\mathcal{Y}}_t$-monomials of $E_t(m)$ have coefficient $\lambda(t)\notin (t-1){\mathbb{Z}}[t^{\pm}]$. As $F_t(\tilde{Y}_{i,0})$ has no dominant ${\mathcal{Y}}_t$-monomial other than $\tilde{Y}_{i,0}$, we have the same property for $\pi_+(F_t(\tilde{Y}_{i,0}))\in\mathfrak{K}^{\infty}$ and $\pi_+(F_t(\tilde{Y}_{i,0}))=E(Y_{i,0})\in {\mathcal{Y}}$. As ${\mathcal{Y}}$ is a subalgebra of ${\mathcal{Y}}^{\infty}$ we get $\pi_+(E_t(m))\in{\mathcal{Y}}$ with the help of the defining formula. The diagram is commutative because $\pi_+\circ s_l=s_l\circ \pi_+$ and $\pi_+(F_t(\tilde{Y}_{i,0}))=E(Y_{i,0})$. It is proved by Frenkel, Reshetikhin in [@Fre] that $\chi_q$ is injective, so $\chi_{q,t}$ is injective. Let us show that $\chi_{q,t}$ verifies the three properties: 1\) By definition we have $\chi_{q,t}(M)=E_t(m)$. But $s_l(F_t(\tilde{Y}_{i,0}))=F_t(\tilde{Y}_{i,l})\in\tilde{D}(\tilde{Y}_{i,l})$. In particular $s_l(F_t(\tilde{Y}_{i,0}))$ is of the form $\tilde{Y}_{i,l}+\underset{m'<Y_{i,l}}{\sum}\lambda_{m'}(t)m'$ and we get the property for $E_t(m)$ by multiplication. 2\) We have $s_l(F_t(\tilde{Y}_{i,0}))=E_t(\tilde{Y}_{i,l})\in\mathfrak{K}_t^{\infty}$ and $\mathfrak{K}_t^{\infty}$ is a subalgebra of ${\mathcal{Y}}_t^{\infty}$, so $\text{Im}(\chi_{q,t})\subset\mathfrak{K}_t^{\infty}$. 3\) If we set $L=\text{max}\{l/\underset{i\in I}{\sum}x_{i,l}(M_1)>0\}$, $m_1=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M_1)}$, $m_2=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M_2)}$, we have: $$E_t(m_1)=\overset{\rightarrow}{\underset{l\leq L}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{x_{i,l}(M_1)}\text{ , }E_t(m_2)=\overset{\rightarrow}{\underset{l\geq L}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{x_{i,l}(M_2)}$$ and in particular: $$E_t(m_1m_2)=E_t(m_1)E_t(m_2)$$ Finally let $f:\text{Rep}_t\rightarrow{\mathcal{Y}}_t^{\infty}$ be a ${\mathbb{Z}}[t^{\pm}]$-linear homomorphism which verifies properties 1,2,3. We saw that the only element of $\mathfrak{K}_t^{\infty}$ with highest weight monomial $\tilde{Y}_{i,l}$ is $s_l(F_t(\tilde{Y}_{i,0}))$. In particular we have $f(X_{i,l})=E_t(Y_{i,l})$. Using property 3, we get for $M\in\text{Rep}_t$ a monomial : $$f(M)=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}f(X_{i,l})^{u_{i,l}(m)}=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}=\chi_{q,t}(M)$$ Quantization of the Grothendieck Ring {#quanta} ------------------------------------- In this section we see that $\chi_{q,t}$ allows us to define a deformed algebra structure on $\text{Rep}_t$ generalizing the quantization of [@Nab]. The point is to show that $\text{Im}(\chi_{q,t})$ is a subalgebra of $\mathfrak{K}_t^{\infty}$. ### Generators of $\mathfrak{K}_t^{\infty,f}$ Recall the definition of $\mathfrak{K}_t^{\infty,f}$ in section \[consp\]. For $m\in B$, all monomials of $E_t(m)$ are in $\{mA_{i_1,l_1}^{-1}...A_{i_K,l_K}^{-1}/k\geq 0,l_k\geq L\}$ where $L=\text{min}\{l\in{\mathbb{Z}},\exists i\in I,u_{i,l}(m)>0\}$. So it follows from lemma \[fini\] that $E_t(m)\in\mathfrak{K}_t^{\infty}$ has only a finite number of dominant ${\mathcal{Y}}_t$-monomials, that is to say $E_t(m)\in\mathfrak{K}_t^{\infty,f}$. \[diago\] The ${\mathbb{Z}}[t^{\pm}]$-module $\mathfrak{K}_t^{\infty,f}$ is freely generated by the $E_t(m)$: $$\mathfrak{K}_t^{\infty, f}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{(B)}$$ [[*Proof:*]{}]{}The $E_t(m)$ are ${\mathbb{Z}}[t^{\pm}]$-linearly independent and we saw $E_t(m)\in\mathfrak{K}_t^{\infty, f}$. It suffices to prove that the $E_t(m)$ generate the $F_t(m)$: let us look at $m_0\in B$ and consider $L=\text{min}\{l\in{\mathbb{Z}},\exists i\in I,u_{i,l}(m_0)>0\}$. In the proof of lemma \[fini\] we saw there is only a finite dominant monomials in $\{m_0A_{i_1,l_1}^{-v_{i_1,l_1}}...A_{i_R,l_R}^{-v_{i_R,l_R}}/R\geq 0,i_r\in I,l_r\geq L\}$. Let $m_0>m_1>...>m_D\in B$ be those monomials with a total ordering compatible with the partial ordering. In particular, for $0\leq d\leq D$ the dominant monomials of $E_t(m_d)$ are in $\{m_d,m_{d+1},...,m_D\}$. So there are elements $(\lambda_{d,d'}(t))_{0\leq d,d'\leq D}$ of ${\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m_d)=\underset{d\leq d'\leq D}{\sum}\lambda_{d,d'}(t)F_t(m_{d'})$$ We have $\lambda_{d,d'}(t)=0$ if $d'<d$ and $\lambda_{d,d}(t)=1$. We have a triangular system with $1$ on the diagonal, so it is invertible in ${\mathbb{Z}}[t^{\pm}]$. ### Construction of the quantization $\mathfrak{K}_t^{\infty,f}$ is a subalgebra of $\mathfrak{K}_t^{\infty}$. [[*Proof:*]{}]{}It suffices to prove that for $m_1,m_2\in B$, $E_t(m_1)E_t(m_2)$ has only a finite number of dominant ${\mathcal{Y}}_t$-monomials. But $E_t(m_1)E_t(m_2)$ has the same monomials as $E_t(m_1m_2)$. It follows from proposition \[diago\] that $\chi_{q,t}$ is a ${\mathbb{Z}}[t^{\pm}]$-linear isomorphism between $\text{Rep}_t$ and $\mathfrak{K}_t^{\infty,f}$. So we can define: The associative deformed ${\mathbb{Z}}[t^{\pm}]$-algebra structure on $\text{Rep}_t$ is defined by: $$\forall \lambda_1,\lambda_2\in\text{Rep}_t,\lambda_1*\lambda_2=\chi_{q,t}^{-1}(\chi_{q,t}(\lambda_1)\chi_{q,t}(\lambda_2))$$\[star\] ### Examples: $sl_2$-case We make explicit computation of the deformed multiplication in the $sl_2$-case: In the $sl_2$-case, the deformed algebra structure on $\text{Rep}_t={\mathbb{Z}}[X_l,t^{\pm}]_{l\in{\mathbb{Z}}}$ is given by: $$X_{l_1}*X_{l_2}*...*X_{l_m}=X_{l_1}X_{l_2}...X_{l_m}\text{ if $l_1\leq l_2\leq ...\leq l_m$}$$ $$X_{l}*X_{l'}=t^{\gamma}X_{l}X_{l'}=t^{\gamma}X_{l'}*X_l\text{ if $l>l'$ and $l\neq l'+2$}$$ $$X_{l}*X_{l-2}=t^{-2}X_{l}X_{l-2}+t^{\gamma}(1-t^{-2})=t^{-2}X_{l-2}*X_{l}+(1-t^{-2})$$ where $\gamma\in{\mathbb{Z}}$ is defined by $\tilde{Y}_l\tilde{Y}_{l'}=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l$. [[*Proof:*]{}]{}For $l\in{\mathbb{Z}}$ we have the $q,t$-character of the fundamental representation $X_l$: $$\chi_{q,t}(X_l)=\tilde{Y}_l+\tilde{Y}_{l+2}^{-1}=\tilde{Y}_l(1+t\tilde{A}_{l+1}^{-1})$$ The first point of the proposition follows immediately from the definition of $\chi_{q,t}$. For example, for $l,l'\in{\mathbb{Z}}$ we have: $$\chi_{q,t}(X_lX_{l'})=\chi_{q,t}(X_{\text{min}(l,l')})\chi_{q,t}(X_{\text{max}(l,l')})$$ In particular if $l\leq l'$, we have $X_l*X_{l'}=X_lX_{l'}$. Suppose now that $l>l'$ and introduce $\gamma\in{\mathbb{Z}}$ such that $\tilde{Y}_l\tilde{Y}_{l'}=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l$. We have: $$\chi_{q,t}(X_l)\chi_{q,t}(X_l')=\tilde{Y}_l(1+t\tilde{A}_{l+1}^{-1})\tilde{Y}_{l'}(1+t\tilde{A}_{l'+1}^{-1})$$ $$=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l+t^{\gamma +1}\tilde{Y}_{l'}\tilde{Y}_l\tilde{A}_{l+1}^{-1}+t^{\gamma +1+2\delta_{l,l'+2}}\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_l+t^{\gamma+ 2}\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_{l}\tilde{A}_{l+1}^{-1}$$ $$=t^{\gamma}\chi_{q,t}(X_{l'}X_l)+t^{\gamma +1}(t^{2\delta_{l,l'+2}}-1)\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_l$$ If $l\neq l'+2$ we get $X_l*X_{l'}=t^{\gamma}X_{l'}*X_l$. If $l=l'+2$, we have: $$\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_{l'+2}=t^{-1}\tilde{Y}_{l'+2}^{-1}\tilde{Y}_{l'+2}=t^{-1}$$ But $t^2\tilde{Y}_l\tilde{Y}_{l-2}=\tilde{Y}_{l-2}\tilde{Y}_l$, so $X_l*X_{l-2}=t^{-2}X_{l-2}*X_l+t^{-2}(t^2-1)$. Note that $\gamma$ were computed in section \[varva\]. We see that the new ${\mathbb{Z}}[t^{\pm}]$-algebra structure is not commutative and not even twisted polynomial. An involution of the Grothendieck ring {#invo} -------------------------------------- In this section we construct an antimultiplicative involution of the Grothendieck ring $\text{Rep}_t$. The construction is motivated by the point view adopted in this article : it is just replacing $c_{|l|}$ by $-c_{|l|}$. In the $ADE$-case such an involution were introduced Nakajima [@Nab] with different motivations. ### An antihomomorphism of $\mathcal{H}$ There is a unique ${\ensuremath{\mathbb{C}}}$-linear isomorphism of $\mathcal{H}$ which is antimultiplicative and such that: $$\overline{c_m}=-c_m\text{ , }\overline{a_i[r]}=a_i[r]\text{ ($m>0,i\in I,r\in{\mathbb{Z}}-\{0\}$)}$$ Moreover it is an involution. [[*Proof:*]{}]{}It suffices to show it is compatible with the defining relations of $\mathcal{H}$ ($i,j\in I,m,r\in{\mathbb{Z}}-\{0\}$): $$\overline{[a_i[m],a_j[r]]}=\overline{a_i[m]a_j[r]}-\overline{a_j[r]a_i[m]}=-[a_i[m],a_j[r]]$$ $$\overline{\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}}=-\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}$$ For the last assertion, we have $\overline{\overline{c_m}}=c_m$ and $\overline{\overline{a_i[r]}}=a_i[r]$, and an algebra morphism which fixes the generators is the identity. It can be naturally extended to an antimultiplicative ${\ensuremath{\mathbb{C}}}$-isomorphism of $\mathcal{H}_h$. \[stbl\] The ${\mathbb{Z}}$-subalgebra ${\mathcal{Y}}_u\subset \mathcal{H}_h$ verifies $\overline{{\mathcal{Y}}_u}\subset{\mathcal{Y}}_u$. [[*Proof:*]{}]{}It suffices to check on the generators of ${\mathcal{Y}}_u$ ($R\in\mathfrak{U},i\in I,l\in{\mathbb{Z}}$): $$\overline{t_R}=\text{exp}(\underset{m>0}{\sum}h^{2m}R(q^m)(-c_m))=t_{-R}$$ $$\overline{\tilde{Y}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}[y_i[-m],y_i[m]])\tilde{Y}_{i,l} =t_{-\tilde{C}_{i,i}(q)(q_i-q_i^{-1})}\tilde{Y}_{i,l}\in{\mathcal{Y}}_u$$ $$\overline{\tilde{Y}_{i,l}}^{-1}=(\overline{\tilde{Y}_{i,l}})^{-1}=t_{\tilde{C}_{i,i}(q)(q_i-q_i^{-1})}\tilde{Y}_{i,l}^{-1}\in{\mathcal{Y}}_u$$ ### Involution of ${\mathcal{Y}}_t$ As for $R,R'\in\mathfrak{U}$, we have $\pi_0(R)=\pi_0(R')\Leftrightarrow\pi_0(-R)=\pi_0(-R')$, the involution of ${\mathcal{Y}}_u$ (resp. of $\mathcal{H}_h$) is compatible with the defining relations of ${\mathcal{Y}}_t$ (resp. $\mathcal{H}_t$). We get a ${\mathbb{Z}}$-linear involution of ${\mathcal{Y}}_t$ (resp. of $\mathcal{H}_t$). For $\lambda,\lambda'\in{\mathcal{Y}}_t,\alpha\in{\mathbb{Z}}$, we have: $$\overline{\lambda.\lambda'}=\overline{\lambda'}.\overline{\lambda}\text{ , }\overline{t^{\alpha}\lambda}=t^{-\alpha}\overline{\lambda}$$ Note that in ${\mathcal{Y}}_u$ for $i\in I,l\in{\mathbb{Z}}$: $$\overline{\tilde{A}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}[a_i[-m],a_i[m]]c_m)\tilde{A}_{i,l}=t_{(-q_i^2+q_i^{-2})}\tilde{A}_{i,l}$$ So in ${\mathcal{Y}}_t$ we have $\overline{\tilde{A}_{i,l}}=\tilde{A}_{i,l}$ and $\overline{\tilde{A}_{i,l}^{-1}}=\tilde{A}_{i,l}^{-1}$. ### The involution of deformed bimodules For $i\in I$, the ${\mathcal{Y}}_{i,u}\subset \mathcal{H}_h$ verifies $\overline{{\mathcal{Y}}_{i,u}}\subset{\mathcal{Y}}_{i,u}$. [[*Proof:*]{}]{}First we compute for $i\in I,l\in{\mathbb{Z}}$: $$\overline{\tilde{S}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q_i^{-m}-q_i^{m}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q_i^{m}-q_i^{-m}}q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{[a_i[-m],a_i[m]]}{-(q_i^{-m}-q_i^{m})^2}c_m)\tilde{S}_{i,l}=t_{\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}}}\tilde{S}_{i,l}\in{\mathcal{Y}}_{i,u}$$ Now for $\lambda\in{\mathcal{Y}}_u$, we have $\overline{\lambda.\tilde{S}_{i,l}}=t_{\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}}}\tilde{S}_{i,l}\overline{\lambda}$. But it is in ${\mathcal{Y}}_{i,u}$ because $\overline{\lambda}\in{\mathcal{Y}}_u$ (lemma \[stbl\]) and ${\mathcal{Y}}_{i,u}$ is a ${\mathcal{Y}}_u$-subbimodule of $\mathcal{H}_h$ (lemma \[currents\]). In $\mathcal{H}_t$ we have $\overline{\tilde{S}_{i,l}}=t\tilde{S}_{i,l}$ because $\pi_0(\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}})=1$. As said before we get a ${\mathbb{Z}}$-linear involution of ${\mathcal{Y}}_{i,t}$ such that: $$\overline{\lambda \tilde{S}_{i,l}}=t\tilde{S}_{i,l}\overline{\lambda}$$ We introduced such an involution in [@Her01]. With this new point of view, the compatibility with the relation $\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}=t^{-1}\tilde{S}_{i,l+r_i}$ is a direct consequence of lemma \[currents\] and needs no computation; for example: $$\overline{\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}}=t\tilde{S}_{i,l}\tilde{A}_{i,l-r_i}=t^3\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}=t^2\tilde{S}_{i,l+r_i}$$ $$\overline{t^{-1}\tilde{S}_{i,l+r_i}}=t\overline{\tilde{S}_{i,l+r_i}}=t^2\tilde{S}_{i,l+r_i}$$ ### The induced involution of $\text{Rep}_t$ For $i\in I$, the subalgebra $\mathfrak{K}_{i,t}\subset{\mathcal{Y}}_t$ verifies $\overline{\mathfrak{K}_{i,t}}\subset\mathfrak{K}_{i,t}$. [[*Proof:*]{}]{}Suppose $\lambda\in\mathfrak{K}_{i,t}$, that is to say $S_{i,t}(\lambda)=0$. So $\overline{(t^2-1)S_{i,t}(\lambda)}=0$ and: $$\underset{l\in{\mathbb{Z}}}{\sum}(\overline{\tilde{S}_{i,l}\lambda}-\overline{\lambda\tilde{S}_{i,l}})=0\Rightarrow t\underset{l\in{\mathbb{Z}}}{\sum}(\overline{\lambda}\tilde{S}_{i,l}-\tilde{S}_{i,l}\overline{\lambda})=0$$ So $t(1-t^2)S_{i,t}(\overline{\lambda})=0$ and $\overline{\lambda}\in\mathfrak{K}_{i,t}$. Note that $\chi\in{\mathcal{Y}}_t$ has the same monomials as $\overline{\chi}$, that is to say if $\chi=\underset{m\in A}{\sum}\lambda(t)m$ and $\overline{\chi}=\underset{m\in A}{\sum}\mu(t)m$, we have $\lambda(t)\neq 0\Leftrightarrow\mu(t)\neq 0$. In particular we can naturally extend our involution to an antimultiplicative involution on ${\mathcal{Y}}_t^{\infty}$. Moreover we have $\overline{\mathfrak{K}_t^{\infty}}\subset \mathfrak{K}_t^{\infty}$ and $\overline{\mathfrak{K}_t^{\infty,f}}=\overline{\text{Im}(\chi_{q,t})}\subset \text{Im}(\chi_{q,t})$. So we can define: The ${\mathbb{Z}}$-linear involution of $\text{Rep}_t$ is defined by: $$\forall \lambda\in\text{Rep}_t\text{ , }\overline{\lambda}=\chi_{q,t}^{-1}(\overline{\chi_{q,t}(\lambda)})$$ Analogues of Kazhdan-Lusztig polynomials ---------------------------------------- In this section we define analogues of Kazhdan-Lusztig polynomials (see [@kalu]) with the help of the antimultiplicative involution of section \[invo\] in the same spirit Nakajima did for the $ADE$-case [@Nab]. Let us begin we some technical properties of the action of the involution on monomials. ### Invariance of monomials {#invmon} We recall that the ${\mathcal{Y}}_t^A$-monomials are products of the $\tilde{A}_{i,l}^{-1}$ ($i\in I, l\in{\mathbb{Z}}$). For $M$ a ${\mathcal{Y}}_t$-monomial and $m$ a ${\mathcal{Y}}_t^A$-monomial there is a unique $\alpha(M,m)\in{\mathbb{Z}}$\[alpham\] such that $\overline{t^{\alpha(M,m)}Mm}=t^{\alpha(M,m)}\overline{M}m$. [[*Proof:*]{}]{}Let $\beta\in{\mathbb{Z}}$ such that $\overline{m}=t^{\beta}m$. We have $\overline{Mm}=\overline{m}\overline{M}=t^{\beta+\gamma}\overline{M}m$ where $\gamma\in 2{\mathbb{Z}}$ (section \[notuil\]). So it suffices to prove that $\beta\in 2{\mathbb{Z}}$. Let us compute $\beta$. Let $\pi_+(m)=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}A_{i,l}^{-v_{i,l}}$. In ${\mathcal{Y}}_u$ we have $\pi_+(m)\pi_-(m)=t_R\pi_-(m)\pi_+(m)$ where $\pi_0(R)=\beta$ and: $$R(q)=\underset{i,j\in I,r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}v_{j,r'}\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_j[-l]]}{c_l}$$ where for $l>0$ we set $\frac{[a_i[l],a_j[-l]]}{c_l}=B_{i,j}(q^l)(q^{l}-q^{-l})\in{\mathbb{Z}}[q^{\pm}]$ which is antisymmetric. For $i=j$, we have the term: $$\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}v_{i,r'}\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_i[-l]]}{c_l}$$ $$=\underset{l>0}{\sum}(\underset{\{r,r'\}\subset{\mathbb{Z}}, r\neq r'}{\sum}v_{i,r}(m)v_{i,r'}(m)(q^{l(r-r')}+q^{l(r'-r)})+\underset{r\in{\mathbb{Z}}}{\sum}v_{i,r}(m)^2)\frac{[a_i[l],a_i[-l]]}{c_l}$$ It is antisymmetric, so it has no term in $q^0$. So $\pi_0(R)=\pi_0(R')$ where $R'$ is the sum of the contributions for $i\neq j$: $$\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}(m)v_{j,r'}(m)\underset{l>0}{\sum}q^{lr-lr'}(\frac{[a_i[l],a_j[-l]]}{c_l}+\frac{[a_j[l],a_i[-l]]}{c_l})$$ $$=2\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}(m)v_{j,r'}(m)\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_j[-l]]}{c_l}$$ In particular $\pi_0(R')\in 2{\mathbb{Z}}$. For $M$ a ${\mathcal{Y}}_t$-monomial denote $A^{\text{inv}}_M=\{t^{\alpha(m,M)}Mm/\text{$m$ ${\mathcal{Y}}_t^A$-monomial}\}$\[ainv\]. In particular for $m'\in A^{\text{inv}}_M$ we have $\overline{m'}{m'}^{-1}=\overline{M}M^{-1}$. ### The polynomials For $M$ a ${\mathcal{Y}}_t$-monomial, denote $B^{\text{inv}}_M=t^{{\mathbb{Z}}}B\cap A^{\text{inv}}_M\label{binv}$. \[expol\] For $m\in t^{{\mathbb{Z}}}B$ there is a unique $L_t(m)\in\mathfrak{K}_t^{\infty}$\[tltm\] such that: $$\overline{L_t(m)}=(\overline{m}m^{-1})L_t(m)$$ $$E_t(m)=L_t(m)+\underset{m'<m, m' \in B^{\text{inv}}_m}{\sum}P_{m',m}(t)L_t(m')$$ where $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$. Those polynomials $P_{m',m}(t)$ are called analogues to Kazhdan-Lusztig polynomials and the $L_t(m)$ ($m\in B$) for a canonical basis of $\mathfrak{K}_t^{f, \infty}$. Such polynomials were introduced by Nakajima [@Nab] for the $ADE$-case. [[*Proof:*]{}]{}First consider $\overline{F_t(m)}$: it is in $\mathfrak{K}_t^{\infty}$ and has only one dominant ${\mathcal{Y}}_t$-monomial $\overline{m}$, so $\overline{F_t(m)}=\overline{m}m^{-1}F_t(m)$. Let be $m=m_L>m_{L-1}>...>m_0$ the finite set $t^{{\mathbb{Z}}}D(m)\cap B^{\text{inv}}_m$ (see lemma \[ordrep\]) with a total ordering compatible with the partial ordering. Note that it follows from section \[invmon\] that for $L\geq l\geq 0$, we have $\overline{m_l}m_l^{-1}=\overline{m}m^{-1}$. We have $E_t(m_0)=F_t(m_0)$ and so $\overline{E_t(m_0)}=\overline{m_0}m_0^{-1}E_t(m_0)$. As $B_{m_0}^{\text{inv}}=\{m_0\}$, we have $L_t(m_0)=E_t(m_0)$. We suppose by induction that the $L_t(m_l)$ ($L-1\geq l\geq 0$) are uniquely and well defined. In particular $m_l$ is of highest weight in $L_t(m_l)$, $\overline{L_t(m_l)}=\overline{m_l}m_l^{-1}L_t(m_l)=\overline{m}m^{-1}L_t(m_l)$, and we can write: $$\tilde{D}_t(m_L)\cap \mathfrak{K}_t^{\infty}={\mathbb{Z}}[t^{\pm}]F_t(m_L)\oplus\underset{0\leq l\leq L-1}{\bigoplus}{\mathbb{Z}}[t^{\pm}]L_t(m_l)$$ In particular consider $\alpha_{l,L}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m)=F_t(m)+\underset{l<L}{\sum}\alpha_{l,L}(t)L_t(m_l)$$ We want $L_t(m)$ of the form : $$L_t(m)=F_t(m)+\underset{l<L}{\sum}\beta_{l,L}(t)L_t(m_l)$$ The condition $\overline{L_t(m)}=\overline{m}m^{-1}mL_t(m)$ means that the $\beta_{l,L}(t)$ are symmetric. The condition $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$ means $\alpha_{l,L}(t)-\beta_{l,L}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$. So it suffices to prove that those two conditions uniquely define the $\beta_{l,L}(t)$: let us write $\alpha_{l,L}(t)=\alpha_{l,L}^+(t)+\alpha_{l,L}^0(t)+\alpha_{l,L}^-(t)$ (resp. $\beta_{l,L}(t)=\beta_{l,L}^+(t)+\beta_{l,L}^0(t)+\beta_{l,L}^-(t)$) where $\alpha_{l,L}^{\pm}(t)\in t^{\pm}{\mathbb{Z}}[t^{\pm}]$ and $\alpha_{l,L}^0(t)\in{\mathbb{Z}}$ (resp. for $\beta$). The condition $\alpha_{l,L}(t)-\beta_{l,L}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$ means $\beta_{l,L}^0(t)=\alpha_{l,L}^0(t)$ and $\beta_{l,L}^-(t)=\alpha_{l,L}^-(t)$. The symmetry of $\beta_{l,L}(t)$ means $\beta_{l,L}^+(t)=\beta_{l,L}^-(t^{-1})=\alpha_{l,L}^-(t^{-1})$. ### Examples for ${\mathfrak{g}}=sl_2$ {#exltcalc} In this section we suppose that ${\mathfrak{g}}=sl_2$. \[calcex\] Let $m\in t^{{\mathbb{Z}}}B$ such that $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$. Then $L_t(m)=F_t(m)$. Moreover: $$E_t(m)=L_t(m)+\underset{m'<m/m'\in B_m^{\text{inv}}}{\sum}t^{-R(m')}L_t(m')$$ where $R(m')\geq 1$ is given by $\pi_+(m'm^{-1})=A_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}$. In particular for $m'\in B^{\text{inv}}_m$ such that $m'<m$ we have $P_{m',m}(t)=t^{-R(m')}$. [[*Proof:*]{}]{}Note that a dominant monomial $m'<m$ verifies $\forall l\in{\mathbb{Z}}, u_l(m')\leq 1$ and appears in $E_t(m)$. We know that $\tilde{D}_m\cap\mathfrak{K}_t=\underset{m'\in t^{{\mathbb{Z}}}D_m\cap B^{\text{inv}}_m}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m')$. We can introduce $P_{m',m}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m)=F_t(m)+\underset{m'\in t^{{\mathbb{Z}}}D_m\cap B^{\text{inv}}-\{m\} }{\sum}P_{m',m}(t)F_t(m')$$ So by induction it suffices to show that $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$. $P_{m',m}(t)$ is the coefficient of $m'$ in $E_t(m)$. A dominant ${\mathcal{Y}}_t$-monomial $M$ which appears in $E_t(m)$ is of the form: $$M=m(m_1...m_{R+1})^{-1}m_1t\tilde{A}_{l_1}^{-1}m_2t\tilde{A}_{l_2}^{-1}m_3...t\tilde{A}_{l_R}^{-1}m_{R+1}$$ where $l_1<...<l_R\in{\mathbb{Z}}$ verify $\{l_r+2,l_r-2\}\cap\{l_1,...,l_{r-1},l_{r+1},...,l_R\}$ is empty, $u_{l_r-1}(m)=u_{l_r+1}(m)=1$ and we have set $m_r=\underset{l_{r-1}<l\leq l_r}{\overset{\rightarrow}{\prod}}\tilde{Y}_l^{u_l(m)}$. Such a monomial appears one time in $E_t(m)$. In particular $P_{m',m}(t)=t^{\alpha}$ where $\alpha\in{\mathbb{Z}}$ is given by $M=t^{\alpha}m'$ that is to say $\overline{M}M^{-1}=t^{-2\alpha}{m'}^{-1}m'=t^{-2\alpha}m^{-1}m$. So we compute: $\overline{M}M^{-1}=t^{-2R}\overline{m_{R+1}}\tilde{A}_{l_R}^{-1}\overline{m_R}...\tilde{A}_{l_1}^{-1}\overline{m_1}(\overline{m}_1^{-1}...\overline{m}_{R+1}^{-1})\overline{m}m_{R+1}^{-1}\tilde{A}_{l_R}m_R^{-1}...\tilde{A}_{l_1}m_1^{-1}(m_1...m_{R+1})m^{-1} \\=t^{-2R}t^{4R}\tilde{A}_{l_R}^{-1}...\tilde{A}_{l_1}^{-1}\overline{m}\tilde{A}_{l_R}...\tilde{A}_{l_1}m^{-1} \\=t^{2R}\tilde{A}_{l_R}^{-1}...\tilde{A}_{l_1}^{-1}\tilde{A}_{l_R}...\tilde{A}_{l_1}\overline{m}m^{-1}=t^{2R}\overline{m}m^{-1}$ Let us look at another example $m=\tilde{Y}_0^2\tilde{Y}_2$. We have: $$E_t(m)=L_t(m)+t^{-2}L_t(m')$$ where $m'=t\tilde{Y}_0^2\tilde{Y}_2\tilde{A}_1^{-1}\in B_{m}^{\text{inv}}$ and: $$L_t(m)=F_t(\tilde{Y}_0)F_t(\tilde{Y}_0\tilde{Y}_2)=\tilde{Y}_0(1+t\tilde{A}_1^{-1})\tilde{Y}_0\tilde{Y}_2(1+t\tilde{A}_3^{-1}(1+t\tilde{A}_1^{-1}))$$ $$L_t(m')=F_t(m')=t\tilde{Y}_0^2\tilde{Y}_2\tilde{A}_1^{-1}(1+t\tilde{A}_1^{-1})$$ Indeed the dominant monomials appearing in $E_t(m)$ are $m$ and $\tilde{Y}_0t\tilde{A}_1^{-1}\tilde{Y}_0\tilde{Y}_2+\tilde{Y}_0^2t\tilde{A}_1^{-1}\tilde{Y}_2=(1+t^{-2})m'$. In particular: $P_{m',m}(t)=t^{-2}$. ### Example in non-simply laced case {#exnons} We suppose that $C=\begin{pmatrix}2 & -2\\-1 & 2\end{pmatrix}$ and $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$. The formulas for $E_t(\tilde{Y}_{2,0})$ and $E_t(\tilde{Y}_{1,5})$ are given is section \[fin\]. We have: $$E_t(m)=L_t(m)+t^{-1}L_t(m')$$ where $m'=t\tilde{Y}_{2,0}\tilde{Y}_{1,5}\tilde{A}_{2,2}^{-1}\tilde{A}_{1,4}^{-1}\in B_m^{\text{inv}}$ and: $$L_t(m')=F_t(m')=t\tilde{Y}_{2,0}\tilde{Y}_{1,5}\tilde{A}_{2,2}^{-1}\tilde{A}_{1,4}^{-1}(1+t\tilde{A}_{1,2}^{-1}(1+t\tilde{A}_{2,4}^{-1}(1+t\tilde{A}_{1,6}^{-1})))$$ Indeed the dominant monomials appearing in $E_t(m)$ are $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$ and $\tilde{Y}_{2,0}t\tilde{A}_{2,2}^{-1}t\tilde{A}_{1,4}^{-1}\tilde{Y}_{1,5}=t^{-1}m'$. In particular $P_{m',m}(t)=t^{-1}$. Questions and conjectures {#quest} ========================= Positivity of coefficients {#acase} -------------------------- \[cpaconj\] If ${\mathfrak{g}}$ is of type $A_n$ ($n\geq 1$), the coefficients of $\chi_{q,t}(Y_{i,0})$ are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$. [[*Proof:*]{}]{}We show that for all $i\in I$ the hypothesis of proposition \[cpfacile\] for $m=Y_{i,0}$ are verified; in particular the property $Q$ of section \[qn\] will be verified. Let $i$ be in $I$. For $j\in I$, let us write $E(Y_{i,0})=\underset{m\in B_j}{\sum}\lambda_j(m)E_j(m)\in\mathfrak{K}_j$ where $\lambda_j(m)\in{\mathbb{Z}}$. Let $D$ be the set $D=\{\text{monomials of $E_j(m)$ }/j\in I,m\in B_j,\lambda_j(m)\neq 0\}$. It suffices to prove that for $j\in I$, $m\in B_j\cap D\Rightarrow u_j(m)\leq 1$ (because proposition \[cpfacile\] implies that for all $i\in I$, $F_t(\tilde{Y}_{i,0})=\pi^{-1}(E(Y_{i,0}))$). As $E(Y_{i,0})=F(Y_{i,0})$, $Y_{i,0}$ is the unique dominant ${\mathcal{Y}}$-monomial in $E(Y_{i,0})$. So for a monomial $m\in D$ there is a finite sequence $\{m_0=Y_{i,0},m_1,...,m_R=m\}$ such that for all $1\leq r\leq R$, there is $r'<r$ and $j\in I$ such that $m_{r'}\in B_j$ and for $r'<r''\leq r$, $m_{r''}$ is a monomial of $E_j(m_{r'})$ and $m_{r''}m_{r''-1}^{-1}\in\{A_{j,l}^{-1}/l\in{\mathbb{Z}}\}$. Such a sequence is said to be adapted to $m$. Suppose there is $j\in I$ and $m\in B_j\cap D$ such that $u_j(m)\geq 2$. So there is $m'\leq m$ in $D\cap B_j$ such that $u_j(m)=2$. So we can consider $m_0\in D$ such that there is $j_0\in I$, $m_0\in B_{j_0}$, $u_{j_0}(m)\geq 2$ and for all $m'<m_0$ in $D$ we have $\forall j\in I,m'\in B_j\Rightarrow u_j(m')\leq 1$. Let us write: $$m_0=Y_{j_0,q^{l}}Y_{j_0,q^{m}}\underset{j\neq j_0}{\prod}m_0^{(j)}$$ where for $j\neq j_0$, $m_0^{(j)}=\underset{l\in{\mathbb{Z}}}{\prod}Y_{j,l}^{u_{j,l}(m_0)}$. In a finite sequence adapted to $m_0$, a term $Y_{j_0,q^{l}}$ or $Y_{j_0,q^{m}}$ must come from a $E_{j_0+1}(m_1)$ or a $E_{j_0-1}(m_1)$. So for example we have $m_1<m_0$ in $D$ of the form $m_1=Y_{j_0,q^{m}}Y_{j_0+1,q^{l-1}}\underset{j\neq j_0,j_0+1}{\prod}m_1^{(j)}$. In all cases we get a monomial $m_1<m_0$ in $D$ of the form: $$m_1=Y_{j_1,q^{m_1}}Y_{j_1+1,q^{l_1}}\underset{j\neq j_1,j_1+1}{\prod}m_1^{(j)}$$ But the term $Y_{j_1+1,q^{l-1}}$ can not come from a $E_{j_1}(m_2)$ because we would have $u_{j_1}(m_2)\geq 2$. So we have $m_2<m_1$ in $D$ of the form: $$m_2=Y_{j_2,q^{m_2}}Y_{j_2+2,q^{l_2}}\underset{j\neq j_2,j_2+1,j_2+2}{\prod}m_2^{(j)}$$ This term must come from a $E_{j_2-1}, E_{j_2+3}$. By induction, we get $m_N<m_0$ in $D$ of the form : $$m_N=Y_{1,q^{m_N}}Y_{n,q^{l_N}}\underset{j\neq 1,..,n}{\prod}m_N^{(j)}=Y_{1,q^{m-N}}Y_{n,q^{l_N}}$$ It is a dominant monomial of $D\subset D_{Y_{i,0}}$ which is not $Y_{i,0}$. It is impossible (proof of lemma \[copieun\]). An analog result is also geometrically proved by Nakajima for the $ADE$-case in [@Nab] (it is also algebraically for $AD$-cases proved in [@Nac]). Those results and the explicit formulas in $n=1,2$-cases (see section \[fin\]) suggest: \[conun\] The coefficients of $F_t(\tilde{Y}_{i,0})=\chi_{q,t}(Y_{i,0})$ are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$. In particular for $m\in B$, the coefficients of $E_t(m)$ would be in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$; moreover $\chi_{q,t}(Y_{i,0})$ and $\chi_q(Y_{i,0})$ would have the same monomials, the $t$-algorithm would stop and $\text{Im}(\chi_{q,t})\subset{\mathcal{Y}}_t$. At the time he wrote this paper the author does not know a general proof of the conjecture. However a case by case investigation seems possible: the cases $G_2, B_2, C_2$ are checked in section \[fin\] and the cases $F_4, B_n, C_n$ ($n\leq 10$) have been checked on a computer. So a combinatorial proof for series $B_n, C_n$ ($n\geq 2$) analog to the proof of proposition \[cpaconj\] would complete the picture. Decomposition in irreducible modules ------------------------------------ The proposition \[calcex\] suggests: \[condeux\] For $m\in B$ we have $\pi_+(L_t(m))=L(m)$. In the $ADE$-case the conjecture \[condeux\] is proved by Nakajima with the help of geometry ([@Nab]). In particular this conjecture implies that the coefficients of $\pi_+(L_t(m))$ are non negative. It gives a way to compute explicitly the decomposition of a standard module in irreducible modules, because the conjecture \[condeux\] implies: $$E(m)=L(m)+\underset{m'<m}{\sum}P_{m',m}(1)L(m')$$ In particular we would have $P_{m',m}(1)\geq 0$. In section \[exltcalc\] we have studied some examples: -In proposition \[calcex\] for ${\mathfrak{g}}=sl_2$ and $m\in B$ such that $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$: we have $\pi_+(L_t(m))=F(m)=L(m)$ and: $$E(m)=\underset{m'\in B/m'\leq m}{\sum}L(m')$$ -For ${\mathfrak{g}}=sl_2$ and $m=\tilde{Y}_0^2\tilde{Y}_2$: we have $\pi_+(L_t(m))=F(Y_0)F(Y_0Y_2)=L(m)$ and: $$E(Y_0^2Y_2)=L(Y_0^2Y_2)+L(Y_0)$$ Note that $L(Y_0^2Y_2)$ has two dominant monomials $Y_0^2Y_2$ and $Y_0$ because $Y_0^2Y_2$ is irregular (lemma \[dominl\]). -For $C=B_2$ and $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$. The $\pi_+(L_t(\tilde{Y}_{2,0}\tilde{Y}_{1,5}))$ has non negative coefficients and the conjecture implies $E(Y_{2,0}Y_{1,5})=L(Y_{2,0}Y_{1,5})+L(Y_{1,1})$. Further applications and generalizations ---------------------------------------- We hope to address the following questions in the future: ### Iterated deformed screening operators Our presentation of deformed screening operators as commutators leads to the definition of iterated deformed screening operators. For example in order 2 we set: $$\tilde{S}_{j,i,t}(m)=[\underset{l\in{\mathbb{Z}}}{\sum}\tilde{S}_{j,l},S_{i,t}(m)]$$ ### Possible generalizations Some generalizations of the approach used in this article will be studied: a\) the theory of $q$-characters at roots of unity ([@Fre3]) suggests a generalization to the case $q^N=1$. b\) in this article we decided to work with ${\mathcal{Y}}_t$ which is a quotient of ${\mathcal{Y}}_u$. The same construction with ${\mathcal{Y}}_u$ will give characters with an infinity of parameters of deformation $t_r=\text{exp}(\underset{l>0}{\sum}h^{2l}q^{lr}c_l)$ ($r\in{\mathbb{Z}}$). c\) our construction is independent of representation theory and could be established for other generalized Cartan matrices (in particular for twisted affine cases). Appendix {#fin} ======== There are 5 types of semi-simple Lie algebra of rank 2: $A_1\times A_1$, $A_2$, $C_2$, $B_2$, $G_2$ (see for example [@Kac]). In each case we give the formula for $E(1),E(2)\in\mathfrak{K}$ and we see that the hypothesis of proposition \[cpfacile\] is verified. In particular we have $E_t(\tilde{Y}_{1,0})=\pi^{-1}(E(1)),E_t(\tilde{Y}_{2,0})=\pi^{-1}(E(2))\in\mathfrak{K}_t$. Following [@Fre], we represent the $E(1),E(2)\in\mathfrak{K}$ as a $I\times{\mathbb{Z}}$-oriented colored tree. For $\chi\in\mathfrak{K}$ the tree $\Gamma_{\chi}$ is defined as follows: the set of vertices is the set of ${\mathcal{Y}}$-monomials of $\chi$. We draw an arrow of color $(i,l)$ from $m_1$ to $m_2$ if $m_2=A_{i,l}^{-1}m_1$ and if in the decomposition $\chi=\underset{m\in B_i}{\sum}\mu_m L_i(m)$ there is $M\in B_i$ such that $\mu_M\neq 0$ and $m_1,m_2$ appear in $L_i(M)$. Then we give a formula for $E_t(\tilde{Y}_{1,0}),E_t(\tilde{Y}_{2,0})$ and we write it in $\mathfrak{K}_{1,t}$ and in $\mathfrak{K}_{2,t}$. $A_1\times A_1$-case -------------------- The Cartan matrix is $C=\begin{pmatrix}2 & 0\\0 & 2 \end{pmatrix}$ and $r_1=r_2=1$ (note that in this case the computations keep unchanged for all $r_1,r_2$). $$\xymatrix{Y_{1,0} \ar[d]^{1,1} \\Y_{1,2}^{-1}}\text{ and }\xymatrix{Y_{2,0} \ar[d]^{2,1} \\Y_{2,2}^{-1}}$$ $$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1})=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+\tilde{Y}_{1,2}^{-1}\in\mathfrak{K}_{2,t}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,2}^{-1})=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,1}^{-1})\in\mathfrak{K}_{2,t}$$ $$=\tilde{Y}_{2,0}+\tilde{Y}_{2,2}^{-1}\in\mathfrak{K}_{1,t}$$ $A_2$-case ---------- The Cartan matrix is $C=\begin{pmatrix}2 & -1\\-1 & 2 \end{pmatrix}$. It is symmetric, $r_1=r_2=1$: $$\xymatrix{Y_{1,0} \ar[d]^{1,1} \\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,2} \\Y_{2,3}^{-1}} \text{ and } \xymatrix{Y_{2,0} \ar[d]^{2,1} \\Y_{2,2}^{-1}Y_{1,1}\ar[d]^{1,2} \\Y_{1,3}^{-1}}$$ $$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,3}^{-1})=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})+\tilde{Y}_{2,3}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,2}^{-1})\in\mathfrak{K}_{2,t}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,2}^{-1}Y_{1,1}+Y_{1,3}^{-1})=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,1}^{-1})+\tilde{Y}_{1,3}^{-1}\in\mathfrak{K}_{2,t}$$ $$=\tilde{Y}_{2,0}+:\tilde{Y}_{2,2}^{-1}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,2}^{-1})\in\mathfrak{K}_{1,t}$$ $C_2,B_2$-case -------------- The two cases are dual so it suffices to compute for the Cartan matrix $C=\begin{pmatrix}2 & -2\\-1 & 2 \end{pmatrix}$ and $r_1=1$, $r_2=2$. $$\xymatrix{Y_{1,0} \ar[d]^{1,1} \\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,3} \\Y_{2,5}^{-1}Y_{1,4}\ar[d]^{1,5} \\Y_{1,6}^{-1}} \text{ and } \xymatrix{Y_{2,0}\ar[d]^{2,2} \\Y_{2,4}^{-1}Y_{1,1}Y_{1,3}\ar[d]^{1,4} \\Y_{1,1}Y_{1,5}^{-1}\ar[d]^{1,2} \\Y_{1,3}^{-1}Y_{1,5}^{-1}Y_{2,2}\ar[d]^{2,4} \\Y_{2,6}^{-1}}$$ $$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,5}^{-1}Y_{1,4}+Y_{1,6}^{-1})$$ $$=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})+:\tilde{Y}_{2,5}^{-1}\tilde{Y}_{1,4}:(1+t\tilde{A}_{1,5}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,3}^{-1})+\tilde{Y}_{1,6}^{-1}\in\mathfrak{K}_{2,t}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,4}^{-1}Y_{1,1}Y_{1,3}+Y_{1,1}Y_{1,5}^{-1}+Y_{1,3}^{-1}Y_{1,5}^{-1}Y_{2,2}+Y_{2,6}^{-1})$$ $$=\tilde{Y}_{2,0}+:\tilde{Y}_{2,4}^{-1}\tilde{Y}_{1,1}\tilde{Y}_{1,3}:(1+t\tilde{A}_{1,4}^{-1}+t^2\tilde{A}_{1,4}^{-1}\tilde{A}_{1,2}^{-1})+\tilde{Y}_{2,6}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,2}^{-1})+:\tilde{Y}_{1,1}\tilde{Y}_{1,5}^{-1}:+:\tilde{Y}_{1,3}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{2,2}:(1+t\tilde{A}_{2,4}^{-1})\in\mathfrak{K}_{2,t}$$ $G_2$-case ---------- The Cartan matrix is $C=\begin{pmatrix}2 & -3\\-1 & 2 \end{pmatrix}$ and $r_1=1$, $r_2=3$. ### First fundamental representation $$\xymatrix{Y_{1,0} \ar[d]^{1,1} \\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,4} \\Y_{2,7}^{-1}Y_{1,4}Y_{1,6}\ar[d]^{1,7} \\Y_{1,4}Y_{1,8}^{-1}\ar[d]^{1,5} \\Y_{1,6}^{-1}Y_{1,8}^{-1}Y_{2,5}\ar[d]^{2,8} \\Y_{2,11}^{-1}Y_{1,10}\ar[d]^{1,11} \\Y_{1,12}^{-1}}$$ $$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,7}^{-1}Y_{1,4}Y_{1,6}+Y_{1,4}Y_{1,6}+Y_{1,6}^{-1}Y_{1,8}^{-1}Y_{2,5}+Y_{2,11}^{-1}Y_{1,10}+Y_{1,12}^{-1})$$ $$=\tilde{Y}_{1,0}(1+\tilde{A}_{1,1}^{-1}) +:\tilde{Y}_{2,7}^{-1}\tilde{Y}_{1,4}\tilde{Y}_{1,6}:(1+t\tilde{A}_{1,7}^{-1}+t^2\tilde{A}_{1,7}^{-1}\tilde{A}_{1,5}^{-1}) +:\tilde{Y}_{2,11}^{-1}\tilde{Y}_{1,10}:(1+t\tilde{A}_{1,11}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0} +:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,4}^{-1}) +:\tilde{Y}_{1,4}\tilde{Y}_{1,6}: +:\tilde{Y}_{1,6}^{-1}\tilde{Y}_{1,8}^{-1}\tilde{Y}_{2,5}:(1+t\tilde{A}_{2,8}^{-1})+:\tilde{Y}_{1,12}^{-1}:\in\mathfrak{K}_{2,t}$$ ### Second fundamental representation $$\xymatrix{&Y_{2,0} \ar[d]^{2,3}& \\&Y_{2,6}^{-1}Y_{1,5}Y_{1,3}Y_{1,1}\ar[d]^{1,6}& \\&Y_{1,7}^{-1}Y_{1,3}Y_{1,1}\ar[d]^{1,4}& \\&Y_{2,4}Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,1}\ar[ld]^{1,2}\ar[rd]^{2,7}& \\Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,3}^{-1}Y_{2,4}Y_{2,2}\ar[d]^{2,5}\ar[rd]^{2,7}&&Y_{2,10}^{-1}Y_{1,9}Y_{1,1}\ar[ld]^{1,2}\ar[d]^{1,10} \\Y_{2,4}Y_{2,8}^{-1}\ar[d]^{2,7}&Y_{2,2}Y_{2,10}^{-1}Y_{1,9}Y_{1,3}^{-1}\ar[ld]^{2,5}\ar[rd]^{1,10}&Y_{1,11}^{-1}Y_{1,1}\ar[d]^{1,2} \\Y_{2,8}^{-1}Y_{2,10}^{-1}Y_{1,9}Y_{1,7}Y_{1,5}\ar[rd]^{1,10}&&Y_{2,2}Y_{1,11}^{-1}Y_{1,3}^{-1}\ar[ld]^{2,5} \\&Y_{2,8}^{-1}Y_{1,11}^{-1}Y_{1,7}Y_{1,5}\ar[d]^{1,8}& \\&Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,5}\ar[d]^{1,6}& \\&Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,7}^{-1}Y_{2,6}\ar[d]^{2,9}& \\&Y_{2,12}^{-1}&}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0} +Y_{2,6}^{-1}Y_{1,5}Y_{1,3}Y_{1,1} +Y_{1,7}^{-1}Y_{1,3}Y_{1,1}0 +Y_{2,4}Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,1} +Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,3}^{-1}Y_{2,4}Y_{2,2}$$ $$+Y_{2,10}^{-1}Y_{1,9}Y_{1,1} +Y_{2,4}Y_{2,8}^{-1}+Y_{2,2}Y_{2,10}^{-1}Y_{1,9}Y_{1,3}^{-1} +Y_{1,11}^{-1}Y_{1,1} +Y_{2,8}^{-1}Y_{2,10}^{-1}Y_{1,9}Y_{1,7}Y_{1,5} +Y_{2,2}Y_{1,11}^{-1}Y_{1,3}^{-1}$$ $$+Y_{2,8}^{-1}Y_{1,11}^{-1}Y_{1,7}Y_{1,5} +Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,5}+Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,7}^{-1}Y_{2,6}+Y_{2,12}^{-1})$$ We use the following relations to write $E_t(\tilde{Y}_{2,0})$ in $\mathfrak{K}_{1,t}$ and in $\mathfrak{K}_{2,t}$: $\tilde{A}_{1,2}\tilde{A}_{2,7}=\tilde{A}_{2,7}\tilde{A}_{1,2}$, $\tilde{A}_{2,5}\tilde{A}_{2,7}=\tilde{A}_{2,7}\tilde{A}_{2,5}$, $\tilde{A}_{1,2}\tilde{A}_{1,10}=\tilde{A}_{1,10}\tilde{A}_{1,2}$, $\tilde{A}_{2,5}\tilde{A}_{1,10}=\tilde{A}_{1,10}\tilde{A}_{2,5}$. $$E_t(\tilde{Y}_{2,0})=\tilde{Y}_{2,0} +:\tilde{Y}_{2,6}^{-1}\tilde{Y}_{1,5}\tilde{Y}_{1,3}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,6}^{-1}(1+t\tilde{A}_{1,4}^{-1}(1+t\tilde{A}_{1,2}^{-1}))) +:\tilde{Y}_{2,10}^{-1}\tilde{Y}_{1,9}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,2}^{-1})(1+t\tilde{A}_{1,10}^{-1})$$ $$+:\tilde{Y}_{2,4}\tilde{Y}_{2,8}^{-1}: +:\tilde{Y}_{2,8}^{-1}\tilde{Y}_{2,10}^{-1}\tilde{Y}_{1,9}\tilde{Y}_{1,7}\tilde{Y}_{1,5}:(1+t\tilde{A}_{1,10}^{-1}(1+t\tilde{A}_{1,8}^{-1}(1+t\tilde{A}_{1,6}^{-1}))) +\tilde{Y}_{2,12}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,3}^{-1}) +:\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,3}\tilde{Y}_{1,1}: +:\tilde{Y}_{2,4}\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{1,1}:(1+t\tilde{A}_{2,7}^{-1})$$ $$+:\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{1,3}^{-1}\tilde{Y}_{2,4}\tilde{Y}_{2,2}:(1+t\tilde{A}_{2,7}^{-1})(1+t\tilde{A}_{2,5}^{-1}) +:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,1}: +:\tilde{Y}_{2,2}\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,3}^{-1}:(1+t\tilde{A}_{2,5}^{-1})$$ $$+:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,9}^{-1}\tilde{Y}_{1,5}: +:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,9}^{-1}\tilde{Y}_{1,7}^{-1}\tilde{Y}_{2,6}:(1+t\tilde{A}_{2,9}^{-1})\in\mathfrak{K}_{2,t}$$ Notations {#notations .unnumbered} ========= ------------------------------------------------------------------------------------- -------------------------------------------------------- --- $A$ set of ${\mathcal{Y}}$-monomials p $A_t$ set of ${\mathcal{Y}}_t$-monomials p $A_m^{\text{inv}}, B_m^{\text{inv}}$ set of ${\mathcal{Y}}_t$-monomials p $\overset{\infty}{A}_t$ product module p $\alpha$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p $\alpha(m)$ character p $a_i[m]$ element of $\mathcal{H}$ p $\tilde{A}_{i,l}, \tilde{A}_{i,l}^{-1}$ elements of ${\mathcal{Y}}_u$ or ${\mathcal{Y}}_t$ p $A_{i,l}, A_{i,l}^{-1}$ elements of ${\mathcal{Y}}$ p $B$ a set of ${\mathcal{Y}}$-monomials p $B_i$, $B_J$ a set of ${\mathcal{Y}}$-monomials p $(B_{i,j})$ symmetrized Cartan matrix p $\beta$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p $(C_{i,j})$ Cartan matrix p $(\tilde{C}_{i,j})$ inverse of $C$ p $c_r$ central element of $\mathcal{H}$ p $d$ bicharacter p $D_{m,K}, D_m$ set of monomials p $\tilde{D}_m$ submodule of ${\mathcal{Y}}_t^{\infty}$ p $E_i(m)$ element of $\mathfrak{K}_i$ p $E_{i,t}(m)$ element of $\mathfrak{K}_{i,t}$ p $E_{i,t}^m$ map p $E(m)$ element of $\mathfrak{K}$ p $E_t(m)$ element of $\mathfrak{K}_{t}^{\infty}$ p $F_i(m)$ element of $\mathfrak{K}_i$ p $F_{i,t}(m)$ element of $\mathfrak{K}_{i,t}$ p $F(m)$ element of $\mathfrak{K}$ p $F_t(m)$ element of $\mathfrak{K}_{t}^{\infty}$ p $\gamma$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p $\mathcal{H}$ Heisenberg algebra p $\mathcal{H}^+, \mathcal{H}^-$ subalgebras of $\mathcal{H}$ p $\mathcal{H}_h$ formal series in $\mathcal{H}$ p $\mathcal{H}_t$ quotient of $\mathcal{H}_h$ p $\mathcal{H}_t^+, \mathcal{H}_t^-$ subalgebras of $\mathcal{H}_t$ p $\mathfrak{K}_i, \mathfrak{K}_J, \mathfrak{K}$ subrings of ${\mathcal{Y}}$ p $\mathfrak{K}_{i,t}, \mathfrak{K}_{J,t}, \mathfrak{K}_t$ subrings of ${\mathcal{Y}}_t$ p $\mathfrak{K}_{i,t}^{\infty}, \mathfrak{K}_{J,t}^{\infty}, \mathfrak{K}_t^{\infty}$ subrings of ${\mathcal{Y}}_t^{\infty}$ p $\chi_q$ morphism of $q$-characters p $\chi_{q,t}$ morphism of $q,t$-characters p $L_i(m)$ element of $\mathfrak{K}_i$ p $L_t(m)$ element of $\mathfrak{K}_t^{\infty}$ p $:m:$ monomial in $A$ p $\tilde{m}$ monomial in $A_t$ p $N,N_t,\mathcal{N},\mathcal{N}_t$ characters, bicharacters p $P(n)$ property of $n\in{\ensuremath{\mathbb{N}}}$ p ------------------------------------------------------------------------------------- -------------------------------------------------------- --- --------------------------------------------------------- ---------------------------------------------------------------- --- $\pi$ map p $\pi_r$ map to ${\mathbb{Z}}$ p $\pi_+,\pi_-$ endomorphisms of $\mathcal{H}_h$, $\mathcal{H}_t$ p $q$ complex number p $Q(n)$ property of $n\in{\ensuremath{\mathbb{N}}}$ p $\text{Rep}$ Grothendieck ring p $\text{Rep}_t$ deformed Grothendieck ring p $s(m_r)_J,s(m_r)$ sequences of ${\mathbb{Z}}[t^{\pm}]$ p $S_i$ screening operator p $\tilde{S}_{i,l}$ screening current p $S_{i,t}$ $t$-screening operator p $t$ central element of ${\mathcal{Y}}_t$ p $t_R$ central element of ${\mathcal{Y}}_u$ p $u_{i,l}$ multiplicity of $Y_{i,l}$ p $u_i$ sum of the $u_{i,l}$ p $\mathfrak{U}$ subring of ${\ensuremath{\mathbb{Q}}}(q)$ p ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ quantum affine algebra p ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})$ Cartan algebra p $X_{i,l}$ element of $\text{Rep}$ p $y_i[m]$ element of $\mathcal{H}$ p $Y_{i,l}, Y_{i,l}^{-1}$ elements of ${\mathcal{Y}}$ p $\tilde{Y}_{i,l},\tilde{Y}_{i,l}^{-1}$ elements of ${\mathcal{Y}}_u$ or ${\mathcal{Y}}_t$ p ${\mathcal{Y}}$ subalgebra of $\mathcal{H}_h$ p ${\mathcal{Y}}_t$ quotient of ${\mathcal{Y}}_u$ p ${\mathcal{Y}}_t^+, {\mathcal{Y}}_t^-$ subalgebras of $\mathcal{H}_t$ p ${\mathcal{Y}}_u$ subalgebra of $\mathcal{H}_h$ p ${\mathcal{Y}}_{i,t}$ ${\mathcal{Y}}_t$-module p ${\mathcal{Y}}_{i,u}$ ${\mathcal{Y}}_u$-module p ${\mathcal{Y}}_t^{\infty}, {\mathcal{Y}}_t^{A, \infty}$ submodules of $\overset{\infty}{A}_t$ p ${\mathcal{Y}}_t^A, {\mathcal{Y}}_t^{A,K}$ submodules of ${\mathcal{Y}}_t$ p $z$ indeterminate p $::$ endomorphism of $\mathcal{H}, \mathcal{H}_h, {\mathcal{Y}}_u, {\mathcal{Y}}_t$ p $*$ deformed multiplication p --------------------------------------------------------- ---------------------------------------------------------------- --- [99]{} , [*Groupes et algèbres de Lie*]{} [Chapitres IV-VI, Hermann (1968)]{} , [*Quantum affine algebras and their representations*]{} [in Representations of groups (Banff, AB, 1994),59-78, CMS Conf. Proc, 16, Amer. Math. Soc., Providence, RI (1995)]{} , [*A Guide to Quantum Groups*]{} [Cambridge University Press, Cambridge (1994)]{} , [*Deformations of $W$-algebras associated to simple Lie algebras*]{} [Comm. Math. Phys. 197, no. 1, 1–32 (1998)]{} , [*The $q$-Characters of Representations of Quantum Affine Algebras and Deformations of $W$-Algebras*]{} [Recent Developments in Quantum Affine Algebras and related topics, Cont. Math., vol. 248, pp 163-205 (1999)]{} , [*Combinatorics of $q$-Characters of Finite-Dimensional Representations of Quantum Affine Algebras*]{} [Comm. Math. Phy., vol 216, no. 1, pp 23-57 (2001)]{} , [*The $q$-characters at roots of unity*]{} [Adv. Math. 171, no. 1, 139–167 (2002)]{} , [*t-analogues des opérateurs d’écrantage associés aux q-caractères*]{} [Int. Math. Res. Not., vol. 2003, no. 8, pp 451-475 (2003)]{} , [*Infinite dimensional Lie algebras*]{} [3rd Edition, Cambridge University Press (1990)]{} , [*Representations of Coxeter Groups and Hecke Algebras*]{} [Inventiones math. 53, pp. 165-184 (1979)]{} , [*$t$-Analogue of the $q$-Characters of Finite Dimensional Representations of Quantum Affine Algebras*]{} [“Physics and Combinatorics”, Proc. Nagoya 2000 International Workshop, World Scientific, pp 181-212 (2001)]{} , [*Quiver Varieties and $t$-Analogs of $q$-Characters of Quantum Affine Algebras*]{} [Preprint arXiv:math.QA/0105173]{} , [*t-analogs of q-characters of quantum affine algebras of type $A_n$, $D_n$*]{} [Preprint arXiv:math.QA/0204184]{} , [*Représentations des groupes quantiques*]{} [Séminaire Bourbaki exp. no. 744, Astérisque 201-203, 443-83, SMF (1992)]{} , [*Perverse Sheaves and Quantum Grothendieck Rings*]{} [Preprint arXiv:math.QA/0103182]{}

ArXiv

--- abstract: 'In this paper, we investigate the modeling power of contextualized embeddings from pre-trained language models, e.g. BERT, on the E2E-ABSA task. Specifically, we build a series of simple yet insightful neural baselines to deal with E2E-ABSA. The experimental results show that even with a simple linear classification layer, our BERT-based architecture can outperform state-of-the-art works. Besides, we also standardize the comparative study by consistently utilizing a hold-out development dataset for model selection, which is largely ignored by previous works. Therefore, our work can serve as a BERT-based benchmark for E2E-ABSA.[^1]' author: - | Xin Li[^1^]{}, Lidong Bing[^2^]{}, Wenxuan Zhang[^1^]{} and Wai Lam[^1^]{}\ [[^1^]{}Department of Systems Engineering and Engineering Management\ The Chinese University of Hong Kong, Hong Kong]{}\ [[^2^]{}R&D Center Singapore, Machine Intelligence Technology, Alibaba DAMO Academy]{}\ [`{lixin,wxzhang,wlam}@se.cuhk.edu.hk`]{}\ [`[email protected]`]{}\ bibliography: - 'emnlp-ijcnlp-2019.bib' title: 'Exploiting BERT for End-to-End Aspect-based Sentiment Analysis[^2]' --- Introduction ============ Aspect-based sentiment analysis (ABSA) is to discover the users’ sentiment or opinion towards an aspect, usually in the form of explicitly mentioned aspect terms [@mitchell-etal-2013-open; @zhang-etal-2015-neural] or implicit aspect categories [@wang-etal-2016-attention], from user-generated natural language texts [@liu2012sentiment]. The most popular ABSA benchmark datasets are from SemEval ABSA challenges [@pontiki-etal-2014-semeval; @pontiki-etal-2015-semeval; @pontiki-etal-2016-semeval] where a few thousand review sentences with gold standard aspect sentiment annotations are provided. Table \[tab:problem\_settings\] summarizes three existing research problems related to ABSA. The first one is the original ABSA, aiming at predicting the sentiment polarity of the sentence towards the given aspect. Compared to this classification problem, the second one and the third one, namely, Aspect-oriented Opinion Words Extraction (AOWE) [@fan-etal-2019-target] and End-to-End Aspect-based Sentiment Analysis (E2E-ABSA) [@ma-etal-2018-joint; @schmitt-etal-2018-joint; @li2019unified; @li2017learning; @li2019learning], are related to a sequence tagging problem. Precisely, the goal of AOWE is to extract the aspect-specific opinion words from the sentence given the aspect. The goal of E2E-ABSA is to jointly detect aspect terms/categories and the corresponding aspect sentiments. Many neural models composed of a task-agnostic pre-trained word embedding layer and task-specific neural architecture have been proposed for the original ABSA task (i.e. the aspect-level sentiment classification) [@tang-etal-2016-aspect; @wang-etal-2016-attention; @chen-etal-2017-recurrent-attention; @liu-zhang-2017-attention; @ma2017interactive; @ma2018targeted; @majumder-etal-2018-iarm; @li-etal-2018-transformation; @he-etal-2018-exploiting; @xue-li-2018-aspect; @wang-etal-2018-target; @fan-etal-2018-multi; @huang-carley-2018-parameterized; @lei2019human; @li2019exploiting; @zhang2019aspect][^3], but the improvement of these models measured by the accuracy or F1 score has reached a bottleneck. One reason is that the task-agnostic embedding layer, usually a linear layer initialized with Word2Vec [@mikolov2013distributed] or GloVe [@pennington-etal-2014-glove], only provides context-independent word-level features, which is insufficient for capturing the complex semantic dependencies in the sentence. Meanwhile, the size of existing datasets is too small to train sophisticated task-specific architectures. Thus, introducing a context-aware word embedding[^4] layer pre-trained on large-scale datasets with deep LSTM [@mccann2017learned; @peters-etal-2018-deep; @howard-ruder-2018-universal] or Transformer [@radford2018improving; @radford2019language; @devlin-etal-2019-bert; @lample2019cross; @yang2019xlnet; @dong2019unified] for fine-tuning a lightweight task-specific network using the labeled data has good potential for further enhancing the performance. @xu-etal-2019-bert [@sun-etal-2019-utilizing; @song2019attentional; @yu2019adapting; @rietzler2019adapt; @huang2019syntax; @hu2019learning] have conducted some initial attempts to couple the deep contextualized word embedding layer with downstream neural models for the original ABSA task and establish the new state-of-the-art results. It encourages us to explore the potential of using such contextualized embeddings to the more difficult but practical task, i.e. E2E-ABSA (the third setting in Table \[tab:problem\_settings\]).[^5] Note that we are not aiming at developing a task-specific architecture, instead, our focus is to examine the potential of contextualized embedding for E2E-ABSA, coupled with various simple layers for prediction of E2E-ABSA labels.[^6] In this paper, we investigate the modeling power of BERT [@devlin-etal-2019-bert], one of the most popular pre-trained language model armed with Transformer [@vaswani2017attention], on the task of E2E-ABSA. Concretely, inspired by the investigation of E2E-ABSA in @li2019unified, which predicts aspect boundaries as well as aspect sentiments using a single sequence tagger, we build a series of simple yet insightful neural baselines for the sequence labeling problem and fine-tune the task-specific components with BERT or deem BERT as feature extractor. Besides, we standardize the comparative study by consistently utilizing the hold-out development dataset for model selection, which is ignored in most of the existing ABSA works [@tay2018learning]. Model ===== In this paper, we focus on the aspect term-level End-to-End Aspect-Based Sentiment Analysis (E2E-ABSA) problem setting. This task can be formulated as a sequence labeling problem. The overall architecture of our model is depicted in Figure \[fig:architecture\]. Given the input token sequence $\mathrm{\bf x} = \{x_1, \cdots, x_T \}$ of length $T$, we firstly employ BERT component with $L$ transformer layers to calculate the corresponding contextualized representations $H^{L} = \{h^L_1, \cdots, h^L_T\} \in \mathbb{R}^{T \times \mathrm{dim}_h}$ for the input tokens where $\mathrm{dim}_h$ denotes the dimension of the representation vector. Then, the contextualized representations are fed to the task-specific layers to predict the tag sequence $\mathrm{\bf y} = \{y_1, \cdots, y_T\}$. The possible values of the tag $y_t$ are `B`-{`POS,NEG,NEU`}, `I`-{`POS,NEG,NEU`}, `E`-{`POS,NEG,NEU`}, `S`-{`POS,NEG,NEU`} or `O`, denoting the beginning of aspect, inside of aspect, end of aspect, single-word aspect, with positive, negative or neutral sentiment respectively, as well as outside of aspect. ![Overview of the designed model.[]{data-label="fig:architecture"}](architecture_crop.pdf){width="50.00000%"} BERT as Embedding Layer ----------------------- Compared to the traditional Word2Vec- or GloVe-based embedding layer which only provides a single context-independent representation for each token, the BERT embedding layer takes the sentence as input and calculates the token-level representations using the information from the entire sentence. First of all, we pack the input features as $H^0 = \{e_1,\cdots,e_T\}$, where $e_t$ ($t\in[1, T]$) is the combination of the token embedding, position embedding and segment embedding corresponding to the input token $x_t$. Then $L$ transformer layers are introduced to refine the token-level features layer by layer. Specifically, the representations $H^{l} = \{h^l_1, \cdots, h^l_T\}$ at the $l$-th ($l \in [1, L]$) layer are calculated below: $$\label{eq:bert} H^l = \text{Transformer}_l(H^{l-1})$$ We regard $H^L$ as the contextualized representations of the input tokens and use them to perform the predictions for the downstream task. Design of Downstream Model -------------------------- After obtaining the BERT representations, we design a neural layer, called E2E-ABSA layer in Figure \[tab:problem\_settings\], on top of BERT embedding layer for solving the task of E2E-ABSA. We investigate several different design for the E2E-ABSA layer, namely, linear layer, recurrent neural networks, self-attention networks, and conditional random fields layer. #### Linear Layer The obtained token representations can be directly fed to linear layer with softmax activation function to calculate the token-level predictions: $$P(y_t|x_t) = \text{softmax}(W_o h^L_t + b_o)$$ where $W_o \in \mathbb{R}^{\mathrm{dim}_h \times |\mathcal{Y}|}$ is the learnable parameters of the linear layer. #### Recurrent Neural Networks Considering its sequence labeling formulation, Recurrent Neural Networks (RNN) [@elman1990finding] is a natural solution for the task of E2E-ABSA. In this paper, we adopt GRU [@cho-etal-2014-learning], whose superiority compared to LSTM [@hochreiter1997long] and basic RNN has been verified in @jozefowicz2015empirical. The computational formula of the task-specific hidden representation $h^{\mathcal{T}}_t \in \mathbb{R}^{\mathrm{dim}_h}$ at the $t$-th time step is shown below: $$\small \begin{split} \begin{bmatrix} r_t \\ z_t \end{bmatrix} &= \sigma(\textsc{Ln}(W_x h^L_t)+\textsc{Ln}(W_h h^{\mathcal{T}}_{t-1})) \\ n_t &= \text{tanh}(\textsc{Ln}(W_{xn} h^L_t)+r_t * \textsc{Ln}(W_{hn} h^{\mathcal{T}}_{t-1})) \\ h^{\mathcal{T}}_{t} &= (1-z_t) * n_t + z_t * h^{\mathcal{T}}_{t-1} \end{split}$$ where $\sigma$ is the sigmoid activation function and $r_t$, $z_t$, $n_t$ respectively denote the reset gate, update gate and new gate. $W_x,W_h \in \mathbb{R}^{2\mathrm{dim}_h \times \mathrm{dim}_h}$, $W_{xn}, W_{hn} \in \mathbb{R}^{\mathrm{dim}_h \times \mathrm{dim}_h}$ are the parameters of GRU. Since directly applying RNN on the output of transformer, namely, the BERT representation $h^L_t$, may lead to unstable training [@chen-etal-2018-best; @liu2019fine], we add additional layer-normalization [@ba2016layer], denoted as $\textsc{Ln}$, when calculating the gates. Then, the predictions are obtained by introducing a softmax layer: $$\label{output} p(y_t|x_t) = \text{softmax}(W_o h^{\mathcal{T}}_t + b_o)$$ #### Self-Attention Networks With the help of self attention [@cheng-etal-2016-long; @lin2017structured], Self-Attention Network [@vaswani2017attention; @shen2018disan] is another effective feature extractor apart from RNN and CNN. In this paper, we introduce two SAN variants to build the task-specific token representations $H^{\mathcal{T}}=\{h^{\mathcal{T}}_1,\cdots,h^{\mathcal{T}}_T\}$. One variant is composed of a simple self-attention layer and residual connection [@he2016deep], dubbed as “SAN”. The computational process of SAN is below: $$\begin{split} H^{\mathcal{T}} &= \textsc{Ln}(H^{L}+\textsc{Slf-Att} (Q, K, V)) \\ Q, K, V &= H^{L} W^{Q}, H^{L}W^{K}, H^{L} W^{V} \end{split}$$ where <span style="font-variant:small-caps;">Slf-Att</span> is identical to the self-attentive scaled dot-product attention [@vaswani2017attention]. Another variant is a transformer layer (dubbed as “TFM”), which has the same architecture with the transformer encoder layer in the BERT. The computational process of TFM is as follows: $$\begin{split} \hat{H}^{L} &= \textsc{Ln}(H^{L}+\textsc{Slf-Att} (Q, K, V)) \\ H^{\mathcal{T}} &= \textsc{Ln}(\hat{H}^{L}+\textsc{Ffn}(\hat{H}^{L})) \end{split}$$ where <span style="font-variant:small-caps;">Ffn</span> refers to the point-wise feed-forward networks [@vaswani2017attention]. Again, a linear layer with softmax activation is stacked on the designed SAN/TFM layer to output the predictions (same with that in Eq(\[output\])). [ll|c|c|c|c]{} & Train & Dev & Test & Total\ & \# sent & 2741 & 304 & 800 & 4245\ & \# aspect & 2041 & 256 & 634 & 2931\ & \# sent & 3490 & 387 & 2158 & 6035\ & \# aspect & 3893 & 413 & 2287 & 6593\ #### Conditional Random Fields Conditional Random Fields (CRF) [@lafferty2001conditional] is effective in sequence modeling and has been widely adopted for solving the sequence labeling tasks together with neural models [@huang2015bidirectional; @lample-etal-2016-neural; @ma-hovy-2016-end]. In this paper, we introduce a linear-chain CRF layer on top of the BERT embedding layer. Different from the above mentioned neural models maximizing the token-level likelihood $p(y_t|x_t)$, the CRF-based model aims to find the globally most probable tag sequence. Specifically, the sequence-level scores $s(\mathrm{\bf x},\mathrm{\bf y})$ and likelihood $p(\mathrm{\bf y}|\mathrm{\bf x})$ of $\mathrm{\bf y} = \{y_1,\cdots,y_T\}$ are calculated as follows: $$ \begin{split} s(\mathrm{\bf x},\mathrm{\bf y}) &= \sum^T_{t=0} M^A_{y_t,y_{t+1}} + \sum^T_{t=1} M^P_{t, y_t} \\ p(\mathrm{\bf y}|\mathrm{\bf x}) &= \text{softmax}(s(\mathrm{\bf x},\mathrm{\bf y})) \end{split}$$ where $M^A \in \mathbb{R}^{|\mathcal{Y}| \times |\mathcal{Y}|}$ is the randomly initialized transition matrix for modeling the dependency between the adjacent predictions and $M^P \in \mathbb{R}^{T \times |\mathcal{Y}|}$ denote the emission matrix linearly transformed from the BERT representations $H^L$. The softmax here is conducted over all of the possible tag sequences. As for the decoding, we regard the tag sequence with the highest scores as output: $$\mathrm{\bf y}^* = \arg\max_{\mathrm{\bf y}} s(\mathrm{\bf x},\mathrm{\bf y})$$ where the solution is obtained via Viterbi search. ![Performances on the Dev set of `REST`.[]{data-label="fig:overfit"}](overfit_v4.pdf){width="40.00000%"} Experiment ========== Dataset and Settings -------------------- We conduct experiments on two review datasets originating from SemEval [@pontiki-etal-2014-semeval; @pontiki-etal-2015-semeval; @pontiki-etal-2016-semeval] but re-prepared in @li2019unified. The statistics are summarized in Table \[tab:dataset\]. We use the pre-trained “bert-base-uncased” model[^7], where the number of transformer layers $L=12$ and the hidden size $\mathrm{dim}_h$ is 768. For the downstream E2E-ABSA component, we consistently use the single-layer architecture and set the dimension of task-specific representation as $\mathrm{dim}_h$. The learning rate is 2e-5. The batch size is set as 25 for `LAPTOP` and 16 for `REST`. We train the model up to 1500 steps. After training 1000 steps, we conduct model selection on the development set for very 100 steps according to the micro-averaged F1 score. Following these settings, we train 5 models with different random seeds and report the average results. We compare with **Existing Models**, including tailor-made E2E-ABSA models [@li2019unified; @luo-etal-2019-doer; @he-etal-2019-interactive], and competitive **LSTM-CRF** sequence labeling models [@lample-etal-2016-neural; @ma-hovy-2016-end; @liu2018empower]. Main Results ------------ From Table \[tab:main\_results\], we surprisingly find that only introducing a simple token-level classifier, namely, BERT-Linear, already outperforms the existing works without using BERT, suggesting that BERT representations encoding the associations between arbitrary two tokens largely alleviate the issue of context independence in the linear E2E-ABSA layer. It is also observed that slightly more powerful E2E-ABSA layers lead to much better performance, verifying the postulation that incorporating context helps to sequence modeling. Over-parameterization Issue --------------------------- Although we employ the smallest pre-trained BERT model, it is still over-parameterized for the E2E-ABSA task (110M parameters), which naturally raises a question: does BERT-based model tend to overfit the small training set? Following this question, we train BERT-GRU, BERT-TFM and BERT-CRF up to 3000 steps on `REST` and observe the fluctuation of the F1 measures on the development set. As shown in Figure \[fig:overfit\], F1 scores on the development set are quite stable and do not decrease much as the training proceeds, which shows that the BERT-based model is exceptionally robust to overfitting. Finetuning BERT or Not ---------------------- We also study the impact of fine-tuning on the final performances. Specifically, we employ BERT to calculate the contextualized token-level representations but kept the parameters of BERT component unchanged in the training phase. Figure \[fig:finetune\] illustrate the comparative results between the BERT-based models and those keeping BERT component fixed. Obviously, the general purpose BERT representation is far from satisfactory for the downstream tasks and task-specific fine-tuning is essential for exploiting the strengths of BERT to improve the performance. ![Effect of fine-tuning BERT.[]{data-label="fig:finetune"}](finetuning_v3.pdf){width="40.00000%"} Conclusion ========== In this paper, we investigate the effectiveness of BERT embedding component on the task of End-to-End Aspect-Based Sentiment Analysis (E2E-ABSA). Specifically, we explore to couple the BERT embedding component with various neural models and conduct extensive experiments on two benchmark datasets. The experimental results demonstrate the superiority of BERT-based models on capturing aspect-based sentiment and their robustness to overfitting. [^1]: Our code is open-source and available at: <https://github.com/lixin4ever/BERT-E2E-ABSA> [^2]: The work described in this paper is substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Code: 14204418). [^3]: Due to the limited space, we can not list all of the existing works here, please refer to the survey [@zhou2019deep] for more related papers. [^4]: In this paper, we generalize the concept of “word embedding” as a mapping between the word and the low-dimensional word representations. [^5]: Both of ABSA and AOWE assume that the aspects in a sentence are given. Such setting makes them less practical in real-world scenarios since manual annotation of the fine-grained aspect mentions/categories is quite expensive. [^6]: @hu-etal-2019-open introduce BERT to handle the E2E-ABSA problem but their focus is to design a task-specific architecture rather than exploring the potential of BERT. [^7]: https://github.com/huggingface/transformers

ArXiv

--- abstract: 'For any two configurations of ordered points ${{\mathbf p}}=({{\mathbf p}}_{1},\cdots,{{\mathbf p}}_{N})$ and ${{\mathbf q}}=({{\mathbf q}}_{1},\cdots,{{\mathbf q}}_{N})$ in Euclidean space ${{\mathbb E}}^d$ such that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$, there exists a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension $2d$; Bezdek and Connelly used this to prove the Kneser-Poulsen conjecture for the planar case. In this paper, we show that this construction is optimal in the sense that for any $d \ge 2$ there exists configurations of $(d+1)^2$ points ${{\mathbf p}}$ and ${{\mathbf q}}$ in ${{\mathbb E}}^d$ such that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ but there is no continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension less than $2d$. The techniques used in our proof are completely elementary.' address: - | Department of Computer Science\ National University of Singapore\ Singapore - | Department of Mathematics\ National University of Singapore\ Singapore 119076 - | Department of Mathematics\ National University of Singapore\ Singapore 119076 author: - 'Holun Cheng, Ser Peow Tan and Yidan Zheng' title: On continuous expansions of configurations of points in Euclidean space --- \[section\] \[thm\][Lemma]{} \[thm\][Conjecture]{} \[thm\][Corollary]{} \[thm\][Addendum]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} \[thm\][[**Example**]{}]{} \[thm\][[ **Question**]{}]{} [^1] Introduction and statement of results. {#s:intro} ====================================== Let ${{\mathbb E}}^d$ be the Euclidean space of dimension $d \ge 2$, where we identify and represent the points of ${{\mathbb E}}^d$ by their position vectors. ${{\mathbb E}}^d$ is endowed with the standard inner product ${{\mathbf u}}\cdot {{\mathbf v}}$ and norm $|{{\mathbf u}}|=\sqrt{{{\mathbf u}}.{{\mathbf u}}}$. Suppose that $d<f$, then ${{\mathbb E}}^f \cong {{\mathbb E}}^d \times {{\mathbb E}}^{f-d}$ and we have the standard projections $\pi_1: {{\mathbb E}}^f \rightarrow {{\mathbb E}}^d$ and $\pi_2: {{\mathbb E}}^f \rightarrow {{\mathbb E}}^{f-d}$ given by $$\pi_1(u_1,\ldots,u_f)=(u_1, \ldots,u_d), \qquad \pi_2(u_1,\ldots,u_f)=(u_{d+1}, \ldots,u_{f}),$$ and the standard inclusion $\iota: {{\mathbb E}}^d \rightarrow {{\mathbb E}}^f$ given by $$\iota({{\mathbf u}})=\iota(u_1,\ldots,u_d)=(u_1,\ldots,u_d,0\ldots,0).$$ Note that $\pi_1 \circ \iota=id$ on ${{\mathbb E}}^d$, and for ${{\mathbf u}}, {{\mathbf v}}\in {{\mathbb E}}^f$, $$\begin{aligned} {{\mathbf u}}&=& (\pi_1({{\mathbf u}}), \pi_2({{\mathbf u}})), \\ {{\mathbf u}}\cdot{{\mathbf v}}&=& \pi_1({{\mathbf u}})\cdot \pi_1({{\mathbf v}})+\pi_2({{\mathbf u}})\cdot \pi_2({{\mathbf v}}), \label{eqn:dotprod}\\ | {{\mathbf u}}|^2 &=& | \pi_1({{\mathbf u}})|^2+| \pi_2({{\mathbf u}}) |^2. \label{eqn:norms}\end{aligned}$$ Let ${{\mathbf p}}=({{\mathbf p}}_{1},\cdots,{{\mathbf p}}_{N})$ and ${{\mathbf q}}=({{\mathbf q}}_{1},\cdots,{{\mathbf q}}_{N})$ be two configurations of $N$ ordered points in ${{\mathbb E}}^{d}$, where ${{\mathbf p}}_{i},{{\mathbf q}}_{i}\in{{\mathbb E}}^{d}$ for $i=1,\ldots,N$, and suppose $f > d$. (Expansions in ${{\mathbb E}}^d$) ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ if $$| {{\mathbf p}}_i-{{\mathbf p}}_j|\le| {{\mathbf q}}_i-{{\mathbf q}}_j|, \qquad 1 \le i<j \le N.$$ (Continuous expansions in ${{\mathbb E}}^f$) We say that there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ if there exists a family of continuous functions (*continuous motions*) $${{\mathbf{f}}}_i:[0,1]\longrightarrow\mathbb{E}^{f}, \quad i=1, \ldots N$$ such that for $1 \le i <j \le N$ and $0 \le t_1<t_2 \le 1$, 1. ${{\mathbf{f}}}_i(0)=\iota({{\mathbf p}}_i)$, ${{\mathbf{f}}}_i(1)=\iota({{\mathbf q}}_i)$; 2. $| {{\mathbf{f}}}_{i}(t_1)-{{\mathbf{f}}}_{j}(t_1)| \le | {{\mathbf{f}}}_{i}(t_2)-{{\mathbf{f}}}_{j}(t_2)|$. Note that if there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$, then ${{\mathbf q}}$ is necessarily an expansion of ${{\mathbf p}}$ in ${{\mathbb E}}^d$, but an expansion may not admit a continuous expansion in the same or a higher dimension. The following result by R. Alexander [@Alex] shows that any expansion admits a continuous expansion in twice the dimension. \[thm:continuous\][@Alex], see also [@BezCon]. Suppose that ${{\mathbf q}}=({{\mathbf q}}_1, \ldots {{\mathbf q}}_N)$ is an expansion of ${{\mathbf p}}=({{\mathbf p}}_1, \ldots {{\mathbf p}}_N)$ in ${{\mathbb E}}^d$. Then the family of functions ${{\mathbf{f}}}_i:[0,1] \longrightarrow {{\mathbb E}}^{2d}, \quad i=1, \ldots, N$, given by $${{\mathbf{f}}}_{i}(t)=\left(\frac{{{\mathbf p}}_{i}+{{\mathbf q}}_{i}}{2}+(cos\pi t)\frac{{{\mathbf p}}_{i}-{{\mathbf q}}_{i}}{2}, \,(sin\pi t)\frac{{{\mathbf p}}_{i}-{{\mathbf q}}_{i}}{2}\right)$$ is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d}$. We reproduce the proof here for completeness. Clearly, ${{\mathbf{f}}}_i$ is continuous for $i=1, \ldots, N$ and ${{\mathbf{f}}}_i(0)=\iota({{\mathbf p}}_i)$, ${{\mathbf{f}}}_i(1)=\iota({{\mathbf q}}_i)$. Expanding, $4| {{\mathbf{f}}}_{i}(t)-{{\mathbf{f}}}_{j}(t)|^{2}$ $$=|({{\mathbf p}}_{i}-{{\mathbf p}}_{j})-({{\mathbf q}}_{i}-{{\mathbf q}}_{j})|^{2}+|({{\mathbf p}}_{i}-{{\mathbf p}}_{j})+({{\mathbf q}}_{i}-{{\mathbf q}}_{j})|^{2}+2(\cos\pi t)(| {{\mathbf p}}_{i}-{{\mathbf p}}_{j}|^{2}-| {{\mathbf q}}_{i}-{{\mathbf q}}_{j}|^{2}).$$ Since ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$, $| {{\mathbf p}}_{i}-{{\mathbf p}}_{j}|^{2}-| {{\mathbf q}}_{i}-{{\mathbf q}}_{j}|^{2}\le0$ for all $ i \neq j$. Therefore $| {{\mathbf{f}}}_{i}(t)-{{\mathbf{f}}}_{j}(t)|$ is non-decreasing on $[0,1]$. Bezdek and Connelly used the above in [@BezCon], together with results of Csikós [@Csi] to prove the Kneser-Poulsen conjecture [@Kne] for the plane. More specifically, they showed that if there is a piecewise analytic expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in dimension $d+2$, then the Kneser-Poulsen conjecture holds for balls centered at ${{\mathbf p}}$ and ${{\mathbf q}}$, that is, the volume of the union of the balls $B({{\mathbf p}}_i,r_i)$ is less than or equal to the volume of the union of the balls $B({{\mathbf q}}_i,r_i)$, where $r_i>0$. Similarly, the same method shows that the conjecture holds if the number of balls $N \le d+3$, generalizing a result of Gromov in [@Gro]. This raises the question, as pointed out in [@BezCon], of whether it is possible to find continuous expansions in dimensions less than $2d$ for all expansions ${{\mathbf q}}$ of ${{\mathbf p}}$ in dimension $d$. If so, then the approach of Bezdek and Connelly can be applied to prove the Kneser-Poulsen conjecture in more general settings. Our main result is a negative answer to this question, specifically, we have: \[thm:main\](Main Theorem) There exists configurations ${{\mathbf p}}=({{\mathbf p}}_1, \ldots {{\mathbf p}}_N)$ in ${{\mathbb E}}^d$ with expansions ${{\mathbf q}}=({{\mathbf q}}_1, \ldots {{\mathbf q}}_N)$, where $N=(d+1)^2$, which do not admit continuous expansions in dimensions less than $2d$. [*Remark:*]{} The example we construct is in fact the same as that constructed independently by Belk and Connelly in [@BelCon], and in both cases, based on the example constructed in [@BezCon] for the planar case. However, our proof is more elementary and uses only basic linear algebra and some simple rigidity results. Indeed, our proof shows that away from the endpoints, any continuous expansion cannot be embedded into dimension less than $2d$ at any time $t \in (0,1)$. The configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ are built from the $(d+1)$ vertices ${{\mathbf v}}_0, \ldots, {{\mathbf v}}_d$ of the regular $d$-simplex $\sigma_d \subset {{\mathbb E}}^d$, together with the vertices of the inward and outward flaps associated to the faces of $\sigma_d$, specifically, each face $F^i$ ($i=0. \ldots,d$) of $\sigma_d$ may be pushed orthogonally towards or away from the center of $\sigma_d$ by a distance $s>0$, to obtain flaps $F^i_{inw}$ and $F^i_{out}$ respectively (note that Belk and Connelly had a slightly different definition for flaps in [@BelCon]). The configuration ${{\mathbf p}}$ consists of the vertices of $\sigma_d$ and of the inward flaps $F^i_{inw}$ and the configuration ${{\mathbf q}}$ consists of the corresponding vertices of $\sigma_d$ and of the outward flaps $F^i_{out}$. Note that each flap has $d$ vertices so that ${{\mathbf p}}$ and ${{\mathbf q}}$ consists of $(d+1)^2$ points. The rest of the paper will be devoted to explaining this construction (§\[s:simplex\]), showing that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ (§\[s:expansion\]), and proving that there is no continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ for $f <2d$ (§\[s:proof\]). [*Acknowledgements.*]{} This work arose from an undergraduate honors project of the third author under the supervision of the first and second authors. The authors are grateful to Jean-Marc Schlenker for helpful conversations, and also for bringing their attention to [@BelCon] arising from his correspondence with R. Connelly. Regular simplices with flaps {#s:simplex} ============================ Let $\sigma :=\sigma_d \subset {{\mathbb E}}^d$ be the regular simplex with vertices ${{\mathbf u}}_i$, $i=0, \ldots, d$ and center at the origin $O$ such that $| {{\mathbf u}}_i| =1$ for all $i$ (see figure 1). Then $$\label{eqn:innerproductofnorms} {{\mathbf u}}_i \cdot {{\mathbf u}}_j=-\frac{1}{d}, \quad i \neq j$$ {width="70.00000%"} see for example Coxeter [@Cox], or Parks and Wills [@ParWil] for an elementary proof. Denote by $F^i$ the face of $\sigma$ which does not contain the vertex ${{\mathbf u}}_i$. Then the norm of $F^i$, the outward facing unit normal ${{\mathbf n}}_i$ to $F^i$ is the vector $-{{\mathbf u}}_i$. Fix $s>0$. For each face $F^i$, $i=0, \ldots, d$, define the outward $i$th flap of depth $s$ to be $F^i$ translated by $s{{\mathbf n}}_i=-{{\mathbf u}}_i$, that is, $$F^i_{out}:=F^i -s{{\mathbf u}}_i.$$ Similarly, the inward $i$th flap of depth $s$ is given by $$F^i_{inw}:=F^i +s{{\mathbf u}}_i.$$ Each flap has $d$ vertices and if we denote the vertices of $F^i_{out}$ by ${{\mathbf{c}}}^i_j$ and those of $F^i_{inw}$ by ${{\mathbf{b}}}^i_j$, where $j\neq i$ (see figure 2 for the case when $d=2$ and $3$), then we have, for $i,j\in\{0, \ldots, d\}$, $i \neq j$, $$\begin{aligned} {{\mathbf{c}}}^i_j &=& {{\mathbf u}}_j+s{{\mathbf n}}_i = {{\mathbf u}}_j-s{{\mathbf u}}_i \\ {{\mathbf{b}}}^i_j &=& {{\mathbf u}}_j-s{{\mathbf n}}_i = {{\mathbf u}}_j+s{{\mathbf u}}_i\end{aligned}$$ {width="80.00000%"} The configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ we are interested in consists of the vertices of the regular simplex with inward and outward flaps respectively, defined by $$\label{eqn:pandq} {{\mathbf p}}=\{{{\mathbf u}}_i\} \cup \{{{\mathbf{b}}}^i_j\}, \quad {{\mathbf q}}=\{{{\mathbf u}}_i\} \cup \{{{\mathbf{c}}}^i_j\}, \quad i,j \in \{0,1,\ldots,d\},\quad i \neq j$$ where ${{\mathbf p}}$ and ${{\mathbf q}}$ are ordered so that the correspondence between the elements from the indexing is preserved. We have: \[thm:main2\] Suppose that ${{\mathbf p}}$ and ${{\mathbf q}}$ are configurations in ${{\mathbb E}}^d$ consisting of the vertices of the regular simplex with inward and outward flaps defined as in (\[eqn:pandq\]). Then 1. ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ in ${{\mathbb E}}^d$; 2. there does not exist a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^f$ for $f <2d$. We will prove (a) in the next section and (b) in the following section. We note that although (a) was claimed in [@BelCon], no proof was given, we give a proof here for completeness. Also our proof of (b) is independent of, and more elementary than that given in [@BelCon]. Proof that ${{\mathbf q}}$ is an expansion of ${{\mathbf p}}$ {#s:expansion} ============================================================= We only need to consider the distances between vertices on $\sigma$ and vertices on the flaps, or between vertices on the flaps. In the first case, we have $$| {{\mathbf u}}_k- {{\mathbf{b}}}^i_j| =| {{\mathbf u}}_k- {{\mathbf{c}}}^i_j|, \quad \hbox{if}\quad k \neq i,$$ since ${{\mathbf u}}_k \subset F^i$, and $$| {{\mathbf u}}_i- {{\mathbf{b}}}^i_j| < | {{\mathbf u}}_i- {{\mathbf{c}}}^i_j|$$ since by reflecting on the face $F^i$, we see there is a broken path from ${{\mathbf u}}_i$ to ${{\mathbf{b}}}^i_j$ of length $| {{\mathbf u}}_k- {{\mathbf{c}}}^i_j|$. The argument works if we replace $\sigma$ by any simplex. In the second case, we have, for $i \neq j$, $k \neq l$, $$\begin{aligned} | {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 &=& | ({{\mathbf u}}_j+s{{\mathbf u}}_i)-({{\mathbf u}}_l+s{{\mathbf u}}_k) |^2 \\ ~ &=& | {{\mathbf u}}_j-{{\mathbf u}}_l |^2+2s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)+s^2| {{\mathbf u}}_i-{{\mathbf u}}_k |^2 \\ | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2 &=& | {{\mathbf u}}_j-{{\mathbf u}}_l |^2-2s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)+s^2| {{\mathbf u}}_i-{{\mathbf u}}_k |^2 \\ \Longrightarrow \quad | {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 &-& | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2= 4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)\end{aligned}$$ If $i=k$, or $j=l$, or $i,j,k,l$ are all distinct, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)=0$ by (\[eqn:innerproductofnorms\]) so that $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|= | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|.$$ If $i=l$ or $j=k$, then $$4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf u}}_i-{{\mathbf u}}_k)=4s(\frac{1}{d}-1)<0$$ by (\[eqn:innerproductofnorms\]), hence in all cases, $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|\leq | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|.$$ [*Remark:*]{} In the case where we start with any simplex instead of $\sigma_d$, then $$| {{\mathbf{b}}}_j^i-{{\mathbf{b}}}_l^k|^2 - | {{\mathbf{c}}}_j^i-{{\mathbf{c}}}_l^k|^2= 4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i).$$ Again, if $i=k$, or $j=l$, or $i,j,k,l$ are all distinct, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i)=0$, and if $i=l$ or $j=k$, then $4s({{\mathbf u}}_j-{{\mathbf u}}_l)\cdot({{\mathbf n}}_k-{{\mathbf n}}_i)<0$, so Theorem \[thm:main2\](a) holds if we replace the regular simplex with any simplex. Proof that there is no continuous expansion in dimension $<2d$ {#s:proof} =============================================================== The main tools we use are some basic linear algebra as described in §\[s:intro\], and the fact that the configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ contain several sub-configurations which are rigid under continuous expansion since the pair-wise distances are preserved in the sub-configurations. We first outline the strategy of our proof, note that it suffices to show that there is no continuous expansion in dimension $2d-1$. 1. We will assume for a contradiction that there exists a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d-1}$; 2. we construct for each face $F^k$ a displacement vector function $${{\mathbf{d}}}_k:[0,1] \longrightarrow {{\mathbb E}}^{2d-1} \cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1};$$ such that ${{\mathbf{d}}}_k(t)$ is orthogonal to $F^k$ and $| {{\mathbf{d}}}_k(t)| =s$ for all $t \in [0,1]$; 3. show that there is some $t_0 \in [0,1]$ such that the projection $\pi_2({{\mathbf{d}}}_k(t_0))$ to ${{\mathbb E}}^{d-1}$ is non-zero for all $k\in \{0,1, \ldots,d\}$; 4. show that the set $\{{{\mathbf w}}_k=\pi_2({{\mathbf{d}}}_k(t_0))\}\subset {{\mathbb E}}^{d-1}$ consists of pairwise obtuse vectors; 5. show that this is not possible to give the required contradiction. \(I) Consider ${{\mathbb E}}^{2d-1}\cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1}$ and define the projections $\pi_1:{{\mathbb E}}^{2d-1} \rightarrow {{\mathbb E}}^d$ and $\pi_2:{{\mathbb E}}^{2d-1} \rightarrow {{\mathbb E}}^{d-1}$ and the inclusion $\iota: {{\mathbb E}}^d \rightarrow {{\mathbb E}}^{2d-1}$ as in §\[s:intro\]. Suppose that there is a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ in ${{\mathbb E}}^{2d-1} \cong {{\mathbb E}}^d \times {{\mathbb E}}^{d-1}$. Let ${{\mathbf{f}}}_k,~~ {{\mathbf{g}}}_j^i:[0,1]\rightarrow {{\mathbb E}}^{2d-1}$, $i,j,k \in \{0, \ldots, d\}$, $i \neq j$, be the continuous motions of ${{\mathbf u}}_k$ and ${{\mathbf{b}}}_j^i$ respectively which define the continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$. Since $\sigma_d$ is rigid, we may assume without loss of generality that ${{\mathbf u}}_k$ remains stationary throughout the motion, that is $$\label{eqn:ffkt} {{\mathbf{f}}}_k(t) \equiv \iota({{\mathbf u}}_k), \quad k=0, \ldots, d.$$ We also have $$\label{eqn:ggji} {{\mathbf{g}}}_j^i(0)=\iota({{\mathbf{b}}}_j^i)=\iota({{\mathbf u}}_j+s{{\mathbf u}}_i), ~~ {{\mathbf{g}}}_j^i(1)=\iota({{\mathbf{c}}}_j^i)=\iota({{\mathbf u}}_j-s{{\mathbf u}}_i).$$ (II) We will need the following: \[prop:parallelogram\] Suppose that $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4), ({{\mathbf v}}_1,{{\mathbf v}}_2,{{\mathbf v}}_3,{{\mathbf v}}_4) \subset {{\mathbb E}}^n$ are configurations such that $$\label{eqn:parallelogram} |{{\mathbf u}}_i-{{\mathbf u}}_j|=|{{\mathbf v}}_i-{{\mathbf v}}_j| \quad \hbox{ for all} \quad i \neq j.$$ If $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$ is a parallelogram, then $({{\mathbf v}}_1,{{\mathbf v}}_2,{{\mathbf v}}_3,{{\mathbf v}}_4)$ is also a parallelogram and $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)\cong ({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$. Let ${{\mathbf w}}$ and ${{\mathbf w}}'$ be the midpoints of $({{\mathbf u}}_2,{{\mathbf u}}_4)$ and $({{\mathbf v}}_2,{{\mathbf v}}_4)$ respectively. We have $\triangle({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_4)\cong \triangle({{\mathbf v}}_1,{{\mathbf v}}_2, {{\mathbf v}}_4)$, hence $|{{\mathbf w}}-{{\mathbf u}}_1|=|{{\mathbf w}}'-{{\mathbf v}}_1|$ (see figure 3). Similarly, $\triangle({{\mathbf u}}_2,{{\mathbf u}}_3, {{\mathbf u}}_4)\cong \triangle({{\mathbf v}}_2,{{\mathbf v}}_3, {{\mathbf v}}_4)$, so $|{{\mathbf u}}_3-{{\mathbf w}}|=|{{\mathbf v}}_3-{{\mathbf w}}'|$. Also, by (\[eqn:parallelogram\]) $|{{\mathbf v}}_3-{{\mathbf v}}_1|=|{{\mathbf u}}_3-{{\mathbf u}}_1|$ and since $({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$ is a parallelogram, $|{{\mathbf u}}_3-{{\mathbf u}}_1|=|{{\mathbf u}}_3-{{\mathbf w}}|+|{{\mathbf w}}-{{\mathbf u}}_1|$. Hence $$|{{\mathbf v}}_3-{{\mathbf v}}_1|=|{{\mathbf u}}_3-{{\mathbf u}}_1|=|{{\mathbf u}}_3-{{\mathbf w}}|+|{{\mathbf w}}-{{\mathbf u}}_1|=|{{\mathbf v}}_3-{{\mathbf w}}'|+|{{\mathbf w}}'-{{\mathbf v}}_1|.$$ ![The configurations (${{\mathbf u}}_1$,${{\mathbf u}}_2$,${{\mathbf u}}_3$,${{\mathbf u}}_4$) and (${{\mathbf v}}_1$,${{\mathbf v}}_2$,${{\mathbf v}}_3$,${{\mathbf v}}_4$)[]{data-label="fig:figure_3"}](./figure_3.png){width="80.00000%"} Hence, ${{\mathbf v}}_1, {{\mathbf w}}'$ and ${{\mathbf v}}_3$ are collinear, and $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)$ lies on a plane with the diagonal from ${{\mathbf v}}_1$ to ${{\mathbf v}}_3$ bisecting the diagonal from ${{\mathbf v}}_2$ to ${{\mathbf v}}_4$. A similar argument shows that the diagonal from ${{\mathbf v}}_2$ to ${{\mathbf v}}_4$ bisects the diagonal from ${{\mathbf v}}_1$ to ${{\mathbf v}}_3$, so that $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)$ is a parallelogram. Now (\[eqn:parallelogram\]) implies $({{\mathbf v}}_1, {{\mathbf v}}_2, {{\mathbf v}}_3, {{\mathbf v}}_4)\cong ({{\mathbf u}}_1,{{\mathbf u}}_2, {{\mathbf u}}_3,{{\mathbf u}}_4)$. Now, for distinct $i,j,k \in \{0, \ldots, d\}$, consider the continuous family of configurations $({{\mathbf{f}}}_i(t), {{\mathbf{f}}}_j(t), {{\mathbf{g}}}_j^k(t), {{\mathbf{g}}}_i^k(t))$, $t \in [0,1]$. By assumption, this is a continuous expansion, but the pairwise distances between points in the initial configuration $$({{\mathbf{f}}}_i(0), {{\mathbf{f}}}_j(0), {{\mathbf{g}}}_j^k(0), {{\mathbf{g}}}_i^k(0))=((\iota({{\mathbf u}}_i), \iota({{\mathbf u}}_j), \iota({{\mathbf u}}_j+s{{\mathbf u}}_k), \iota({{\mathbf u}}_i+s{{\mathbf u}}_k))$$ and those of the final configuration $$({{\mathbf{f}}}_i(1), {{\mathbf{f}}}_j(1), {{\mathbf{g}}}_j^k(1), {{\mathbf{g}}}_i^k(1))=((\iota({{\mathbf u}}_i), \iota({{\mathbf u}}_j), \iota({{\mathbf u}}_j-s{{\mathbf u}}_k), \iota({{\mathbf u}}_i-s{{\mathbf u}}_k))$$ are equal since they form congruent rectangles. Since the initial configuration describes a rectangle, it follows from proposition \[prop:parallelogram\] that all intermediate configurations are congruent rectangles. Hence, $${{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j)={{\mathbf{g}}}_i^k(t)-\iota({{\mathbf u}}_i), \quad \forall~~ i \neq j \neq k \neq i.$$ We can define ${{\mathbf{d}}}_k(t): [0,1] \rightarrow {{\mathbb E}}^{2d-1}$ by $${{\mathbf{d}}}_k(t):={{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j), \quad \hbox{for any} \quad j \neq k,$$ then $$|{{\mathbf{d}}}_k(t)|=|{{\mathbf{g}}}_j^k(t)-\iota({{\mathbf u}}_j)|=|{{\mathbf{g}}}_j^k(0)-\iota({{\mathbf u}}_j)|=s$$ and ${{\mathbf{d}}}_k(t) \cdot (\iota({{\mathbf u}}_j-{{\mathbf u}}_i))=0$ for all $i \neq j \neq k \neq i$, hence, ${{\mathbf{d}}}_k(t)$ is orthogonal to $\iota(F^k)$, since $\{(\iota({{\mathbf u}}_j-{{\mathbf u}}_i))\}$, $i \neq j \neq k \neq i$ spans $\iota(F^k)$. \(III) For $k=0, \ldots, d$, let $$\pi_1({{\mathbf{d}}}_k(t)):={{\mathbf v}}_k(t)\in {{\mathbb E}}^d, \quad \pi_2({{\mathbf{d}}}_k(t)):={{\mathbf w}}_k(t) \in {{\mathbb E}}^{d-1}$$ so that ${{\mathbf{d}}}_k(t)=({{\mathbf v}}_k(t), {{\mathbf w}}_k(t))$. Since ${{\mathbf{d}}}_k(t).\iota({{\mathbf u}}_i-{{\mathbf u}}_j)={{\mathbf v}}_k(t).({{\mathbf u}}_i-{{\mathbf u}}_j)=0$ for all $i \neq j \neq k \neq i$, ${{\mathbf v}}_k(t)$ is orthogonal to $F^k \subset {{\mathbb E}}^d$, so ${{\mathbf v}}_k(t)=a_k(t){{\mathbf u}}_k$, $a_k(t) \in {{\mathbb R}}$, and furthermore, $|a_k(t)| \le s$ since $|{{\mathbf v}}_k(t)|^2+|{{\mathbf w}}_k(t)|^2=|{{\mathbf{d}}}_k(t)|^2=s^2$ by (\[eqn:norms\]). By the intermediate value theorem, since $a_k(0)=s$ and $a_k(1)=-s$, $a_k(t)$ takes all values in $[-s,s]$, so in particular, there exists some $t_0 \in [0,1]$ such that $a_k(t_0)=0$ , so that ${{\mathbf v}}_k(t_0)={\mathbf 0}$. Hence $|{{\mathbf w}}_k(t_0)|^2=s^2$, in particular, ${{\mathbf w}}_k(t_0) \neq {\mathbf 0}$ (in fact, we only need that $|a_k(t_0)|<s$ to get ${{\mathbf w}}_k(t_0) \neq {\mathbf 0}$). Now for $i \neq j \neq k \neq i$, we have $\triangle(\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(0),{{\mathbf{g}}}_i^k(0)) \cong \triangle(\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(1),{{\mathbf{g}}}_i^k(1))$ since $${{\mathbf{g}}}_i^j(0)-\iota({{\mathbf u}}_i)=\iota(s{{\mathbf u}}_j), \quad {{\mathbf{g}}}_i^k(0)-\iota({{\mathbf u}}_i)=\iota(s{{\mathbf u}}_k),$$ $${{\mathbf{g}}}_i^j(1)-\iota({{\mathbf u}}_i)=\iota(-s{{\mathbf u}}_j), \quad {{\mathbf{g}}}_i^k(1)-\iota({{\mathbf u}}_i)=\iota(-s{{\mathbf u}}_k),$$ so all the triangles $ \triangle (\iota({{\mathbf u}}_i),{{\mathbf{g}}}_i^j(t),{{\mathbf{g}}}_i^k(t))$, $t \in [0,1]$ are congruent. In particular, $$\label{eqn:dkdotdj} ({{\mathbf{g}}}_i^k(t)-\iota({{\mathbf u}}_i))\cdot ({{\mathbf{g}}}_i^j(t)-\iota({{\mathbf u}}_i))={{\mathbf{d}}}_k(t) \cdot {{\mathbf{d}}}_j(t)={{\mathbf{d}}}_k(0) \cdot {{\mathbf{d}}}_j(0)=s{{\mathbf u}}_k \cdot s{{\mathbf u}}_j=-\frac{s^2}{d}$$ for all $t \in [0,1]$ by (\[eqn:innerproductofnorms\]). Now using ${{\mathbf v}}_k(t_0)={\mathbf 0}$ and applying (\[eqn:dotprod\]) to (\[eqn:dkdotdj\]) gives, $$\label{eqn:wkdotwj} -\frac{s^2}{d}={{\mathbf{d}}}_k(t_0)\cdot {{\mathbf{d}}}_j(t_0)={{\mathbf v}}_k(t_0) \cdot {{\mathbf v}}_j(t_0)+{{\mathbf w}}_k(t_0)\cdot {{\mathbf w}}_j(t_0)={{\mathbf w}}_k(t_0)\cdot {{\mathbf w}}_j(t_0)$$ for all $j \neq k$. In particular, we see that ${{\mathbf w}}_j(t_0) \neq {\mathbf 0}$ for all $j=0, \ldots, d$ (again, we really only need that $|a_k(t_0)|<s$ to obtain this conclusion). \(IV) We need to show that ${{\mathbf w}}_i(t_0)\cdot {{\mathbf w}}_j(t_0)<0$ for all distinct $i, j \in \{0, \ldots, d\}$. Recall that ${{\mathbf{d}}}_i(t)=({{\mathbf v}}_i(t), {{\mathbf w}}_i(t))=(a_i(t){{\mathbf u}}_i, {{\mathbf w}}_i(t))$. Since ${{\mathbf w}}_i(t_0) \neq {\mathbf 0}$ and by (\[eqn:norms\]) $$s^2=|{{\mathbf{d}}}_i(t_0)|^2=|{{\mathbf v}}_i(t_0)|^2+|{{\mathbf w}}_i(t_0)|^2=|a_i(t_0)|^2+|{{\mathbf w}}_i(t_0)|^2$$ we have $$\label{eqn:aitlessthans} -s< a_i(t_0)<s, \quad \hbox{for all} \quad i=0, \ldots,d.$$ Now by (\[eqn:innerproductofnorms\]), for $i \neq j$, $${{\mathbf{d}}}_i(t_0) \cdot {{\mathbf{d}}}_j(t_0)={{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0) +{{\mathbf w}}_i(t_0) \cdot {{\mathbf w}}_j(t_0).$$ ${{\mathbf{d}}}_i(t_0) \cdot {{\mathbf{d}}}_j(t_0)= {{\mathbf{d}}}_i(0)\cdot {{\mathbf{d}}}_j(0)=-\frac{s^2}{d}$ and $${{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0)=a_i(t_0)a_j(t_0){{\mathbf u}}_i\cdot {{\mathbf u}}_j=-\frac{a_i(t_0)a_j(t_0)}{d},$$ where by (\[eqn:aitlessthans\]), $|{{\mathbf v}}_i(t_0) \cdot {{\mathbf v}}_j(t_0)|< \frac{s^2}{d}$. It follows that ${{\mathbf w}}_i(t_0)\cdot {{\mathbf w}}_j(t_0)<0$ for all distinct $i, j \in \{0,\ldots, d\}$. [*Remark:*]{} In proving the conclusion in (IV) holds, we only really require that the outward normals ${{\mathbf n}}_i$, $i=0, \ldots, d$ of $\sigma_d$ are pairwise obtuse, that is, ${{\mathbf n}}_i \cdot {{\mathbf n}}_j<0$ for all distinct $i,j \in \{0, \ldots, d\}$. Hence we may replace the regular simplex with one for which the above holds. \(V) Recall that ${{\mathbf u}}_1, {{\mathbf u}}_2 \in {{\mathbb E}}^n$ are obtuse if ${{\mathbf u}}_1 \cdot {{\mathbf u}}_2 <0$. The lemma below states that we cannot have a collection of $n+2$ pairwise obtuse vectors in ${{\mathbb E}}^n$. \[lem:obtuse\] For any set $\{{{\mathbf u}}_1, \ldots, {{\mathbf u}}_{n+2}\}$ of $n+2$ vectors in ${{\mathbb E}}^n$, ${{\mathbf u}}_i\cdot {{\mathbf u}}_j \ge 0$ for some $i \neq j$, that is, the vectors cannot be all pairwise obtuse. We prove by induction on the dimension $n$. The result is clearly true when $n=1$ since for any 3 vectors ${{\mathbf u}}_1, {{\mathbf u}}_2, {{\mathbf u}}_3 \in {{\mathbb E}}^1$, either at least one of the vectors is $\mathbf 0$, or two are in the same direction so have positive dot product. Assume the lemma is true for $n$ and suppose for a contradiction that there exists ${{\mathbf u}}_1, \ldots, {{\mathbf u}}_{n+3} \in {{\mathbb E}}^{n+1}$ that are all pairwise obtuse. Without loss of generality, we may assume that none of ${{\mathbf u}}_i$ are zero, and that ${{\mathbf u}}_{n+3}=(-1,0, \ldots, 0)$. Write ${{\mathbb E}}^{n+1}\cong {{\mathbb E}}^1 \times {{\mathbb E}}^{n}$ and consider the projections $\pi_1:{{\mathbb E}}^{n+1} \rightarrow {{\mathbb E}}^1$ and $\pi_2:{{\mathbb E}}^{n+1} \rightarrow {{\mathbb E}}^n$ respectively as in §\[s:intro\]. For $i=1, \ldots, n+2$, let ${{\mathbf v}}_i:=\pi_1({{\mathbf u}}_i) \in {{\mathbb E}}^1 \cong {{\mathbb R}}$, ${{\mathbf w}}_i:=\pi_2({{\mathbf u}}_i) \in {{\mathbb E}}^n$, see figure 4. Note that ${{\mathbf v}}_i >0$ since ${{\mathbf u}}_i \cdot {{\mathbf u}}_{n+3}<0$, so ${{\mathbf v}}_i \cdot {{\mathbf v}}_j>0$ for $i,j \in \{1, \ldots, n+2\}$. Then we have, from (\[eqn:dotprod\]), for distinct $i, j \in \{1, \ldots, n+2\}$, $${{\mathbf u}}_i \cdot {{\mathbf u}}_j={{\mathbf v}}_i\cdot {{\mathbf v}}_j +{{\mathbf w}}_i \cdot {{\mathbf w}}_j.$$ ![Projection of ${{\mathbb E}}^{n+1}$ vectors into ${{\mathbb E}}^{n}$ space []{data-label="fig:figure_4"}](./figure_4.png){width="100.00000%"} By assumption, ${{\mathbf u}}_i \cdot {{\mathbf u}}_j <0$, and ${{\mathbf v}}_i\cdot {{\mathbf v}}_j >0$ from the above, so $${{\mathbf w}}_i\cdot {{\mathbf w}}_j <0.$$ Hence $\{{{\mathbf w}}_1, \ldots, {{\mathbf w}}_{n+2}\}$ is a collection of pairwise obtuse vectors in ${{\mathbb E}}^{n}$ contradicting the induction hypothesis. Applying lemma \[lem:obtuse\] to the set $\{{{\mathbf w}}_0, {{\mathbf w}}_1, \ldots, {{\mathbf w}}_{d}\} \subset {{\mathbb E}}^{d-1}$ in (IV) we get the required contradiction which concludes the proof of Theorem \[thm:main2\] from which Theorem \[thm:main\] follows. [**Concluding remarks.**]{} The method of proof above works if we construct ${{\mathbf p}}$ and ${{\mathbf q}}$ from any simplex in ${{\mathbb E}}^d$ whose pairwise norms are obtuse. It also shows that any intermediate configuration in a continuous expansion from ${{\mathbf p}}$ to ${{\mathbf q}}$ cannot be embedded in a space of dimension less than $2d$. An interesting open question is, for each $d$, what is the smallest number of points in the configurations ${{\mathbf p}}$ and ${{\mathbf q}}$ for which there is no continuous expansion in ${{\mathbb E}}^{2d-1}$. We have shown that $N=(d+1)^2$ suffices, but this may not be optimal. Finally, it is also interesting to ask if we can find configurations ${{\mathbf p}}$, and expansions ${{\mathbf q}}$ of ${{\mathbf p}}$ such that the continuous expansion given by Theorem \[thm:continuous\] is essentially, up to some trivial motions, the only continuous expansion in dimension $2d$. [1]{} R. Alexander, [*Lipschitzian mappings and total mean curvature of polyhedral surfaces. I*]{}. Trans. Amer. Math. Soc. 288 (1985), no. 2, 661–678. M. Belk, R. Connelly, [*Making contractions continuous: a problem related to the Kneser-Poulsen conjecture*]{}. Preprint (2007). K. Bezdek, R. Connelly, [*Pushing disks apartthe Kneser-Poulsen conjecture in the plane.*]{} J. Reine Angew. Math. 553 (2002), 221236. H.S.M. Coxeter, [*Regular polytopes*]{}. Second edition The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963 xx+321 pp. B. Csikós, [*On the volume of the union of balls*]{}. Discrete Comput. Geom. 20 (1998), no. 4, 449–461. M. Gromov, [*Monotonicity of the volume of intersection of balls*]{}. Geometrical aspects of functional analysis (1985/86), 1–4, Lecture Notes in Math., 1267, Springer, Berlin, 1987. M. Kneser, [*Einige Bemerkungen über das Minkowskische Flächenmass*]{}. Arch. Math. (Basel) 6 (1955), 382–390. H. R.Parks, D.C. Wills, [*An elementary calculation of the dihedral angle of the regular $n$-simplex*]{}. Amer. Math. Monthly 109 (2002), no. 8, 756–758. [^1]: The second author is partially supported by the National University of Singapore academic research grant R-146-000-133-112

ArXiv

--- abstract: 'We performed a radio recombination line (RRL) survey to construct a high-mass star-forming region (HMSFR) sample in the Milky Way based on the all-sky Wide-Field Infrared Survey Explorer (*All-WISE*) point source catalog. The survey was observed with the Shanghai 65m Tianma radio telescope (TMRT) covering 10 hydrogen RRL transitions ranging from H98$\alpha$ to H113$\alpha$ (corresponding to the rest frequencies of 4.5$-$6.9 GHz) simultaneously. Out of 3348 selected targets, we identified an HMSFR sample consisting of 517 sources traced by RRLs, a large fraction of this sample (486) locate near the Galactic plane ($|$*b*$|$ $<$ 2$\degr$). In addition to the hydrogen RRLs, we also detected helium and carbon RRLs towards 49 and 23 sources respectively. We cross-match the RRL detections with the 6.7 methanol maser sources built up in previous works for the same target sample, as a result, 103 HMSFR sources were found to harbor both emissions. In this paper, we present the HMSFR catalog accompanied by the measured RRL line properties and a correlation with our methanol maser sample, which is believed to tracer massive stars at earlier stages. The construction of an HMSFR sample consisting of sources in various evolutionary stages indicated by different tracers is fundamental for future studies of high-mass star formation in such regions.' author: - 'Hong-Ying Chen' - Xi Chen - 'Jun-Zhi Wang' - 'Zhi-Qiang Shen' - Kai Yang bibliography: - 'ref.bib' title: 'A 4-6 GHz Radio Recombination Line Survey in the Milky Way[^1]' --- Introduction\[1\] ================= Formation of high-mass stars in the giant molecular clouds, though intensively studied, remains mysterious (see review papers, e.g., @ZY2007 [@tan2014]). To reveal the intrinsic of high-mass star formation (HMSF) at the very early stage, the fundamental and vital step is to construct a complete sample of high-mass star-forming regions (HMSFRs). Ultra-compact H regions (UCHRs) ($<$ 0.1 pc) are hot ionized gas surrounding an exciting central high-mass star. Such regions are excited by an early O$-$B star from which the ultra-violet photons are strong enough to ionize neutral hydrogen. H regions spread widely at a Galactic scale and have strong luminosity across multiple wavebands (ultraviolet, visible, infra-red and radio), therefore, they are ideal tracers of HMSFRs. H region surveys in the Milky Way were firstly studied in visible wavelengths [@sharpless1953; @sharpless1959; @gum1955; @rodgers1960]. However, the extinction in the optical largely limited the capability of such researches. The dust-free radio observations are therefore needed to construct a more complete sample of Galactic H regions. In 1965, radio recombination line (RRL) was firstly detected by @HM1965 from M 17 and Orion A. Its thin optical depth in centimeter wavelengths makes it an optimal tracer of H regions. RRL surveys were then performed in the next few decades, e.g. @MH1967 [@wilson1970; @reifenstein1970; @downes1980; @CH1987] and @lockman1989. The properties of the Galactic RRLs, such as their spatial distribution, line widths, LSR velocities and intensities are probes of the morphological, chemical and dynamical information of the Milky Way (see @anderson2011). Thus, RRL is important in a range of astrophysical topics, such as the Galactic structure (e.g. @HH2015 [@downes1980; @AB2009]) and metallicity gradient across the Galactic disk which helps understanding the Galactic chemical evolution (GCE) [@wink1983; @shaver1983; @quireza2006; @balser2011]. More recent RRL surveys were performed with high-sensitivity facilities (e.g. @liu2013 [@alves2015; @anderson2011; @anderson2014]). In particular, the recent Green Bank Telescope (GBT) H Region Discovery Survey (HRDS) detected 603 discrete RRL components from 448 targets which were considered to be H regions, thus doubled the number of known Galactic H regions [@anderson2011]. With the demonstration that H regions can be reliably identified by their mid-infrared (MIR) morphology, @anderson2014 extended the HRDS sample to $\sim~8000$ candidate sources based on the *all-sky Wide-Field Infrared Survey Explorer* (*WISE*) MIR images (hereafter the catalog). The catalog contains $\sim~1500$ confirmed H regions with observed RRL data in the literature, it is the most complete sample of H regions to date. The *WISE* data have four MIR bands: 3.4 $\mu$m, 4.6 $\mu$m, 12 $\mu$m and 22 $\mu$m, with angular resolutions of 6$\arcsec$.1, 6$\arcsec$.4, 6$\arcsec$.5 and 12$\arcsec$, respectively, which are sensitive to HMSFRs. Its complete sky coverage and up-to-date database provide an optimal target sample for identifying HMSFR candidates. To further extended the HMSFR sample traced by RRLs beyond the catalog, we conducted an RRL survey with the Shanghai 65m Tianma Radio Telescope (TMRT) based on the *WISE* point source catalog rather than the *WISE* MIR images. Since as H regions form and evolve they will expand, selecting targets from the point source catalog will make our sample to include more compact, and therefore younger sources. Comparing to other single-dish RRL surveys, we concentrate more on the correlation and association with methanol masers to signpost different periods of star-forming processes. Class methanol maser is a powerful tracer of the hot molecular cloud phase of HMSFR [@minier2003; @ellingsen2006; @xu2008], when there is significant mass accretion. H region generally appears in more evolved phases of star formation, well before the main sequence [@walsh1998; @BM1996]. As suggested by @churchwell2002, due to beam-blending and thick optical depth, the densest and earliest H regions are heavily obscured, more extended detectable UCH regions probably are only formed until the central star reaches main sequence, and no longer accreting significant mass. By cross-matching the RRL and class methanol maser samples, the evolutionary stages of their hosts may be specified more accurately. Therefore, simultaneous observation for both the RRLs and 6.7 GHz methanol masers were conducted to investigate their associations. Notably, due to beam dilution, RRL emissions from dense UCH regions at earlier stages will be undetectable, thus our detected RRL sources will trace more extended and evolved H regions. Previous studies have demonstrated that 6.7 GHz methanol masers can be excited in the UCH regions, including both extended and compact sources identified by radio continuum data (e.g @hu2016). Since RRL-detected UCH regions are generally more evolved than those without RRL emissions, RRL researches will be helpful for identifying which methanol maser sources are at more evolutionary stages. In this paper, we report the RRL detections with the measured line parameters, as well as the results of a correlation with the 6.7 GHz methanol maser sample towards the same target sample built by [@yang2017; @yang2019]. Section \[2\] describes the sample selection and observations. Section \[f3\] presents the results of the survey followed by a discussion in section \[4\]. We summarize our main conclusions in Section \[5\].\ Observations and Data Reduction\[2\] ==================================== Source Selection\[2.1\] ----------------------- RRLs and 6.7 GHz methanol masers were observed simultaneously in our survey. The targets were selected with the following methodology: firstly, a cross-matching was applied between the 6.7 GHz methanol maser catalog created by the Methanol Multi beam (MMB) Survey conducted with the Parkes telescope [@caswell2010; @caswell2011; @green2010; @green2012; @breen2015], and the *All-WISE* point source catalog. As a result, there are 502 MMB maser sources which have a *WISE* counterpart with a spatial offset within 7$\arcsec$. We only kept 473 sources with *WISE* data available from all four bands. A magnitude and color-color analysis was then applied to those 473 sources (see @yang2017 for details). 73$\%$ of those sources fell in the color region with well-constrained *WISE* color criteria: \[3.4\] $<$ 14 mag; \[4.6\] $<$ 12 mag; \[12\] $<$ 11 mag; \[22\] $<$ 5.5 mag; \[3.4\] - \[4.6\] $>$ 2, and \[12\] - \[22\] $>$ 2. To avoid repetition, we excluded sources locating in the MMB survey region ($20\degr < l < 186\degr$ and $|$*b*$|$ $>$ 2$\degr$). Due to the limitation of observing range, we also excluded those with a declination below $-30 \degr$. In total, 3348 *WISE* point sources were selected searching for RRLs and methanol maser emissions. In this sample, 1473 sources are located at a high Galactic latitude region with $|$*b*$|$ $>$ 2$\degr$ and 1875 sources fall within $\pm 2\degr$ of the Galactic Plane. Among the selected targets, @yang2017 [@yang2019] detected 6.7 GHz methanol masers from 241 sources, 209 of them are near the Galactic Plane where $|$*b*$|$ $<$ 1$\degr$.\ Observation & Data Reduction\[2.2\] ----------------------------------- The observations were performed between 2015 September and 2018 January with the 65m TMRT in Shanghai, China [@yang2017; @yang2019]. A cryogenically cooled C-band receiver (4-8 GHz) with two orthogonal polarizations was employed in this survey. We used an FPGA-based spectrometer Digital Backend System (DIBAS) (VEGAS; @bussa2012) to receive and record the signals. A total of 16 spectral windows were applied in the observations, each has 16384 channels and a bandwidth of 23.4 MHz supplying a velocity resolution of $\sim$ 0.1 km s$^{-1}$ at 4.5 GHz. Ten of the spectral windows were set up to cover hydrogen RRL transitions spanning from H98$\alpha$ to H113$\alpha$ as specified in Table \[t1\]. In addition to RRLs, the 6.7 GHz methanol maser (CH$_3$OH) emission line, 4.8 GHz H$_2$CO emission and absorption lines, as well as the 4.7 and 6.0 GHz excited-state OH maser transitions were also observed. The observed 6.7 GHz methanol maser results are reported in @yang2017 [@yang2019]. This paper mainly focuses on RRL detection and a cross-match between RRLs and methanol masers. In the observations, the system temperature is about 20 $\sim$  30 K and the main beam efficiency of the TMRT is $\sim~60\%$. The beam has a full width at half-maximum ([FWHM]{}) of $\sim~3\arcmin - 4\arcmin$ at the frequencies of RRLs. There is an uncertainty of $<~20\%$ in the detected flux densities for the sources estimated from the observed variation of the calibrators. [500pt]{}\[c\][&gt;p[2.5cm]{}|&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[2.7cm]{}&gt;p[1.3cm]{}]{} Window Number& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\ $\nu_o$ (MHz) & 4497.78 & 4593.09 & 4618.79 & 4758.11 & 4829.66 & 4874.16 & 5008.92 & 5148.7\ Line Name & H113$\alpha$ & H$_2^{13}$CO & H112$\alpha$ & OH & H$_2$CO & H110$\alpha$ & H109$\alpha$ & H108$\alpha$\ Window Number& 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\ $\nu_o$ (MHz) & 6016.75 & 6032.92 & 6049.08 & 6106.85 & 6289.14 & 6478.76 & 6672.30 (6676.08) & 6881.49\ Line Name & OH & OH & OH & H102$\alpha$ & H101$\alpha$ & H100$\alpha$ & CH$_3$OH $\&$ H99$\alpha$ & H98$\alpha$\ \ [cccccccc]{} 654 & 517 & 103 & 137 & 49 & 23 & 5 & 5\ A position-switching mode was applied in the observations. Each source was observed with two ON-OFF cycles, both the ON- and OFF-positions in each cycle take $\sim~2$ minutes. To ensure data reliability, sources with bad data quality (eg. radio frequency interferences (RFI), high noise level or abnormal signals) were re-observed with many more cycles. For each source, we primarily set the OFF-point to (0.0$\degr$, $-0.4\degr$) away from the ON-position in (R.A., Decl.). Sources with OFF-positions showing RRL emission were re-observed with a different OFF-position to exclude the background emissions. In total, the observations took $\sim$ 700 hours of observing time excluding calibrations. The data were processed with the [GILDAS/CLASS]{}[^2] software package [@pety2005; @gildas2013]. Adjacent Hn$\alpha$ transitions with large quantum numbers (n $> 50$) have similar line properties, such as line intensity and [FWHM]{} line width (see @balser2006). Therefore, the spectra of the observed 10 transitions through H98$\alpha$ to H113$\alpha$ can be averaged to achieve a higher S/N. After averaging over the 10 RRL transitions at two polarizations, a typical 3-$\sigma$ sensitivity of $\sim$ 7 mK per channel was achieved for the majority of our sources. Due to different rest frequencies, the 10 RRL transitions have different velocity resolutions in their individual spectrum. When averaging over the spectral windows, the [GILDAS/CLASS]{} software will resample the data with the coarsest resolution of them. Depending on the spectral quality, we typically smooth the averaged spectrum over 5 to 30 channels to have a spectral resolution of $\sim$ 0.4 to $\sim$ 2.4 km s$^{-1}$. After that, depending on the background fluctuations, the spectral baselines were subtracted by a first or multi-order ($<$ 4) polynomial fitting. Then the line profiles were fitted with a Gaussian model. For the multi-component sources, the number of Gaussian components was decided via a visual inspection. Notably, it is somehow hard to disentangle the blended line components which have very close central frequencies, so some of the blended sources may be identified as single emission with wide line width (see Section \[3.1\]). The Gaussian fitting results are reported in Section \[3\].\ Result\[3\] =========== RRL Detections\[3.1\] --------------------- A summary of RRL detections in this work and 6.7 GHz methanol maser detections from @yang2017 [@yang2019] are presented in Table \[t2\]. Out of the 3348 targets, we detected hydrogen (H) RRL emissions from 527 positions, corresponding to a detection rate of 15.7%. Excluding the potential Planetary Nebula (PNe) and Supernova Remnant (SNR) (see Sec. \[3.2\]), we built a sample of 517 HMSFRs based on the RRL detections. The derived line parameters and source information for the H RRLs from the HMSFR sample are listed in Table \[t3\]. The spectra for all the detected RRLs (including helium (He) and carbon (C) RRLs; see Section \[4.3\]) are given in Appendix A. Amongst the 517 HMSFR candidates, 488 of them reside within $|b| < 2 \degr$, only 28 are from higher Galactic latitude regions ($|b| > 2 \degr$). Combining with the 240 methanol maser sample in @yang2017 [@yang2019], excluding one associated with a potential SNR, there are 654 HMSFR sources traced by RRLs and/or methanol maser listed in Table \[t4\]. We cross-matched our detected sources with the catalog consisting of $\sim$ 8000 sources. Due to the lack of radio observations, only $\sim$ 1500 sources in their sample were confirmed to be “Known” H regions (denoted as “K" sources in Table 2 of @anderson2014). For targets which spatially associate with multiple sources in the catalog, “Known” sources would be designated preferentially. For targets associate with multiple “Known” sources or other types of sources, the closest counterpart would be designated. There are 467 HMSFR candidates in our sample which are associated with at least one of sources within a separation of 3$\arcmin$ (corresponding to the beam size of TMRT) plus the radius of the source. Amongst them, 425 were classified as “known” H regions, we thus confirmed the other 42 sources being H regions. For the sources included in the catalog, we label their type accordingly in Column (11) of Table \[t3\]. There are also 3 PNe and 4 SNR candidate sources associated with sources, due to the extended morphology of H regions, these targets may be overlapped with the sources along the line of sight (LOS). There are 133 sources showing multiple (typically two or three) H RRL emission components. Figure \[f1\]a shows an example spectrum for such sources. The multiple RRL components may arise from nearby H regions within the TMRT beam or overlapped H regions along the LOS. Diffused ionized gas leaked from nearby HMSFRs may also cause multi-components in the RRL spectra [@zavagno2007; @anderson2010; @OK1997]. Some components have line emissions with close peak velocities causing confusion with a wide line width or non-Gaussian profile. An example of this is presented in Figure \[f1\]b. Notably, RRL detections at adjacent observing positions with similar peak velocities are possibly from the same extended H region. Moreover, H regions with large angular size may have leaked RRL emission detected by the side lobe of the telescope when observing its nearby target points. Those RRL components have similar line velocities but much weaker line intensities comparing to the emission from the central position of the source. There are 5 RRL sources which were potentially detected by the side lobe, we label those emissions as possible duplicated sources by “SL" and the name of the real source in Column (11) in Table \[t3\]. Our RRL detections have peak intensities ranging from 0.01 to 2 K with an average value of 0.07 K, and an integrated intensity from 0.1 to 58 K$\cdot$km s$^{-1}$ with an average of 1.9 K$\cdot$km s$^{-1}$. Among the 517 HMSFR candidates, there are 12 weak sources, labeled with “?" in column (11) in Table \[t3\], which have a line intensity only slightly above our 3$\sigma$ detection threshold, no accurate Gaussian fitting results can be achieved for them. For those weak sources, we only give their peak intensity in Table \[t3\]. Further observations with longer integration time are required for them to get a higher S/N. Figure \[f1\]c illustrates an example of these weak sources. In addition to the 12 weak sources, for multi-component sources containing such weak line component(s), we only provide Gaussian fitting results for the stronger components. In addition to H RRL transitions, the 23.4 MHz spectral windows (see Section \[2.2\]) also simultaneously cover the rest frequencies of the nearby He and C RRLs, they have the same intrinsic quantum numbers with the H RRLs within each spectral window. In the velocity domain, He and C RRLs typically have a constant velocity offset with respect to H RRLs of $-$122 km s$^{-1}$ and $-$149 km s$^{-1}$, respectively. The derived line parameters of He and C RRLs are given in Table \[t5\]. Figure \[f2\] shows an example source with all the three atomic RRLs. In total, we found 49 He RRLs and 23 C RRLs in the observed sample (see Section \[4.3\] for further discussions).\ Other Sources\[3.2\] -------------------- In addition to HMSFRs, ionized gas associated with other astrophysical objects such as planetary Nebula (PNe) and Supernova Remnant (SNR) can also produce RRLs. We performed a matching analysis for our sample with the [SIMBAD]{}[^3] catalog to exclude previously known PNe and SNR sources with a positional criterion of $< 3 \arcmin$. We mark sources with an explicit PNe or SNR identifier as “PNe" or “SNR" in Tables \[t6\] and \[t7\], respectively. To maximize the reliability of our HMSFR sample, for those candidate PNe/SNR sources, which may be associated with both PNe/SNR and H regions or with PNe/SNR located near the edge of the detecting beam ($\sim3 \arcmin$ away), we remove them from the final HMSFR sample and denote them as “PNe?"/“SNR?" in Table \[t6\] / \[t7\]. Notably, though spatially associated with PNe or SNR, the RRLs detected from those target positions may originated from H regions along the LOS. PNe usually has an expanding shell of ionized gas ejected from red giant stars. It is the main contamination of H region samples traced by RRLs. RRLs from PNe usually have wider line width (typically 30 $\sim$  50 km s$^{-1}$) than those from H regions (typical line width of 20 $\sim$  30 km s$^{-1}$) due to their expansion (see @garay1989 [@balser1997]). A typical RRL spectrum from PNe with a line width of $\sim$ 60 km s$^{-1}$ is shown in Figure \[f3\]. The existence of unknown PNe may cause a bias in the statistical analysis of the line width distribution for H regions (see Section \[4.2.2\]). There are 2 sources (G84.913$-$3.505; G85.946$-$3.488) which are explicitly associated with the well-built PNe NGC 7027, as well as 3 possible PNe sources. The full list of PNe candidates is given in Table \[t6\]. SNRs are non-thermal radio sources, and generally have only weak RRLs with wide line width. @liu2019 suggested that the broad line width (mostly $>$ 50 km s$^{-1}$) of the RRLs toward SNRs implies high temperature or turbulent motions of the plasma. Former studies indicated that stimulated emission may be a possible origin for RRLs from SNRs (see @liu2019 and references therein). The potentially similar radio morphology of SNR and H regions is the major confusion when disentangling these two samples [@anderson2017]. There are 5 potential SNR candidates in our sample, all these sources are spatially associated with multiple sources thus no clear identifier can be designated. We list the potential SNR sources in Table \[t7\]. Figure \[f4\] shows an example of RRL spectra from SNR. There is one potential SNR source (G28.532+0.129) which also exhibit methanol maser emission, however, it has a small spatial offset (57.41$\arcsec$) to a possible SNR source G28.5167+0.1333, and has a very wide RRL line width of 70.2 km s$^{-1}$, thus there may have both HMSFR and SNR sources along the LOS of this target. To minimize contamination, this source was excluded from the HMSFR sample and classified as a potential SNR.\ Discussion\[4\] =============== Association and Correlation with Methanol Masers\[4.1\] ------------------------------------------------------- In addition to our RRL emissions, there are 241 6.7-GHz class methanol maser sources detected towards the same sample with TMRT [@yang2017; @yang2019] including one towards a potential SNR (G28.532+0.129; see Section \[3.2\]). The majority of the maser sample (224/241) are close to the Galactic Plane ($|\textit{b}| < 2\degr$). 6.7 GHz methanol masers are believed to appear at an earlier stage of star formation, while H regions typically exist at more evolved stages (see Section \[2.1\]). Thus the association of 6.7 GHz methanol masers and RRLs would be helpful on discriminating HMSFRs at different evolutionary stages [@walsh1998; @jordan2017]. Our survey observed RRLs and methanol masers simultaneously with the same pointing positions, providing the most accurate results of cross-matching the signals. In addition, combing data from different surveys would bring systematic error in the cross-matching caused by inconsistent sensitivities and resolutions. Since we mainly focus on building up a cross-matched HMSFR sample, to reach higher reliability, we do not combine our date with the literature for the analysis in this paper. The 6.7-GHz class methanol maser signal is observed and received by the 15$^{th}$ window in our survey (see Table \[t1\]). Out of the 517 HMSFR sample with RRL detections, there are 103 (20.1$\%$) sources associated with methanol masers, meanwhile, 43.2$\%$ (103/240) methanol maser sources (excluding one potential SNR) exhibit RRL emissions. The remaining 137 methanol maser sources without RRLs may correspond to a younger evolutionary stage compared to those with RRLs. We label the sources showing both emission features in Column (12) in Table \[t3\]. Notably, selecting targets from the point source catalog will bias the sample to include younger sources, and therefore may increase the number of sources associated with both RRL and maser. As a comparison, @anderson2011 found that only $\sim$ 10% (46/448) of their H region sample associated with methanol maser. The correlation of LSR velocities between RRLs and methanol masers is shown in Figure \[f5\]. The majority of the associated RRL and methanol emissions seem to have fairly similar velocities, which intrinsically represent the systemic motion of the sources. As shown in Figure \[f5\], there is one source (G25.395+0.033) which has a large offset between the $V_{\rm LSR}$ of RRL and maser. Due to the commonly extended morphology of H regions, RRL and maser emissions detected from the same target position but with a large velocity offset may from different sources along the LOS, or different regions in the same extended cluster in our $\sim 3~\arcmin$ beam. As the processes of star formation at early stages are poorly understood, comparing the physical properties of HMSFR in different evolutionary stages will help us to study the formation and early evolution of massive star in such sources. We briefly discuss the spatial and intensity distributions of the three sub-sampled sources in the following sections, a more detailed discussion on their physical properties will be given in our future works.\ ### Distance and Galactocentric Distance Distributions\[4.1.1\] Figure \[f6\] shows the normalized distance and galactocentric distance ($R_{{\rm Gal}}$) distributions of RRL-only sources, maser-only sources, and sources associated with both tracers. The average distances with standard deviations of the three samples are 5.62 $\pm$ 2.25, 5.72 $\pm$ 3.69 kpc, and 6.63 $\pm$ 3.03 respectively. As shown in Figure \[f6\]a, the majority of our sources are located at $\lesssim$ 8kpc, and there is no significant difference between the distance distribution of the three samples. This is consistent with previous studies that HMSFRs at different stages are distributed similarly with distance (e.g. @urquhart2014). The sources are more dispersed in the term of $R_{{\rm Gal}}$. As shown in Figure \[f6\]b, although the three sub-samples have similar average $R_{{\rm Gal}}$ values (6.25 $\pm$ 2.09, 6.95 $\pm$ 1.86 and 5.70 $\pm$ 1.77 kpc for RRL-only sources, maser-only sources, and sources with both RRL and maser, respectively), both RRL-only sources and sources associated with both tracers seem to be more adequate near the Galactic center. This can be explained by the fact that the thin gas at the outer Galaxy makes ionized hydrogen hard to be formed, and vice versa. As illustrated by Figure \[f6\]b, three peaks can be seen in the distribution of $R_{{\rm Gal}}$ at $\sim$ 3$-$4, 6$-$7 and 8 kpc. The peak at 3$-$4 kpc for RRL-only sources and sources with both tracers may associate with the 4-kpc molecular ring [@dame2001]. The peak at 6$-$7 kpc appears in the $R_{{\rm Gal}}$ distribution for all sources may coincident with the northern segment of Sagittarius arm. The sources at the 8-kpc peak may be mostly associated with the local Sagittarius–Carina arm.\ ### RRL and Maser Line Intensity Distributions\[4.1.2\] Figure \[f7\] presents the normalized distributions of peak line intensity corrected by distance of our samples, which is defined by the peak line intensity times the square of Bayesian distance (see Section \[4.2.1\] for more information). For the RRL sources, the peak line intensity used here are the main beam temperature (T$_{\rm mb}$) of the RRLs. The RRL-only sources have a mean RRL intensity of 2.88 $\pm$ 0.70 mK kpc$^2$, which is lower than the mean intensity of 3.29 $\pm$ 0.60 mK kpc$^2$ of the sources associated with both RRL and maser. Meanwhile, the maser-only sources and sources associated with both tracers have similar mean maser intensity of 1.95 $\pm$ 0.83 Jy kpc$^2$ and 2.11 $\pm$ 0.8 Jy kpc$^2$, respectively. @urquhart2014 performed similar analyses and found that clumps with methanol maser sources have lower bolometric luminosity than those with H regions. The authors suggested that this is because earlier stage massive stars with methanol masers are more heavily embedded than those at later stages without methanol maser. Although there is no significant difference between our maser line intensity distribution of maser-only and sources associated with both RRL and maser (Figure \[f7\]b), as shown in Figure \[f7\]a, RRL-only sources have lower RRL line intensities than that of sources exhibiting methanol maser on average, and there is a lack of maser detection for RRL sources below 2.0 mK kcp$^2$. As illustrated by @ouyang2019, H regions with methanol masers appear to have higher electron temperature and emission measure than those without, therefore methanol masers are more likely to be produced in regions with high gas densities and hence have a higher detection rate at more luminous H regions.\ Galactic Distribution --------------------- ### RRL Spatial Distribution\[4.2.1\] Figure \[f8\] represents the Galactic latitude and longitude distributions of the H-RRL-only sources, methanol-maser-only sources, sources associated with both tracers, and He RRL sources (see Section \[4.3\] for more details). As shown in this figure, a large fraction of the sources associated with both signals locate near the Galactic Plane, only 3 of them with a Galactic latitude larger than $\pm$ 2$\degr$, reflecting a sparse abundance in such regions. The LSR velocities of the RRLs can be used to calculate the distance of its host H regions using the Bayesian distance calculator built by @reid2016[^4]. @reid2016 combines various types of distance information (spiral arm mode, kinematic distance, Galactic latitude, and parallax source) in a Bayesian approach, and fits the combined probability density function with multiple Gaussian components. The distance and error are estimated by the peak probability density and width of the Gaussian fitted component with maximum integrated probability density. The kinematic distances augmented by H absorption spectra were used to resolve the near/far ambiguity for sources within the Solar circle. A user-adjustable prior probability (from 0 to 1) that the source is beyond the tangent point was set to the default value of 0.5. The result distances with error are presented in Column (9) and (10) in Table \[t3\]. In Figure \[f9\], we compare the velocity distribution along Galactic longitude of our HMSFRs to the longitude-velocity diagram retrieved from @vallee2008. For sources exhibiting multiple RRL components with different peak velocities, we only use the peak velocity of the strongest component to do the analysis. As shown in this figure, most of the sources located in the first quadrant and our data align with the model curves well in the first and second quadrants.\ ### Line Width Distribution\[4.2.2\] Figure \[f10\] shows the RRL line width distribution of our H region sample excluding those potential PNe, SNR and weak sources. These sources have an average line width with a standard deviation of 23.6 $\pm$ 2.0 km s$^{-1}$, which is very close to that of the catalog (22.3 $\pm$ 5.3 km s$^{-1}$). Thermal broadened RRL has a line profile with an [FWHM]{} width proportional to T$^{1/2}\nu_o$ [@BS1972], where T is the temperature of the host H region, $\nu_o$ is the rest central frequency. For a typical RRL-hosting source with T $\sim$ 5000 to 13000 K across the Galaxy [@balser2015], the corresponding thermal-broadened line width is $\sim$ 15.3 to 24.6 km s$^{-1}$. In addition to thermal broadening, pressure broadening is also significant for RRLs at centimeter wavelengths [@keto2008]. The ratio between pressure-broadened and thermal broadened line width is proportional to $n_e N^7$, where $n_e$ is the electron density and $N$ is the principal quantum number of the RRL transition [@BS1972; @grim1974; @keto1995; @keto2008]. As illustrated by @keto2008, since pressure broadening is proportional to the density of host H region and less significant at high frequencies, comparing the line widths of RRLs measured with high-resolution interferometer in various wavelengths can be used to measure the electron density of the natal H region. RRLs with a line width narrower than 15 km s$^{-1}$ are likely from comparably cold, sparse H regions and are broadened purely by thermal, thus they can be used to probe the temperature of the host H regions. There are 69 sources in our sample containing H RRLs with a line width of $<$ 15 km s$^{-1}$, for thermally broadened RRL source with such narrow line width, the host H regions have an upper limit to the temperature of $\lesssim 5000$ K. According to @shaver1970 and @shaver1979, RRLs with such narrow line width are from cold nebulae. Another possible explanation for the narrow line width is that the observed RRL line width is underestimated due to radiative transfer effect caused by strong free-free emission, which typically has a larger optical depth. We detected 51 sources with a very broad line width ($>$ 35 km s$^{-1}$). Out of these sources, 22 have a line width broader than 40 km s$^{-1}$. RRL with a line width $>$ 40 km s$^{-1}$ may be very dense H regions. For these RRLs, there may also exist large-scale motions around the central young stars (@sewilo2004 [@ridge2001; @RB2001] and references therein) such as nebular expansion, rotation of the inner parts of the accretion disk, infall of matter, shocks, bipolar jets or photo evaporating flows. Some of those broad line sources are possibly previously unknown PNe or SNR sources. As mentioned in Section \[3.2\], since PNe generally have larger line width due to expansion, their existence may affect the RRL line width distribution of our sample. For example, one of the sources, G28.393+0.085, with the widest line width in our sample was identified as a known H region (G028.394+00.076) in @anderson2011. However, its RRL emission with extremely large line width is potentially from an unknown PNe or SNR source along the LOS. In addition, blended multiple RRL components with very close peak velocities may also cause confusion with a single wide line component.\ Helium and Carbon RRLs\[4.3\] ----------------------------- ### Helium RRL Detections\[4.3.1\] In addition to H RRLs, there are 49 sources also exhibit He RRLs. Their spatial distribution is shown in Figure \[f8\]. He RRLs are believed to have the same origin with H RRLs. The atomic heliums are ionized by the UV photons emitted by a central O6 or hotter star [@mezger1978; @roshi2017]. Since higher ionizing energy is needed for He RRLs, they are generally from stars that are more massive than those emit H RRL only. He RRLs are usually weaker than H RRLs, they can only be detected from sources with strong H RRL intensities. In our sample, the mean peak temperature of H RRL from the sources with He RRL emissions is 374.7 mK, which is $\sim$ 8 times brighter than those without He RRL detections. The detected He RRLs have a mean peak temperature of 31.6 mK and an average value of $T_{p {\rm He}}$/$T_{p {\rm H}}$ $\sim$ 0.1, where $T_{p {\rm He}}$ and $T_{p {\rm H}}$ denote the peak temperatures of He and H RRLs. Due to sensitivity limitation, He RRLs with peak temperature lower than $\sim$ 7 mK are below our detection threshold, this results in a lower limit to the $T_{p {\rm H}}$ of 70 mK for He RRL to be detected. Figure \[f11\] shows the plot between $T_{p {\rm He}}$ and $T_{p {\rm H}}$. For sources without He RRL detections, the expected $T_{p {\rm He}}$ values are calculated by multiplying their $T_{p {\rm H}}$ by 0.1. There are 76 sources which have an H RRL peak temperature above 70 mK (black dots above the red dashed line in Figure \[f11\]) but had no He RRL detections, showing a low abundance of He$^+$. We averaged over the H-RRL-only sources with $T_{p {\rm H}}$ above and below 70 mK, as shown in Figure \[f12\]. After averaging, He RRLs can be seen with a $S_{i {\rm He}}$ to $S_{i {\rm H}}$ ratio of $\sim$ 0.01 and $\sim$ 0.02 for sources above and below 70 mK, respectively, where $S_{i {\rm He}}$ and $S_{i {\rm H}}$ denote the integrated intensities of He and H RRLs. These values are much lower than the mean $S_{i {\rm He}}$ to $S_{i {\rm H}}$ ratio of 0.06 for sources with He RRL detections. This fact may suggest that in addition to the selecting effect caused by the sensitivity limit, the non-detections of He RRLs for the H-RRL-only sources may be mostly originated from the low abundance of He$^+$. Those sources are possibly being ionized by a less massive, later OB-type star. That is, helium is under ionized with respect to hydrogen.\ ### Distance and Galactocentric Distance Distribution Figure \[f13\] shows the normalized distance distribution of the 76 H-RRL-only sources with $T_{p {\rm H}}$ $>$ 70 mK and 380 sources (excluding the weak sources, and for multi-component sources, only the strongest emission were taken into consideration.) with $T_{p {\rm H}}$ $<$ 70 mK comparing to the 49 sources with He RRL. No significant difference in the distance distributions were found for the two sub-samples in this figure. This fact may support the argument that the non-detections of He RRLs are mainly caused by the low abundance of He$^+$ rather than sensitivity limit caused by distance. Figure \[f14\] shows the normalized $R_{{\rm Gal}}$ distribution density of the 49 He RRL sources and the 76 H-RRL-only sources with $T_{p {\rm H}}$ above 70 mK. As shown in this figure, sources without He RRL emissions locate nearer to the Galactic center than those with He RRLs on average. Under the above assumption that He RRL sources are being ionized by a more massive star with respect to those without He RRL, we suggest that more massive SFRs locate at longer distances on average. This is contradicting to previous studies that due to higher gas density, there is a concentration of mass in the inner Galaxy (e.g., @green2011 [@casassus2000; @lepine2011]). A more plausible explanation for the lower abundance of ionized He is line-blanketing effect caused by higher metallicity in such regions. There is a known negative metallicity radial gradient along the Galactic disk [@HW1999], the higher metal content in the atmosphere of OB-type stars near the Galactic Center will cause line-blanketing and reduce the number of He-ionizing photons that escape the star. This may result in a lower abundance of ionized He in such regions. Nevertheless, both Figures \[f13\] and \[f14\] may suffer from statistical bias due to limited sample size.\ ### He$^+$ Abundance along the Galactic Plane\[4.3.3\] H and He RRLs with high principal quantum numbers in the radio act similarly, so their line intensity ratio y$^+$ can be used to diagnose the abundance ratio between $^4$He$^+$ and H$^+$ [@wenger2013; @balser2006]. y$^+$ is defined as the following: $${\rm y}^+ = \frac{T_{p {\rm He}^+}\Delta \nu_{{\rm He}^+}}{T_{p {\rm H}^+}\Delta \nu_{{\rm H}^+}}$$ where $T_{p {\rm He}^+}$ and $T_{p {\rm H}^+}$ are the peak line temperatures of He and H RRLs, $\Delta \nu_{{\rm He^+}}$ and $\Delta \nu_{{\rm H^+}}$ are the line widths of them, respectively. The distribution of y$^+$ values along the Galactic Plane for the sources exhibiting He RRLs is presented in Figure \[f15\]. Our sample has a mean y$^+$ value with a standard deviation of 0.062 $\pm$ 0.029, this value is similar to that of the catalog (0.068 $\pm$ 0.023) measured in [@wenger2013]. Our sample shows no significant trend on y$^+$ with $R_{{\rm Gal}}$, as presented by the black fitting line in Figure \[f15\], a small positive slope of 0.002 $\pm$ 0.015 is derived for the sources. There is a very low correlation coefficient of $\sim$ 0.09 between y$^+$ and $R_{{\rm Gal}}$, and a large standard error of $\pm$ 0.015 to the fitting line slope, both showing weak dependence of y$^+$ on $R_{{\rm Gal}}$. This is consistent with earlier studies that y$^+$ has a negative or no obvious gradient with $R_{{\rm Gal}}$ through our Galaxy (see @balser2001 for a review). A weak increasing trend was also found on y$^+$ (y$^+$ = 0.0035 $\pm$ 0.0016 kpc$^{-1}$) in [@wenger2013], however, they pointed out that this result only weakly constrains the actual y$^+$ gradient due to the large uncertainties in their data. Due to the limited sample size, our analysis may also suffer from statistical bias. There are two sources with prominently high y$^+$ values ($>$ 0.15) (G23.563+0.008 and G84.722$-$1.248; see Figure \[f15\]), as concluded by @balser2001, possible origins for the high y$^+$ value are mass loss of helium near the surface of the massive star or overestimated abundance due to the radiative transfer effect.\ ### MIR Color Distribution\[4.3.4\] Due to the emission from the dust heated by the central star, one would expect higher luminosities at longer MIR wavelength from more massive SFRs, thus there may exist a difference between the color-color distribution of H regions with different masses of their exciting stars. Figure \[f16\] shows the *WISE* \[12\] $-$ \[22\] vs. \[3.4\] $-$ \[4.6\] and \[4.6\] $-$ \[12\] vs. \[3.4\] $-$ \[4.6\] color distributions of sources with and without He RRL detections (above 70 mK). However, there is no significant separation found between the color distributions of these two sub-samples. This may suggest that *WISE* colors are insensitive to mass variation of UCH regions.\ ### Carbon RRLs\[4.3.5\] Carbon RRLs are from cooler gas in photo-dissociation regions (PDRs) or diffuse gas ionized by the interstellar UV radiation (see @alves2015 and references therein). For a target position with both H and C RRLs, if the C RRL is emitted from a cold gas with a different velocity to the H region, it can have a shifted velocity offset with respect to the H RRLs [@alves2012]. There are 23 sources showing C RRL in our sample, 3 of them are from the nearby Orion Molecular Cloud Complex (G208.894$-$19.313; G213.706$-$12.602; G213.885$-$11.832). The line properties of the 23 C RRLs are presented in Table \[t5\]. C RRLs are essential to the studies of morphology and physical properties of its host PDR (e.g., @RA2001 [@roshi2002; @wenger2013]). For example, the non-thermal component of the carbon RRL line width can be used to diagnose the magnetic field in the host PDR [@roshi2007; @balser2016]. For a thermally broadened C RRL, assuming a typical PDR temperature of $\sim$ 10$^3$ K, a line width of $\sim$ km s$^{-1}$ is expected. However, the C RRLs in our sample have an average line width of 8.9 km s$^{-1}$. The large average line width of C RRLs may indicate that there may exist non-thermal turbulence in the PDRs in our sample (@roshi2007 and @barrett1964).\ Summary\[5\] ============ \(1) Using the TMRT, we performed a Galactic RRL survey in C band toward 3348 targets, selected from the *WISE* point source catalog. Excluding 5 potential PNe and 5 potential SNR candidates, we built a sample of 517 HMSFRs traced by RRL. The peak flux densities of the detected hydrogen RRLs are in a range of $\sim$ 10 to 1900 mK. Though the majority of the sources have a line width within 15 $\sim$ 35 km s$^{-1}$, there are 82 of them show very narrow line width characteristic and 30 of them have line width larger than 35 km s$^{-1}$. Our sample further expanded the H region sample on the basis of previous surveys. \(2) Within the detected H region sample, 103 sources also harbor 6.7 GHz methanol masers emissions. Combining the sources traced by RRL and/or maser, we built up a sample of 654 HMSFRs, providing fundamental information to study the high-mass star formation evolutionary stages. According to the argument that methanol maser appears earlier than the formation of H region, our sources may be associated with HMSFR at various star-forming sequences. By comparing the physical properties of the RRL-only sources, maser-only sources and sources associated with both tracers, we found no significant difference in distance and $R_{{\rm Gal}}$ distribution of the three sub-samples. A slightly higher maser association rate was found for more luminous RRL sources. \(3) In addition to H RRL, we also detected He RRLs from 49 sources which may associate with more massive HMSFRs, no significant gradient on the He/H abundance along the galactocentric distance was found from the 49 He RRL sources. \(4) A sample of 23 C RRLs were also built in this survey, which provides a promising sample for future studies on the physical properties of PDRs surrounding H regions, the wide average line width of C RRLs may indicate non-thermal turbulence in their host PDRs.\ Acknowledgement {#acknowledgement .unnumbered} =============== We are thankful for the assistance from the operators of the TMRT during the observations, the funding and support from China Scholarship Council (CSC) (File No.201704910999), Science and Technology Facilities Council (STFC) and the University of Manchester. This work was supported by the National Natural Science Foundation of China (11590781, 11590783, 11590784 and 11873002) , and Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2019). [p[1.8cm]{}p[1.6cm]{}p[1.5cm]{}p[1cm]{}p[0.6cm]{}p[1.6cm]{}p[1.6cm]{}p[1.6cm]{}p[0.5cm]{}p[0.5cm]{}p[1.5cm]{}p[0.6cm]{}p[3cm]{}]{} G18.059+2.035 & 18:16:27.60 & $-$12:14:45.7 & 22/04/16 & 26 & 0.53(0.05) & 19.4(2.0) & 19.7(0.8) & 1.69 & 0.24 & & & (K) G018.426+01.922\ G18.559+2.029 & 18:17:26.97 & $-$11:48:32.0 & 15/12/17 & 49 & 1.26(0.05) & 24.4(1.1) & 32.7(0.5) & 1.77 & 0.26 & & & (K) G018.426+01.922 / IRAS 18146-1148\ G18.915+2.027 & 18:18:08.75 & $-$11:29:45.2 & 15/12/17 & 31 & 0.42(0.04) & 12.7(1.3) & 33.1(0.5) & 1.77 & 0.26 & & & (K) G018.426+01.922 / IRAS 18153-1131\ G20.495+0.157 & 18:27:54.06 & $-$10:58:37.3 & 04/07/16 & 52 & 1.10(0.04) & 19.7(0.9) & 26.1(0.4) & 14.01 & 0.31 & & & (K) G020.481+00.168\ G20.749$-$0.112 & 18:29:21.28 & $-$10:52:38.5 & 04/07/16 & 263 & 7.54(0.06) & 26.9(0.2) & 54.5(0.1) & 3.46 & 0.20 & & & (K) G020.728-00.105 / Kes 68\ G20.762$-$0.064 & 18:29:12.19 & $-$10:50:36.0 & 04/07/16 & 243 & 5.52(0.06) & 21.3(0.2) & 53.5(0.1) & 3.45 & 0.20 & & yes & (K) G020.728-00.105 / Kes 68\ G21.919$-$0.324 & 18:32:19.23 & $-$09:56:19.9 & 04/07/16 & 41 & 0.85(0.05) & 19.5(1.4) & 83.2(0.6) & 5.27 & 0.32 & & & (K) G021.884-00.318 / IRAS 18294-0959\ G22.355+0.066 & 18:31:44.05 & $-$09:22:17.3 & 15/07/16 & 39 & 0.80(0.05) & 19.1(1.3) & 84.7(0.6) & 5.41 & 0.37 & & yes & (K) G022.357+00.064\ G22.396+0.334 & 18:30:50.75 & $-$09:12:39.8 & 15/07/16 & 21 & 0.27(0.03) & 12.1(1.6) & 82.9(0.7) & 5.38 & 0.40 & & & (C) G022.424+00.337\ [lllllllll]{}\ G18.059+2.035 & G18.559+2.029 & G18.915+2.027 & G20.495+0.157 & G20.749$-$0.112 & G21.919$-$0.324 & G22.396+0.334 & G22.397+0.300 & G22.551$-$0.522\ G22.835$-$0.438 & G22.873$-$0.258 & G22.877$-$0.432 & G22.953$-$0.358 & G23.009$-$0.379 & G23.039$-$0.641 & G23.096$-$0.413 & G23.172$-$0.183 & G23.240$-$0.114\ G23.241$-$0.481 & G23.271$-$0.139 & G23.315$-$0.184 & G23.323$-$0.294 & G23.338$-$0.213 & G23.351$-$0.139 & G23.386$-$0.130 & G23.402+0.450 & G23.416$-$0.108\ G23.428$-$0.231 & G23.431$-$0.519 & G23.458+0.066 & G23.458$-$0.016 & G23.465+0.115 & G23.473$-$0.212 & G23.490$-$0.028 & G23.538$-$0.004 & G23.563+0.008\ G23.601$-$0.015 & G23.696+0.167 & G23.740+0.157 & G23.771+0.149 & G23.823+0.135 & G23.868$-$0.117 & G23.873+0.086 & G23.899$-$0.268 & G23.900+0.520\ G23.929+0.499 & G23.959+0.405 & G23.964+0.168 & G23.995$-$0.097 & G24.010+0.503 & G24.113$-$0.176 & G24.132+0.123 & G24.191$-$0.036 & G24.235$-$0.223\ G24.273$-$0.137 & G24.283$-$0.009 & G24.323+0.047 & G24.349+0.020 & G24.351$-$0.269 & G24.359+0.127 & G24.393+0.013 & G24.425+0.243 & G24.426+0.351\ G24.427+0.122 & G24.443$-$0.228 & G24.470+0.464 & G24.479$-$0.250 & G24.499+0.390 & G24.519$-$0.111 & G24.520$-$0.565 & G24.546$-$0.245 & G24.554+0.503\ G24.564$-$0.308 & G24.615+0.421 & G24.626$-$0.101 & G24.639$-$0.030 & G24.730+0.153 & G24.775+0.118 & G24.811+0.056 & G24.818$-$0.108 & G24.820+0.158\ G24.826$-$0.073 & G24.848$-$0.102 & G24.865+0.145 & G24.910+0.037 & G25.141$-$0.400 & G25.156$-$0.273 & G25.229+0.175 & G25.356+0.263 & G25.383$-$0.377\ G25.392$-$0.131 & G25.664$-$0.120 & G26.087$-$0.055 & G26.327+0.307 & G26.331+0.134 & G26.353+0.010 & G26.374+0.246 & G26.526+0.381 & G26.579$-$0.120\ G26.580+0.080 & G26.854$-$0.077 & G26.902$-$0.306 & G26.983$-$0.228 & G27.028+0.283 & G27.069+0.072 & G27.180$-$0.004 & G27.185$-$0.082 & G27.977+0.078\ G28.063$-$0.085 & G28.150+0.169 & G28.188$-$0.212 & G28.231+0.039 & G28.264$-$0.182 & G28.280$-$0.154 & G28.291+0.010 & G28.328$-$0.075 & G28.342+0.101\ G28.393+0.085 & G28.413+0.145 & G28.439+0.035 & G28.452+0.002 & G28.577$-$0.333 & G28.579+0.144 & G28.585+3.712 & G28.688$-$0.278 & G28.692+0.028\ G28.747+0.270 & G28.757+0.059 & G28.852+3.701 & G28.855$-$0.219 & G28.920$-$0.228 & G28.928+0.019 & G29.119+0.029 & G29.414+0.185 & G29.609+0.197\ G29.725+0.074 & G29.780$-$0.260 & G29.833$-$0.261 & G29.873+0.032 & G29.887$-$0.005 & G29.887$-$0.779 & G29.939$-$0.870 & G29.941$-$0.071 & G30.030$-$0.383\ G30.103$-$0.079 & G30.136$-$0.228 & G30.145$-$0.067 & G30.197+0.309 & G30.301$-$0.203 & G30.338$-$0.251 & G30.339$-$0.174 & G30.365+0.288 & G30.392+0.121\ G30.446$-$0.359 & G30.464+0.033 & G30.587$-$0.125 & G30.604+0.176 & G30.610+0.235 & G30.624$-$0.107 & G30.652$-$0.204 & G30.667$-$0.332 & G30.668+0.063\ G30.672+0.014 & G30.693+0.228 & G30.735$-$0.295 & G30.741$-$0.195 & G30.769+0.105 & G30.810+0.046 & G30.810+0.314 & G30.846$-$0.075 & G30.857+0.004\ G30.902$-$0.035 & G30.912+0.020 & G30.920+0.088 & G30.927+0.351 & G30.945+0.158 & G31.036+0.236 & G31.041$-$0.232 & G31.101+0.265 & G31.122+0.063\ G31.375+0.483 & G31.496+0.177 & G31.508$-$0.164 & G31.544$-$0.043 & G31.554$-$0.101 & G31.611+0.151 & G31.677+0.245 & G32.010$-$0.323 & G33.031+0.084\ G33.086+0.001 & G33.265+0.066 & G33.430$-$0.016 & G33.548+0.021 & G34.033$-$0.024 & G34.185+0.114 & G34.286+0.129 & G34.366$-$0.058 & G34.515+0.066\ G34.530$-$1.087 & G34.546+0.535 & G34.712$-$0.595 & G34.719$-$0.678 & G35.067$-$1.569 & G35.291+0.808 & G35.360$-$1.781 & G35.442$-$0.018 & G35.467+0.138\ G35.500$-$0.021 & G35.579$-$0.031 & G35.603$-$0.203 & G35.615$-$0.951 & G35.681$-$0.176 & G35.823$-$0.202 & G36.454$-$0.187 & G37.359$-$0.074 & G37.593$-$0.124\ G37.659+0.119 & G37.669$-$0.093 & G37.769$-$0.263 & G37.800$-$0.372 & G39.312$-$0.216 & G39.537$-$0.378 & G39.882$-$0.346 & G40.445+2.528 & G40.495+2.570\ “G40.545+2.596 & G40.592+ 2.509 & G41.355+0.406 & G43.177$-$0.008 & G43.181$-$0.056 & G43.262$-$0.045 & G44.241+0.152 & G45.124+0.136 & G45.525+0.012\ ” G45.541$-$0.016 & G48.628+0.214 & G48.632$-$0.587 & G48.652$-$0.315 & G48.655$-$0.728 & G48.742$-$0.512 & G48.888$-$0.410 & G48.923$-$0.445 & G48.946$-$0.331\ G48.961$-$0.396 & G49.025$-$0.526 & G49.028$-$0.217 & G49.072$-$0.327 & G49.224$-$0.334 & G49.268$-$0.337 & G49.341$-$0.337 & G49.368$-$0.303 & G49.391$-$0.235\ G49.406$-$0.372 & G49.461$-$0.551 & G49.958+0.126 & G50.042+0.260 & G50.094$-$0.677 & G50.817+0.242 & G51.341+0.065 & G51.371$-$0.045 & G51.383$-$0.007\ G52.234+0.759 & G52.260$-$0.521 & G52.355$-$0.588 & G52.399$-$0.936 & G52.540$-$0.927 & G52.921$-$0.621 & G53.575+0.069 & G54.110$-$0.081 & G59.474$-$0.185\ G59.582$-$0.147 & G63.153+0.442 & G75.840+0.367 & G75.841+0.425 & G76.659+1.922 & G77.973+2.236 & G78.161+1.871 & G78.231+0.905 & G78.259$-$0.017\ G78.377+1.020 & G78.405+0.609 & G78.633+0.979 & G78.641+0.672 & G78.662+0.266 & G78.670+0.184 & G78.697+1.234 & G78.728+0.946 & G78.840+0.695\ G78.873+0.754 & G78.881+1.427 & G78.901+0.661 & G79.024+2.449 & G79.027+0.436 & G79.075+3.462 & G79.128+2.278 & G79.170+0.396 & G79.207+2.146\ G79.246+0.451 & G79.312$-$0.654 & G79.330$-$0.800 & G79.362$-$0.131 & G79.385$-$1.564 & G79.393+1.782 & G79.545$-$1.057 & G79.561$-$0.766 & G79.699+1.019\ G79.843+0.890 & G79.854$-$1.495 & G79.877+2.476 & G79.886+2.552 & G79.887$-$1.481 & G79.900+1.111 & G79.998$-$1.454 & G80.820+0.405 & G80.859$-$0.083\ G80.865+0.342 & G80.865+0.420 & G80.939$-$0.127 & G81.111$-$0.145 & G81.250+1.123 & G81.252+0.982 & G81.264$-$0.136 & G81.266+0.931 & G81.337+0.824\ G81.341+0.759 & G81.435+0.704 & G81.469+0.023 & G81.512+0.030 & G81.525+0.218 & G81.548+0.095 & G81.582+0.103 & G81.601+0.291 & G81.663+0.465\ G81.683+0.541 & G81.685$-$0.040 & G81.722+0.021 & G81.840+0.917 & G81.876+0.734 & G81.898+0.809 & G81.918$-$0.010 & G82.069$-$0.309 & G82.186+0.100\ G82.278+2.209 & G82.434+1.785 & G84.586$-$1.111 & G84.638$-$1.140 & G84.649$-$1.089 & G84.707$-$0.270 & G84.708$-$1.285 & G84.716$-$0.848 & G84.722$-$1.248\ G84.724$-$1.138 & G84.753+0.253 & G84.773$-$1.046 & G84.826$-$1.137 & G84.835$-$1.187 & G84.841$-$1.085 & G84.852+3.697 & G84.854$-$0.744 & G84.856$-$0.500\ G84.870$-$1.073 & G84.929$-$1.095 & G84.941$-$1.126 & G84.941$-$1.162 & G85.019$-$1.131 & G85.021$-$0.157 & G85.081$-$0.215 & G85.082$-$1.159 & G85.112$-$1.207\ G85.171$-$1.169 & G85.481$-$1.176 & G92.670+3.072 & G94.442$-$5.478 & G107.222$-$0.893 & G108.763$-$0.948 & G110.081+0.081 & G111.526+0.803 & G111.567+0.752\ G113.603$-$0.616 & G118.038+5.108 & G123.035$-$6.355 & G123.050$-$6.310 & G126.645$-$0.786 & G133.690+1.113 & G133.716+1.207 & G133.718+1.137 & G133.750+1.198\ G133.948+1.065 & G134.004+1.144 & G134.219+0.721 & G134.239+0.639 & G134.279+0.856 & G134.469+0.431 & G136.918+1.067 & G137.585+1.351 & G138.297+1.556\ G138.327+1.570 & G149.383$-$0.361 & G150.525$-$0.930 & G169.174$-$0.921 & G173.615+2.732 & G182.339+0.249 & G206.573$-$16.362 & G208.675$-$19.191 & G208.724$-$19.192\ G208.726$-$19.232 & G208.760$-$19.216 & G208.792$-$19.243 & G208.824$-$19.256 & G208.894$-$19.313 & G209.184$-$19.494 & G213.752$-$12.616 & G213.885$-$11.832 &\ \ G20.762$-$0.064 & G22.355+0.066 & G23.010$-$0.410 & G23.185$-$0.380 & G23.271$-$0.256 & G23.389+0.185 & G23.436$-$0.184 & G23.653$-$0.143 & G23.680$-$0.189\ G23.899+0.065 & G23.965$-$0.110 & G24.313$-$0.154 & G24.328+0.144 & G24.362$-$0.146 & G24.485+0.180 & G24.528+0.337 & G24.633+0.153 & G24.790+0.084\ G24.943+0.074 & G25.177+0.211 & G25.346$-$0.189 & G25.395+0.033 & G25.709+0.044 & G26.545+0.423 & G28.147$-$0.004 & G28.287$-$0.348 & G28.320$-$0.012\ G28.609+0.017 & G28.804$-$0.023 & G28.832$-$0.250 & G28.862+0.066 & G29.320$-$0.162 & G29.835$-$0.012 & G29.927+0.054 & G30.004$-$0.265 & G30.250$-$0.232\ G30.403$-$0.297 & G30.419$-$0.232 & G30.536$-$0.004 & G30.589$-$0.043 & G30.662$-$0.139 & G30.789+0.232 & G30.807+0.080 & G30.810$-$0.050 & G30.823+0.134\ G30.866+0.114 & G30.897+0.163 & G30.959+0.086 & G30.973+0.562 & G30.980+0.216 & G31.076+0.458 & G31.159+0.058 & G31.221+0.020 & G31.237+0.067\ G31.413+0.308 & G31.579+0.076 & G32.118+0.090 & G32.798+0.190 & G32.992+0.034 & G33.092$-$0.073 & G33.143$-$0.088 & G33.393+0.010 & G33.638$-$0.035\ G34.411+0.235 & G35.141$-$0.750 & G35.194$-$1.725 & G35.398+0.025 & G35.578+0.048 & G37.479$-$0.105 & G37.602+0.428 & G38.076$-$0.266 & G38.119$-$0.229\ G38.202$-$0.068 & G38.255$-$0.200 & G38.258$-$0.074 & G41.121$-$0.107 & G42.692$-$0.129 & G43.076$-$0.078 & G43.089$-$0.011 & G43.148+0.013 & G43.178$-$0.519\ G43.890$-$0.790 & G45.454+0.060 & G48.905$-$0.261 & G48.991$-$0.299 & G49.466$-$0.408 & G50.779+0.152 & G52.199+0.723 & G53.618+0.036 & G59.498$-$0.236\ G75.770+0.344 & G78.882+0.723 & G78.969+0.541 & G79.736+0.991 & G80.862+0.383 & G81.752+0.591 & G81.871+0.779 & G84.951$-$0.691 & G84.984$-$0.529\ G97.527+3.184 & G111.532+0.759 & G173.596+2.823 & G213.706$-$12.602 & & & & &\ \ G16.872$-$2.154 & G16.883$-$2.186 & G17.021$-$2.402 & G173.482+2.446 & G173.617+2.883 & G174.205$-$0.069 & G183.349$-$0.575 & G20.234+0.085 & G20.363$-$0.014\ G20.926$-$0.050 & G21.023$-$0.063 & G21.370$-$0.226 & G213.752$-$12.615 & G22.050+0.211 & G24.148$-$0.009 & G24.634$-$0.323 & G25.256$-$0.446 & G25.410+0.105\ G25.498+0.069 & G25.613+0.226 & G25.649+1.050 & G25.837$-$0.378 & G26.421+1.686 & G26.598$-$0.024 & G26.623$-$0.259 & G26.645+0.021 & G27.220+0.261\ G27.222+0.136 & G27.287+0.154 & G27.725+0.037 & G27.784+0.057 & G27.795$-$0.277 & G28.180$-$0.093 & G28.393+0.085 & G28.843+0.494 & G29.281$-$0.330\ G29.941$-$0.070 & G30.370+0.483 & G30.770$-$0.804 & G30.788+0.203(R) & G30.819+0.273 & G30.972$-$0.141 & G31.253+0.003(L) & G31.253+0.003(R) & G32.045+0.059\ G32.773$-$0.059 & G32.828$-$0.315 & G33.229$-$0.018 & G33.322$-$0.364 & G33.425$-$0.315 & G33.641$-$0.228 & G33.726$-$0.119 & G34.096+0.018 & G34.229+0.133\ G34.757+0.025 & G34.789$-$1.392 & G34.974+0.365 & G35.149+0.809 & G35.197$-$0.729 & G35.225$-$0.360 & G35.247$-$0.237 & G35.792$-$0.174 & G36.137+0.564\ G36.634$-$0.203 & G36.705+0.096 & G36.833$-$0.031 & G36.919+0.483 & G37.043$-$0.035 & G37.430+1.517 & G37.554+0.201 & G37.763$-$0.215 & G38.598$-$0.213\ G38.933$-$0.361 & G39.100+0.491 & G39.387$-$0.141 & G40.282$-$0.220 & G40.425+0.700 & G40.597$-$0.719 & G40.622$-$0.138 & G40.964$-$0.025 & G41.307$-$0.169\ G42.035+0.191 & G43.037$-$0.453 & G43.808$-$0.080 & G45.070+0.124 & G45.360$-$0.598 & G45.493+0.126 & G45.804$-$0.356 & G49.043$-$1.079 & G49.265+0.311\ G49.537$-$0.904 & G49.599$-$0.249 & G50.034+0.581 & G51.678+0.719 & G52.663$-$1.092 & G52.922+0.414 & G53.022+0.100 & G53.141+0.071 & G53.485+0.521\ G54.371$-$0.613 & G56.963$-$0.234 & G58.775+0.647 & G59.436+0.820 & G59.634$-$0.192 & G59.785+0.068 & G59.833+0.672 & G62.310+0.114 & G69.543$-$0.973\ G71.522$-$0.385 & G73.063+1.796 & G74.098+0.110 & G75.010+0.274 & G76.093+0.158 & G78.122+3.633 & G81.794+0.911 & G82.308+0.729 & G84.193+1.439\ G85.394$-$0.023 & G89.930+1.669 & G90.921+1.487 & G94.609$-$1.790 & G98.036+1.446 & G99.070+1.200 & G108.184+5.518 & G108.758$-$0.986 & G109.839+2.134\ G109.868+2.119 & G110.196+2.476 & G111.256$-$0.770 & G121.329+0.639 & G123.035$-$6.355 & G123.050$-$6.310 & G124.015$-$0.027 & G134.029+1.072 & G136.859+1.165\ G137.068+3.002 & G149.076+0.397 & & & & & & &\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[0.6cm]{}p[3cm]{}]{} G20.749$-$0.112 & 18:29:21.28 & $-$10:52:38.5 & 04/07/16 & 20 & 0.70(0.06) & 33.4(3.5) & $-$68.4(1.3) & 3.46 & 0.20 & He & & (K) G020.728-00.105 / Kes 68\ G23.428$-$0.231 & 18:34:48.38 & $-$08:33:22.0 & 21/08/16 & 15 & 0.24(0.06) & 15.2(3.8) & $-$20.0(1.9) & 5.93 & 0.23 & He & & (K) G023.423-00.216\ G23.436$-$0.184 & 18:34:39.21 & $-$08:31:40.4 & 21/08/16 & 14 & 0.17(0.04) & 10.8(2.1) & $-$24.6(1.3) & 5.87 & 0.24 & He & yes & (K) G023.458-00.179\ G23.473$-$0.212 & 18:34:49.22 & $-$08:30:28.2 & 21/08/16 & 9 & 0.17(0.04) & 17.6(4.6) & $-$19.8(2.0) & 5.86 & 0.24 & He & & (K) G023.458-00.179\ G23.563+0.008 & 18:34:12.06 & $-$08:19:36.6 & 06/08/16 & 13 & 0.30(0.04) & 21.6(3.6) & $-$34.1(1.3) & 5.60 & 0.40 & He & & (K) G023.572-00.020\ G24.479$-$0.250 & 18:36:49.63 & $-$07:37:55.3 & 31/08/16 & 15 & 0.16(0.03) & 10.0(1.7) & $-$26.7(1.1) & 5.83 & 0.24 & He & & (K) G024.493-00.219\ G24.790+0.084 & 18:36:12.46 & $-$07:12:10.8 & 05/09/16 & 28 & 0.50(0.06) & 16.4(2.1) & $-$15.1(0.9) & 6.04 & 0.23 & He & yes & (K) G024.844+00.093\ G25.346$-$0.189 & 18:38:12.83 & $-$06:50:00.8 & 05/09/16 & 17 & 0.40(0.05) & 22.4(3.5) & $-$64.9(1.7) & 3.83 & 0.29 & He & yes & (K) G025.382-00.151 / IRAS 18354-0652\ G25.392$-$0.131 & 18:38:05.54 & $-$06:45:58.3 & 06/09/16 & 22 & 0.61(0.11) & 26.5(6.8) & $-$22.1(2.1) & 8.88 & 0.31 & He & & (K) G025.382-00.151\ & & & & 23 & 0.20(0.06) & 7.9(3.3) & $-$55.7(1.0) & & & C & &\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[3.5cm]{}]{} G30.035$-$0.002 & 18:46:09.37 & $-$02:34:44.4 & 17/09/16 & 47 & 1.32(0.07) & 26.3(1.6) & 97.1(0.6) & 6.82 & 1.31 & PNe? & PN G030.0+00.0 / (K) G030.022 / IRAS 18436-0239\ G78.911+0.792 & 20:29:08.19 & +40:15:26.5 & 16/09/16 & 18 & 0.40(0.04) & 20.8(2.9) & 10.7(1.3) & 2.72 & 0.97 & PNe? & PN Sd 1 / (K) G078.886+00.709\ G78.931+0.722 & 20:29:29.59 & +40:13:55.0 & 16/09/16 & 32 & 1.05(0.07) & 30.7(2.2) & 17.2(1.0) & 3.07 & 0.87 & PNe? & PN Sd 1 / (K) G078.886+00.709\ G84.913$-$3.505 & 21:07:00.13 & +42:13:01.8 & 14/12/17 & 35 & 2.24(0.05) & 58.7(1.5) & 25.5(0.7) & 1.28 & 0.07 & PNe & NGC 7027\ G84.946$-$3.488 & 21:07:03.27 & +42:15:12.1 & 15/12/17 & 38 & 2.23(0.09) & 55.6(2.4) & 22.4(1.0) & 1.28 & 0.07 & PNe & NGC 7027\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[3.5cm]{}]{} G30.726+0.103 & 18:47:02.74 & $-$01:54:56.2 & 18/08/16 & 75 & 1.86(0.01) & 23.5(0.4) & 117.8(0.4) & 7.19 & 0.82 & SNR? & SNR G030.3+00.7 / (K) G030.796+00.183 / IRAS 18445-0158\ & & & & 51 & 1.42(0.01) & 26.2(0.4) & 90.2(0.4) & & & &\ & & & & 36 & 0.91(0.01) & 23.7(0.4) & 42.0(0.4) & & & &\ G79.831+1.280 & 20:29:53.77 & +41:17:18.8 & 08/09/16 & 40 & 1.10(0.06) & 25.6(1.7) & -16.9(0.7) & 5.28 & 0.63 & SNR? & SNR G079.8+01.2 / (K) G080.362+01.212\ G28.532+0.129 & 18:42:56.49 & $-$03:51:21.7 & 19/09/16 & 26 & 0.83(0.06) & 30.1(2.1) & 101.0(1.0) & 8.13 & 0.31 & SNR? & MAGPIS SNR? G28.5167+0.1333 / (K) G028.581+00.145\ & & & & 19 & 1.45(0.08) & 70.2(4.6) & 26.0(2.0) & & & &\ G28.565+0.021 & 18:43:23.21 & $-$03:52:32.3 & 19/09/16 & 86 & 2.59(0.06) & 28.3(0.8) & 96.9(0.3) & 8.09 & 0.37 & SNR? & SNR G028.56+00.00 / (K) G028.607+00.019\ & & & & 19 & 0.34(0.05) & 16.4(2.5) & 39.5(1.1) & & & &\ G27.102+0.024 & 18:40:41.50 & $-$05:10:32.5 & 09/09/16 & 42 & 1.41(0.07) & 31.1(1.8) & 92.9(0.7) & 8.55 & 0.39 & SNR? & MAGPIS SNR? G27.1333+0.0333\ **Appendix A:** RRL spectra for the HMSFR sample.\ \ \ \ \ \ **Appendix B:** RRL spectra for the previously known and potential PNe sources.\ \ **Appendix C:** RRL spectra for the potential SNR sources.\ \ [^1]: a machine readable catalog accompanies this paper [^2]: https://www.iram.fr/IRAMFR/GILDAS/ [^3]: http://simbad.u-strasbg.fr/simbad/ [^4]: http://bessel.vlbi-astrometry.org/bayesian

ArXiv

INTRODUCTION ============ The physics of compact objects is entering a particularly exciting phase, as new instruments can now yield unprecedented observations. For example, there is evidence that the Rossi X-ray Timing Explorer has identified the innermost stable circular orbit around an accreting neutron star [@zsss98]. Also, the new generation of gravitational wave detectors under construction, including LIGO, VIRGO, GEO and TAMA, promise to detect, for the first time, gravitational radiation directly (see, e.g., [@t95]). In order to learn from these observations (and, in the case of the gravitational wave detectors, to dramatically increase the likelihood of detection), one has to predict the observed signal from theoretical modeling. The most promising candidates for detection by the gravitational wave laser interferometers are the coalescences of black hole and neutron star binaries. Simulating such mergers requires self-consistent, numerical solutions to Einstein’s field equations in 3 spatial dimensions, which is extremely challenging. While several groups, including two “Grand Challenge Alliances” [@gc], have launched efforts to simulate the coalescence of compact objects (see also [@on97; @wmm96]), the problem is far from being solved. Before Einstein’s field equations can be solved numerically, they have to be cast into a suitable initial value form. Most commonly, this is done via the standard 3+1 decomposition of Arnowitt, Deser and Misner (ADM, [@adm62]). In this formulation, the gravitational fields are described in terms of spatial quantities (the spatial metric and the extrinsic curvature), which satisfy some initial constraints and can then be integrated forward in time. The resulting “$\dot g - \dot K$” equations are straightforward, but do not satisfy any known hyperbolicity condition, which, as it has been argued, may cause stability problems in numerical implementations. Therefore, several alternative, hyperbolic formulations of Einstein’s equations have been proposed [@fr94; @bmss95; @aacy96; @pe96; @f96; @acy98]. Most of these formulations, however, also have disadvantages. Several of them introduce a large number of new, first order variables, which take up large amounts of memory in numerical applications and require many additional equations. Some of these formulations require taking derivatives of the original equations, which may introduce further inaccuracies, in particular if matter sources are present. It has been widely debated if such hyperbolic formulations have computational advantages [@texas95]; their performance has yet to be compared directly with that of the original ADM equations. Accordingly, it is not yet clear if or how much the numerical behavior of the ADM equations suffers from their non-hyperbolicity. In this paper, we demonstrate by means of a numerical experiment and a direct comparison that the standard implementation of the ADM system of equations, consisting of evolution equations for the bare metric and extrinsic curvature variables, is more susceptible to numerical instabilities than a modified form of the equations based on a conformal decomposition as suggested by Shibata and Nakamura [@sn95]. We will refer to the standard, “$\dot g - \dot K$” form of the equations as “System I” (see Section \[sys1\] below). We follow Shibata and Nakamura and modify these original ADM equations by factoring out a conformal factor and introducing a spatial field of connection functions (“System II”, see Section \[sys2\] below). The conformal decomposition separates “radiative” variables from “nonradiative” ones in the spirit of the “York-Lichnerowicz” split [@l44; @y71]. With the help of the connection functions, the Ricci tensor becomes an elliptic operator acting on the components of the conformal metric. The evolution equations can therefore be reduced to a set of wave equations for the conformal metric components, which are coupled to the evolution equations for the connection functions. These wave equations reflect the hyperbolic nature of general relativity, and can also be implemented numerically in a straight-forward and stable manner. We evolve low amplitude gravitational waves in pure vacuum spacetimes, and directly compare Systems I and II for both geodesic slicing and harmonic slicing. We find that System II is not only more appealing mathematically, but performs far better numerically than System I. In particular, we can evolve low amplitude waves in a stable fashion for hundreds of light travel timescales with System II, while the evolution crashes at an early time in System I, independent of gauge choice. We present these results in part to alert developers of 3+1 general relativity codes, many of whom currently employ System I, that a better set of equations may exist for numerical implementation. The paper is organized as follows. In Section \[sec2\], we present the basic equations of both Systems I and II. We briefly discuss our numerical implementation in Section \[sec3\], and present numerical results in Section \[sec4\]. In Section \[sec5\], we summarize and discuss some of the implications of our findings. BASIC EQUATIONS {#sec2} =============== System I {#sys1} -------- We write the metric in the form $$ds^2 = - \alpha^2 dt^2 + \gamma_{ij} (dx^i + \beta^i dt)(dx^j + \beta^j dt),$$ where $\alpha$ is the lapse function, $\beta^i$ is the shift vector, and $\gamma_{ij}$ is the spatial metric. Throughout this paper, Latin indices are spatial indices and run from 1 to 3, whereas Greek indices are spacetime indices and run from 0 to 3. The extrinsic curvature $K_{ij}$ can be defined by the equation $$\label{gdot1} \frac{d}{dt} \gamma_{ij} = - 2 \alpha K_{ij},$$ where $$\frac{d}{dt} = \frac{\partial}{\partial t} - {\cal L}_{\beta}$$ and where ${\cal L}_{\beta}$ denotes the Lie derivative with respect to $\beta^i$. The Einstein equations can then be split into the Hamiltonian constraint $$\label{ham1} R - K_{ij}K^{ij} + K^2 = 2 \rho,$$ the momentum constraint $$\label{mom1} D_j K^{j}_{~i} - D_i K = S_i,$$ and the evolution equation for the extrinsic curvature $$\label{Kdot1} \frac{d}{dt} K_{ij} = - D_i D_j \alpha + \alpha ( R_{ij} - 2 K_{il} K^l_{~j} + K K_{ij} - M_{ij} )$$ Here $D_i$ is the covariant derivative associated with $\gamma_{ij}$, $R_{ij}$ is the three-dimensional Ricci tensor $$\begin{aligned} \label{ricci} R_{ij} & = & \frac{1}{2} \gamma^{kl} \Big( \gamma_{kj,il} + \gamma_{il,kj} - \gamma_{kl,ij} - \gamma_{ij,kl} \Big) \\[1mm] & & + \gamma^{kl} \Big( \Gamma^m_{il} \Gamma_{mkj} - \Gamma^m_{ij} \Gamma_{mkl} \Big), \nonumber\end{aligned}$$ and $R$ is its trace $R = \gamma^{ij} R_{ij}$. We have also introduced the matter sources $\rho$, $S_i$ and $S_{ij}$, which are projections of the stress-energy tensor with respect to the unit normal vector $n_{\alpha}$, $$\begin{aligned} \rho & = & n_{\alpha} n_{\beta} T^{\alpha \beta}, \nonumber \\[1mm] S_i & = & - \gamma_{i\alpha} n_{\beta} T^{\alpha \beta}, \\[1mm] S_{ij} & = & \gamma_{i \alpha} \gamma_{j \beta} T^{\alpha \beta}, \nonumber\end{aligned}$$ and have abbreviated $$M_{ij} \equiv S_{ij} + \frac{1}{2} \gamma_{ij}(\rho - S),$$ where $S$ is the trace of $S_{ij}$, $S = \gamma^{ij} S_{ij}$. The evolution equations (\[gdot1\]) and (\[Kdot1\]) together with the constraint equations (\[ham1\]) and (\[mom1\]) are equivalent to the Einstein equations, and are commonly referred to as the ADM form of the gravitational field equations [@adm62; @footnote1]. We will call these equations System I. This system is widely used in numerical relativity calculations (e.g. [@aetal98; @cetal98]), even though its mathematical structure is not simple to characterize and may not be ideal for computation. In particular, the Ricci tensor (\[ricci\]) is not an elliptic operator: while the last one of the four terms involving second derivatives, $\gamma^{kl}\gamma_{ij,kl}$, is an elliptic operator acting on the components of the metric, the elliptic nature of the whole operator is spoiled by the other three terms involving second derivatives. Accordingly, the system as a whole does not satify any known hyperbolicity condition (see also the discussion in [@f96]). Therefore, to establish existence and uniqueness of solutions to Einstein’s equations, most mathematical analyses rely either on particular coordinate choices or on different formulations. System II {#sys2} --------- Instead of evolving the metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$, we can evolve a conformal factor and the trace of the extrinsic curvature separately (“York-Lichnerowicz split” [@l44; @y71]). Such a split is very appealing from both a theoretical and computational point of view, and has been widely applied in numerical axisymmetric (2+1) calculations (see, e.g., [@e84]). More recently, Shibata and Nakamura [@sn95] applied a similar technique in a three-dimensional (3+1) calculation. Adopting their notation, we write the conformal metric as $$\tilde \gamma_{ij} = e^{- 4 \phi} \gamma_{ij}$$ and choose $$e^{4 \phi} = \gamma^{1/3} \equiv \det(\gamma_{ij})^{1/3},$$ so that the determinant of $\tilde \gamma_{ij}$ is unity. We also write the trace-free part of the extrinsic curvature $K_{ij}$ as $$A_{ij} = K_{ij} - \frac{1}{3} \gamma_{ij} K,$$ where $K = \gamma^{ij} K_{ij}$. It turns out to be convenient to introduce $$\tilde A_{ij} = e^{- 4 \phi} A_{ij}.$$ We will raise and lower indices of $\tilde A_{ij}$ with the conformal metric $\tilde \gamma_{ij}$, so that $\tilde A^{ij} = e^{4 \phi} A^{ij}$ (see [@sn95]). Taking the trace of the evolution equations (\[gdot1\]) and (\[Kdot1\]) with respect to the physical metric $\gamma_{ij}$, we find [@footnote2] $$\label{phidot2} \frac{d}{dt} \phi = - \frac{1}{6} \alpha K$$ and $$\label{Kdot2} \frac{d}{dt} K = - \gamma^{ij} D_j D_i \alpha + \alpha(\tilde A_{ij} \tilde A^{ij} + \frac{1}{3} K^2) + \frac{1}{2} \alpha (\rho + S),$$ where we have used the Hamiltonian constraint (\[ham1\]) to eliminate the Ricci scalar from the last equation. The tracefree parts of the two evolution equations yield $$\label{gdot2} \frac{d}{dt} \tilde \gamma_{ij} = - 2 \alpha \tilde A_{ij}.$$ and $$\begin{aligned} \label{Adot2} \frac{d}{dt} \tilde A_{ij} & = & e^{- 4 \phi} \left( - ( D_i D_j \alpha )^{TF} + \alpha ( R_{ij}^{TF} - S_{ij}^{TF} ) \right) \nonumber \\[1mm] & & + \alpha (K \tilde A_{ij} - 2 \tilde A_{il} \tilde A^l_{~j}).\end{aligned}$$ In the last equation, the superscript $TF$ denotes the trace-free part of a tensor, e.g. $R_{ij}^{TF} = R_{ij} - \gamma_{ij} R/3$. Note that the trace $R$ could again be eliminated with the Hamiltonian constraint (\[ham1\]). Note also that $\tilde \gamma_{ij}$ and $\tilde A_{ij}$ are tensor densities of weight $-2/3$, so that their Lie derivative is, for example, $${\cal L}_{\beta} \tilde A_{ij} = \beta^k \partial_k \tilde A_{ij} + \tilde A_{ik} \partial_j \beta^k + \tilde A_{kj} \partial_i \beta^k - \frac{2}{3} \tilde A_{ij} \partial_k \beta^k.$$ The Ricci tensor $R_{ij}$ in (\[Adot2\]) can be written as the sum $$R_{ij} = \tilde R_{ij} + R_{ij}^{\phi}.$$ Here $R_{ij}^{\phi}$ is $$\begin{aligned} R^{\phi}_{ij} & = & - 2 \tilde D_i \tilde D_j \phi - 2 \tilde \gamma_{ij} \tilde D^l \tilde D_l \phi \nonumber \\[1mm] & & + 4 (\tilde D_i \phi)(\tilde D_j \phi) - 4 \tilde \gamma_{ij} (\tilde D^l \phi) (\tilde D_l \phi),\end{aligned}$$ where $\tilde D_i$ is the derivative operator associated with $\tilde \gamma_{ij}$, and $\tilde D^i = \tilde \gamma^{ij} \tilde D_j$. The “tilde” Ricci tensor $\tilde R_{ij}$ is the Ricci tensor associated with $\tilde \gamma_{ij}$, and could be computed by inserting $\tilde \gamma_{ij}$ into equation (\[ricci\]). However, we can bring the Ricci tensor into a manifestly elliptic form by introducing the “conformal connection functions” $$\label{cgsf} \tilde \Gamma^i \equiv \tilde \gamma^{jk} \tilde \Gamma^{i}_{jk} = - \tilde \gamma^{ij}_{~~,j},$$ where the $\tilde \Gamma^{i}_{jk}$ are the connection coefficients associated with $\tilde \gamma_{ij}$, and where the last equality holds because $\tilde \gamma = 1$. In terms of these, the Ricci tensor can be written [@footnote3] $$\begin{aligned} \tilde R_{ij} & = & - \frac{1}{2} \tilde \gamma^{lm} \tilde \gamma_{ij,lm} + \tilde \gamma_{k(i} \partial_{j)} \tilde \Gamma^k + \tilde \Gamma^k \tilde \Gamma_{(ij)k} + \nonumber \\[1mm] & & \tilde \gamma^{lm} \left( 2 \tilde \Gamma^k_{l(i} \tilde \Gamma_{j)km} + \tilde \Gamma^k_{im} \tilde \Gamma_{klj} \right).\end{aligned}$$ The principal part of this operator, $\tilde \gamma^{lm} \tilde \gamma_{ij,lm}$, is that of a Laplace operator acting on the components of the metric $\tilde \gamma_{ij}$. It is obviously elliptic and diagonally dominant (as long as the metric is diagonally dominant). All the other second derivatives of the metric appearing in (\[ricci\]) have been absorbed in the derivatives of the connection functions. At least in appropriately chosen coordinate systems (for example $\beta^i = 0$), equations (\[gdot2\]) and (\[Adot2\]) therefore reduce to a coupled set of nonlinear, inhomogeneous wave equations for the conformal metric $\tilde \gamma_{ij}$, in which the gauge terms $K$ and $\tilde \Gamma^i$, the conformal factor $\exp(\phi)$, and the matter terms $M_{ij}$ appear as sources. Wave equations not only reflect the hyperbolic nature of general relativity, but can also be implemented numerically in a straight-forward and stable manner. The same method has often been used to reduce the four-dimensional Ricci tensor $R_{\alpha\beta}$ [@c62] and to bring Einstein’s equations into a symmetric hyperbolic form [@fm72]. Note that the connection functions $\tilde \Gamma^i$ are pure gauge quantities in the sense that they could be chosen, for example, to vanish by a suitable choice of spatial coordinates (“conformal three-harmonic coordinates”, compare [@sy78]). The $\tilde \Gamma^i$ would then play the role of “conformal gauge source functions” (compare [@c62; @fm72]). Here, however, we impose the gauge by choosing the shift $\beta^i$, and evolve the $\tilde \Gamma^i$ with equation (\[Gammadot2\]) below. Similarly, $K$ is a pure gauge variable (and could be chosen to vanish by imposing maximal time slicing). An evolution equation for the $\tilde \Gamma^i$ can be derived by permuting a time derivative with the space derivative in (\[cgsf\]) $$\begin{aligned} \frac{\partial}{\partial t} \tilde \Gamma^i & = & - \frac{\partial}{\partial x^j} \Big( 2 \alpha \tilde A^{ij} \nonumber \\[1mm] & & - 2 \tilde \gamma^{m(j} \beta^{i)}_{~,m} + \frac{2}{3} \tilde \gamma^{ij} \beta^l_{~,l} + \beta^l \tilde \gamma^{ij}_{~~,l} \Big).\end{aligned}$$ It turns out to be essential for the numerical stability of the system to eliminate the divergence of $\tilde A^{ij}$ with the help of the momentum constraint (\[mom1\]), which yields $$\begin{aligned} \label{Gammadot2} \frac{\partial}{\partial t} \tilde \Gamma^i & = & - 2 \tilde A^{ij} \alpha_{,j} + 2 \alpha \Big( \tilde \Gamma^i_{jk} \tilde A^{kj} - \nonumber \\[1mm] & & \frac{2}{3} \tilde \gamma^{ij} K_{,j} - \tilde \gamma^{ij} S_j + 6 \tilde A^{ij} \phi_{,j} \Big) + \nonumber \\[1mm] & & \frac{\partial}{\partial x^j} \Big( \beta^l \tilde \gamma^{ij}_{~~,l} - 2 \tilde \gamma^{m(j} \beta^{i)}_{~,m} + \frac{2}{3} \tilde \gamma^{ij} \beta^l_{~,l} \Big).\end{aligned}$$ We now consider $\phi$, $K$, $\tilde \gamma_{ij}$, $\tilde A_{ij}$ and $\tilde \Gamma^i$ as fundamental variables. These can be evolved with the evolution equations (\[phidot2\]), (\[Kdot2\]), (\[gdot2\]), (\[Adot2\]), and (\[Gammadot2\]), which we call System II. Note that obviouly not all these variables are independent. In particular, the determinant of $\tilde \gamma_{ij}$ has to be unity, and the trace of $\tilde A_{ij}$ has to vanish. These conditions can either be used to reduce the number of evolved quantities, or, alternatively, all quantities can be evolved and the conditions can be used as a numerical check (which is what we do in our implementation). NUMERICAL IMPLEMENTATION {#sec3} ======================== In order to compare the properties of Systems I and II, we implemented them numerically in an identical environment. We integrate the evolution equations with a two-level, iterative Crank-Nicholson method. The iteration is truncated after a certain accuracy has been achieved. However, we iterate at least twice, so that the scheme is second order accurate. The gridpoints on the outer boundaries are updated with a Sommerfeld condition. We assume that, on the outer boundaries, the fundamental variables behave like outgoing, radial waves $$Q(t,r) = \frac{G(\alpha t - e^{2 \phi} r)}{r}.$$ Here $Q$ is any of the fundamental variables (except for the diagonal components of $\tilde \gamma_{ij}$, for which the radiative part is $Q = \tilde \gamma_{ii} - 1$), and $G$ can be found by following the characteristic back to the previous timestep and interpolating the corresponding variable to that point (see also [@sn95]). We found that a linear interpolation is adequate for our purposes. We impose octant symmetry in order to minimize the number of gridpoints, and impose corresponding symmetry boundary conditions on the symmetry plains. Unless noted otherwise, the calculations presented in this paper were performed on grids of $(32)^3$ gridpoints, and used a Courant factor of $1/4$. The code has been implemented in a parallel environment on SGI Power ChallengeArray and SGI CRAY Origin2000 computer systems at NCSA using DAGH [@dagh] software for parallel processing. RESULTS {#sec4} ======= Initial Data ------------ For initial data, we choose a linearized wave solution (which is then evolved with the full nonlinear systems I and II). Following Teukolsky [@t82], we construct a time-symmetric, even-parity $L=2$, $M = 0$ solution. The coefficients $A, B$ and $C$ (see equation (6) in [@t82]) are derived from a function $$F(t,r) = {\cal A}\, (t \pm r) \, \exp(- \lambda (t \pm r)^2).$$ Unless noted otherwise, we present results for an amplitude ${\cal A} = 10^{-3}$ and a wavelength $\lambda = 1$. The outer boundary conditions are imposed at $x, y, z = 4$. We evolve these initial data for zero shift $$\beta^i = 0,$$ and compare the performance of Systems I and II for both geodesic and harmonic slicing. Geodesic Slicing ---------------- In geodesic slicing, the lapse is unity $$\alpha = 1.$$ Since the acceleration of normal observers satisfies $a_a = D_a \ln \alpha = 0$, these observers follow geodesics. The energy content of even a small, linear wave packet will therefore focus these observers, and even after the wave has dispersed, the observers will continue to coast towards each other. Since $\beta^i = 0$, normal observers are identical to coordinate observers, hence geodesic slicing will ultimately lead to the formation of a coordinate singularity even for arbitrarily small waves. The timescale for the formation of this singularity can be estimated from equation (\[Kdot2\]) with $\alpha = 1$ and $\beta^i = 0$. The $\tilde A_{ij}$, which can be associated with the gravitational waves, will cause $K$ to increase to some finite value, say $K_0$ at time $t_0$, even if $K$ was zero initially. After roughly a light crossing time, the waves will have dispersed, and the further evolution of $K$ is described by $\partial_t K \sim K^2/3$, or $$\label{K_approx} K \sim \frac{3 K_0}{3 - K_0(t - t_0)}$$ (see [@sn95]). Obviously, the coordinate singularity forms at $t \sim 3/K_0 + t_0$ as a result of the nonlinear evolution. We can now evolve the wave initial data with Systems I and II and compare how well they reproduce the formation of the coordinate singularity. In Figure 1, we show $K$ at the origin ($x = y = z = 0$) as a function of time both for System I (dashed line) and System II (solid line). We also plot the approximate analytic solution (\[K\_approx\]) as a dotted line, which we have matched to the System I solution with values $K_0 = 0.00518$ and $t_0 = 10$. For these values, equation (\[K\_approx\]) predicts that the coordinate singularity appears at $t \sim 590$. In the insert, we show a blow-up of System II for early times. It can be seen very clearly how the initial wave content lets $K$ grow from zero to the “seed” value $K_0$. Once the waves have dispersed, System II approximately follows the solution (\[K\_approx\]) up to fairly late times. System I, on the other hand, crashes long before the coordinate singularity appears. In Figure 2, we compare the extrinsic curvature component $K_{zz}$ evaluated at the origin. The noise around $t \sim 8$, which is present in the evolutions of both systems, is caused by reflections of the initial wave off the outer boundaries. It is obvious from these plots that System II evolves the equations stably to a fairly late time, at which the integration eventually becomes inaccurate as the coordinate singularity approaches. We stopped this calculation when the iterative Crank-Nicholson scheme no longer converged after a certain maximum number of iterations. It is also obvious that System I performs extremely poorly, and crashes at a very early time, well before the coordinate singularity. It is important to realize that the poor performance of System I is [*not*]{} an artifact of our numerical implementation. For example, the ADM code currently being used by the Black Hole Grand Challenge Alliance, is based on the equations of System I, and also crashes after a very similar time [@r98] (see also [@aetal98], where a run with a much smaller initial amplitude nevertheless crashes earlier than our System II). This shows that the code’s crashing is intrinsic to the equations and slicing, and not to our numerical implementation. Harmonic Slicing ---------------- Since geodesic slicing is known to develop coordinate singularities for generic, nontrivial initial data, it is obviously not a very good slicing condition. We therefore also compare the two Systems using harmonic slicing. In harmonic slicing, the coordinate time $t$ is a harmonic function of the coordinates $\nabla^{\alpha} \nabla_{\alpha} t = 0$, which is equivalent to the condition $$\Gamma^0 \equiv g^{\alpha\beta} \Gamma^0_{\alpha\beta} = 0$$ (where the $\Gamma^{\alpha}_{\beta\gamma}$ are the connection coefficients associated with the four-dimensional metric $g_{\alpha\beta}$). For $\beta^i = 0$, the above condition reduces to $$\partial_t \alpha = - \alpha^2 K.$$ Inserting (\[phidot2\]), this can be written as $$\partial_t (\alpha e^{-6 \phi}) = 0 \mbox{~~~or~~~} \alpha = C(x^i) e^{6 \phi},$$ where $C(x^i)$ is a constant of integration, which depends on the spatial coordinates only. In practice, we choose $C(x^i) = 1$. In Figure 3, we show results for the same initial data as in the last section. Obviously, both Systems do much better for this slicing condition. System I crashes much later than in geodesic slicing (after about 40 light crossing times, as opposed to about 10 for geodesic slicing), but it still crashes. System II, on the other hand, did not crash after even over 100 light crossing times. We never encountered a growing instability that caused the code to crash. SUMMARY AND CONCLUSION {#sec5} ====================== We numerically implement two different formulations of Einstein’s field equations and compare their performance for the evolution of linear wave initial data. System I is the standard set of ADM equations for the evolution of $\gamma_{ij}$ and $K_{ij}$. In System II, we conformally decompose the equations and introduce connection functions. The conformal decomposition naturally splits “radiative” variables from “nonradiative” ones, and the connection functions are used to bring the Ricci tensor into an elliptic form. These changes are appealing mathematically, but also have a striking numerical consequence: System II performs far better than System I. It is interesting to note that most earlier axisymmetric codes (e.g. [@e84]) also relied on a decomposition similar to that of System II. Much care was taken to identify radiative variables and to integrate those variables as opposed to the raw metric components. It is surprising that this experience was abandoned in the development of most 3+1 codes, which integrate equations equivalent to System I. These codes have been partly successful [@cetal98], but obvious problems remain, as for example the inability to integrate low amplitude waves for arbitrarily long times. While efforts have been undertaken to stabilize such codes with the help of appropriate outer boundary conditions [@aetal98; @rar98], our findings point to the equations themselves as the fundamental cause of the problem, and not to the outer boundaries. Obviously, boundary conditions as employed in the perturbative approach in [@aetal98; @rar98] or in the characteristic approach in [@bglswi96] are still needed for accuracy – but our results clearly suggest that they are not needed for stability [@footnote4]. Some of the recently proposed hyperbolic systems are very appealing in that they bring the equations in a first order, symmetric hyperbolic form, and that all characteristics are physical (i.e., are either at rest with respect to normal observers or travel with the speed of light) [@aacy96; @acy98]. These properties may be very advantageous for numerical implementations, in particular at the boundaries (both outer boundaries and, in the case of black hole evolutions, inner “apparent horizon” boundaries). Some of these systems have also been implemented numerically, and show stability properties very similar to our System II [@cs98]. Our System II, on the other hand, uses fewer variables than most of the hyperbolic formulations, and does not take derivatives of the equations, which may be advantageous especially when matter sources are present. This suggests that a system similar to System II may be a good choice for evolving interior solutions and matter sources, while one may want to match to one of the hyperbolic formulations for a better treatment of the boundaries. The mathematical structure of System II is more appealing than that of System I, and these improvements are reflected in the numerical behavior. We therefore conclude that the mathematical structure has a very deep impact on the numerical behavior, and that the ability to finite difference the standard “$\dot g - \dot K$” ADM equations may not be sufficient to warrant a stable evolution. It is a pleasure to thank A. M. Abrahams, L. Rezzolla, J. W. York and M. Shibata for many very useful conversations. We would also like to thank H. Friedrich for very valuable comments, and S. A. Hughes for a careful checking of our code. Calculations were performed on SGI CRAY Origin2000 computer systems at the National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign. This work was supported by NSF Grant AST 96-18524 and NASA Grant NAG 5-3420 at Illinois, and the NSF Binary Black Hole Grand Challenge Grant Nos. NSF PHYS 93-18152, NSF PHY 93-10083 and ASC 93-18152 (ARPA supplemented). W. Zhang, Z. P. Smale, T. E. Stohmayer and J. H. Swank, Astrophys. J. [**500**]{}, L171 (1998). K. Thorne, in [*Proceedings of the Seventeenth Texas Symposium on Relativistic Astrophysics and Cosmology*]{}, edited by H. Böhringer, G. E. Morfill and J. E. Trümper (Annals of the New York Academy of Sciences, Vol. 759, New York, 1995). Information about the Binary Black Hole Grand Challenge can be found at [www.npac.syr.edu/projects/bh/]{}, and about the Binary Neutron Star Grand Challenge at [wugrav.wustl.edu/Relativ/nsgc.html]{} K. Oohara and T. Nakamura, in [*Relativistic Gravitation and Gravitational Radiation*]{}, edited by J.-A. Marck and J.-P. Lasota (Cambridge University Press, Cambridge, 1997). J. R. Wilson and G. J. Mathews, Phys. Rev. Lett. [**75**]{}, 4161 (1995); J. R. Wilson, G. J. Mathews and P. Marronetti, Phys. Rev. D [**54**]{}, 1317 (1996). R. Arnowitt, S. Deser and C. W. Misner, in [*Gravitation: An Introduction to Current Research*]{}, edited by L. Witten (Wiley, New York, 1962). S. Frittelli and O. Reula, Commun. Math. Phys. [**166**]{}, 221 (1994). C. Bona, J. Massó, E. Seidel and J. Stela, Phys. Rev. Lett. [**75**]{}, 600 (1995). A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, Jr., Phys. Rev. Lett. [**75**]{}, 3377 (1996). M. H. P. M. van Putten and D. M. Eardley, Phys. Rev. D [**53**]{}, 3056 (1996). H. Friedrich, Class. Quantum Gravit. [**13**]{}, 1451 (1996). A. Anderson, Y. Choquet-Bruhat and J. W. York, Jr., to appear in Topol. Methods in Nonlinear Analysis (also gr-qc/9710041). For example at the Third Texas Workshop on 3-dimensional Numerical Relativity of the Binary Black Hole Grand Challenge, held at Austin, Texas, 1995 (Proceedings can be obtained from Richard Matzner, Center for Relativity, University of Texas at Austin, Texas). M. Shibata and T. Nakamura, Phys. Rev. D [**52**]{}, 5428 (1995). A. Lichnerowicz, J. Math. Pure Appl. [**23**]{}, 37 (1944). J. W. York, Jr., Phys. Rev. Lett. [**26**]{}, 1656 (1971). Note, however, that Arnowitt, Deser and Misner [@adm62] wrote these equations in terms of the conjugate momenta $\pi_{ij}$ instead of the extrinsic curvature $K_{ij}$. A. M. Abrahams [*et. al.*]{} (The Binary Black Hole Grand Challenge Alliance) Phys. Rev. Lett. [**80**]{}, 1812 (1998); G. B. Cook [*et. al.*]{} (The Binary Black Hole Grand Challenge Alliance) Phys. Rev. Lett. [**80**]{}, 2512 (1998). For example: J. M. Bardeen and T. Piran, Phys. Rep. [**96**]{}, 205 (1983); C. R. Evans, PhD thesis, University of Texas at Austin (1984); A. M. Abrahams and C. R. Evans, Phys. Rev. D [**37**]{}, 318 (1988); A. M. Abrahams, G. B. Cook, S. L. Shapiro and S. A. Teukolsky, Phys. Rev. D [**49**]{}, 5153 (1994). Note also that ${\cal L}_{\beta} \phi = \beta^i \partial_i \phi + \partial_i \beta^i / 6$. Shibata and Nakamura [@sn95] use a similar auxiliary variable $F_i = \tilde \gamma_{ij,j}$ to eliminate some second derivatives from the Ricci tensor. T. De Donder, [*La gravifique einsteinienne*]{} (Gauthier-Villars, Paris, 1921); C. Lanczos, Phys. Z. [**23**]{}, 537 (1922); Y. Choquet-Bruhat, in [*Gravitation: An Introduction to Current Research*]{}, edited by L. Witten (Wiley, New York, 1962). A. Fischer and J. Marsden, Comm. Math. Phys., [**28**]{}, 1 (1972). L. Smarr and J. W. York, Jr., Phys. Rev. D [**17**]{}, 1945 (1978). M. Parashar and J. C. Brown, in [*Proceedings of the International Conference for High Performance Computing*]{}, eds. S. Sahni, V. K. Prasanna and V. P. Bhatkar (Tata McGraw-Hill, New York, 1995), also [www.caip.rutgers.edu/$\sim$parashar/DAGH/]{} S. A. Teukolsky, Phys. Rev. D [**26**]{}, 745 (1982). L. Rezzolla, talk presented at the Binary Black Hole Grand Challenge Alliance’s meeting at the University of Pittsburgh, April 1998. M. E. Rupright, A. M. Abrahams and L. Rezzolla, Phys. Rev. D [**58**]{}, 044005 (1998); L. Rezzolla, A. M. Abrahams, R. A. Matzner, M. Rupright and S. L. Shapiro, 1998, submitted. N. Bishop, R. Gomez, L. Lehner, B. Szilagyi, J. Winicour and R. Isaacson, Phys. Rev. Lett. [**76**]{}, 4303 (1996). Alternatively, the outer boundary conditions can be completely removed by a conformal rescaling; see, for example, P. Hübner, Phys. Rev. D [**53**]{}, 701 (1996) and P. Hübner, 1998, submitted (also gr-qc/9804065). G. B. Cook, M. S. Scheel, private communication.

ArXiv

--- author: - | M. Krech\ Fachbereich Physik, BUGH Wuppertal, 42097 Wuppertal, Germany title: 'Short - time scaling behavior of growing interfaces' --- Introduction ============ Analytic Theory =============== Monte-Carlo Results =================== Acknowledgment ============== The author gratefully acknowledges partial financial support of this work through the Heisenberg program of the Deutsche Forschungsgemeinschaft.

ArXiv

--- abstract: 'Many applications involving complex multi-task problems such as disaster relief, logistics and manufacturing necessitate the deployment and coordination of heterogeneous multi-agent systems due to the sheer number of tasks that must be executed simultaneously. A fundamental requirement for the successful coordination of such systems is leveraging the specialization of each agent within the team. This work presents a Receding Horizon Planning (RHP) framework aimed at scheduling tasks for heterogeneous multi-agent teams in a robust manner. In order to allow for the modular addition and removal of different types of agents to the team, the proposed framework accounts for the capabilities that each agent exhibits (e.g. quadrotors are agile and agnostic to rough terrain but are not suited to transport heavy payloads). An instantiation of the proposed RHP is developed and tested for a search and rescue scenario. Moreover, we present an abstracted search and rescue simulation environment, where a heterogeneous team of agents is deployed to simultaneously explore the environment, find and rescue trapped victims, and extinguish spreading fires as quickly as possible. We validate the effectiveness of our approach through extensive simulations comparing the presented framework with various planning horizons to a greedy task allocation scheme.' address: - 'Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: {emamy, sean.t.wilson, magnus}@gatech.edu).' - 'Siemens Corpororate Technology, Princeton, NJ 08540, USA (e-mail: {mathias.hakenberg, ulrich.muenz}@siemens.com)' author: - Yousef Emam - Sean Wilson - Mathias Hakenberg - Ulrich Munz - Magnus Egerstedt bibliography: - 'ifacconf.bib' title: 'A Receding Horizon Scheduling Approach for Search & Rescue Scenarios' --- Scheduling Algorithms, Optimization Problems, Multiagent Systems, Robotics, Search and Rescue. Introduction {#sec:intro} ============ Multi-robot systems are well suited to solve complex tasks in dynamic and dangerous environments due to their redundancy, ability to operate in parallel, and system level fault tolerance to individual failure as highlighted in [@brambilla2013swarm; @sahin2004swarm]. Multi-Robot Task Allocation (MRTA) deals with the assignment of agents to tasks in order to achieve an overall system goal within the constraints of the deployment setting. Therefore, in order to leverage the potential multi-robot systems have to successfully operate in dynamic and dangerous environments to solve complex problems such as disaster response, search and rescue, environmental monitoring, and automated warehousing, effective methods for solving the MRTA problem are needed ([@gerkey2004formal]). The MRTA problem becomes more complex when introducing morphological or behavioral heterogeneity within a deployed multi-robot system ([@swarmanoid2013]). However, this additional complexity comes with the benefit of improving the overall system efficiency by leveraging the strengths of individual robots within the collective. For example, in a Search and Rescue (SaR) scenario, quadrotors, which are quick and agile, are better suited for scouting and surveying while ground robots are better suited for debris clearing and resource extraction. By leveraging these strengths efficiently and allocating tasks appropriately the heterogeneous system could out perform a system comprised of only aerial or ground robots. Moreover, in many scenarios the MRTA problem is accompanied by timing constraints where tasks must be performed sequentially, e.g. a robot must wait for a delivery before transporting the delivered package. Problems of this type are typically referred to as scheduling problems and involve an additional complexity. This class of problems can be solved with Mixed Integer Linear Programs (MILP) that attempt to schedule all the tasks at once, however, this approach suffers from an exponential complexity as noted in [@gombolay2013fast]. Additionally, when deploying a multi-agent system in dynamic environments, the system must be able to respond and reschedule tasks when unavoidable and inevitable environmental disturbances occur. Lastly and specific to the SaR operation following a natural disaster (e.g. wildfires, earthquakes, hurricanes) is that the team of agents must cover the targeted area and rescue victims amongst other tasks within a short-time window. This is due to the drastic decrease in the likelihood of victims surviving after $48$ hours as highlighted in [@48hours]. Therefore, the SaR problem can be cast as an instance of heterogeneous multi-agent system scheduling with the objective of minimizing the time of completion of all tasks (i.e. the makespan). Inspired by the work in [@rhp], this paper proposes a Receding Horizon Planning framework to solve the heterogeneous MRTA scheduling problem and demonstrates its effectiveness in a SaR application. Inspired by Model Predictive Control, at fixed time intervals, the proposed framework detailed in Section \[sec:rhp\] schedules tasks for each agent up to a pre-defined time horizon and leverages a heuristic to estimate the cost to go for each schedule. As such, this framework does not suffer from the exponential complexity caused by the scheduling of the tasks and is robust to changes in the environment. Moreover, a specialized version of the framework for the SaR application is presented in Section \[sec:useCase\] along with a simulation environment for an abstracted SaR scenario. To validate the effectiveness of the proposed approach, Section \[sec:experiments\] presents extensive experimentation comparing the proposed Receding Horizon Planning based approach to a greedy scheduling scheme. Literature Review {#sec:litReview} ================= As mentioned in Section \[sec:intro\], the topic of coordination for multi-agent teams in SaR scenarios following natural disasters falls under the umbrella of Multi-Robot Task Allocation (MRTA). A comprehensive taxonomy of existing methods for MRTA can be found in [@gerkey2004formal]. As highlighted in [@khamis2015multi], most existing approaches can be categorized as decentralized or centralized approaches. There are two main advantages to decentralized approaches: their robustness to varying team sizes and communication failures, and their scalability with respect to the size of the agent fleet thanks to the computational burden being shared amongst the agents. Moreover, market-based approaches such as [@coalitions; @dias2006market; @vig2006market], attempt to combine the benefits of centralized and decentralized methods by having the computational burden shared between a central entity and the remainder of the fleet. For example, in [@coalitions], the authors suggest a protocol where agents communicate their respective capabilities and use this information to form the coalitions in a decentralized fashion. As such, these methods are able to generate better solutions than fully decentralized approaches while maintaining a certain level of scalability. However, since SaR scenarios typically involve a bounded number of agents and little computational constraints, scalability with respect to the size of the team is of no concern; thus making fully-centralized approaches more suitable for this application. Moreover, the topic of task allocation specifically pertaining to SaR scenarios is well-studied, most notably, by the participants of RoboCup SaR Agent Simulation competition as highlighted in [@Sheh2016]. The competition setup is as follows. A heterogeneous team is to be deployed to extinguish fires and rescue victims. Specifically, there are three types of agents: ambulances which rescue victims, fire brigades which extinguish fires and police units which remove the road blockades enabling the two other type of agents to reach their desired targets faster. The state-of-the-art task allocation strategy utilized by winning teams such as MRL in the competition is K-Means clustering of the Fires/Victims followed by a cluster to agent assignment using the Hungarian Algorithm which runs in polynomial time. We refer the reader to [@mrl19] for details. Inspired by the RoboCup competition, the motivation behind the development of the new simulation environment presented in Section \[sec:useCase\] is two-fold. First, the proposed scenario can be seen as a generalized version of the competition’s scenario. Specifically, instead of having a fixed number of types of agents each associated with a single class of tasks (e.g. police units only capable of removing road blockades), through characterizing each agent through the capabilities it exhibits, the proposed simulation framework allows for the modular addition of agent types and the collaboration of a heterogeneous sub-team of agents in achieving a single task. For example, given any two agents and their potentially different capacities to transport water, in the proposed scenario, they can indeed collaborate to extinguish a target fire. Moreover, since not all SaR scenarios are identical in nature, the proposed simulation environment frames the problem as a dynamic set of pick and place tasks where victims and resources are to be delivered to target locations. Based on this abstracted view of SaR problems, the Receding Horizon Planning framework presented in this paper aims to leverage the strengths of centralized scheduling approaches while keeping the problem size tractable and remaining robust to changes occurring in the environment. The robustness property is obtained through the repeated generation of schedules at fixed time intervals; whereas tractability of the problem size is obtained through only scheduling tasks up to a certain time horizon and leveraging a load-balancing Linear Program as a heuristic to estimate the cost-to-go. Consequently solutions produced by the proposed framework are not guaranteed global optimality since the schedules are of finite horizon. However, we present extensive empirical evidence demonstrating the effectiveness of the proposed approach in Section \[sec:experiments\]. In the next section, the proposed Receding Horizon Planning framework is introduced and presented in detail. The Receding Horizon Planner {#sec:rhp} ============================ In this section, we present an extended version of the Receding Horizon Planner (RHP), first presented in [@rhp], aimed at solving the Single-Task robots, Single-Robot tasks, Time-extended Assignment (ST-SR-TA) problem, as defined in [@gerkey2004formal], for heterogeneous teams of agents. This problem class involves building a schedule of tasks for each agent that minimizes a given cost function and is strongly ${\mathcal}{N}{\mathcal}{P}$-hard as highlighted in [@brucker1999scheduling]. The brute-force approach for solving this class of problems is to enumerate all possible schedules and choose the one with the smallest associated cost. In its simplest form, the process of generating all possible schedules is done through iteratively assigning one of the remaining tasks to each agent’s schedule. As such, a set of partial schedules (i.e. schedules that do not include all tasks) will be generated to which the process is applied again. Similarly to Branch and Bound ([@lawler1966branch]), this process can be depicted as a tree-search where each partial schedule is associated to a node ${\mathcal}N_i$, and partial schedules generated through subsequent assignments are depicted as children of that node. However, given the set of tasks ${\mathcal}T$ and the set of agents ${\mathcal}A$, the number of possible schedules grows with $\bigO(|\mathcal{A}|^{|\mathcal{T}|})$, rendering the enumeration of all schedules intractable. A scalable alternative is the greedy approach, which solely considers the next best option given the current partial schedule. However, this approach suffers from a reduced performance of the overall system in terms of optimality. The RHP is a task allocation scheme inspired by Model Predictive Control (MPC) which, at fixed time intervals, computes the optimal schedule for a limited number of tasks and leverages a heuristic to estimate the cost of executing the remaining tasks. Thus, the size of the optimization problem remains constant with respect to the number of tasks and can be adjusted to the computational resources available. With regard to this variable look-ahead time, the receding horizon approach is a superset of the greedy algorithms (zero look-ahead) and the full-blown optimization (infinite look-ahead). Similarly to MPC, the number of assignments planned by the RHP is larger than the number of assignments that are executed. This creates an overlap between the consecutive optimization cycles, which reduces the loss in optimality due to the neglected future operations. The cyclic nature of this scheme allows for the incorporation of the current system state into the optimization. This feedback loop – as in classic control – provides robustness against disturbances and model deviations. ![Example decision tree generated by the Receding Horizon Planner.[]{data-label="fig:DecisionTree"}](figures/DecisionTree.pdf){width="0.7\columnwidth"} As introduced in [@rhp], The RHP utilizes a branch-and-bound method to generate the optimal schedule up to the desired time-horizon. As illustrated by Figure \[fig:DecisionTree\], each node $\mathcal{N}_i$ in the search tree corresponds to a partial schedule and the addition of a new assignment of a task to an agent creates a new node. To enable an efficient exploration of the tree, the cost $J(\cdot)$ at node $\mathcal{N}_i$ is decomposed into an accumulated cost value $g(\cdot)$ and a remaining cost estimation $h(\cdot)$ $$J({\mathcal}N_i) = g({\mathcal}N_i) + h({\mathcal}N_i). \label{eq:CostDecomposition}$$ The accumulated cost evaluates the already assigned operations and is a measure of the consumption of resources. Since each new task assignment increases the consumption of resources, the accumulated cost increases monotonically $$g({\mathcal}N_j) \geq g({\mathcal}N_i), \label{eq:AccumulatedCost}$$ where ${\mathcal}{N}_j \in Children({\mathcal}{N}_i)$. The second summand in  is a lower-bound estimate of the remaining efforts to reach the overall goal. As each new task assignment reduces the outstanding efforts the cost of the remaining tasks must decay $$h({\mathcal}N_j) \leq h({\mathcal}N_i). \label{eq:RemainingCost}$$ Moreover, the estimate $h$ must provide a lower bound of the true remaining cost at each step, therefore satisfying $$g({\mathcal}N_j) - g({\mathcal}N_i) \geq h({\mathcal}N_i) - h({\mathcal}N_j). \label{eq:LowerBoundCondition}$$ We will show later, how such a lower bound estimation can be obtained by relaxation of the integer constraints. If $h$ is chosen such that  holds we can conclude that $$J({\mathcal}N_j) \geq J({\mathcal}N_i), \label{eq:TotalCostEstimate}$$ the cost of each node increases as the tree grows. This allows to stop the further exploration of a branch if at any time $J({\mathcal}N_i) \geq J_{\text{opt}}$ (i.e. the cost of node ${\mathcal}N_i$ is larger than the cost of a known solution). This strategy is guaranteed to find the optimal solution on the tree. To eliminate symmetries and thus reduce the number of nodes to be explored in a tree, only one resource (agent) is chosen for the set of offspring-nodes that are generated from any node in the tree. The agent is chosen as $$a^{\ast} = \operatorname*{arg\,min}_{t \in {\mathcal}T , a \in {\mathcal}A} \ (y_{ta} + T_{ta}). \label{eq:NextAgent}$$ The indices $t$ and $a$ tally the available tasks ${\mathcal}T$ and agents ${\mathcal}A$ respectively. $T_{ta}$ is the duration of task $t$ if performed by agent $a$ and $y_{ta}$ is the potential start time for agent $a$ on task $t$. In other words, out of all agents, the RHP chooses the one with the earliest completion-time of all tasks. Once the agent is decided upon, all potential tasks that satisfy $$y_{ta} = \min_{t \in {\mathcal}T } (y_{ta^{\ast}} + T_{ta^{\ast}}), \label{eq:NextStates}$$ are considered as next nodes. That is, we include all tasks that can be started, before the earliest task can be finished. This again reduces the search space in the tree exploration without affecting the optimality of the solution. The main contribution of this paper is to extend the range of applications of the RHP to scenarios in which multiple agents, each exhibiting different capabilities, are needed to complete a single task or vice-versa. The wildfires in our SaR scenario represent the former type, where the combined effort of multiple agents is needed to extinguish a fire. The rescue operations represent the second type, where a single agent can carry multiple survivors. The decision space in the tree search inherently includes the various agent capabilities and teaming scenarios, if this is properly described in the set of dispatching rules for each agent. These dispatching rules describe which possible tasks an agent can do, given its current location and occupation, and to what amount the agent can contribute to the overall task, i.e. its capacity. In the context of the SaR scenario, the capacity corresponds to the mundane load capacity for water or victims of each agent. While the implementation of these dispatching rules is straightforward, the challenge for the tree search is again limiting the search space. To avoid the exploration of all possible combinations of capacities to complete a task, we make use of the heuristic $h(\cdot)$ to guide the tree search. To achieve this, we include a high level load balancing into the heuristic, which breaks the required effort for one large task down into a set of sub-tasks that can be handled by individual agents. The general approach is to first group all agents with equivalent capabilities into different classes and then minimize the latest finishing time among all agents under the following constraints (i) \[constraints:first\] The effort for each task is distributed among the different agent classes (ii) \[constraints:second\] The efforts from (\[constraints:first\]) for each class are distributed among the individual member agents of this class The detailed application to the SaR use-case is described below. {width="\textwidth"}   {width="\textwidth"}   {width="\textwidth"}   Use Case: Search And Rescue {#sec:useCase} =========================== Problem Setup ------------- Inspired by the RoboCup Search and Rescue Agent competition, we setup the SaR scenario as follows. A fire breaks out in a forest near a city and starts spreading. Victims are hidden within the city and forest themselves. The objective is to locate and save all victims and extinguish all fires simultaneously using a heterogeneous team of agents as quickly as possible. Note that the fires also grow and spread, therefore solely focusing on rescuing the victims is commonly a sub-optimal strategy. Moreover, the victims are initially not visible to the agents. Therefore, exploring the map is also of paramount importance. Each fire is represented as a circle of radius proportional to its health and requires a specific amount of water that is also proportional to its health. Upon reaching $100\%$ health, a fire then spread to a nearby territory which is illustrated through the creation of another circle in that location. In order to keep the problem setup as general as possible, we frame the scenario as a pick and place problem where agents need to repeatedly go back to the base and deliver water/victims to the fires/hospital. Moreover, we solely consider each agent’s capabilities when considering it for a given task. As such, we allow for the modular addition or removal of agents from the setup as discussed in the next subsection. The Heterogeneous Team ---------------------- Teams deployed for SaR are typically heterogeneous due to the variety of the tasks at hand and terrains to be navigated. Therefore, we chose to create teams composed of $4$ types of agents: Ground Units, Helicopters, Drones and Autonomous Ground Vehicles (AGVs). It is important to note that our approach is agnostic to the specific types of agents and the number existing types. In fact, the proposed approach solely considers the number of agents of each type, and each type’s capabilities (as defined in [@gerkey2004formal]). We restrict the agent capabilities we consider to two categories. The first type deals with the mobility of the agents (e.g. what is the velocity of the agent when navigating in the forest?), and the second type considers what action the agent can perform once at the desired location (e.g. can the agent “pickup” a victim?). A tabulation of the capabilities of the $4$ types of agents is presented in Table \[tab:agentCaps\]. As such, each agent type’s specific capabilities can be accounted for explicitly by the proposed framework in the process of generating feasible schedules. Moreover, it is worth noting that additional capabilities can be modularly added since the framework solely considers the capabilities required by each task in the process of generating new nodes. In the next subsection, we present the load-balancing Linear Program used to estimate the cost-to-go. Estimation of the cost-to-go ---------------------------- We formulate the cost-to-go required for the RHP as a Linear Program (LP), which added to the accumulated cost provides a lower-bound on the total cost of each node. The objective of the LP is to minimize the makespan $s$, which is the time of completion of the last task. In order to compute the makespan, one must be able to estimate the time taken to complete each task (e.g. rescuing a victim). This is a non-trivial problem, since the time of completion of a task by an agent is dependant upon the previous assignment of the agent. This difficulty also arises in the travelling salesman problem, where the time to travel to a given city depends on the last destination of the salesman. In order to overcome this difficulty, we assume that the distances between the victims/fires are negligible compared to their distances to the base. Therefore by lower-bounding all distances between the targets and the base, we can obtain a “tight” lower-bound on the amount of time a given agent takes to complete a trip to any target $t_{Type(a)}$ given the agent’s velocity. The decision variable for the LP are - the number of assignments of task $t$ to the agents of class $c$, denoted by $n_{c,t}$ - the number of assignments of task $t$ to the individual agent $j$ denoted by $m_{j,t}$ - the total makespan denoted by $s$ With these constraints stacked into a vector $$x^{\text{T}} = [n_{c,t}\,,\,m_{j,t}\,,\, s]$$ the LP is formulated as follows \[eq:heuristicLP\] $$\begin{aligned} \min_{x} ~& \; \; [0\, \ldots\, 0\, 1] x \label{eq:lp:a}\\ \operatorname*{subject\,to}~~& \smashoperator{\sum_{c \in {\mathcal}C}} C_{c}^{(t)}n_{c,t} \geq R_{t} \: \forall t \in {\mathcal}T \label{eq:lp:b} \\ & y_j + \smashoperator{\sum_{t \in {\mathcal}T}} T_t^{(c)}m_{j,t} \leq s \:\: \forall j \in {\mathcal}A \label{eq:lp:c}\\ & \sum_{j \in \mathcal{A}}B_j^{(c)} m_{j,t} = n_{c,t} \:\: \forall c,t \in {\mathcal}C \otimes {\mathcal}T \label{eq:lp:d},\end{aligned}$$ where ${\mathcal}T$, ${\mathcal}C$ and ${\mathcal}A$ denote the set of all tasks, agent types and individual agents respectively. The program aims to minimize the makespan (i.e. the time of completion of the last task). Moreover, constraint ensures that the required effort for accomplishing task $t$ denoted by $R_{t}$ is matched by the agent fleet. The effort provided by the agent fleet is computed through summing the effort provided by each agent type. The capacity of agent type $c$ for task $t$ is denoted by $C_{c}^{(t)}$. Additionally, the makespan is computed in constraint , where $y_j$ denotes the completion of time of agent $j$’s current schedule and the execution time of task $t$ for class $c$ is denoted by $T_{t}^{(c)}$. Lastly, constraint ensures that the sum of contributions of each individual agent of a given type matches the total contribution of that type, where $B_{j}^{(c)}$ indicates if agent $j$ is of type $c$. In the next section, we present experimental results demonstrating the effectiveness of the SaR specialized Receding Horizon Planner presented in this section compared to a greedy scheduling approach. ---------------------- --------- ------- ------- ------ Types G. Unit Heli. Drone AGV \[0.5ex\] Water Cap. 1 5 0 2 Rescue Cap. 1 4 0 4 Move Forest 0.1 0.5 0.40 0.20 Move City 0.1 0.5 0.40 0.00 ---------------------- --------- ------- ------- ------ : The capabilities of the $4$ types of agents: Ground Units, Helicopters, Drones and AGVs. Note for example how AGVs are unable to navigate through the forest or how the capacity for carrying water units or victims vary depending on the type of agent.[]{data-label="tab:agentCaps"} Experiments {#sec:experiments} =========== In this section, we present empirical results validating the use of a receding horizon through simulations and experiments on the Robotarium, a remotely accessible swarm accessible testbed ([@robotarium]). The setup of the experiments is as follows. The team is composed of 6 ground units, 3 helicopter units, 14 drones and 3 AGVs. Moreover, in each experiment $10$ initially hidden victims are randomly placed in the forest and city along with $3$ fires that were also initialized at random positions. However, the number of total fires generated in a given experiment may differ depending on the planner being used. This is due to the fact that the fires grow and spread over time and therefore require even more resources to extinguish. This is an accurate depiction of many real-world SaR scenarios and serves to emphasize that the time of completion of tasks is an important measure in such scenarios. The path-action planning algorithm details the execution of the generated schedules and is implemented as follows. First, it generates each agent’s path to its corresponding task, then ensures that the agent takes the corresponding required action for the task if feasible. For example, once an agent tasked with rescuing a victim reaches its location, it will take the “pick-up” action if and only if the agent’s maximum capacity for the number of victims is not already reached. Since path planning is not the focus of this work, we assume all agents except the AGVs possess single-integrator dynamics and use proportional controllers to guide them to their targets. However, since the AGVs are presented as robots (GRITSBot X) in the Robotarium experiments and can indeed collide, we implemented multi-agent A\* to generate way-points the agents can follow to their targets. Moreover, since we do not explicitly check for collision-avoidance in the trajectories between way-points, we also utilize Control Barrier Functions (CBFs) to instantaneously ensure collision-avoidance at all times as described in [@AmesBarriers; @ames2014]. This is achieved through solving a Quadratic Program at each point in time that generates a minimally altered trajectory for the agents relative to their nominal trajectory to ensure collision-avoidance. A run of $20$ simulated experiments with randomized initial conditions were run to compare several planning depths of the RHP and a greedy scheduling approach. The greedy scheduling scheme used for bench-marking the proposed RHP is a one-step look-ahead planning approach. Specifically, at each scheduling iteration, each idle agent is assigned to the task that it is closest to. The mean, median and variance of the makespans of each of the scheduling approaches over all experiments are presented in Table \[tab:results\]. As shown in the table, the mean makespan decreases significantly when the RHP is used. However, the rate of improvement decreases at higher planning depths, which highlights the trade-off between computing time and solution quality. Moreover, to demonstrate the applicability of the RHP onto real systems, $10$ experiments were conducted on the Robotarium testbed comparing the RHP using a planning depth of $10$ to the greedy scheduler. The Robotarium’s robots (GRITSBot X) were used instead of the three simulated AGVs as depicted in Figure \[fig:robotariumExp\]. To obtain the desired frequency of operation on the Robotarium ($\sim100$ Hz), a separate computing node was used to run the scheduling algorithms, and the schedules were transmitted to the agents using a publisher-subscriber protocol. The results of the experiments are presented in Table \[tab:resultsR\]. Indeed, the use of the RHP reduces the makespan, thus validating the applicability of the proposed scheduling approach. ![Search and Rescue experiment on the Robotarium using 3 GRITSBot X as the AGVs. The description of all entities are as in Figure  \[fig:sim\].[]{data-label="fig:robotariumExp"}](figures/roboExp.png){width="40.00000%"} ------------------ ------- -------- ---------- Types Mean Median $\sigma$ \[0.5ex\] Greedy 78.78 53.03 10.63 RHP10 42.54 40.34 9.54 RHP15 40.09 38.56 7.95 RHP20 39.87 36.84 9.39 ------------------ ------- -------- ---------- : Results over $20$ simulations comparing the mean, median and standard deviation of the makespans of the Receding Horizon Planner with different planning depths and greedy scheduling scheme.[]{data-label="tab:results"} ------------------ ------- -------- ---------- Types Mean Median $\sigma$ \[0.5ex\] Greedy 46.70 45.06 9.64 RHP10 39.00 36.72 9.09 ------------------ ------- -------- ---------- : Results over $10$ Robotarium experiments comparing the mean, median and variance of the makespans of the Receding Horizon Planner with planning depth $10$ and the greedy scheduling scheme.[]{data-label="tab:resultsR"} Conclusion {#sec:conclusion} ========== This paper introduces a task allocation framework capable of scheduling tasks for heterogeneous teams of agents in a manner that is tractable and robust to changes in the environment and in the agent fleet. This was achieved through repeatedly scheduling solely up to a fixed horizon and leveraging a load-balancing Linear Program for the estimation of the cost to go. Moreover, a simulation framework for an abstracted Search and Rescue scenario inspired by the RoboCup Search and Rescue Agent Simulation competition was presented along with a specialized formulation of the Receding Horizon Planning approach. Experimental results showcase the efficacy of the proposed scheduling method in extensive multi-agent simulations and experiments on the Robotarium.

ArXiv

--- abstract: 'We discuss the time evolution of quotations of stocks and commodities and show that corrections to the orthodox Bachelier model inspired by quantum mechanical time evolution of particles may be important. Our analysis shows that traders tactics can interfere as waves do and trader’s strategies can be reproduced from the corresponding Wigner functions. The proposed interpretation of the chaotic movement of market prices imply that the Bachelier behaviour follows from short-time interference of tactics adopted (paths followed) by the rest of the world considered as a single trader and the Ornstein-Uhlenbeck corrections to the Bachelier model should qualitatively matter only for large time scales. The famous smithonian invisible hand is interpreted as a short-time tactics of whole the market considered as a single opponent. We also propose a solution to the currency preference paradox.' author: - | Edward W. Piotrowski\ Institute of Theoretical Physics, University of Białystok,\ Lipowa 41, Pl 15424 Białystok, Poland\ e-mail: <[email protected]>\ Jan Sładkowski\ Institute of Physics, University of Silesia,\ Uniwersytecka 4, Pl 40007 Katowice, Poland\ e-mail: <[email protected]> title: Quantum diffusion of prices and profits --- Introduction ============ We have formulated a new approach to quantum game theory [@1]-[@3] that is suitable for description of market transactions in term of supply and demand curves [@4]-[@8]. In this approach quantum strategies are vectors (called states) in some Hilbert space and can be interpreted as superpositions of trading decisions. Tactics or moves are performed by unitary transformations on vectors in the Hilbert space (states). The idea behind using quantum games is to explore the possibility of forming linear combination of amplitudes that are complex Hilbert space vectors (interference, entanglement [@3]) whose squared absolute values give probabilities of players actions. It is generally assumed that a physical observable (e.g energy, position), defined by the prescription for its measurement, is represented by a linear Hermitian operator. Any measurement of an observable produces an eigenvalue of the operator representing the observable with some probability. This probability is given by the squared modulus of the coordinate corresponding to this eigenvalue in the spectral decomposition of the state vector describing the system. This is often an advantage over classical probabilistic description where one always deals directly with probabilities. The formalism has potential applications outside physical laboratories [@4]. Strategies and not the apparatus or installation for actual playing are at the very core of the approach. Spontaneous or institutionalized market transactions are described in terms of projective operation acting on Hilbert spaces of strategies of the traders. Quantum entanglement is necessary (non-trivial linear combinations of vectors-strategies have to be formed) to strike the balance of trade. This approach predicts the property of undividity of attention of traders (no cloning theorem) and unifies the English auction with the Vickrey’s one attenuating the motivation properties of the later [@5]. Quantum strategies create unique opportunities for making profits during intervals shorter than the characteristic thresholds for an effective market (Brown motion) [@5]. On such market prices correspond to Rayleigh particles approaching equilibrium state. Although the effective market hypothesis assumes immediate price reaction to new information concerning the market the information flow rate is limited by physical laws such us the constancy of the speed of light. Entanglement of states allows to apply quantum protocols of super-dense coding [@6] and get ahead of “classical trader”. Besides, quantum version of the famous Zeno effect [@4] controls the process of reaching the equilibrium state by the market. Quantum arbitrage based on such phenomena seems to be feasible. Interception of profitable quantum strategies is forbidden by the impossibility of cloning of quantum states. There are apparent analogies with quantum thermodynamics that allow to interpret market equilibrium as a state with vanishing financial risk flow. Euphoria, panic or herd instinct often cause violent changes of market prices. Such phenomena can be described by non-commutative quantum mechanics. A simple tactics that maximize the trader’s profit on an effective market follows from the model: [*accept profits equal or greater than the one you have formerly achieved on average*]{} [@7].\ The player strategy $|\psi\rangle$[^1] belongs to some Hilbert space and have two important representations $\langle q|\psi\rangle\negthinspace$ (demand representation) and $\langle p|\psi\rangle\negthinspace$ (supply representation) where $q$ and $p$ are logarithms of prices at which the player is buying or selling, respectively [@4; @8]. After consideration of the following facts: - error theory: second moments of a random variable describe errors - M. Markowitz’s portfolio theory - L. Bachelier’s theory of options: the random variable $q^{2} + p^{2}$ measures joint risk for a stock buying-selling transaction ( and Merton & Scholes works that gave them Nobel Prize in 1997) we have defined canonically conjugate Hermitian operators (observables) of demand $\mathcal{Q}_k$ and supply $\mathcal{P}_k$ corresponding to the variables $q$ and $p$ characterizing strategy of the k-th player. This led us to the definition of the observable that we call [*the risk inclination operator*]{}: $$H(\mathcal{P}_k,\mathcal{Q}_k):=\frac{(\mathcal{P}_k-p_{k0})^2}{2\,m}+ \frac{m\,\omega^2(\mathcal{Q}_k-q_{k0})^2}{2}\,, \label{hamiltonian}$$ where $p_{k0}\negthinspace:=\negthinspace\frac{ \phantom{}_k\negthinspace\langle\psi|\mathcal{P}_k|\psi\rangle_k } {\phantom{}_k\negthinspace\langle\psi|\psi\rangle_k}\,$, $q_{k0}\negthinspace:=\negthinspace\frac{ \phantom{}_k\negthinspace\langle\psi|\mathcal{Q}_k|\psi\rangle_k } {\phantom{}_k\negthinspace\langle\psi|\psi\rangle_k}\,$, $\omega\negthinspace:=\negthinspace\frac{2\pi}{\theta}\,$. $ \theta$ denotes the characteristic time of transaction [@7; @8] which is, roughly speaking, an average time spread between two opposite moves of a player (e. g. buying and selling the same commodity). The parameter $m\negthinspace>\negthinspace0$ measures the risk asymmetry between buying and selling positions. Analogies with quantum harmonic oscillator allow for the following characterization of quantum market games. One can introduce the constant $h_E$ that describes the minimal inclination of the player to risk, $ [\mathcal{P}_k,\mathcal{Q}_k]=\frac{i}{2\pi}h_E$. As the lowest eigenvalue of the positive definite operator $H$ is $\frac{1}{2}\frac{h_E}{2\pi} \omega$, $h_E$ is equal to the product of the lowest eigenvalue of $H(\mathcal{P}_k,\mathcal{Q}_k) $ and $2\theta$. $2\theta $ is in fact the minimal interval during which it makes sense to measure the profit. Let us consider a simple market with a single commodity $\mathfrak{G}$. A consumer (trader) who buys this commodity measures his/her profit in terms of the variable $\mathfrak{w}\negthinspace=\negthinspace-\mathfrak{q}$. The producer who provides the consumer with the commodity uses $\mathfrak{w}\negthinspace=\negthinspace-\mathfrak{p}$ to this end. Analogously, an auctioneer uses the variable $\mathfrak{w}\negthinspace=\negthinspace\mathfrak{q}$ (we neglect the additive or multiplicative constant brokerage) and a middleman who reduces the store and sells twice as much as he buys would use the variable $\mathfrak{w}\negthinspace=\negthinspace-2\hspace{.1em}\mathfrak{p}-\mathfrak{q}$. Various subjects active on the market may manifest different levels of activity. Therefore it is useful to define a standard for the “canonical” variables $\mathfrak{p}$ and $\mathfrak{q}$ so that the risk variable [@8] takes the simple form $\tfrac{\mathfrak{p}^2}{2}\negthinspace+\negthinspace\tfrac{\mathfrak{q}^2}{2}$ and the variable $\mathfrak{w}$ measuring the profit of a concrete market subject dealing in the commodity $\mathfrak{G}$ is given by $$u\,\mathfrak{q}+v\,\mathfrak{p}+\mathfrak{w}(u,v)=0\,, \label{rowprosrad}$$ where the parameters $u$ and $v$ describe the activity. The dealer can modify his/her strategy $|\psi\rangle$ to maximize the profit but this should be done within the specification characterized by $u$ and $v$. For example, let us consider a fundholder who restricts himself to purchasing realties. From his point of view, there is no need nor opportunity of modifying the supply representation of his strategy because this would not increase the financial gain from the purchases. One can easily show by recalling the explicit form of the probability amplitude $|\psi\rangle\negthinspace\in\negthinspace\mathcal{L}^2$ that the triple $(u,v,|\psi\rangle)$ describes properties of the profit random variable $\mathfrak{w}$ gained from trade in the commodity $\mathfrak{G}$. We will use the Wigner function $W(p,q)$ defined on the phase space $(p,q)$ $$\begin{aligned} W(p,q)&:=& h^{-1}_E\int_{-\infty}^{\infty}e^{i\hslash_E^{-1}p x} \;\frac{\langle q+\frac{x}{2}|\psi\rangle\langle\psi|q-\frac{x}{2}\rangle} {\langle\psi|\psi\rangle}\; dx\\ &=& h^{-2}_E\int_{-\infty}^{\infty}e^{i\hslash_E^{-1}q x}\; \frac{\langle p+\frac{x}{2}|\psi\rangle\langle\psi|p-\frac{x}{2}\rangle} {\langle\psi|\psi\rangle}\; dx,\end{aligned}$$ to measure the (pseudo-)probabilities of the players behaviour implied by his/her strategy $|\psi\rangle$ (the positive constant $h_E=2\pi\hslash_E$ is the dimensionless economical counterpart of the Planck constant discussed in the previous section [@4; @8]). Therefore if we fix values of the parameters $u$ and $v$ then the probability distribution of the random variable $\mathfrak{w}$ is given by a marginal distribution $W_{u,v}(w)dw$ that is equal to the Wigner function $W(p,q)$ integrated over the line $u\,p+v\,q+w=0$: $$W_{u,v}(w):=\iint\displaylimits_{\mathbb{R}^2}W(p,q)\,\delta(u\, q\negthinspace+\negthinspace v\,p\,\negthinspace+\negthinspace w,0) \,dpdq\, , \label{defiont}$$ where the Dirac delta function is used to force the constraint ( $\delta(u\, q\negthinspace+\negthinspace v\,p\,\negthinspace+\negthinspace w,0)$). The above integral transform $(W\negthinspace:\negthinspace\mathbb{R}^2 \negthinspace\rightarrow\negthinspace \mathbb{R})\longrightarrow (W\negthinspace:\negthinspace\mathbb{P}^2\negthinspace\rightarrow\negthinspace\mathbb{R})$ is known as the Radon transform [@9]. Let us note that the function $W_{u,v}(w)$ is homogeneous of the order -1, that is $$W_{\lambda u,\lambda v}(\lambda w)=|\lambda|^{-1}W_{u,v}(w)\,.$$ Some special examples of the (pseudo-) measure $W_{u,v}(w)dw$ where previously discussed in [@4; @8; @10]. The squared absolute value of a pure strategy in the supply representation is equal to $W_{0,1}(p)$ ($|\langle p|\psi\rangle|^2=W_{0,1}(p)$) and in the demand representation the relation reads $|\langle q|\psi\rangle|^2=W_{1,0}(q)$. It is positive definite in these cases for all values of $u$ and $v$. If we express the variables $u$ and $v$ in the units $\hslash_E^{-\frac{1}{2}}$ then the definitions of $W(p,q)$ and $W_{u,v}$ lead to the following relation between $W_{u,v}(w)$ i $\langle p|\psi\rangle$ or $\langle q|\psi\rangle$ for both representations[^2] [@11]: $$W_{u,v}(w)\frac{1}{2\pi|v|}\Bigl|\int_{-\infty}^\infty\negthinspace\negthinspace \text{e}^{\frac{\text{i}}{2v}(up^2+2pw)}\langle p|\psi\rangle\, dp\,\Bigr|^2. \label{radonpsi}$$ The integral representation of the Dirac delta function $$\delta(uq\negthinspace+\negthinspace vp\negthinspace+\negthinspace w,0)=\frac{1}{2\pi}\int^\infty_{-\infty}\text{e}^{ \text{i}k(uq+vp+w)}dk \label{hraddel}$$ helps with finding the reverse transformation to $(\ref{defiont})$. The results is: $$W(p,q)=\frac{1}{4\pi^2}\iiint\displaylimits_{\mathbb{R}^3}\cos(uq\negthinspace+\negthinspace vp\negthinspace+\negthinspace w)\, W_{u,v}(w)\,dudvdw\,. \label{odwrrado}$$ Traders using the same strategy (or single traders that can adapt their moves to variable market situations) can form sort of “tomographic pictures” of their strategies by measuring profits from trading in the commodity $\mathfrak{G}$. These pictures would be influenced by various circumstances and characterized by values of $u$ and $v$. These data can be used for reconstruction of the respective strategies expressed in terms the Wigner functions $W(p,q)$ according to the formula $(\ref{odwrrado})$. Example: marginal distribution of an adiabatic strategy ------------------------------------------------------- Let us consider the Wigner function of the $n$-th excited[^3] state of the harmonic oscillator [@12]$$W_n(p,q)dpdq=\frac{(-1)^n}{\pi\hslash_E}\thinspace e^{-\frac{2H(p,q)}{\hslash_E\omega}} L_n\bigl(\frac{4H(p,q)}{\hslash_E\omega}\bigr)dpdq\,,$$ where $L_{n}$ is the $n$-th Laguerre polynomial. We can calculate (cf the definition $(\ref{defiont})$) marginal distribution corresponding to a fixed risk strategy (that is the associated risk is not a random variable). We call such a strategy an adiabatic strategy [@4]. The identity [@13] $$\int_{-\infty}^\infty \text{e}^{\text{i}kw-\frac{k^2}{4}}\,L_n \Bigl(\frac{k^2}{2}\Bigr)\,dk=\frac{2^{n+1}\sqrt{\pi}}{n!}\,\text{e}^{-w^2}H^2_n(w)\, ,$$ where $H_n(w)$ are the Hermite polynomials, Eq $(\ref{hraddel})$ and the generating function for the Laguerre polynomials, $\frac{1}{1-t}\thinspace \text{e}^\frac{xt}{t-1}=\sum_{n=0}^\infty L_n(x)\,t^n$ lead to $$W_{n,u,v}(w)=\frac{2^n}{\sqrt{\pi(u^2+v^2)}\,n!}\,\,\text{e}^{-\frac{w^2}{u^2+v^2}} \,H^2_n\Bigl(\frac{w}{\sqrt{u^2+w^2}}\Bigr)=|\langle w|\psi_n\rangle|^2 \label{hradwkw}.$$ This is the squared absolute value of the probability amplitude expressed in terms of the variable $w$. It should be possible to interpret Eq $(\ref{hradwkw})$ in terms of stochastic interest rates but this outside the scope of the present paper. Canonical transformations ========================= Let us call those linear transformations $(\mathcal{P},\mathcal{Q})\negthinspace\rightarrow\negthinspace(\mathcal{P'},\mathcal{Q'})$ of operators $\mathcal{P}$ and $\mathcal{Q}$ that do not change their commutators $\mathcal{P}\mathcal{Q}\negthinspace-\negthinspace\mathcal{Q}\hspace{.1em}\mathcal{P}$ canonical. The canonical transformations that preserve additivity of the supply and demand components of the risk inclination operator $\tfrac{\mathcal{P}^2}{2m}+\tfrac{m\mathcal{Q}^2}{2}\ $ [@4; @8] can be expressed in the compact form $$\begin{pmatrix} \mathcal{P}\\\mathcal{Q} \end{pmatrix}\begin{pmatrix} \tfrac{\text{Re}\,z}{z\overline{z}}&\text{Im}\,z\vspace{.5ex}\\ -\tfrac{\text{Im}\,z}{z\overline{z}}&\text{Re}\,z \end{pmatrix} \begin{pmatrix} \mathcal{P'}\\\mathcal{Q'} \end{pmatrix}\,, \label{hradcytcyt}$$ where $z\negthinspace\in\negthinspace\overline{\mathbb{C}}$ is a complex parameter that is related to the risk asymmetry parameter $m$, $m\negthinspace=\negthinspace z\overline{z}$. Changes in the absolute value of the parameter $z$ correspond to different proportions of distribution of the risk between buying and selling transactions. Changes in the phase of the parameter $z$ may result in mixing of supply and demand aspects of transactions. For example, the phase shift $\tfrac{\pi}{4}$ leads to the new canonical variables $\mathcal{P'}=\mathcal{Y}:=\tfrac{1}{\sqrt{2}}\,(\mathcal{P} \negthinspace-\negthinspace\mathcal{Q})$ and $\mathcal{Q'}=\mathcal{Z}:=\tfrac{1}{\sqrt{2}}\,(\mathcal{P}\negthinspace +\negthinspace\mathcal{Q})$. The new variable $\mathcal{Y}$ describes arithmetic mean deviation of the logarithm of price from its expectation value in trading in the asset $\mathfrak{G}$. Accordingly, the new variable $\mathcal{Z}$ describes the profit made in one buying-selling cycle in trading in the asset $\mathfrak{G}$. Note that the normalization if forced by the requirement of canonicality of transformations. In the following we will use the Schrödinger-like picture for description of strategies. Therefore strategies will be functions of the variable $y$ being the properly normalized value of the logarithm of the market price of the asset in question. The dual description in terms of the profit variable $z$ is also possible and does not require any modification due to the symmetrical form of the risk inclination operator $H(\mathcal{Y},\mathcal{Z})$ [@4; @8]. The player’s strategy represents his/her actual position on the market. To insist on a distinction, we will define tactics as the way the player decides to change his/her strategy according to the acquired information, experience and so on. Therefore, in our approach, strategies are represented by vectors in Hilbert space and tactics are linear transformations acting on strategies (not necessary unitary because some information can drastically change the players behaviour!) Diffusion of prices =================== Classical description of the time evolution of a logarithm of price of an asset is known as the Bachelier model. This model is based on the supposition that the probability density of the logarithm of price fulfills a diffusion equation with an arbitrage forbidding drift. Therefore we will suppose that the (quantum) expectation value of the arithmetic mean of the logarithm of price of an asset $E(\mathcal{Y})$ is a random variable described by the Bachelier model. So the price variable $y$ has the properties of a particle performing random walk that can be described as Brown particle at large time scales $t$ and as Rayleigh particle at short time scales $\gamma$ [@16]. The superposition of these two motions gives correct description of the behaviour of the random variable $y$. It seems that the parameters $t$ and $\gamma $ should be treated as independent variables because the first one parameterizes evolution of the “market equilibrium state” and the second one parameterizes the “quantum” process of reaching the market equilibrium state [@7; @17]. Earlier [@14], we have introduced canonical portfolios as equivalence classes of portfolios having assets with equal proportions. An external observer describes the moves performed by the portfolio manager as a draw in the following lottery. Let $p_{n}, n=1,...,N$ be the probability of the purchase of $w_{n}$ units of the $n$-th asset. Our analysis lead us to Gibbs-like probability distribution: $$p_{n}\left( c_{0},\dots ,c_{N}\right) = \frac{ \exp \left( \beta c_{n}w_{n} \right) } {\sum _{k=0}^{N}\exp \left( \beta c_{k}w_{k} \right) }\label{pkanon}.$$ The coefficient $c_{n}$ denotes the present relative price of a unit of the asset $\mathfrak{G}_{n}$, $c_{n}=\frac{u_{n}}{\overline{u}_{n}}$ where $u_{n}$ is the present price of the $n$-th asset and $\overline{u}_{n}$ its price at the moment of drawing. Now let us consider an analogue of canonical Gibbs distribution function $$\text{e}^{-\gamma H(\mathcal{Y},\mathcal{Z})}, \label{htakryk}$$ where we have denoted the Lagrange multiplier by $\gamma$ instead of the more customary $\beta$ for later convenience. The analysis performed in Ref. [@1; @15] allows to interpret $(\ref{htakryk})$ as non-unitary tactics leading to a new strategy[^4]: $$\text{e}^{-\gamma H(\mathcal{Y},\mathcal{Z})}|\psi\rangle = |\psi ' \rangle. \label{dzialanie}$$ Therefore the parameter $\gamma$ can be interpreted as the inverse of the temperature ($\beta \sim (temperature)^{-1}$) of a canonical portfolio that represents strategies of traders having the same risk inclination (cf Ref.[@14]). These traders adapt such tactics that the resulting strategy form a ground state of the risk inclination operator $ H(\mathcal{Y},\mathcal{Z})$ (that is they aim at the minimal eigenvalue). We call tactics characterized by constant inclination to risk, $E(H(\mathcal{P},\mathcal{Q}))={const}$ and maximal entropy thermal tactics. Regardless of the possible interpretations, adoption of the tactics $(\ref{htakryk})$ means that traders have in view minimization of the risk (within the available information on the market). It is convenient to adopt such a normalization (we are free to fix the Lagrange multiplier) of the operator of the tactics so that the resulting strategy is its fixed point. This normalization preserves the additivity property, $ \mathcal{R}_{\gamma_1+\gamma_2}\negthinspace=\negthinspace \mathcal{R}_{\gamma_2}\mathcal{R}_{\gamma_1} $ and allows consecutive (iterative) implementing of the tactics. The operator representing such thermal tactics takes the form ($\omega\negthinspace=\negthinspace\hslash_E\negthinspace=\negthinspace1$) $$\mathcal{R}_\gamma:=\text{e}^{-\gamma (H(\mathcal{Y},\mathcal{Z})-\frac{1}{2})}\, .$$ Note that the operator $H(\mathcal{Y},\mathcal{Z})-\frac{1}{2}$ annihilate the minimal risk strategy (remember that the minimal eigenvalue is $\frac{1}{2}$). The integral representation of the operator $\mathcal{R}_\gamma$ (heat kernel) acting on strategies $\langle y|\psi\rangle\negthinspace\in\negthinspace\mathcal{L}^2$ reads: $$\langle y|\mathcal{R}_\gamma\psi\rangle=\int_{-\infty}^{\infty} \negthinspace\negthinspace \mathcal{R}_\gamma(y,y') \langle y'|\psi\rangle dy', \label{forhradof}$$ where (the Mehler formula [@18]) $$\mathcal{R}_\gamma(y,y')=\tfrac{1}{\sqrt{\pi(1-\text{e}^{-2\gamma})}}\,\,\text{e}^{-\frac{y^2- {y'}^2}{2^{\vphantom{2}}}-\frac{(\text{e}^{-\gamma}y-y')^2}{1-\text{e}^{-2\gamma}}}\,.$$ $\mathcal{R}_\gamma(y,y')$ gives the probability density of Rayleigh particle changing its velocity from $y'$ to $y$ during the time $\gamma$. Therefore the fixed point condition for the minimal risk strategy takes the form $$\int_{-\infty}^{\infty}\mathcal{R}_\gamma(y,y')\, \text{e}^{\frac{y^2-{y'}^2}{2}}dy'=1\,.$$ &gt;From the mathematical point of view, the tactics $\mathcal{R}_\gamma$ is simply an Ornstein-Uhlenbeck process. It is possible to construct such a representation of the Hilbert space $\mathcal{L}^2$ so that the fixed point of the thermal tactics corresponds to a constant function. This is convenient because the “functional” properties are “shifted” to the probability measure $\widetilde{dy}\negthinspace:=\negthinspace \tfrac{1}{\sqrt{\pi}}\,\text{e}^{-y^2}\negthinspace dy$. After the transformation $\mathcal{L}^2(dy)\negthinspace\rightarrow\negthinspace \mathcal{L}^2(\widetilde{dy})$, proper vectors of the risk inclination operator are given by Hermite polynomials (the transformation in question reduces to the multiplication of vectors in $\mathcal{L}^2$ by the function $\sqrt[4]{\pi}\,\text{e}^{\tfrac{y^2}{2}}$). Now Eq $(\ref{forhradof})$ takes the form: $$\widetilde{\langle y|\mathcal{R}_\gamma\psi\rangle}\int_{-\infty}^{\infty}\negthinspace\negthinspace \widetilde{\mathcal{R}}_\gamma(y,y')\, \widetilde{\langle y'|\psi\rangle}\, \widetilde{dy'}\,,$$ where $$\widetilde{\mathcal{R}}_\gamma(y,y'):\tfrac{1}{\sqrt{1-\text{e}^{-2\gamma}}}\,\text{e}^{{y'^2}- \frac{(\text{e}^{-\gamma}y-y')^2}{1-\text{e}^{-2\gamma}}}\,.$$ In this way we get the usual description of the Ornstein-Uhlenbeck process in terms of the kernel $\widetilde{\mathcal{R}}_\gamma(y,y')$ being a solution to the Fokker-Planck equation [@19]. “Classical” picture of quantum diffusion ======================================== Let us consider the integral kernel of one-dimensional exponent of the Laplace operator $\text{e}^{-\frac{\gamma}{2}\, \frac{\partial^2}{\partial y^2}}\negthinspace$ representing the fundamental solution of the diffusion equation $$\frac{\partial f(y,\gamma)}{\partial \gamma}=\frac{1}{2}\, \frac{\partial^2 f(y,\gamma)}{\partial y^2}\,\,.$$ The kernel takes the following form $$\mathcal{R}^0_\gamma(y,y'):=\tfrac{1}{\sqrt{2\pi\gamma}}\, \text{e}^{-\frac{(y-y')^2}{2\gamma}}\,,$$ and the appropriate measure invariant with respect to $\mathcal{R}^0_\gamma(y,y')$ reads: $$dy_0:=\tfrac{1}{\sqrt{\pi\gamma}}\, \text{e}^{-\frac{y^2}{2\gamma}}dy\,.$$ The corresponding stochastic process is known as the Wiener-Bachelier process. In physical applications the variables $y$ and $\gamma$ are interpreted as position and time, respectively (Brownian motion). Let us define the operators $\mathcal{X}_k$ acting on $\mathcal{L}^2$ as multiplications by functions $x_k(y(\gamma_k))$ for successive steps $k\negthinspace=\negthinspace1,\ldots,n$ such that $-\tfrac{\gamma}{2}\negthinspace\leq\negthinspace\gamma_1\negthinspace\leq \negthinspace\ldots\negthinspace\leq\negthinspace\gamma_n\negthinspace \leq\negthinspace\frac{\gamma}{2}$. The corresponding (conditional) Wiener measure $dW^\gamma_{y,y'}$ for $ y\negthinspace=\negthinspace y(-\tfrac{\gamma}{2})$ and $y'\negthinspace=\negthinspace y(\tfrac{\gamma}{2})$ is given by the operator $$\int\negthinspace\prod\limits_{k=1}^n x_k(y(\gamma_k))\,dW^\gamma_{y,y'}:=\Bigl( \text{e}^{-\tfrac{\gamma_1+\gamma/2}{2}\tfrac{\partial^2}{\partial y^2}} \mathcal{X}_1 \text{e}^{-\tfrac{\gamma_2-\gamma_1}{2}\tfrac{\partial^2}{\partial y^2}} \mathcal{X}_2\cdots\mathcal{X}_n \text{e}^{-\tfrac{\gamma/2-\gamma_n}{2}\tfrac{\partial^2}{\partial y^2}} \Bigr)(y,y')\,.$$ If the operators $\mathcal{X}_k$ are constant ($x_k(y(\gamma_k))\negthinspace\equiv\negthinspace1$) then $$\int dW^\gamma_{y,y'}=\mathcal{R}^0_\gamma(y,y')\,.$$ The Wiener measure allows to rewrite the integral kernel of the thermal tactics in the form [@18] $$\mathcal{R}_\gamma(y,y')=\int\mathcal{T}^{\prime }\,\text{e}^{-\int \limits_{-\gamma/2}^{\gamma/2} \frac{y^2(\gamma')-1}{2}\,\,d\gamma'}dW^\gamma_{y,y'} \label{feynmankac}$$ known as the Feynman-Kac formula where $\mathcal{T}^{\prime }$ is the anti-time ordering operator. According to the quantum interpretation of path integrals [@20] we can expand the exponent function in Eq $\ref{feynmankac}$ to get “quantum” perturbative corrections to the Bachelier model that result interference[^5] of all possible classical scenarios of profit changes in time spread $\gamma$, cf [@21].[^6] These quantum corrections are unimportant for short time intervals $\gamma\negthinspace\ll\negthinspace 1$ and the Ornstein-Uhlenbeck process resembles the Wiener-Bachelier one. This happens, for example, for “high temperature” thermal tactics and for disorientated markets (traders)[^7]. In effect, due to the cumulativity of dispersion during averaging for normal distribution $\eta(x,\sigma^2)$ $$\int_{-\infty}^\infty\negthinspace\negthinspace\eta(x\negthinspace+\negthinspace y,\sigma^2_1)\, \eta(y,\sigma^2_2)\,dy=\eta(x, \sigma^2_1\negthinspace+\sigma^2_2)$$ the whole quantum random walk parameterized by $\gamma$ can be incorporated additively into the mobility parameter of the classical Bachelier model. This explains changes in mobility of the logarithm of prices in the Bachelier model that follow, for example, from changes in the tactics temperature or received information. Therefore the intriguing phenomenon of market prices evolution can be interpreted in a reductionistic way as a quantum process. In this case the Bachelier model is a consequence of a short-time tactics adopted by the smithonian invisible hand (under the perfect concurrence assumption all other traders can be considered as an abstract trader dealing with any single real trader) [@4]. From the quantum point of view the Bachelier behaviour follows from short-time interference of tactics adopted (paths followed) by the rest of the world considered as a single trader. Collected information about the market results after time $\gamma\negthinspace\ll\negthinspace1$ in the change of tactics that should lead the trader the strategy being a ground state of the risk inclination operator (localized in the vicinity of corrected expectation value of the price of the asset in question). This should be done in such a way that the actual price of the asset is equal to the expected price corrected by the risk-free rate of return (arbitrage free martingale)[@22]. Both interpretations of the chaotic movement of market prices imply that Ornstein-Uhlenbeck corrections to the Bachelier model should qualitatively matter only for large $\gamma$ scales. An attentive reader have certainly noticed that we have supposed that the drift of the logarithm of the price of an asset must be a martingale (that is typical of financial mathematics [@22]). Now suppose that we live in some imaginary state where the ruler is in a position to decree the exchange rate between the local currency $\mathfrak{G}$ and some other currency $\mathfrak{G'}$. The value of the logarithm of the price of $\mathfrak{G}$ (denoted by $\mathfrak{n}$) is proportional to the result of measurement of position of a one dimensional Brown particle. Any owner of $\mathfrak{G}$ will praise the ruler for such policy and prefer $\mathfrak{G}$ to $\mathfrak{G'}$ because the the price of $\mathfrak{G}$ in units of $\mathfrak{G'}$ will, on average, raise (the process $\exp \mathfrak{n}$ is sub-martingale). For the same reasons a foreigner will be content with preferring $\mathfrak{G'}$ to $\mathfrak{G}$. This currency preference paradoxical property of price drifts suggest that the common assumption about logarithms of assets prices being a martingale should be carefully analyzed prior to investment. If one measures future profits from possessing $\mathfrak{G}$ with the anticipated change in quotation of $\mathfrak{n}$ then the paradox is solved and expectation values of the profits from possessing $\mathfrak{G}$ or $\mathfrak{G'}$ are equal to zero. Therefore the common reservations on using of logarithms of exchange rates as martingales to avoid the Siegel’s paradox is fulfilled [@22] (cf Bernoulli’s solution to the Petersburg paradox [@23]). Note that if we suppose that the price of an asset and not its logarithms is a martingale then the proposed model of quantum price diffusion remains valid if we suppose that the observer’s reference system drifts with a suitably adjusted constant velocity (in logarithm of price variable).\ Final remarks ============= We have proposed a model of price movements that is inspired by quantum mechanical evolution of physical particles. The main novelty is to use complex amplitudes whose squared modules describe the probabilities. Therefore such phenomena as interference of tactics (strategies) are possible. The analysis shows the movement of market prices imply that the Bachelier behaviour follows from short-time interference of tactics adopted by the rest of the world considered as a single trader and the Ornstein-Uhlenbeck corrections to the Bachelier model should qualitatively matter only for large time scales. Roughly speaking, traders dealing in the asset $\mathfrak{G}$ act as a sort of (quantum) tomograph and their strategies can be reproduced from the corresponding Wigner functions in a way analogous to the mathematical tomography used in medicine. Therefore we can speculate about possibilities using the experience acquired in medicine, geophysics and radioastronomy to investigate intricacies of supply and demand curves.\ [**Acknowledgments**]{} This paper has been supported by the [**Polish Ministry of Scientific Research and Information Technology**]{} under the (solicited) grant No [**PBZ-MIN-008/P03/2003**]{}. [99]{} D. Meyer, [*Quantum strategies*]{}, [*Phys. Rev. Lett. *]{}[**82**]{} (1999) 1052. J. Eisert, M. Wilkens, and M. Lewenstein, [*Quantum games and quantum strategies*]{}, [*Phys. Rev. Lett.*]{} [**83**]{} (1999) 3077. E. W. Piotrowski, J. Sładkowski, [*The next stage: quantum game theory*]{}, in [*“Progress in Mathematical Physics Research”*]{}, Nova Science Publishers, Inc. (2004); quant-ph/0308027. E. W. Piotrowski and J. Sładkowski, [*Quantum market games*]{}, [*Physica*]{} [**A 312**]{} (2002) 208; quant-ph/0104006. H. E. Stanley et al, [*Quantifying economic fluctuations*]{}, [*Physica A*]{} 302 (2001) 126. M. A. Nielsen, I. L. Chuang, [*Quantum Computation and Quantum Information*]{}, Cambridge University Press, Cambridge (2000). E. W. Piotrowski and J. Sładkowski, [*The Merchandising Mathematician Model*]{}, [*Physica*]{} [**A 318**]{} (2003) 496, cond-mat/0102174. E. W. Piotrowski and J. Sładkowski, [*Quantum-like approach to financial risk: quantum anthropic principle*]{}, [*Acta Phys. Pol. *]{}[**B32**]{} (2001) 3873; quant-ph/0110046. S. Helgason, [*The Radon transform*]{}, Birkhäuser, Boston (1999). E. W. Piotrowski and J. Sładkowski, [*Quantum bargaining games*]{}, [*Physica A*]{} 308 (2002) 391; quant-ph/0107140. V. I. Man’ko and R. V. Mendes, [*Non-commutative time frequency tomography*]{}, [*Phys. Lett. *]{}[**A 263**]{} (1999) 53; physics/9712022. V. I. Tatarskii, Uspiekhi Fiz. Nauk [**139**]{} (1983) 587. V. I. Man’ko, [*Conventional quantum mechanics without wave function and density matrix*]{}, in [*New perspectives in quantum mechanics*]{}, eds. S. Hacyan et al, AIP (1999). E. W. Piotrowski and J. Sładkowski, [*The thermodynamics of portfolios*]{}, [*Acta Phys. Pol. *]{}[[**B**]{} 32 (2001) 597]{}. E. W. Piotrowski, J. Sładkowski and J. Syska, [*Interference of quantum strategies*]{}, [*Physica* ]{}[**A 318**]{} (2003) 516; quant-ph/0205087. N. G. van Kampen, [*Stochastic Processes in Physics and Chemistry*]{}, Elsevier, New York (1983). E. E. Haven, [*A discussion on embedding the Black-Sholes option pricing model in a quantum physics setting*]{}, [*Physica A*]{} 304 (2002) 507. J. Glimm and A. Jaffe, [*Quantum Physics. A Functional Integral Point of View*]{}, Springer-Verlag, New York (1981). , Dordrecht, Amsterdam (1997). H. Kleinert, [*Path integrals in quantum mechanics, statistics and polyner physics*]{}, World Scientific, Singapore (1995). M. Kac, [*Probability and Related Topics in Physical Sciences*]{}, Interscience, New York (1959). M. P. Taylor, [*The economics of exchange rates*]{}, [*Journ. of Econ. Lit.*]{} [**33**]{}(1995) 13. M.H. DeGroot, [*Optimal Statistical Decisions*]{}, McGraw Hill, New York (1970). [^1]: We use the standard Dirac notation. The symbol $|\ \rangle$ with a letter $\psi$ in it denoting a vector parameterized by $\psi$ is called a [*ket*]{}; the symbol $\langle\ |\negthinspace$ with a letter in it is called a [*bra*]{}. Actually a [*bra*]{} is a dual vector to the corresponding [*ket*]{}. Therefore scalar products of vectors take the form $\langle \phi |\psi\rangle\negthinspace$ ([*bracket*]{}) and the expectation value of an operator $A$ in the state $|\psi\rangle\negthinspace$ is given by $\langle \psi |A\psi\rangle\negthinspace$. [^2]: One must remember that switching roles of $p$ and $q$ must be accompanied by switching $u$ with $v$ [^3]: Eigenvalues of the operator $H(\mathcal{P}_k,\mathcal{Q}_k)$ can be parameterized by natural numbers including 0. The $n-th$ eigenvalue is equal to $n +\frac{1}{2}$ in units of $\hslash_E$. The lowest eigenvalue state is called the ground state; the others are called exited states. [^4]: If the numbers $c_{n}w_{n}$ are eigenvalues of some bounded below Hermitian operator $H$ then we get the statistical operator $\frac{\text{e}^{ -\beta H}}{Tr \text{e}^{ -\beta H} }$. The expectation value of any observable $\mathcal{X}$ is given by $\langle\mathcal{X}\rangle_H :=\frac{Tr \mathcal{X}\text{e}^{ -\beta H}}{Tr \text{e}^{ -\beta H }}$. [^5]: Roughly speaking path intragrals sum up all possible ways of evolution (“paths”) with phases (weights) resulting from interaction. [^6]: Note that in the probability theory one measures risk associated with a random variable by squared standard deviation. According to this we could define the complex profit operator $\mathcal{A}:=\tfrac{1}{\sqrt{2}}\,(\mathcal{Y}+\text{i}\,\mathcal{Z})$. The appropriate risk operator would take the form $H(\mathcal{A}^\dag,\mathcal{A}) =\mathcal{A}^\dag\mathcal{A} +\tfrac{1}{2}$. [^7]: That is the parameter $\gamma$ is very small (but positive).

ArXiv

--- abstract: | A model with a singular forward scattering amplitude for particles with opposite spins in d spatial dimensions is proposed and solved by using the bosonization transformation. This interacting potential leads to the spin-charge separation. Thermal properties at low temperature for this Luttinger liquid are discussed. Also, the explicit form of the single-electron Green function is found; it has square-root branch cut. New fermion field operators are defined; they describe holons and spinons as the elementary excitations. Their single particle Green functions possess pseudoparticle properties. Using these operators the spin-charge separated Hamiltonian for an ideal gases of holons and spinons is derived and reflects an inverse (fermionization) transformation.\ PACS Nos.71.10.+x, 71.27.+a address: | (a) Institute of Theoretical Physics, Warsaw University, ul. Hoża 69, 00-681 Warszawa, Poland\ (b) Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland author: - 'Krzysztof Byczuk$^{a}$ and Jozef Spałek$^{b,a}$ [^1]' title: 'Spin-Charge Separated Luttinger Liquid in Arbitrary Spatial Dimensions ' --- It was suggested [@and1] that the properties of normal state of high-temperature superconductors are properly described by Luttinger liquid, where the spin and charge degrees of freedom are separated. In one-dimensional systems this phenomenon is well understood [@rew]. However, in two and three dimensions the present understanding of the spin-charge separation is rather poor. In this letter we formulate and solve exactly a d-dimensional model exhibiting the spin-charge separation, as well as discuss its thermal and dynamic properties. A natural approach to study spin-charge decoupling phenomena is the bosonization transformation, generalized recently to the multidimensional space situation [@hal]. Here we adopt the operator version of the bosonization developed in Ref.[@how]. The starting assumption in this method is the existence of the Fermi surface (FS) defined as a collection of points at which the momentum distribution function has singularities at zero temperature ($T=0$). These points are parameterized by vectors $\bf S$ and ${\bf T}$, which label a finite and a locally flat (rectangular in shape) mesh of grid points on FS with spacing $\Lambda \ll k_F$ between them [@how; @hal]. Introducing coarse-grained density fluctuation operators $J_{\sigma}({\bf S},{\bf q})$, defined in boxes centered at each FS point and having surface area $\Lambda^{d-1}$ and the thicknesses $\lambda/2$ both above and below it, one can transform the effective Hamiltonian for interacting fermions into an effective Hamiltonian for free bosons. Explicitly, it takes the general form $$H = \frac{1}{2} \sum_{{\bf S}, {\bf T}} \sum_{\bf q} \sum_{\sigma \sigma'} \Gamma_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q}) J_{\sigma}({\bf S}, {\bf q})J_{\sigma'}({\bf T}, - {\bf q}), \label{e7}$$ where $ \Gamma_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q}) = v_F( {\bf S}) \frac{1}{\Omega} \delta_{\sigma, \sigma'} \delta^{d-1}_{{\bf S}, {\bf T}} + \frac{1}{L^d} V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q}) %\label{e8} $ is the positive defined matrix element. The first term corresponds to the kinetic energy part of the original fermionic Hamiltonian with linearized dispersion relation close to FS, whereas the second term is the effective (low-energy) interaction between the particles with spins $\sigma$ and $\sigma'$. The geometrical factor $\Omega = \Lambda^{d-1} (\frac{L}{2 \pi})^d$ depends on the system dimension $d$. The explicit expression for $V_{\sigma \sigma'}({\bf S},{\bf T}, {\bf q})$ is generally derived by transforming out the high energy modes in the fermionic Hamiltonian [@shankar]. Obviously, this procedure can also change the Fermi velocity $v_F({\bf S})$. Therefore, we take $v_F({\bf S})$ as an effective value obtained after removing the high-energy degrees of freedom. As shown below, we can characterize the universal properties of fermions knowing only the asymptotic behavior of the interaction potential $V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q})$ in the thermodynamic limit. If the system is invariant under the time reversal, the interaction part must be explicitly symmetric under this operation, which means that $ V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q}) = V_{\bar{\sigma} \bar{\sigma}'}(-{\bf S}, -{\bf T}, - {\bf q}) $. Furthermore, if FS is also invariant under the reflections ${\bf S} \rightarrow - {\bf S}$ etc., the last condition becomes $ V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q}) = V_{\bar{\sigma} \bar{\sigma}'}({\bf S}, {\bf T}, {\bf q}) $. In that case $V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q})$ depends only on the relative orientation of the spins $\sigma$ and $\sigma'$; there are only two independent components: $V_{\sigma \sigma}$ for parallel spins and $V_{\sigma \bar{\sigma}}$ for antiparallel spins. It is convenient to introduce the symmetric and the antisymmetric combinations: $ V^{c,s}({\bf S},{\bf T},{\bf q}) \equiv \frac{1}{2} (V_{\sigma \sigma}({\bf S}, {\bf T}, {\bf q}) \pm V_{\sigma \bar{\sigma}}({\bf S}, {\bf T}, {\bf q}) ), $ where $c$ and $s$ superscripts corresponds to “$\pm$” signs, respectively. Correspondingly, we define the currents $ J_{c,s} ({\bf S}, {\bf q}) \equiv \frac{1}{\sqrt{2}} ( J_{\uparrow}({\bf S}, {\bf q}) \pm J_{\downarrow}({\bf S}, {\bf q})), %\label{e13} $ which describe the charge and the spin density fluctuations, respectively. Then, the original Hamiltonian (\[e7\]) takes the form $$H = \sum_{\alpha = c,s} \frac{1}{2} \sum_{{\bf S}} v_F({\bf S}) \frac{1}{\Omega} \sum_{{\bf q}} J_{\alpha}({\bf S}, {\bf q}) J_{\alpha}({\bf S}, - {\bf q}) + \frac{1}{L^d} \sum_{{\bf S}, {\bf T}} \sum_{{\bf q}} V^{\alpha}({\bf S}, {\bf T}, {\bf q}) J_{\alpha}({\bf S}, {\bf q}) J_{\alpha}( {\bf T}, -{\bf q}). \label{e15}$$ The $\alpha =c$ term describes the dynamics of the charge density fluctuations in the system, whereas the $\alpha = s$ term deals with the longitudinal spin density fluctuations. There are no terms which mix the degrees of freedom (i.e. $ \sim J_c \cdot J_s $) because the Hamiltonian is assumed to be invariant under the spin flip (i.e. $J_c \rightarrow J_c$, $J_s \rightarrow - J_s$). We can check out that for the noninteracting case, i.e. for $V^{\alpha}\equiv 0$, both the spin and the charge density fluctuations propagate with the same velocity $v_F({\bf S})$. The commutation relation for density fluctuation operators take the following form $ \left[ J_{\alpha}({\bf S}, {\bf q}) , J_{\beta}({\bf T}, {\bf p}) \right] = \delta_{\alpha \beta} \delta^{d-1}_{{\bf S}, {\bf T}} \delta^d_{ {\bf p} + {\bf q},0}\: \Omega \: {\bf q} \cdot \hat{n}_{\bf S}, %\label{e19} $ where $\alpha , \beta = c,s$. Thus, the two branches of density fluctuations are independent of each other. The commutation relations become equivalent to those obeyed by the bosonic harmonic-oscillator creation and annihilation operators after rescaling them by the factor on the right hand side, i.e. by defining the creation ($a_{\alpha}^+$) and the annihilation ($a_{\alpha}$) operators according to $$J_{\alpha}({\bf S}, {\bf q}) = \theta( \hat{n}_{{\bf S}} \cdot {\bf q}) \sqrt{\Omega \hat{n}_{{\bf S}} \cdot {\bf q}} \; a_{\alpha}({\bf S}, {\bf q}) + \theta(- \hat{n} _{\bf S} \cdot {\bf q}) \sqrt{-\Omega \hat{n}_{{\bf S}} \cdot {\bf q}} \; a_{\alpha}^+({\bf S},- {\bf q}),$$ where $\theta(x) $ is the step function. Then $ \left[ a_{\alpha}({\bf S}, {\bf q}), a_{\beta}^+({\bf T}, {\bf p}) \right] = \delta^{d-1}_{{\bf S}, {\bf T}} \delta_{{\bf p},{\bf q}}^d \delta_{\alpha \beta}. $ In terms of the bosonic harmonic-oscillator creation and annihilation operators the Hamiltonian (\[e15\]) is $$H = \sum_{\alpha = c,s} \sum_{{\bf S}, {\bf T}} \frac{1}{2} \sum_{{\bf q}} \left\{ \theta( \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(\hat{n}_{{\bf T}}\cdot {\bf q}) \sqrt{ ({\bf v}_F({\bf S}) \cdot {\bf q}) ({\bf v}_F({\bf T}) \cdot {\bf q})} \; \times \right. \label{e22}$$ $$\left( \delta_{{\bf S}, {\bf T}}^{d-1} + \frac{\Lambda^{d-1}}{(2 \pi)^d} \frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}} \right) a_{\alpha}^+({\bf S},{\bf q}) \: a_{\alpha}({\bf T}, {\bf q}) \; +$$ $$\theta(- \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(-\hat{n}_{{\bf T}}\cdot {\bf q}) \sqrt{ (-{\bf v}_F({\bf S}) \cdot {\bf q}) (-{\bf v}_F({\bf T}) \cdot {\bf q})} \; \times$$ $$\left( \delta_{{\bf S}, {\bf T}}^{d-1} + \frac{\Lambda^{d-1} }{(2 \pi)^d} \frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}} \right) a_{\alpha}^+({\bf S},-{\bf q})\: a_{\alpha}({\bf T},- {\bf q}) \; +$$ $$\theta( \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(-\hat{n}_{{\bf T}}\cdot {\bf q}) \sqrt{ ({\bf v}_F({\bf S}) \cdot {\bf q}) (-{\bf v}_F({\bf T}) \cdot {\bf q})} \left( \frac{\Lambda^{d-1}}{(2 \pi)^d} \frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}} \right) a_{\alpha}({\bf S},{\bf q})\: a_{\alpha}({\bf T}, -{\bf q}) \; +$$ $$\theta(- \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(\hat{n}_{{\bf T}}\cdot {\bf q}) \sqrt{ (-{\bf v}_F({\bf S}) \cdot {\bf q}) ({\bf v}_F({\bf T}) \cdot {\bf q})} \left. \left( \frac{ \Lambda^{d-1}}{(2 \pi)^d} \frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}} \right) a_{\alpha}^+({\bf S},-{\bf q})\: a_{\alpha}^+({\bf T}, {\bf q}) \right\}.$$ We have shown previously [@my; @phd] that a universal behavior of the system takes place and depends on how the interaction part of the effective Hamiltonian (\[e22\]) behaves in the scaling (thermodynamic) limit $\Lambda \rightarrow 0$. If the interaction in this limit has a singular power-law behavior, i.e. $V_{\sigma,\bar{\sigma}}({\bf S},{\bf T},{\bf q})$ scales for ${\bf S} \rightarrow {\bf T}$ as $\Lambda^{d-1} / \Lambda^{\eta}$, then: (i) for $\eta < d-1$ the Landau Fermi liquid (FL) fixed point is stable; (ii) for $\eta > d-1$ the statistical spin liquid (SSL) is the stable fixed point; (iii) the case $\eta = d-1$ leads to the Luttinger liquid (LL) type of behavior. Properties of FL in $d=3$ and $2$ are well known [@lfl]. SSL was discussed in Refs.[@ssl; @my; @phd]. Here we concentrate on the properties of the spin-charge separated LL in an arbitrary spatial dimension. To examine the basic properties of Luttinger liquid we assume the following form of the effective interaction in (\[e7\]): $$V_{\uparrow \uparrow} ({\bf S}, {\bf T}, {\bf q}) = f_{\uparrow \uparrow}({\bf S}, {\bf T}, {\bf q}), %\label{e33}$$ and $$V_{\uparrow \downarrow} = \left\{ \begin{array}{ccc} \frac{1}{\Lambda^{d-1}} \tilde{g}({\bf S}, {\bf q}) & for & {\bf S} = {\bf T} \\ f_{\uparrow \downarrow} ({\bf S}, {\bf T}, {\bf q}) & for & {\bf S} \neq {\bf T} , \end{array} \right. \label{e34}$$ where $f_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q})$ and $\tilde{g}({\bf S}, {\bf q}) $ are nonsingular functions. In this manner, we model the situation with a divergent forward scattering amplitude (taking place for ${\bf S} = {\bf T}$ and ${\bf q}=0$), which leads to the LL fixed point. Also, since we examine only the low-energy limit, we can expand the nonsingular part according to: $ \tilde{g}({\bf S}, {\bf q}) = g_0({\bf S}) + g({\bf S}) \hat{n}_{{\bf S}} \cdot {\bf q} + ... $, and omit the higher-order terms. Substituting this expansion into (\[e34\]) we obtain in the thermodynamic limit ($\Lambda \rightarrow 0$) the Hamiltonian with two branches of free bosons excitations, each with a different form of the kinetic energy. Namely, defining $ v_F^{c,s} ({\bf S}) \equiv v_F({\bf S}) \pm g({\bf S}), $ the Hamiltonian (\[e22\]) simplifies to $$H = \sum_{\alpha = c , s} \sum_{{\bf S}, {\bf q}> 0} ({\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q} ) \; a_{\alpha}^+ ({\bf S}, {\bf q}) \; a_{\alpha}({\bf S}, {\bf q}). \label{e40}$$ The charge and the spin fluctuations propagate in the system with different velocities and express the separation of the corresponding degrees of freedom. One of the velocities diminishes and the other increases. In the extreme case one of them vanishes transforming into a soft mode, signalling a phase transition in the corresponding channel, $c$ or $s$. The character of this transition will be determined by the regular part. The spin-charge decoupling takes always place in the one-dimensional systems of interacting fermions in the low energy limit [@rew]. In a system of higher dimension the potential must be singular. Obviously, this is a simplified model. More appealing form would be $ V_{\uparrow \downarrow}({\bf S}, {\bf T}, {\bf q}) \sim \frac{1}{ |{\bf S} - {\bf T}|^{\eta} + |{\bf q}|^{\eta} }, $ which clearly diverges in the forward direction (for ${\bf q} \rightarrow 0$). Such a singular effective potential in $d=2$ was discussed in Refs.[@and2; @fqhe; @stamp] in the context of the high-temperature superconductors and the fractional quantum Hall effect. Our choice (\[e34\]) is modeled by the most divergent term of this potential, and physically means that the fermions with antiparalel spins interact through the forward scattering processes along the radial FS direction only. Since the Hamiltonian (\[e40\]) is diagonal we can calculate the internal energy of the system and then the specific heat. The energy is $$U = E_0 + \sum_{\alpha=c,s} \sum_{{\bf S}} \sum_{{\bf q}} \frac{ {\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q}} {e^{\beta {\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q}} - 1}, \label{e42}$$ where we utilize the fact that now we are dealing with bosons. $E_0$ is the ground state energy of the initial noninteracting system. This term must be incorporated because in the bosonization procedure the energy is measured with respect to the Fermi energy. Also, the chemical potential $\mu$ does not appear in (\[e42\]) because the number of bosons is not conserved. In other words, those fields describe the system particle-hole excitations. This is also the reason why those bosons cannot condense. For an isotropic system the velocities do not depend on the index labeling the point of FS, i.e. $v_F^{\alpha}({\bf S}) = v_F^{\alpha}$, and we find that $ U = \frac{\pi^2}{6} (k_B T)^2 \left( \frac{L}{2 \pi} \right) ^d \frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1} \sum_{\alpha} \frac{1}{v_F^{\alpha}} + E_0. $ Hence, the specific heat is $ C_V = \frac{1}{L^d} \frac{\partial U}{\partial T} = \frac{\pi^2}{3} (k_B )^2 T \left( \frac{1}{2 \pi} \right) ^d \frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1} \sum_{\alpha} \frac{1}{v_F^{\alpha}}. %\label{e44} $ Introducing the density of states for the charge and the spin excitations on FS: $ \rho_{\alpha}(\epsilon_F) = \left( \frac{1}{2 \pi} \right) ^d \frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1} \frac{1}{v_F^{\alpha}}, $ we have that $ C_V = \frac{\pi^2}{3} (k_B )^2 T \sum_{\alpha} \rho_{\alpha} (\epsilon_F) \equiv \gamma_{LL} T. $ The spin-charge separated liquid has a linear specific heat, with $ \gamma_{LL} \sim \frac{1}{v_F^c} + \frac{1}{v_F^s}. $ The linear specific heat is thus a general characteristic following from the existence of the FS, independently of the statistical properties of the particles. In the limiting case $v_F^c = v_F^s = v_F$ we recover the FL result. Similarly, we find the free energy in the form $ F= E_0 - \frac{\pi^2}{6} (k_BT)^2 \sum_{\alpha} \rho_{\alpha}(\epsilon_F), $ and the low-$T$ entropy $ S = \frac{U-F}{T} = \frac{\pi^2}{3} (k_B)^2 T \sum_{\alpha} \rho_{\alpha}(\epsilon_F), $ which coincides with the specific heat. Thus the low-T thermal properties of the present Luttinger spin-charge separated liquid are very similar to those of free fermions. The only difference is in the form of the density of states at FS. However, the dynamic properties in the LL case are quite unique, as we discuss next. We define the fermion correlation function as $$G^>_{\sigma} ({\bf S}, {\bf x}, t > 0) = < \psi_{\sigma}({\bf S}, {\bf x}, t) \psi_{\sigma}^+ ({\bf S}, 0, 0)>. \label{e50}$$ To derive an explicit form of this function we substitute the bosonized Fermi field operators [@how] $ \psi_{\sigma} ({\bf S}, {\bf x}) = \sqrt{\frac{\Omega}{a}} e^{i {\bf k}_{{\bf S}} \cdot {\bf x}} e^{i \frac{\sqrt{4 \pi}}{\Omega} \phi_{\sigma} ({\bf S},{\bf x})} \hat{O}({\bf S}), $ and utilize the identity $ e^A e^B = :e^{A+B}: e^{<AB - \frac{1}{2}(A^2 + B^2)>}, $ where $::$ means that $:e^{A+B}:$ is a normal ordered product of operators. Then, we have that $ G^>_{\sigma} ({\bf S}, {\bf x}, t) = \frac{\Omega}{a} exp \left( {\frac{4 \pi}{\Omega ^2} \frac{1}{2} G_B^{\sigma}({\bf S}, {\bf x}, t)} \right), $ where $G_B^{\sigma}$ is expressed via the Bose fields, namely $ G_B^{\sigma}({\bf S}, {\bf x}, t) = < (\phi_c({\bf S}, {\bf x}, t) + \sigma \phi_s({\bf S}, {\bf x}, t)) (\phi_c({\bf S}, 0, 0) + \sigma \phi_s({\bf S}, 0, 0))> - <(\phi_c({\bf S}, 0, 0) + \sigma \phi_s({\bf S}, 0, 0))^2 >. $ Next, using the Heisenberg representation for the boson field operators $\phi_{\alpha} ({\bf S},{\bf x},t)$, and subsequently decomposing them into the Fourier components, we find the explicit form of the boson correlation function: $ G_B^{\sigma}({\bf S}, {\bf x}, t) = - \frac{\Omega^2}{4 \pi} \ln \left( \frac{ (\hat{n}_{{\bf S}}\cdot {\bf x} - v_F^c t+ ia) ( \hat{n}_{{\bf S}} \cdot {\bf x} - v_F^s t + ia)}{ (ia)^2} \right) . $ Hence the fermion correlation function is $$G_{\sigma}^> ({\bf S}, {\bf x}, t) = i \Omega \frac{ e^{i {\bf k}_{{\bf S}} \cdot {\bf x}} }{ \sqrt{\hat{n}_{{\bf S}}\cdot {\bf x} - v_F^c t + ia } \sqrt{ \hat{n}_{{\bf S}} \cdot {\bf x} - v_F^s t + ia } }. \label{e55}$$ It is independent of the spin index $\sigma$. We see that instead of a quasiparticle pole, taking place in the FL case, we have now a branch cut ranging from $v_F^s$ to $v_F^c$. This branch cut survives when we transform $G^>_{\sigma}({\bf S}, {\bf x}, t)$ into $G^>_{\sigma}({\bf S}, {\bf k}, \omega)$ as can be easily seen by decomposing the argument into normal and transverse parts: ${\bf k} \cdot {\bf x} = k ( \hat{n}_{{\bf S}} \cdot {\bf x} + {\bf t} \cdot {\bf x})$, where ${\bf t} k$ is the transverse part of ${\bf k}$, and noting that the part $\hat{n}_{{\bf S}} \cdot {\bf x}$ can be transformed in the same manner, as in the $d=1$ case [@and3]. The ${\bf t} \cdot {\bf x}$ part has a trivial form. This means that the analytic character of the LL Green function is the same in both $d=1$ and $d>1$ cases. This universal character follows from the relation $ G^>({\bf S}, {\bf k}, \omega) \equiv G^> ( {\bf S}, \hat{n}_{{\bf S}}\cdot {\bf k}, \omega) \delta_{ {\bf t} \cdot {\bf k}, 0}$. This very significant result tells us also that there are no quasiparticle excitations having a direct relation to the noninteracting particles in this charge-spin separated system. In other words, when we put a single electron forming a wave packet on FS, it dissociates into many wave packets propagating with velocities ranging form $v_F^s$ to $v_F^c$. This nonperturbative result means that there is no one-to-one correspondence between the dynamics of LL liquid system and the system of non-interacting fermions. Note that the distribution function $\bar{n}_{\bf k}$ is in our model situation at $T=0$ a step function with a jump at $k_F$, as in the FL case. The inclusion the nonsingular part of the interaction will change this step distribution for a finite-volume system. The fundamental question arises if we can still define a proper fermionic pseudoparticles in this spin-charge separated liquid. To construct such a state we consider the field operators $\psi_c$ and $\psi_s$, defined through the following fermionization transformation $$\psi_{c,s} ({\bf S}, {\bf x}) = \sqrt{\frac{\Omega}{a}} e^{i {\bf k}_{{\bf S}} \cdot {\bf x}} e^{i \frac{\sqrt{4 \pi}}{\Omega} \phi_{c,s} ({\bf S},{\bf x})} \hat{O}({\bf S}). \label{e58}$$ The operators $\psi_{\alpha}({\bf S},{\bf x})$ obey proper anticommutation relations. By the procedure similar to that employed in deriving (\[e55\]), we now have the correlation function in the new fermionic variables in the form $$G^>_{\alpha} ( {\bf S}, {\bf x}, t>0) \equiv <\psi_{\alpha}({\bf S}, {\bf x}, t) \psi_{\alpha}^+({\bf S},0,0)> = i \Omega \frac{ e^{ i {\bf k}_{{\bf S}} \cdot {\bf x} } }{ \hat{n}_{{\bf S}} \cdot {\bf x} - v_F^{\alpha} t + ia },$$ which has simple poles. Also, one can show that $[ \psi^+_{\alpha} ({\bf S},{\bf x}), \int d {\bf y} J_{\alpha}({\bf S}, {\bf y}) ] = \psi^+_{\alpha}({\bf S}, {\bf x})$, which means that $\psi^+_{\alpha}( {\bf S},{\bf x})$ changes the total number of either charge $(\alpha=c)$ or spin $(\alpha=s)$ of the system at the FS point ${\bf S}$ by one unit. Hence, the operators (\[e58\]) represent new fermionic pseudoparticles for the interacting non-Fermi liquid. Those single-particle excitations are called the [**holons**]{} for $\alpha = c$ and the [**spinons**]{} for $\alpha =s$. They are the only single-particle excitations across FS in our model system. Since they obey the fermion statistics, the contribution to the specific heat is of the fermionic type. In other words, the spin-charge separated liquid is composed of ideal gases of spinons and holons. They represent the exact eigenstates of the system. Effectively, we have the following Hamiltonian for noninteracting fermion pseudoparticles $$H = \sum_{\alpha = c,s} \sum_{{\bf S}} v_F^{\alpha} ({\bf S}) \int d {\bf x} \; \psi^+_{\alpha}({\bf S}, {\bf x}) \left( \frac{\hat{n}_{{\bf S}} \cdot \nabla}{i} \right) \psi_{\alpha}({\bf S}, {\bf x}) . \label{nonham}$$ Since we have treated the interaction between fermions exactly in the thermodynamic limit ($\Lambda \rightarrow 0$), the spectrum of our new noninteracting Hamiltonian (\[nonham\]) is exactly the same as that of the interacting Hamiltonian (\[e7\]). To conclude, the two branches of elementary excitations in this LL can be represented either as boson (collective) or as fermion (single-particle) excitations. Having determined the excitation spectrum of the singular part of interaction, which resulted in the spin-charge separated liquid, we can now include in (\[nonham\]) the nonsingular parts of the interaction, transformed to the fermionic representation (\[e58\]). Since for the former part we determined an exact state of the system, we can now treat the residual interaction among them as a perturbation, i.e. regard the system of interacting holons and spinons as being in one-to-one correspondence to the system of noninteracting holons and spinons. This statement can be proved by referring to the Gell-Mann and Low theorem which means that the evolution operator is well defined in all orders, since the interactions among the holons and the spinons are nonsingular functions. In other words, the adiabatic “switching on” procedure is justified. Here we discuss only a semiclassical approach to the interacting holons and spinons in the bosonic language. The equations of motion for $J_{\alpha}({\bf S},{\bf q})$ operators are $$i \frac{\partial}{\partial t} J_{\alpha}({\bf S}, {\bf q},t) = {\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q} \; J_{\alpha}({\bf S},{\bf q},t) + {\bf q} \cdot \hat{n}_{{\bf S}} \; \Lambda^{d-1} \left( \frac{1}{2\pi} \right)^d \sum_{{\bf T}} f_{\alpha}({\bf S} - {\bf T}) J_{\alpha}({\bf S}, {\bf q},t),$$ where we supposed that the interaction $f_{\alpha}$ does not depend on ${\bf q}$. In the semiclassical approach we take the expectation value of $J_{\alpha}$, i.e. define $ u_{\alpha}({\bf S},{\bf q},t) \equiv < J_{\alpha}({\bf S},{\bf q},t)>$. This quantity measures the shape deformation of FS with respect to the ground state form. Furthermore, focusing our attention on the single Fourier mode $u_{\alpha}(t) = e^{-i \omega t}u_{\alpha}$, we find the integral equation for the collective excitation spectrum amplitude $u_{\alpha} \equiv u_{\alpha} (\tilde{\Omega},{\bf q})$, in the form $$(v_F^{\alpha} q \cos \theta - \omega ) u_{\alpha} (\tilde{\Omega}, q) = q v_F^{\alpha} \cos \theta \int \frac{d \tilde{\Omega}'}{S_d} F_{\alpha}(\tilde{\Omega}, \tilde{\Omega}') u_{\alpha}(\tilde{\Omega}', q), \label{e63}$$ where $S_d = \int d \tilde{\Omega}$, $F_{\alpha}(\tilde{\Omega}, \tilde{ \Omega}') = \rho(\epsilon_F) f_{\alpha}(\tilde{\Omega}, \tilde{\Omega}')$, and $\tilde{\Omega} $ is the solid angle. This is an equation of motion for either the holon ($\alpha =c$) or the spinon ($\alpha=s$) sound-wave amplitudes. The introduced bosons represent the sound waves propagating around FS, here characterized by $\omega$ and ${\bf q}$. Eq.(\[e63\]) gives both the stable solution for collective modes and the solution with the imaginary frequency. Solution of Eq.(\[e63\]) is analogical to that considered in the FL theory [@lfl]; it will not be discussed in detail here. In summary, we presented a model with the spin-charge separation in the space of arbitrary dimensions, as well as have discussed some of its basic properties. The next step would require a careful analysis of the nonsingular part of the interaction for finite-volume systems. In particular, the most important question is to construct a theory of interacting holons and spinons at low energies (in the spirit of the Landau FL), including also the effects coming from the presence of an applied magnetic field. One should also examine the stability of this liquid against charge or spin-density wave formation. The paper was supported by the Committee of Scientific Research (KBN) of Poland. The work was performed in part at Purdue University (U.S.A.), where it was supported by the MISCON Grant No. DE-FG 02-90 ER 45427, and by the NSF Grant No. INT. 93-08323. P.W.Anderson, Y.Ren, in “High-temperature superconductivity”, (Addison-Wesley (1990)), ed. K.S.Bedel et al, pp.3-33. e.g. J.Solyom, Adv.Phys. [**28**]{}, 209 (1979); V.J.Emery, in “High conducting one-dimensional solids”, ed. J.T.Deveeese et al, (Plenum, (1979)) p. 327; F.D.M.Haldane, J.Phys. [**C 14**]{}, 2585 (1981). F.D.M.Haldane, in Proc. of the Int. School of Physics “Enrico Fermi”, Course 121, Varenna 1992, ed. J.R.Shrieffer et al, (North-Holland, New-York (1994)) , p.5. A.Houghton, J.B.Marston, Phys.Rev. [**B 48**]{}, 7790 (1993). see e.g. R.Shankar, Rev.Mod.Phys. [**66**]{}, 129 (1994). K.Byczuk, J.Spałek, Phys.Rev. [**B 51**]{}, 7934 (1995). K.Byczuk, Ph.D. Thesis - Warsaw University (1995) (unpublished). D.Pines, P.Nozieres, “Quantum liquids” vol.1, (W.A.Benjamin (1966)); R.Freeman, Phys.Rev. [**B 18**]{}, 2482 (1978); A.H.Castro Neto, E.Fradkin, Phys.Rev. [**B 49**]{}, 10 877 (1994), ibid. [**51**]{}, 4084 (1995). J.Spałek, W.Wójcik, Phys.Rev. [**B 37**]{}, 1532 (1988); J.Spałek, Physica [**B 163**]{}, 621 (1990). P. W. Anderson, Phys. Rev. Lett. [**64**]{}, 1839 (1990); ibid. [**65**]{}, 2306 (1990); ibid. [**66**]{}, 3226 (1991). A.Stern, B.I.Halperin, (1995) (unpublished), and references therein. P.C.E.Stamp, Phys.Rev.Lett. [**68**]{}, 2180 (1992); D.V.Khveshchenko, et al, Phys.Rev.Lett. [**71**]{}, 2118 (1993). Y.Ren, P.W.Anderson, Phys.Rev. [**B 48**]{}, 16662 (1993). [^1]: E-mail: [email protected], [email protected]

ArXiv

--- address: 'Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA' author: - 'Yuri G. Zarhin' title: Hodge classes on certain hyperelliptic prymians --- Definitions and statements ========================== Throughout this paper $K$ is a field, $\K$ its algebraic closure and $\Gal(K)=\Aut(\K/K)$ the absolute Galois group of $K$. If $X$ is an abelian variety over $\K$ then we write $\End(X)$ for the ring of all its $\K$-endomorphisms ; the notation $1_X$ stands for the identity automorphism of $X$. If $Y$ is an abelian variety over $\K$ then we write $\Hom(X,Y)$ for the corresponding group of all $\K$-homomorphisms. Let $f(x)\in K[x]$ be a polynomial of degree $n\ge 2$ with coefficients in $K$ and without multiple roots, $\RR_f\subset \K_a$ the ($n$-element) set of roots of $f$ and $K(\RR_f)\subset \K_a$ the splitting field of $f$. We write $\Gal(f)=\Gal(f/K)$ for the Galois group $\Gal(K(\RR_f)/K)$ and call it the Galois group of $f(x)$ over $K$; it permutes roots of $f$ and may be viewed as a certain permutation group of $\RR_f$, i.e., as a subgroup of the group $\Perm(\RR_f)\cong\Sn$ of permutation of $\RR_f$. (It is well known that $\Gal(f)$ is transitive if and only if $f$ is irreducible.) Let us put $$g=\left[\frac{n-1}{2}\right].$$ Clearly, $g$ is a nonnegative integer and either $n=2g+1$ or $n=2g+2$. Let us assume that $\fchar(K)\ne 2$. We write $C_{f}$ for the genus $g$ hyperelliptic $K$-curve $y^2=f(x)$ and $J(C_{f})$ for its jacobian. Clearly, $J(C_{f})$ is a $g$-dimensional abelian variety that is defined over $K$. In particular, $J(C_{f})=\{0\}$ if and only if $n=2$. The abelian variety $J(C_{f})$ is an elliptic curve if and only if $n=4$. Let us assume that $K$ is a subfield of the field $\C$ of complex numbers (and $\bar{K}$ is the algebraic closure of $K$ in $\C$). Then one may view $J(C_{f})$ as a complex abelian variety and consider its first rational homology group $\H_1(J(C_{f}),\Q)$ and the Hodge group $\Hdg(J(C_{f}))$ of $J(C_{f})$, which is a certain connected reductive algebraic $\Q$-subgroup of the general linear group $\GL(\H_1(J(C_{f}),\Q))$ [@MumfordSh; @Deligne; @Ribet; @ZarhinIzv; @MZ2]. The canonical principal polarization on $J(C_{f})$ gives rise to the nondegenerate alternating bilinear form $$\H_1(J(C_{f}),\Q) \times \H_1(J(C_{f}),\Q) \to \Q$$ and the corresponding symplectic group $\Sp( \H_1(J(C_{f}),\Q))$ contains $\Hdg(J(C_{f}))$ as a (closed) algebraic $\Q$-subgroup. In addition, $\End(J(C_{f}))$ coincides with the endomorphism ring of the complex abelian variety $J(C_{f})$ and $\End^{0}(J(C_{f}))$ coincides with the centralizer of $\Hdg(J(C_{f}))$ in $\End_{\Q}(\H_1(J(C_{f}),\Q))$ [@MumfordSh; @Ribet; @MZ2]. The following result was obtained by the author in [@ZarhinMRL Th. 2.1], [@ZarhinMMJ Sect. 10]. (See also [@ZarhinPLMS2], [@ZarhinBSMF; @ZarhinL].) \[jacobian\] Suppose that $K\subset \C$, $n \ge 5$ (i.e., $g \ge 2$) and $\Gal(f)=\ST_n$ or the alternating group $\A_n$. Then $\End(J(C_{f}))=\Z$ and $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$. Every Hodge class on each self-product of $J(C_{f})$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture is valid for each self-product of $J(C_{f})$. The assertion that $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$ was not stated explicitly in [@ZarhinMMJ]. However, it follows immediately from the description of the Lie algebra $\mathrm{mt}$ of the corresponding Mumford-Tate group [@ZarhinMMJ p. 429] as the direct sum of the scalars $\Q \mathrm{Id}$ and the Lie algebra of the symplectic group, because the Lie algebra of the Hodge group coincides with the intersection of Lie algebras of the Mumford-Tate group and the symplectic group. (The same arguments prove the equality $\Hdg(J(C_{f}))=\Sp( \H_1(J(C_{f}),\Q))$ for all $f(x)$ that satisfy the conditions of Theorem 10.1 of [@ZarhinMMJ].) Our next result that was obtained in [@ZarhinSh Th. 1.2 and Theorem 2.5] deals with homomorphisms of hyperelliptic jacobians. \[homo\] Suppose that $\fchar(K)\ne 2$, $n\ge 3$ and $m\ge 3$ are integers and let $f(x)$ and $h(x)$ be irreducible polynomials over $K$ of degree $n$ and $m$ respectively. Suppose that $$\Gal(f)=\ST_n, \ \Gal(h)=\ST_m$$ and the corresponding splitting fields $K(\RR_f)$ and $K(\RR_h)$ are linearly disjoint over $K$. Then either $$\Hom(J(C_f),J(C_h))=\{0\}, \ \Hom(J(C_h),J(C_f))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_f)$ and $J(C_h)$ are supersingular abelian varieties. \[homoC\] If $K\subset \C$ then Theorem \[homo\] implies that (under its assumptions) there are no nonzero homomorphisms between complex abelian varieties $J(C_f)$ and $J(C_h)$. The main results of the present paper are the following statements. \[main\] Suppose that $n=2g+2=\deg(f)\ge 8$. Let $\tilde{C}_f \to C_{f}$ be an unramified double cover of complex smooth projective irreducible curves and let $P$ be the corresponding Prym variety, which is a $(g-1)$-dimensional (principally polarized) complex abelian variety. If $\Gal(f)=\ST_n$ then: - $\End(P)=\Z$ or $\Z\oplus \Z$. - Every Hodge class on each self-product of $P$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture holds true for each self-product of $P$. \[main1\] Suppose that $n=2g+2\ge 10$, $K\subset\C$ and $f(x)=(x-a)h(x)$ where $a\in K$ and $h(x)\in K[x]$ is an irreducible degree $n-1$ polynomial with $\Gal(h)=\ST_{n-1}$. Let $\tilde{C}_{f} \to C_{f}$ be an unramified double cover of complex smooth projective irreducible curves and let $P$ be the corresponding Prym variety, which is a $(g-1)$-dimensional (principally polarized) complex abelian variety. Then: - $\End(P)=\Z$ or $\Z\oplus \Z$. - every Hodge class on each self-product of $P$ can be presented as a linear combination of products of divisor classes. In particular, the Hodge conjecture holds true for each self-product of $P$. Our proof is based on the explicit description of Prym varieties of hyperelliptic curves [@MumfordP; @Dal] and our results about Hodge groups of hyperelliptic jacobians mentioned above. If $n=2g+2\le 10$ then $\dim(P)=g-1\le 3$. Notice that if $A$ is a complex abelian varietiy of dimension $\le 3$ then it is well known that every Hodge class on each self-product of $A$ can be presented as a linear combination of products of divisor classes [@MZ2 Th. 0.1(iv)]. In particular, the Hodge conjecture holds true for each self-product of $A$. The paper is organized as follows. In Section \[Galois\] we discuss an elementary construction from Galois theory and apply it in Section \[homohyper\] to homomorphisms of hyperelliptic jacobians. Section \[hyperhodge\] deals with Hodge groups of hyperelliptic jacobians. In Section \[hyperprym\] we discuss hyperelliptic prymians and prove the main results. Galois theory {#Galois} ============= Throughout this Section, $K$ is an arbitrary field and $n\ge 3$ is an integer, $f(x)\in K[x]$ is a degree $n$ irreducible polynomial, whose Galois group $$\Gal(f)=\Gal(K(\RR_f)/K)$$ is the full symmetric group $\Perm(\RR_f)=\ST_n$. If $T\subset \RR_f$ is a non-empty subset then we put $$f_T(x)=\prod_{\alpha\in T}(x-\alpha)\in K(\RR_f)[x].$$ By definition, $$\deg(f_T)=\#(T), \ \RR_{f_T}=T.$$ We view $\Perm(T)$ as a subgroup of $\Perm(\RR_f)=\Gal(f)$ that consists of all permutations that leave invariant every element outside $T$. \[oneT\] Let us consider the subfield $E_0=K(\RR_f)^{\Perm(T)}$ of $\Perm(T)$-invariants. Since $\Perm(T)$ leaves invariant $T=\RR_{f_T}$, $$f_T(x)\in E_0[x].$$ Clearly, $$\Gal(K(\RR_f)/E_0)=\Perm(T)\subset \Perm(\RR_f)=\Gal(\RR_f/K).$$ Let us prove that the splitting field $E_0(\RR_{f_T})=E_0(T)$ of $f_T(x)$ over $E_0$ coincides with $K(\RR_f)$. Indeed, $\Gal(K(\RR_f))/E_0(T))$ consists of all elements of $\Perm(T)=\Gal(K(\RR_f)/E_0)$ that leave invariant every element of $T$. Since every element of $\Perm(T)$ leaves invariant every element of $\RR_f\setminus T$, $\Gal(K(\RR_f))/E_0(T))=\{1\}$, i.e., $K(\RR_f)=E_0(T)=E_0(\RR_{f_T})$. This implies that the Galois group $$\Gal(E_0(\RR_{f_T})/E_0)=\Gal(K(\RR_f)/E_0)=\Perm(T).$$ \[key\] Let $T$ and $S$ be two nonempty disjoint subsets of $\RR_f$. Then there exists a field subextension $E/K \subset K(\RR_f)/K$ that enjoys the following properties. - The Galois group $\Gal(K(\RR_f)/E)$ coincides with the subgroup $$\Gal(T)\times \Gal(S)\subset \Perm(\RR_f)=\Gal(K(\RR_f)/K),$$ which consists of all permutations that leave invariant $T,S$ and every element outside $T \sqcup S$. - Both $f_T(x)$ and $f_S(x)$ lie in $E[x]$, i.e., all their coefficients belong to $E$. - Let $E(\RR_T)=E(T)$ and $E(\RR_S)=E(S)$ be the splitting fields over $E$ of $f_T(x)$ and $f_S(x)$ respectively. Then the natural injective homomorphisms $$\Gal(E(T)/E)\hookrightarrow \Perm(T), \ \Gal(E(S)/E)\hookrightarrow \Perm(S)$$ are group isomorphisms, i.e., $$\Gal(E(T)/E)=\Perm(T), \ \Gal(E(S)/E)= \Perm(S).$$ - $E(T)$ and $E(S)$ are linearly disjoint over $E$. - The compositum $E(T) E(S)$ coincides with $K(\RR_f)$. Recall that $$\RR_{f_T}=T, \ \RR_{f_S}=S.$$ We define $E$ as the subfield $K(\RR_f)^{\Perm(T)\times \Perm(S)}$ of $\Perm(T)\times \Perm(S)$-invariants. Now Galois theory gives us (i). The subgroup $\Perm(T)\times \Perm(S)$ leaves invariant both sets $T=\RR_T$ and $S=\RR_S$. This implies that all the coefficients of $f_T(x)$ and $f_S(x)$ are $[\Perm(T)\times \Perm(S)]$-invariant, i.e., lie in $E$. This proves (ii). Clearly, $$[K(\RR_f):E]=\#(\Perm(T)\times \Perm(S)).$$ Clearly, the subgroup of $\Perm(T)\times \Perm(S)$ that consists of all permutations that act identically on $S$ coincides with $\Perm(T)$. Similarly, the subgroup of $\Perm(T)\times \Perm(S)$ that consists of all permutations that act identically on $T$ coincides with $\Perm(S)$. This implies that $$\Gal(E(T)/E)=[\Perm(T)\times \Perm(S)]/\Perm(S)=\Perm(T),$$ $$\Gal(E(S)/E)=[\Perm(T)\times \Perm(S)]/\Perm(T)=\Perm(S),$$ which proves (iii). This implies that $$[E(T):E]=\#(\Perm(T)), \ [E(S):E]=\#(\Perm(S)).$$ Let $L$ be the compositum $E(T) E(S)$. Clearly, $L$ contains $T$, $S$ and $E$. Therefore $\Gal(K(\RR_f)/L)$ consists of elements of $\Perm(T)\times \Perm(S)=\Gal(K(\RR_f)/E)$ that act identically on $T$ and $T$. Since all elements of $\Perm(T)\times \Perm(S)$ act identically on the complement to $T \sqcup S$, we conclude that $\Gal(K(\RR_f)/L)=\{1\}$, i.e., $$K(\RR_f)=L=E(T) E(S).$$ This proves (v). We also obtain that $$[E(T) E(S):E]=[K(\RR_f):E]=\#(\Perm(T)\times \Perm(S))=[E(T):E] [E(S):E],$$ i.e. $$[E(T) E(S):E]=[E(T):E] [E(S):E],$$ which means that $E(T)/E$ and $E(S)/E$ are linearly disjoint. This proves (iv). Homomorphisms of hyperelliptic jacobians {#homohyper} ======================================== We keep the notation and assumptions of Section \[Galois\]. Also we assume that $\fchar(K)\ne 2$. \[homoG\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and consider the hyperelliptic curves $$C_{f_T}:y^2=f_T(x), \ C_{f_S}:y^2=f_S(x)$$ and their jacobians $J(C_{f_T})$ and $J(C_{f_S})$. Then either $$\Hom(J(C_{f_T}), J(C_{f_S}))=\{0\}, \ \Hom(J(C_{f_S}), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C_{f_S})$ are supersingular abelian varieties. If $\#(T)<3$ (resp. $\#(S)<3$) then $C_{f_T}$ (resp. $C_{f_S}$) has genus zero and $J(C_{f_T})=0$ (resp. $J(C_{f_S})=0$), which implies that there are no nonzero homomorphisms between $J(C_{f_T})$ and $J(C_{f_S})$. So, further we assume that $$n_1:=\#(T)\ge 3, \ n_2:=\#(S)\ge 3.$$ By Lemma \[key\], there exists a field $E$ such that both $f_T(x)$ and $f_S(x)$ lie in $E[x]$, their Galois groups are $\Perm(T)\cong\ST_{n_1}$ and $\Perm(S)\cong\ST_{n_2}$ respectively. In addition, their splitting fields are linearly disjoint over $E$. Now the result follows from Theorem \[homo\] applied to $E, f_T(x), f_S(x)$ instead of $K, f(x), h(x)$. \[odd\] Suppose that $m:=\#(S)=2r+1$ is odd and let $b$ be an arbitrary element of $K$. Let us consider the hyperelliptic curve $C^{b}_{f_S}:y^2=(x-b)f_S(x)$. By Remark \[oneT\], there exists a field $E_0\subset K(\RR_f)$ such that $f_S(x)$ lies in $E_0[x]$ and $\Gal(E_0(\RR_{f_S})/E_0)=\Perm(S)=\ST_m$. Then the standard substitution [@ZarhinPLMS2 p. 25] $$x_1=\frac{1}{x-b}, \ y_1:=\frac{y}{(x-b)^{r+1}}$$ gives us a degree $m$ irreducible polynomial $h(x_1)\in E[x_1]$ such that $$E_0(\RR_h)=E_0(\RR_{f_S}), \ \Gal(E_0(\RR_h)/E)=\Gal(E_0(\RR_{f_S})/E)=\ST_m$$ and $C^{b}_{f_S}$ is $E_0$-birationally isomorphic to the hyperelliptic curve $C_{h}:y_1^2=h(x_1)$. (It is assumed in [@ZarhinPLMS2 p. 25] that $m \ge 5$ but the substitution works for any positive odd $m$.) \[homoGt\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and assume that $\#(S)$ is odd. Let $b$ be an arbitrary element of $K$. Let us consider the hyperelliptic curves $$C_{f_T}:y^2=f_T(x), \ C^{b}_{f_S}:y^2=(x-b)f_S(x)$$ and their jacobians $J(C_{f_T})$ and $J(C^{b}_{f_S})$. Then either $$\Hom(J(C_{f_T}), J(C^{b}_{f_S}))=\{0\}, \ \Hom(J(C^{b}_{f_S}), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C^{b}_{f_S})$ are supersingular abelian varieties. We may assume that both $m_1=\#(T)$ and $m_2=\#(S)$ are, at least, $3$. Let $E$ be as in Lemma \[key\]. In particular, the splitting fields $E(T)$ and $E(S)$ are linearly disjoint over $E$ and $$\Gal(E(T)/E)\cong \ST_{m_1}, \ \Gal(E(S)/E) \cong \ST_{m_2}.$$ Using Remark \[homoGt\] over $E$ (instead of $E_0$), we obtain that there is a degree $m$ irreducible polynomial $h(x)\in E[x]$ such that $$E(\RR_h)=E(\RR_{f_S}), \ \Gal(E(\RR_h)/E)=\Gal(E(\RR_{f_S})/E)=\ST_{m_2}$$ and $C^{b}_{f_S}$ is birationally $E$-isomorphic to the hyperelliptic curve $C_{h}:y^2=h(x)$. Clearly, the jacobians $J(C^{b}_{f_S})$ and $J(C_h)$ are isomorphic. Applying Theorem \[homo\] to $E, f_T(x),h(x)$, we conclude that either $$\Hom(J(C_{f_T}), J(C_h))=\{0\}, \ \Hom(J(C_h)), J(C_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{f_T})$ and $J(C_{f_S})$ are supersingular abelian varieties. Since $J(C^{b}_{f_S})$ and $J(C_h)$ are isomorphic, we are done. \[homoGab\] Let $T$ and $S$ be disjoint nonempty subsets of $\RR_f$ and assume that both $\#(T)$ and $\#(S)$ are odd. Let $a$ and $b$ be arbitrary (not necessarily distinct) elements of $K$. Let us consider the hyperelliptic curves $$C^{a}_{f_T}:y^2=(x-a)f_T(x), \ C^{b}_{f_S}:y^2=(x-b)f_S(x)$$ and their jacobians $J(C^{a}_{f_T})$ and $J(C^{b}_{f_S})$. Then either $$\Hom(J(C^{a}_{f_T}), J(C^{b}_{f_S}))=\{0\}, \ \Hom(J(C^{b}_{f_S}), J(C^{a}_{f_T}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C^{b}_{f_T})$ and $J(C^{a}_{f_S})$ are supersingular abelian varieties. We may assume that both $m_1:=\#(T)$ and $m_2:=\#(S)$ are, at least, $3$. Again, let $E$ be as in Lemma \[key\]. Applying Remark \[homoGt\] two times over $E$ (instead of $E_0$) to the polynomials $(x-a)f_T(x)$ and $(x-b)f_S(x)$, we conclude that there are degree $m$ irreducible polynomials $h_1(x)\in E[x]$ and $h_2 (x)\in E[x]$ such that $$E(\RR_{h_1})=E(T), \ E(\RR_{h_2})=E(S),$$ $$\Gal(E(\RR_{h_1})/E)=\ST_{m_1}, \ \Gal(E(\RR_{h_2})/E)=\ST_{m_2},$$ $C^{a}_{f_T}$ is $E$-birationally isomorphic to $C_{h_1}$ and $C^{b}_{f_T}$ is $E$-birationally isomorphic to $C_{h_2}$. Clearly, $J(C^{a}_{f_T})\cong J(C_{h_1})$ and $J(C^{b}_{f_S})\cong J(C_{h_2})$. Applying Theorem \[homo\] to $E, h_1(x),h_2(x)$, we conclude that either $$\Hom(J(C_{h_1}), J(C_{h_2}))=\{0\}, \ \Hom(J(C_{h_2})), J(C_{h_1}))=\{0\}$$ or $\fchar(K)>0$ and both $J(C_{h_1})$ and $J(C_{h_2})$ are supersingular abelian varieties. The rest is clear. Let $K_2/K$ be the only quadratic subextension of $K(\RR_f)/K$. Clearly, $K_2(\RR_f)=K(\RR_f)$ and the Galois group $\Gal(K_2(\RR_f)/K_2)$ coincides with the alternating group $\A_n$. \[ellipticZ\] Suppose that $\fchar(K)\ne 2$. Let $T \subset \RR_f$ be a $4$-element subset. Let us consider the corresponding elliptic curve $$C_{f_T}:y^2=f_T(x)$$ and its jacobian $J(C_{f_T})$. If $n \ge 8$ then one of the following conditions holds: - $\End(J(C_{f_T}))=\Z$ for all $T$. - $\fchar(K)>0$ and all $J(C_{f_T})$’s are supersingular elliptic curves mutually isomorphic over $\K$. Let $j_T$ be the $j$-invariant of the elliptic curve $J(C_{f_T})$ ([@Tate], [@Knapp Ch. III, Sect. 2]). Clearly, $$j_T \in K(T)\subset K(\RR_f)$$ and $$j_{\sigma T}=\sigma j_T \ \forall \sigma\in \Gal(K(\RR_f)/K)=\Gal(f).$$ Suppose that $J(C_{f_T})$ admits complex multiplication. Then one of the following two conditions holds. - $p=\fchar(K)>0$. Then a classical result of M. Deuring asserts that $j_T$ is [*algebraic*]{}, i.e., lies in a finite field $\F_q$ where $q$ is a power of the prime $p$. (See [@Deuring], [@Oort Sect. 3.2], [@Lang Ch. 13, Sect. 5].) In particular, $K(j_T)/K$ is an abelian field extension. - $\fchar(K)=0$. Then there exists an imaginary quadratic field $k$ such that $\End^0(J(C_{f_T}))=k$. In addition, a classical result of the theory of complex multiplication asserts that $j_T$ is an algebraic number such that the field extension $k(j_T)/k$ is abelian. (See [@Shimura Sect. 5.4], [@Lang Ch. 10, Sect. 3].) Let us consider the overfield $K^{\prime}$ of $K$ that is defined as follows. If $\fchar(K)>0$ then $K^{\prime}=K_2$. If $\fchar(K)=0$ then $K^{\prime}$ is the compositum $K_2 k$ of $K_2$ and the imaginary quadratic field $k$; in particular, $K^{\prime}$ contains $k$. Since $\A_n=\Gal(K^{\prime}(\RR_f)/K^{\prime})$ is simple nonabelian, the field extension $K^{\prime}(\RR_f)/K^{\prime}$ does not contain nontrivial abelian subextensions. However, $j_T \in K^{\prime}(\RR_f)$ and the field (sub)extension $K^{\prime}(j_T)/K^{\prime}$ is abelian. This implies that this subextension is trivial, i.e., $j_T\in K^{\prime}$. This means that for all $\sigma \in \Gal(K^{\prime}(\RR_f)/K^{\prime})=\A_n$ $$j_T=\sigma j_T=j_{\sigma T}.$$ Since $n \ge 8$, the permutatation group $\A_n$ is $4$-transitive and therefore the jacobians $J(C_{f_T})$’s are mutually isomorphic over $\K$ for all $4$-element subsets $T\subset \RR_f$. Let $T_1$ and $T_2$ be two [*disjoint*]{} $4$-element subsets of $\RR_f$. (Since $n\ge 8$, such $T_1$ and $T_2$ do exist.) Applying Theorem \[homoG\] to $T_1$ and $T_2$ (instead of $T$ and $S$) and taking into account that $J(C_{f_{T_1}})$ and $J(C_{f_{T_2}})$ are isomorphic over $\K$ (i.e., $\Hom(J(C_{f_{T_1}}),J(C_{f_{T_2}}))\ne\{0\}$), we conclude that $\fchar(K)>0$ and both $J(C_{f_{T_1}})$ and $J(C_{f_{T_2}}))$ are supersingular elliptic curves. \[ellipticZ1\] Suppose that $\fchar(K)\ne 2$. Let $a$ be an arbitrary element of $K$. Let $T \subset \RR_f$ be a $3$-element subset. Let us consider the corresponding elliptic curve $$C^{a}_{f_T}:y^2=(x-a)f_T(x)$$ and its jacobian $J(C^{a}_{f_T})$. If $n \ge 6$ then one of the following conditions holds: - $\End(J(C^{a}_{f_T}))=\Z$ for all $T$. - $\fchar(K)>0$ and all $J(C^{a}_{f_T})$’s are supersingular elliptic curves mutually isomorphic over $\K$. Let $j_{T,a}$ be the $j$-invariant of the elliptic curve $J(C^{a}_{f_T})$. Clearly, $$j_{T,a} \in K(T)\subset K(\RR_f)$$ and $$j_{\sigma T,a}=\sigma j_{T,a} \ \forall \sigma\in \Gal(K(\RR_f)/K)=\Gal(f).$$ Suppose that $J(C^{a}_{f_T})$ admits complex multiplication. Then, as in the proof of Theorem \[ellipticZ\], there exists an overfield $K^{\prime}\supset K_2$ such that either $K^{\prime}=K_2$ or $K^{\prime}$ is a quadratic extension of $K_2$ and in both cases $K^{\prime}(j_{T,a})\subset K^{\prime}(\RR_f)$ and the field (sub)extension $K^{\prime}(j_{T,a})/K^{\prime}$ is abelian. Again, $\A_n=\Gal(K^{\prime}(\RR_f)/K^{\prime})$ is simple nonabelian and therefore there are no nontrivial abelian subextensions of $K^{\prime}(\RR_f)/K^{\prime}$. This implies that $j_{T,a}\in K^{\prime}$, i.e., for all $\sigma \in \Gal(K^{\prime}(\RR_f)/K^{\prime})=\A_n$ $$j_{T,a}=\sigma j_{T,a}=j_{\sigma T, a}.$$ Since $n \ge 6$, the permutatation group $\A_n$ is $3$-transitive and therefore the jacobians $J(C^{a}_{f_T})$’s are mutually isomorphic over $\K$ for all $3$-element subsets $T\subset \RR_f$. Let $T_1$ and $T_2$ be two [*disjoint*]{} $3$-element subsets of $\RR_f$. (Since $n\ge 6$, such $T_1$ and $T_2$ do exist.) Applying Theorem \[homoGab\] to $T_1,a$ and $T_2,a$ (instead of $T,a$ and $S,b$) and taking into account that $J(C^{a}_{f_{T_1}})$ and $J(C^{a}_{f_{T_2}})$ are isomorphic over $\K$ (i.e., $\Hom(J(C_{f_{T_1}}),J(C_{f_{T_2}}))\ne\{0\}$), we conclude that $\fchar(K)>0$ and both $J(C^{a}_{f_{T_1}})$ and $J(C^{a}_{f_{T_2}}))$ are supersingular elliptic curves. The rest is clear. Hodge groups of hyperelliptic jacobians {#hyperhodge} ======================================= We keep the notation and assumptions of Sections \[Galois\] and \[homohyper\]. Also we assume that $K\subset \C$. We say (as in [@MZ2 Sect. 1.8]) that a complex abelian variety $X$ satisfies property (D) if every Hodge class on each self-product $X^r$ of $X$ can be presented as a linear combination of products of divisor classes. If this condition is satisfied then the Hodge conjecture is true for all $X^r$. Abelian varieties that satisfy (D) are also called [*stably nondegenerate*]{} [@hazamaT]; see also [@murty]. \[DE\] If $Y$ is an elliptic curve over $\C$ with $\End(Y)=\Z$ then it is well known [@MZ2 Th. 0.1(iv)] that $Y$ satisfies (D) and $\Hdg(Y)=\Sp(\H_1(Y,\Q))$. \[nonsimple\] Let $X_1$ and $X_2$ be complex abelian varieties of positive dimension and $X=X_1\times X_2$. Suppose that $$\End(X_1)=\Z, \ \End(X_2)=\Z, \ \Hom(X_1,X_2)=\{0\}.$$ Then: 1. $\End(X_1\times X_2)=\Z\oplus \Z$. 2. If both $X_1$ and $X_2$ satisfy (D) then $\Hdg(X)= \ \Hdg(X_1)\times \Hdg(X_2)$ and $X$ satisfies (D). \(i) is obvious. (ii) follows from [@Hazama Th. 0.1 and Prop. 1.8] (see also Theorem 3.2(i) of [@MZ2]). Since $\End(X_i)=\Z$ and $X_i$ satisfies (D), $$\Hdg(X_i)=\Sp(H_1(X_i,\Q)$$ [@hazamaT; @murty]. (See also [@MZ2 Sect. 1.8].) \[jacprym\] Suppose that $T$ is a subset of $\RR_f$ with $\#(T)\ge 5$. Let us consider the hyperelliptic curve $C_{f_T}:y^2=f_T(x)$ and its jacobian $J(C_{f_T})$. Then $\End(J(C_{f_T}))=\Z$ and $\Hdg(J(C_{f_T}))=\Sp( \H_1(J(C_{f_T}),\Q))$. In addition, $J(C_{f_T})$ satisfies (D). Let us put $m=\#(T)$. We have $m \ge 5$ and $\deg(f_T)=m\ge 5$. By Remark \[oneT\], there exists a (sub)field $$E_0\subset K(\RR_f)\subset\K\subset \C$$ such that $f_T(x)\in E_0(T)$ and the Galois group of $f_T(x)$ over $E_0$ is $\Perm(T)\cong \ST_m$. Now the result follows from Theorem \[jacobian\] applied to $m,E_0, f_T(x)$ instead of $n,K,f(x)$. \[jacfam\] Suppose that $T$ is a subset of $\RR_f$ with $\#(T)\ge 5$. Suppose that $m$ is odd and let $a$ be an arbitrary element of $K$. Let us consider the hyperelliptic curve $C^{a}_{f_T}:y^2=(x-a)f_T(x)$ and its jacobian $J(C^{a}_{f_T})$. Then $\End(J(C^{a}_{f_T}))=\Z$ and $\Hdg(J(C^{a}_{f_T}))=\Sp( \H_1(J(C^{a}_{f_T}),\Q))$. In addition, $J(C^{a}_{f_T})$ satisfies (D). By Remark \[odd\], there exists a field $$E_0\subset K(\RR_f)\subset\K\subset\C$$ and a degree $m$ irreducible polynomial $h(x)\in E_0[x]$ such that $$E(\RR_h)=E(\RR_{f_T}), \ \Gal(E(\RR_h)/E)=\Gal(E(\RR_{f_T})/E)=\ST_m$$ and $C^{a}_{f_S}$ is birationally $E$-isomorphic to to the hyperelliptic curve $C_{h}:y^2=h(x)$. Clearly, the jacobians $J(C^{a}_{f_S})$ and $J(C_h)$ are isomorphic. It follows from Lemma \[jacprym\] applied to $m,E_0,h(x)$ (instead of $n,K,f(x)$) that $\End(J(C_{h}))=\Z$, $\Hdg(J(C_h)=\Sp( \H_1(J(C_h),\Q))$ and $J(C_{h})$ satisfies (D). Since $J(C^{a}_{f_S})$ and $J(C_h)$ are isomorphic, we are done. Prym varieties {#hyperprym} ============== Following [@MumfordP; @Dal], let us give an explicit description of hyperelliptic prymians $P$, assuming that $\fchar(K)\ne 2$. Suppose that $n=2g+2\ge 6$ and $$f(x)\in K[x]\subset \K[x]$$ is a degree $n$ polynomial without multiple roots. Let us split the $n$-element set $\RR_f$ of roots of $f(x)$ into a [*disjoint*]{} union $$\RR_f =\RR_1 \sqcup \RR_2$$ of [*non-empty*]{} sets $\RR_1$ and $\RR_2$ of [*even*]{} cardinalities $n_1$ and $n_2$ respectively. Further we assume that $$n_1 \ge n_2\ge 2.$$ and put $$f_1(x)=\prod_{\alpha\in\RR_1}(x-\alpha), \ f_2(x)=\prod_{\alpha\in\RR_2}(x-\alpha).$$ We have $n_1+n_2=n$ and define nonnegative integers $g_1$ and $g_2$ by $$n_1=2g_1+2, \ n_2=2g_2+2.$$ Clearly, $$g_1+g_2=g-1.$$ Let us consider the hyperelliptic curves $C_{f_1}:y^2=f_1(x)$ and $C_{f_2}:y^2=f_2(x)$ of genus $g_1$ and $g_2$ respectively and corresponding hyperellptic jacobians $J(C_{f_1})$ and $J(C_{f_2})$ of dimension $g_1$ and $g_2$ respectively. Then the prymians $P$ of $C_f: y^2=f(x)$ are just the products $J(C_{f_1})\times J(C_{f_2})$ for all the partitions $\RR_f =\RR_1 \sqcup \RR_2$. Now Theorem \[main\] becomes an immediate corollary of the following statement. \[help\] Suppose that $n=2g+2\ge 8$, $K\subset\C$ and $\Gal(f)=\ST_n$. Let us put $P=J(C_{f_1})\times J(C_{f_2})$. Then: - $\Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}, \ \Hom(J(C_{f_2}),J(C_{f_1}))=\{0\}$. - Suppose that $g_i \ge 1$, i.e., $n_i \ge 4$. Then $$\End(J(C_{f_i}))=\Z, \ \Hdg(J(C_{f_i}))=\Sp( \H_1(J(C_{f_i}),\Q))$$ and $J(C_{f_i})$ satisfies (D). - If $n_2=2$ then $P=J(C_{f_1})$. In particular, $\End(P)=\Z$, $\Hdg(P)=\Sp( \H_1(P,\Q))$ and $P$ satisfies (D). - If $n_2 \ge 4$ then $\End(P)=\Z\oplus\Z$, $$\Hdg(P)=\Hdg(J(C_{f_1}))\times \Hdg(J(C_{f_2}))=\Sp( \H_1(J(C_{f_1}),\Q))\times \Sp( \H_1(J(C_{f_2}),\Q))$$ and $P$ satisfies (D). The assertion (i) follows from Theorem \[homoG\]. If $n_i\ge 6$ then (ii) follows from Lemma \[jacprym\]. Suppose that $n_i=4$. Then $J(C_{f_i})$ is an elliptic curve. It follows from Theorem \[ellipticZ\] that $\End(J(C_{f_i}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. This completes the proof of (ii). Let us prove (iii). If $n_2=2$ then $J(C_{f_2})=0$ and therefore $P=J(C_{f_1})$. Now the assertion follows from (ii). Let us prove (iv). We assume that $$n_1 \ge n_2 \ge 4.$$ By already proven (i) and (ii), $$\End(J(C_{f_1}))=\Z, \ \End(J(C_{f_2}))=\Z, \ \Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}.$$ Now (iv) follows from Theorem \[nonsimple\] applied to $X_1=J(C_{f_1})$, $X_2=J(C_{f_2})$ and $X=P$. Theorem \[main1\] is an immediate corollary of the following statement. \[help1\] Suppose that $n=2g+2\ge 10$, $K\subset\C$ and $f(x)=(x-a)h(x)$ where $a\in K$ and $h(x)\in K[x]$ is an irreducible degree $(n-1)$ polynomial with $\Gal(h)=\ST_{n-1}$. Let us put $P=J(C_{f_1})\times J(C_{f_2})$. Then: - $\Hom(J(C_{f_1}),J(C_{f_2}))=\{0\}, \ \Hom(J(C_{f_2}),J(C_{f_1}))=\{0\}$. - Suppose that $g_i \ge 1$, i.e., $n_i \ge 4$. Then $$\End(J(C_{f_i}))=\Z, \ \Hdg(J(C_{f_i}))=\Sp( \H_1(J(C_{f_i}),\Q))$$ and $J(C_{f_i})$ satisfies (D). - If $n_2=2$ then $P=J(C_{f_1})$. In particular, $\End(P)=\Z$, $\Hdg(P)=\Sp( \H_1(P,\Q))$ and $P$ satisfies (D). - If $n_2 \ge 4$ then $\End(P)=\Z\oplus\Z$, $$\Hdg(P)=\Hdg(J(C_{f_1}))\times \Hdg(J(C_{f_2}))=\Sp( \H_1(J(C_{f_1}),\Q))\times \Sp( \H_1(J(C_{f_2}),\Q))$$ and $P$ satisfies (D). Clearly, $\RR_f=\RR_h \sqcup\{a\}$; in particular, $a$ belongs to precisely one of $\RR_1$ and $\RR_2$. Suppose that $a$ lies in $\RR_j$ and does [*not*]{} belong to $\RR_k$ and put $$T=\RR_k\subset\RR_h, \ S=\RR_j\setminus \{a\}\subset\RR_h.$$ Now the assertion (i) follows from Corollary \[homoGt\] applied to $h(x)$ (instead of $f(x)$). If $n_k\ge 6$ then the assertion (ii) for $J(C_{f_k})$ follows from Lemma \[jacprym\]. Suppose that $n_k=4$, i.e., $J(C_{f_k})$ is an elliptic curve. Then it follows from Theorem \[ellipticZ\] applied to $m=n-1\ge 9$ and $h(x)$ (instead of $f(x)$) that $\End(J(C_{f_k}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. If $n_j\ge 6$ then the assertion (ii) for $J(C_{f_j})$ follows from Lemma \[jacfam\]. Suppose that $n_j=4$, i.e., $J(C_{f_j})$ is an elliptic curve. Then it follows from Theorem \[ellipticZ1\] applied to $m=n-1$ and $h(x)$ (instead of $f(x)$) that $\End(J(C_{f_j}))=\Z$. Now the assertion about its Hodge group and property (D) follows from Example \[DE\]. This ends the proof of (ii). The proof of the remaining assertions (iii) and (iv) goes literally as the proof of the corresponding assertions of Theorem \[help\]. Let us take $K=\Q$ and $f_n(x)=x^n-x-1$. It is known [@SerreGalois p. 42] that $\Gal(f_n)=\ST_n$. Let $a$ be a rational number. Suppose that $n=2g+2$ and let us consider the hyperelliptic genus $g$ curves $C_{f_n}: y^2=f_n(x)$ and $C^{a}_{f_{n-1}}: y^2=(x-a) f_{n-1}(x)$. Then: - If $n=2g+2\ge 8$ then all $(2^{2g}-1)$ Prym varieties $P$ of $C_{f_{n}}$ satisfy (D). Among them there are exactly $n(n-1)/2$ complex abelian varieties with $\End(P)=\Z$; for all others $\End(P)=\Z\oplus \Z$. - If $n=2g+2\ge 10$ then all $(2^{2g}-1)$ Prym varieties $P$ of $C^{a}_{f_{n-1}}$ satisfy (D). Among them there are exactly $n(n-1)/2$ complex abelian varieties with $\End(P)=\Z$; for all others $\End(P)=\Z\oplus \Z$. Let $z_1, \dots , z_n$ be algebraically independent (transcendental) complex numbers and $L=\Q(z_1, \dots , z_n)\subset \C$ the corresponding subfield of $\C$, which is isomorphic to the field of rational functions in $n$ variables over $\Q$. Let $K\subset L$ be the (sub)field of symmetric rational functions. Then $$f(x)=\prod_{i=1}^n (x-z_i)\in K[x], \ \RR_f=\{z_1, \dots , z_n\}, \ \Gal(f)=\ST_n.$$ Suppose that $n=2g+2$ and let us consider the hyperelliptic genus $g$ curve $C_f:y^2=f(x)$. If $g\ge 3$ (i.e., $n\ge 8$) then all $(2^{2g}-1)$ Prym varieties $P$ of $C_{f}$ satisfy (D). Among them there are exactly $n(n-1)/2$ complex abelian varieties with $\End(P)=\Z$; for all others $\End(P)=\Z\oplus \Z$. If $n=6$ (i.e., $g=2$) then all fifteen Prym varieties $P$ are elliptic curves $y^2=\prod_{z\in T}(x-z)$ where $T$ is a $4$-element subset of $\{z_1, \dots , z_6\}$. The algebraic independence of $z_1, \dots , z_6$ implies that the $j$-invariants of these elliptic curves are transcendental numbers and therefore all $P$ have no complex multiplication, i.e., $\End(P)=\Z$. The property (D) and equality $\End(P)=\Z$ for [*general*]{} (not necessarily unramified) Prym varieties $P$ of arbitrary smooth projective curves were proven in [@Biswas]. [99]{} I. Biswas, K. H. Paranjape, [*The Hodge conjecture for general Prym varieties*]{}. J. Algebraic Geometry [**11**]{} (2002), 33–39. S. G. Dalaljan, [*The Prym variety of an unramified double covering of a hyperelliptic curve*]{}. (Russian) Uspehi Mat. Nauk 29 (1974), no. 6(180), 165–166. MR0404270 (53 \#8073). P. Deligne, [*Hodge cycles on abelian varieties*]{} (notes by J.S. Milne). Lecture Notes in Math., vol. [**900**]{} (Springer-Verlag, 1982), pp. 9–100. M. Deuring, [*Die Typen der Multiplikatorenringe elliptischer Funktionenkörper*]{}. Abh. Math. Sem. Hansischen Univ. [**14**]{} (1941), 197–-272. MR0005125 (3,104f). F. Hazama, [*Algebraic cycles on certain abelian varieties and powers of special surfaces*]{}. J. Fac. Sci. Univ. Tokyo Sect. IA Math. [**31**]{} (1985), no. 3, 487–520. F. Hazama, [*Algebraic cycles on nonsimple abelian varieties*]{}. Duke Math. J. [**58**]{} (1989), 31–37. A. Knapp, Elliptic curves. Princeton University Press, Princeton, 1992. S. Lang, Elliptic functions, Second edition. Springer Verlag, New York, 1987. B. Moonen, Yu. G. Zarhin, [*Hodge classed on abelian varieties of low dimension*]{}. Math. Ann. [**315**]{} (1999), 711–733. D. Mumford, [*A note of Shimura’s paper “Discontinuous groups and abelian varieties”*]{}. Math. Ann. [**181**]{} (1969), 345–-351. D. Mumford, [*Prym varieties*]{} I. In: Contributions to Analysis, pp. 325–350, Academic Press, 1974; Selected Papers, vol. I, pp. 545–570, Springer Verlag, New York, 2004. V. Kumar Murty, [*Exceptional Hodge classes on certain abelian varieties*]{}. Math. Ann. 268 (1984), no. 2, 197–206. F. Oort, [*The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field*]{}. J. Pure Appl. Algebra [**3**]{} (1973), 399–408. K. Ribet, [*Hodge classes on certain abelian varieties*]{}. Amer. J. Math. [**105**]{} (1983), 523–538. J.-P. Seree, Topics in Galois Theory. Jones and Bartlett Publishers, Boston-London, 1992. G. Shimura, Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan [**11**]{}, Iwanami Shoten and Princeton University Press, Princeton, 1971. J. Tate, [*Algebraic formulas in arbitrary characteristic*]{}. Appendix 1 to [@Lang]. pp. 299–306. Yu.G. Zarhin, [*Weights of simple Lie algebras in the cohomology of algebraic varieties*]{}. Izv. Akad. Nauk SSSR Ser. Mat. [**48**]{} (1984), 264–304; Math. USSR Izv. [**24**]{} (1985), 245 - 281. Yu.G. Zarhin, [*Hyperelliptic jacobians without complex multiplication*]{}. Math. Res. Letters [**7**]{} (2000), 123–132. Yu.G. Zarhin, [*Very simple $2$-adic representations and hyperelliptic jacobians*]{}. Moscow Math. J. [**2**]{} (2002), issue 2, 403-431. Yu.G. Zarhin, [*Homomorphisms of hyperelliptic Jacobians*]{}. In: Number Theory, Algebra and Algebraic Geometry (Shafarevich Festschrift). Trudy Mat. Inst. Steklov [**241**]{} (2003), 90–104; Proc. Steklov Inst. Math. [**241**]{} (2003), 79–92. Yu.G. Zarhin, [*Non-supersingular hyperelliptic Jacobians*]{}. Bull. Soc. Math. France [**132**]{} (2004), no. 4, 617–634. Yu.G. Zarhin, [*Homomorphisms of abelian varieties*]{}. In: Y. Aubry, G. Lachaud (ed.), Arithmetic, Geometry and Coding Theory (AGCT 2003), Séminaires et Congrés [**11**]{} (2005), 189–215. Yu.G. Zarhin, [*Families of absolutely simple hyperelliptic Jacobians*]{}. Proc. Lond. Math. Soc. (3) [**100**]{} (2010), no. 1, 24–54,

ArXiv

--- abstract: 'We study the effects of an electron-electron effective interaction on the formation of entangled states in a two-qubit system, driven by the coupling of electronic states with vibrational modes. The system is composed by four quantum dots separated in pairs, each pair with one excess electron, which is able to tunnel between the dots. Also, the dots from each pair are coupled with different vibrational modes. The combined action of both, this effective interaction and the electronic tunneling explains not only features on the spectrum and the eigenstates of the Hamiltonian, but also the formation of electronic Bell states by exploiting the quantum dynamics of the system.' author: - 'F. M. Souza' - 'P. A. Oliveira' - 'L. Sanz' title: 'Quantum entanglement driven by electron-nanomechanical coupling' --- Introduction {#sec:intro} ============ For many years, the single-molecule electronics has being an outstanding issue due to its potentiality for future implementations of a cheaper and faster single-electron transistor [@xiang2016; @Ratner05]. There are already many successful examples of single-molecule devices that operate in the Coulomb-blockade regime, presenting transistor-like behavior [@park2002]. To increase further the functionality of a single-electron transistor, it is possible to couple nanomechanical and electronic degrees of freedom [@park2000; @steele20092; @lassagne2009]. It is well known that this kind of interaction plays a signifcant role, bringing a wealth of interesting effects, such as quantum-shuttles in quantum dots (QDs) systems [@gorelik1998; @armour2002; @donarini2005], local cooling [@kepesidis2016], phonon-assisted transport in molecular quantum dot junctions [@walter2013; @Sowa17], and Franck-Condon blockade [@leturcq2009]. Concerning this single electron systems, there is an emerging interest in the interplay between the electronic degrees of freedom, confined inside nanoparticles, and discrete vibrational modes. In particular, this problem has being analyzed considering two different contexts: carbon nanotubes quantum dots and vibrational modes of a cavity coupled with quantum dots. Carbon nanotubes (CNT) become one of the most successful new materials in front of their wide set of direct applications [@dresselhaus2001]. These include the implementation of ultrahigh tunable frequency resonators [@sazonova2004; @deng2016], nanoradios[@jensen2007], and ultrasensitive mass sensors [@chiu2008; @jensen2008]. Moreover, when operated as mechanical ressonators, nanotubes show high quality factors [@lassagne2009; @laird2012; @moser2016] being possible, for instance, to excite, detect and control specifical vibrational modes of a CNT with a current being injected from a scanning tunneling microscopy (STM) tip into a CNT [@leroy2004]. Additionally, it have been reported strong coupling regimes between single-electron tunneling and nanomechanical motion on a suspended nanotube, tuned via electrical gates [@benyamini2014]. Regarding applications in micro and nanoelectronics, carbon nanotubes presented balistic conduction [@laird2015] and Coulomb blockade effect in single and double nanotube based quantum dot devices [@steele2009]. Particularly, it was proposed a mechanically induced “two-qubit" quantum gate and the generation of entanglement between electronic spin states in CNT [@wang2015]. Also, phonons on CNTs showed its potentiality as “flying” qubits for electron spin communications over long distances [@deng2016]. Alternatively, regarding the second experimental context, the coupling between electronic degrees of freedom and vibrational modes can also be accomplished in piezoelectric phononic cavities, based on Bragg mirrors that confine a vibrational mode [@Chen15]. This coupling was experimentally demonstrated in transport measurements as a phonon assisted tunneling in a double quantum dot structure embedded in the cavity. The electron-vibrational mode coupling factor in this quantum dots system is ten times bigger if compared with couplings found on cavity quantum electrodynamics (CQED) and can be enhanced, on demand, by a factor of 20-500 of its regular value [@Chen15]. In front of the advantages cited above, both systems have a great potentiality as a solid state based quantum information processing. In the present work, we explore the properties of a Hamiltonian which describes two electrons on four quantum dots, considering a general model of coupling between electrons and vibrational modes, which applies to both experimental setups. Our proposal is based on charge-qubits, instead of spin, where the states of two qubits are defined depending of the occupation of the quantum dots. We are interested on the interplay between the electron-vibrational mode coupling and the tunneling of electrons between adjacent quantum dots. In Sec. \[sec:model\], by using the unitary transformation of Lang-Firsov, we demonstrate that both couplings are responsible for the apparition of an effective electron-electron interaction. Section \[sec:eigenproblem\] is devoted to the exploration of the signatures of this effective interaction and correlated phenomena on the spectrum and eigenstates of the model. Finally, in Sec. \[sec:dynamics\] , using our previous experience on quantum dynamics on coupled quantum dots [@Oliveira15; @souza2017], we study the formation of Bell states under specific conditions. Section \[sec:summary\] contains our final remarks. Model {#sec:model} ===== Consider a multipartite system with two main parts: the electronic subspace $\mathcal{D}$, consisting on four quantum dots, and the bosonic subspace $\mathcal{V}$, with two devices containing vibrational modes. Vibrational mode $1$ couples with electronic levels of the dots $1$ and $3$, while vibrational mode $2$ couples with dots $2$ and $4$. Tunneling is allowed between dots $1$ ($3$) and $2$ ($4$) so the pair $1-2$ ($3-4$) can be described as a qubit. A possible experimental setup is illustrated in Fig. \[fig:system\], where the bosonic devices are carbon nanotubes. Electronic levels on the quantum dots can be populated by using sources and drains, with additional gates to control the process of tunneling between the dots [@Shinkai07]. The electron-vibrational interaction does not change the electronic population, although creates or destroys vibrational excitations in both devices. Although electrons in different qubits could interact, for instance, by Coulomb interaction [@Hayashi03], we assume this interaction is weak enough to be ignored. ![A possible experimental setup of the system of interest: quantum dots $1$ and $3$ ($2$ and $4$) are coupled with the vibrational (bosonic) mode $1$ ($2$) on carbon nanotubes. Also, the dots 1 (3) and 2 (4) are coupled to each other by tunneling, encoding a qubit.[]{data-label="fig:system"}](figure1){width="0.5\linewidth"} The Hamiltonian that describes the two charged qubits and vibrational modes is given by $$\label{eq:Hgeneral} H=H_{\mathcal{D}} \otimes I_{\mathcal{V}}+I_{\mathcal{D}} \otimes H_{\mathcal{V}}+V_{\mathcal{DV}}.$$ Here $H_{\mathcal{D}}$ and $H_{\mathcal{V}}$ are the free Hamiltonians of the quantum dots and vibrational modes subspaces, respectively, $V_{\mathcal{DV}}$ is the dots-vibrational modes coupling and $I_{\mathcal{D(V)}}$ is the identity matrix for quantum dots ($\mathcal{D}$) or the vibrational modes ($\mathcal{V}$) subspaces. While each pair of dots is a $2$-dimensional subspace, the vibrational mode spans on an infinite dimension subspace $N_v$, where $N_{v}\rightarrow \infty$ for $v$-th subspace ($v=1,2$). The elements of the computational basis have the general form ${\mathinner{|{n_1 n_2 n_3 n_4}\rangle}}_{\mathcal{D}}\otimes{\mathinner{|{N_1 N_2}\rangle}}_{\mathcal{V}}={\mathinner{|{n_1 n_2 n_3 n_4; N_1 N_2}\rangle}}$ with the first four indexes indicating the occupation of the specific dot ($0$-empty and $1$-occupied) and the last two being the population of the vibrational modes. One example is the state ${\mathinner{|{1010,00}\rangle}}$ which represents the situation where dots 1 and 3 contains a single electron each and there is no excitations in the vibrational modes. We proceed to write each term of the Hamiltonian (\[eq:Hgeneral\]). The free Hamiltonian for the four quantum dots subsystem is written as ($\hbar=1$): $$\begin{aligned} \label{eq:Hmol} H_{\mathcal{D}}&=&\left[\sum_{i=1,2}\varepsilon_i N^{\mathcal{D}}_i+\Delta_{12}\left(S_1^\dagger S_2+S_2^\dagger S_1\right)\right]\otimes I^{\otimes 2}\\ &&+ I^{\otimes 2}\otimes\left[\sum_{j=3,4}\varepsilon_j N^{\mathcal{D}}_j+\Delta_{34}\left(S_3^\dagger S_4+S_4^\dagger S_3\right)\right],\nonumber\end{aligned}$$ where $S_{i(j)}^\dagger$ $\left[S_{i(j)}\right]$ are the creation (annihilation) operators for the $i(j)$-th quantum dot and $N^{\mathcal{D}}_{i(j)}=S_{i(j)}^\dagger S_{i(j)}$ ($i=1,2$ while $j=3,4$). The parameters $\varepsilon_{i(j)}$ are the electronic levels for each dot while $\Delta_{12(34)}$ describes the tunneling coupling. If we consider a single vibrational mode per bosonic subsystem, the free Hamiltonian $ H_{\mathcal{V}}$ becomes $$H_{\mathcal{V}} = \omega_1 B_1^\dagger B_1 \otimes I^{\otimes N_2} + I^{\otimes N_1} \otimes \omega_{2} B^\dagger_2 B_2,$$ where $\omega_{1(2)}$ is the energy of the corresponding vibrational mode. Here $B^{\dagger}_{v}$ ($B_v$) creates (annihilates) an excitation in a $v$-th vibrational mode subspace. Now we focus on the term $V_{\mathcal{DV}}$, which provides the electron-vibrational mode coupling. This coupling happens when a single electron on dot $i$ interacts with the vibrational subsystem thus creating or annihilating one excitation in the corresponding vibrational mode. We consider that the electron-vibrational mode coupling is the same for both bosonic modes, with a strength given by $g_v$. The term $V_{\mathcal{DP}}$ is written as $$\begin{aligned} \label{eq:Vmp} &&V_{\mathcal{DV}}=g_1\left(N^{\mathcal{D}}_1\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_3\right) \otimes \left({B}_{1}^\dagger+{B}_{1}\right)\otimes I^{\otimes N_2}\nonumber\\ &&\;\;+g_2\left(N^{\mathcal{D}}_2\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_4\right) \otimes I^{\otimes N_1}\otimes \left({B}_{2}^\dagger+{B}_{2}\right).\end{aligned}$$ As pointed out by Sowa *et. al* [@Sowa17], there can be a phase difference in the coupling parameters $g_v$ given by where $\phi_v=\mathbf{k}_v \cdot \mathbf{d}_v$, with $\mathbf{k}_v$ being the wavevector of the $v$ vibrational mode and $\mathbf{d}_v$ the distance between dots coupled with this specific mode. Here, we assume that the distance between dots inside a molecule is smaller than the vibrational wavelength so the phase difference can be ignored and $g_1=g_2=g$. In order to analyze the action of electron-vibrational mode and tunneling couplings, we apply the Lang-Firsov [@Mahanbook] unitary transformation over the Hamiltonian in Eq.(\[eq:Hgeneral\]) calculating $\bar{H} = e^{S} H e^{-S}$, where $$\begin{aligned} &&S=\alpha_1 \left(N^{\mathcal{D}}_1\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_3\right) \otimes \left(B_1^\dagger - B_1\right)\otimes I^{\otimes N_2} \nonumber \\ &&+\alpha_2 \left(N^{\mathcal{D}}_2\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_4\right) \otimes I^{\otimes N_1}\otimes \left(B_2^\dagger - B_2\right),\end{aligned}$$ with $\alpha_v=\frac{g}{\omega_v}$. This calculation results on a new form for the Hamiltonian written as $$\label{eq:Htrans} \bar{H}=\left(\bar{H}_{\mathcal{D}}+V^{C}_{\mathrm{eff}}\right)\otimes I_{\mathcal{V}}+I_{\mathcal{D}}\otimes H_{\mathcal{V}}+\Delta^T_{\mathcal{DV}},$$ where $$\label{eq:Hmtrans} \bar{H}_{\mathcal{D}}=\sum_{i=1,2}\widetilde{\varepsilon}_i N^{\mathcal{D}}_i\otimes I^{\otimes 2}+I^{\otimes 2}\otimes \sum_{j=3,4}\widetilde{\varepsilon}_j N^{\mathcal{D}}_j,$$ is the transformed Hamiltonian for the dots with $\widetilde{\varepsilon}_{i(j)}$ being an energy level shifted due to the action of the electron-vibrational mode coupling. Specifically, $\widetilde{\varepsilon}_{1(3)}=\varepsilon_{1(3)}-\alpha_1^2\omega_1$ while $\widetilde{\varepsilon}_{2(4)}=\varepsilon_{2(4)}-\alpha_2^2\omega_2$. Apart from this dressed uncoupled electronic Hamiltonian, we want to highlight two new terms on Eq.(\[eq:Htrans\]). The first one can be seen as an effective electron-electron interaction $$\label{eq:Heeeff} V^{C}_{\mathrm{eff}}=-2\alpha_1^2\omega_1N^{\mathcal{D}}_1\otimes N^{\mathcal{D}}_3-2\alpha_2^2\omega_2N^{\mathcal{D}}_2\otimes N^{\mathcal{D}}_4,$$ which cooperates with tunneling in order to generate maximally entangled states. The last term $$\begin{aligned} \label{eq:Hvmvmeff} &&\Delta^T_{\mathcal{DV}}=\left[\left(\Delta_{12} S_1^\dagger S_2\right)\otimes I^{\otimes 2} + I^{\otimes 2}\otimes\left(\Delta_{34}S_3^\dagger S_4\right)\right]\otimes X_{12}\nonumber\\ &&\;\;+\left[\left(\Delta_{12} S_1 S_2^\dagger\right)\otimes I^{\otimes 2} + I^{\otimes 2}\otimes \left(\Delta_{43} S_3 S_4^\dagger\right)\right]\otimes X^{\dagger}_{12},\end{aligned}$$ describes an effective interaction between the two vibrational modes considered on our problem. Here $X_{12}=e^{-\alpha_1\left(B_1 - B^\dagger_1\right)}\otimes e^{-\alpha_2\left(B^\dagger_2- B_2\right)}=D_1(\alpha_1)\otimes D_2(\alpha_2)$, being a tensorial product of displacement operators for the quantum harmonic oscillator [@Scullybook]. The new transformed Hamiltonian, Eq. (\[eq:Htrans\]) and its terms Eqs. (\[eq:Hmtrans\])-(\[eq:Hvmvmeff\]), highlights important effects of the couplings considered on this particular physical system. The first is a shift on the value of the electronic levels which depends on both, the coupling parameter $g$ and $\omega_v$. The second is the effective electron-electron interaction which couples the electrons from different qubits, which is mediated by the electron-vibrational mode coupling. As we discuss below, the effective electron-electron interaction, together with electronic tunneling, is behind the apparition of entangled eigenstates of the full Hamiltonian and the subsequent possibility of generation, by quantum dynamics, of Bell states. Spectral analysis {#sec:eigenproblem} ================= We proceed to explore the characteristics of energy spectrum and eigenstates of the Hamiltonian in Eq. (\[eq:Hgeneral\]). Along with the study of energy spectrum, we are interested on the entanglement properties of the eigenstates. It is well known that Coulomb interaction is behind the formation of entangled states in coupled quantum dots molecule [@Fujisawa11; @Oliveira15], once this interaction couples two single electrons, making viable the encoding of two qubits. In the present problem, we expected the apparition of signatures of the effective electron-electron interaction on the entanglement degree of the eigenstates. To perform our numerical analysis, both basis associated with the vibrational modes are truncated at $N_1=N_2=13$, although both basis for vibrational modes have infinite dimension. This number of computational states is enough to guarantee the accuracy of the calculation of lower energies and eigenstates. To analyze the formation of entangled states, we first build up a density matrix for each eigenstate in the complete basis $\hat{\rho}_l={\mathinner{|{\psi_l}\rangle}}{\mathinner{\langle{\psi_l}|}}$, where ${\mathinner{|{\psi_l}\rangle}}$ is the $l$th eigenstate of Hamiltonian (\[eq:Hgeneral\]). The second step is the calculation of the reduced $4\times 4$ density matrix for the two qubits, by tracing out the degrees of freedom of the vibrational modes so $\hat{\rho}_{\mathcal{D},l}=\mathrm{Tr}_{\mathcal{V}}[\hat{\rho}_l]$. Then, the concurrence, defined by Wootters [@Wootters98], is used as a measurement of entanglement. ![The first $24$ eigenvalues of Hamiltonian (\[eq:Hgeneral\]) varying the detuning $\delta=\varepsilon_1-\varepsilon_2$ considering $\varepsilon_3=\varepsilon_4=0$ and $g=0.5\omega$ and $\Delta_{12}=\Delta_{34}=5\times 10^{-2}\omega$. The energy increases from panel (c) to panel (a) showing the first four states (black solid lines and squares), the next eight states (red dashed lines and circles), and the following twelve states (blue dotted lines and triangles).[]{data-label="fig:eigenproblem"}](figure2){width="0.7\linewidth"} An auxiliary Hermitian operator [@Hill97] $R_l$ is defined as $R_l=\sqrt{\sqrt{\hat{\rho}_{\mathcal{D},l}}\;\widetilde{\hat{\rho}_{\mathcal{D},l}}\sqrt{\hat{\rho}_{\mathcal{D},l}}}$, where $\widetilde{\hat{\rho}_{\mathcal{D},l}}=(\sigma_y \otimes \sigma_y)\hat{\rho}^\star_{\mathcal{D},l}(\sigma_y \otimes \sigma_y)$, is the spin-flipped matrix with $\hat{\rho}^\star_{\mathcal{D},l}$ being the complex conjugate of $\hat{\rho}_{\mathcal{D},l}$. The concurrence is obtained once $C=\mathrm{max}(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$ where $\lambda_k$ ($k=1...4$) are the eigenvalues of the operator $R_l$ in decreasing order. Figures \[fig:eigenproblem\](a) to (c) show the behavior of the first $24$ eigenvalues of the Hamiltonian as a function of detuning $\delta=\varepsilon_1-\varepsilon_2$, considering $\varepsilon_3=\varepsilon_4=0$, a resonance condition for the dots 3 and 4. We consider that both vibrational modes have the same frequency value so $\omega_1=\omega_2=\omega$. The coupling parameters are defined in terms of the frequency $\omega$ being $g=\omega/2$ the electron-vibrational mode coupling and $\Delta_{12}=\Delta_{34}=\Delta/2=\omega/20$ the tunneling rates. The energy spectrum shows the emergence of branches, spanned on an energy interval $\Delta E=\omega$, each with an increasing number of inner states as energy increases: while the first branch (solid black lines) has the four states shown in Fig. \[fig:eigenproblem\] (c) including the ground level, the second branch (dashed red lines) has eight states, as seen in Fig. \[fig:eigenproblem\] (b). The subsequent branch (dotted blue lines) contains twelve eigenstates, shown in Fig. \[fig:eigenproblem\] (a). The branches share some common features. The first is the appearance of anticrossings at $\delta=\pm 0.5$, due to first order transitions that switch only one electron per time, e.g., ${\mathinner{|{1001}\rangle}}_{\mathcal{D}}$ $\leftrightarrow$ ${\mathinner{|{1010}\rangle}}_{\mathcal{D}}$. The second is a little anticrossing arising at $\delta=0$, related to higher order transition processes. For instance, two electrons can start at dots 1 and 4 (state ${\mathinner{|{1001}\rangle}}_{\mathcal{D}}$) ending at dots 2 and 3 (state ${\mathinner{|{0110}\rangle}}_{\mathcal{D}}$)[^1]. For each branch, the inner states have interesting properties concerning entanglement as can be seen from Fig. \[fig:eigenconcurrence\] (a) to (c), where we show the behavior of concurrence as a function of detuning $\delta$. Comparing our findings with results on a previous work [@Oliveira15], some similarities let us to conclude that the effective electron-electron coupling is behind the apparition of dressed Bell states as eigenstates. A signature of this fact is the increasing value of maximally entangled states at $\delta=0$, as shown by the scattered plots: there is one on the black branch \[filled squares on Fig. \[fig:eigenconcurrence\] (a) and Fig. \[fig:eigenproblem\] (c)\], two inside the red branch \[filled and open circles on Fig. \[fig:eigenconcurrence\] (b) and Fig. \[fig:eigenproblem\] (b)\] and three on the blue branch \[filled and open triangles point up and down on Fig. \[fig:eigenconcurrence\] (c) and Fig. \[fig:eigenproblem\] (a)\]. Additionally, some satellite peaks at $\delta=\pm 0.5$ are observed, having a lower value of concurrence. While those secondary peaks are related to first order processes, the sharp peak at zero energy arises from second and higher order transitions. Those features of the eigenstates can be explored theoretically through a straightforward calculation (see Appendix \[ap:dressedbell\] for details) which consists on perform a basis transformation going from the electronic computational four-dimensional basis given by $\left\{{\mathinner{|{1010}\rangle}},{\mathinner{|{0101}\rangle}},{\mathinner{|{1001}\rangle}},{\mathinner{|{0110}\rangle}}\right\}_{\mathcal{D}}$ to an electronic Bell basis ordered as $\{{\mathinner{|{\Psi_{-}}\rangle}},{\mathinner{|{\Phi_{-}}\rangle}},{\mathinner{|{\Psi_{+}}\rangle}},{\mathinner{|{\Phi_{+}}\rangle}}\}_{\mathcal{D}}$, where ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1001}\rangle}}\pm{\mathinner{|{0110}\rangle}}\right)$ and ${\mathinner{|{\Phi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1010}\rangle}}\pm{\mathinner{|{0101}\rangle}}\right)$. For the specific case of equal tunneling rates and $\delta=0$, the calculation shows that the terms of the Hamiltonian regarding ${\mathinner{|{\Psi_{-}}\rangle}}_{\mathcal{D}}$ can be written as a tensorial product given by $$\begin{aligned} \label{eq:HtermsPsi-} H_{\mathrm{with }{\mathinner{|{\Psi_-}\rangle}}}&=&({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}})_{\mathcal{D}}\otimes\big\{\big[\mathbf{{\mathinner{|{00}\rangle}}_{\mathcal{V}}}\big(E_{00}{\mathinner{\langle{00}|}}_{\mathcal{V}}+g{\mathinner{\langle{10}|}}_{\mathcal{V}} +g{\mathinner{\langle{01}|}}_{\mathcal{V}}\big)\big] +\big[\mathbf{{\mathinner{|{01}\rangle}}_{\mathcal{V}}}\big(E_{01}{\mathinner{\langle{01}|}}_{\mathcal{V}}+g{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{02}|}}_{\mathcal{V}}\big)\nonumber\\ &&+\mathbf{{\mathinner{|{10}\rangle}}_{\mathcal{V}}}\big(E_{10}{\mathinner{\langle{10}|}}_{\mathcal{V}}+g{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{20}|}}_{\mathcal{V}}\big)\big] +\big[\mathbf{{\mathinner{|{11}\rangle}}_{\mathcal{V}}}\big(E_{11}{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{21}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{12}|}}_{\mathcal{V}}\big)\nonumber\\ &&+\mathbf{{\mathinner{|{02}\rangle}}_{\mathcal{V}}}\big(E_{02}{\mathinner{\langle{02}|}}_{\mathcal{V}}+...\big)+\mathbf{{\mathinner{|{20}\rangle}}_{\mathcal{V}}}\big(E_{20}{\mathinner{\langle{20}|}}_{\mathcal{V}}+...\big)\big]+...+\mathrm{h.c.}\big\}.\end{aligned}$$ Other terms on Hamiltonian cannot be written as a tensorial product of the form ${\mathinner{|{\psi}\rangle}}{\mathinner{\langle{\psi}|}}_{\mathcal{D}}\otimes\sum \alpha{\mathinner{|{N'_1N'_2}\rangle}}_{\mathcal{V}}{\mathinner{\langle{N_1N_2}|}}$: terms with ${\mathinner{|{\Psi_+}\rangle}}$ are coupled with ${\mathinner{|{\Phi_+}\rangle}}$ by electron-vibrational mode interaction, while elements ${\mathinner{|{\Phi_{+}}\rangle}}$ and ${\mathinner{|{\Phi_{-}}\rangle}}$ are also coupled to each other by tunneling (see Appendix \[ap:dressedbell\]). In the Eq. \[eq:HtermsPsi-\], we use bold type and the square brackets, $[\;]$, to emphasize the new dressed basis $\left\{{\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}\right\}$. The number of eigenstates per branch and the number of maximally entangled molecular states at $\delta=0$ are linked with the dimension of original subspaces with the same value of the sum $N_1+N_2$, as can be seen from Eq.(\[eq:HmatrixBell\]). Although these subspaces are coupled with each other, each branch can be seen as dressed Bell states, with an energy increasing as $N=N_1+N_2$ grows. ![Behavior of the concurrence, $C$, for the electronic part of the first $24$ eigenstates. We use the same physical parameters and the same color, lines and symbol conventions that in Fig. \[fig:eigenproblem\]. Note that the condition $\delta=0$ is related with the apparition of electronic maximally entangled states, with $C=1$.[]{data-label="fig:eigenconcurrence"}](figure3){width="0.8\linewidth"} Dynamical generation of electronic Bell states {#sec:dynamics} ============================================== After studying the properties of the eigenstates of the model, it is worthwhile to explore the preparation of electronic entangled states by quantum dynamics. We again use a numerical approach, considering the general Hamiltonian (\[eq:Hgeneral\]), to simulate the quantum dynamics through the evolution of the density matrix $\rho(t)={\mathinner{|{\psi(t)}\rangle}}{\mathinner{\langle{\psi(t)}|}}$. Tracing out the vibrational degrees of freedom, it results on the electronic reduced density matrix $$\rho_{\mathcal{D}}(t)=\mathrm{Tr}_{\mathcal{V}}[\rho(t)],$$ used to explore the behavior of the electronic part of the system through the calculation of physical properties as populations, fidelity, and concurrence. Initially, we perform a test to check the precision of our calculation together with our findings about the special character of the electronic Bell state ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ as discussed on Sec. \[sec:eigenproblem\]. We consider $\Delta_{12}=\Delta_{34}$ (equal tunneling couplings), detuning $\delta=0$ and the initial state given by $$\rho(0)= \left({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\right)_{\mathcal{D}}\otimes{\mathinner{|{00}\rangle}}_{\mathcal{V}}{\mathinner{\langle{00}|}}.$$ After the calculation of $\rho(t)$ and the $ \rho_{\mathcal{D}}(t)$, we obtain the fidelity of the evolved electronic state with this initial state being $\mathcal{F}(t)=\mathrm{Tr}_{\mathcal{D}}[\rho_{\mathcal{D}}(t)\rho_{\mathcal{D}}(0)]$. Our results shown that the fidelity remains constant with value $\mathcal{F}(t)=1$, showing that, from the electronic point of view, any initial state with $\left({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\right)_{\mathcal{D}}$ as the electronic part, acts as a stationary state. In order to explore the generation of electronic entangled states, we now assume the system is being prepared as $$\rho_0={\mathinner{|{1001}\rangle}}_{\mathcal{D}}{\mathinner{\langle{1001}|}}\otimes{\mathinner{|{00}\rangle}}_{\mathcal{V}}{\mathinner{\langle{00}|}},$$ which is an experimentally feasible initial state, once experimental setups include a set of sources and drains attached to the quantum dots, that can inject electrons. We consider the condition of $\delta=0$ and the same choices of physical parameters used in Figs. \[fig:eigenproblem\]-\[fig:eigenconcurrence\]. In Fig. \[fig:popconcfid\] (a), we show the evolution of the populations of electronic states $\left\{{\mathinner{|{1001}\rangle}},{\mathinner{|{0110}\rangle}},{\mathinner{|{0101}\rangle}},{\mathinner{|{1010}\rangle}}\right\}$. At initial times, the values is given by $P_{1001}=1$ (black line), $P_{0110}=0$ (red line), and $P_{0101}=P_{1010}=0$ (green line) in agreement with the initial condition. As time evolves, $P_{1001}$ and $P_{0110}$ oscillates out of phase between zero (non-occupied) and $1$ (occupied), while $P_{1010}$ and $P_{0101}$ remain oscillating close to zero. This is a promising sign toward the formation of a state similar with ${\mathinner{|{\Psi_{\pm}}\rangle}}$ states on Bell basis, once the population of states ${\mathinner{|{1001}\rangle}}$ and ${\mathinner{|{0110}\rangle}}$ is $0.5$ at $\omega t = 300$. We also plot the concurrence, solid blue line in Fig. \[fig:popconcfid\] (a), which reaches the value $C\approx 1$ when $P_{1001}=P_{0110}$, indicating the actual formation of a maximally entangled electronic state. A similar effect was originally reported in Ref.  in the context of two-qubits coupled via Coulomb interaction. In the present case, this evolution shows that the effective electron-electron interaction mediated by the vibrational modes turns out to be responsible for the coupling between the qubits. In order to give a closer look at the entangled states created by quantum dynamics, we compute the fidelity compared with a pre-defined target Bell state, $\mathcal{F}(t)=\mathrm{Tr}_{\mathcal{D}}[\rho_{\mathcal{D}}(t)\rho_{\mathcal{D}}^{\mathrm{tar}}]$. For this calculation, we use as target state the electronic density matrix $\rho_{\mathcal{D}}^{\mathrm{tar}}={\mathinner{|{\Psi(\varphi)}\rangle}}_{\mathcal{D}}{\mathinner{\langle{\Psi(\varphi)}|}}$, where $${\mathinner{|{\Psi(\varphi)}\rangle}}_{\mathcal{D}} = \frac{1}{\sqrt{2}} [|1001\rangle_{\mathcal{D}}+e^{i\varphi} |0110\rangle_{\mathcal{D}}], \label{eq:psi_fase}$$ where $\varphi$ is a relative phase between the state. Note that if $\varphi=0$ ($\varphi=\pi$) then ${\mathinner{|{\Psi(0)}\rangle}}\equiv{\mathinner{|{\Psi_+}\rangle}}$ (${\mathinner{|{\Psi(\pi)}\rangle}}\equiv{\mathinner{|{\Psi_-}\rangle}}$). In Fig. \[fig:popconcfid\] (b) we show how $\mathcal{F}$ evolves with time for different values of $\varphi$. For $\varphi=0$ the fidelity remains stationary, with a value of $0.5$, while for $\varphi=\pi$, the fidelity shows small oscillations around those value. By setting the $\varphi=\pm \pi/2$, we fulfill our goal of find the correct relative phase of the dynamically created entangled state, once the fidelity for both cases oscillates out of phase between $0$ and $1$. The concurrence maximum values being $1$ let us to conclude that the electronic entangled state alternates between ${\mathinner{|{\Psi(\pi/2)}\rangle}}_{\mathcal{D}}$, at $\omega t=300$ and ${\mathinner{|{\Psi(-\pi/2)}\rangle}}_{\mathcal{D}}$, at $\omega t=600$. This means that the dynamics shows not only the ability of create Bell states related with ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}$ but also an additional ingredient, which is the imprint of a relative phase $\varphi$. Finally in Fig. \[fig:concvsg\] we explore the effect of the electron-vibrational mode coupling parameter, $g$, on the formation of the electronic entangled states, going into a weak coupling regime. We first notice that, by decreasing the value of $g$, we still obtain values of $C\approx 1$ at some evolved time. Second feature is that it becomes clear that the period of oscillations of the formation of the electronic Bell states is governed by this coupling parameter. This shows that the formation of electronic maximally entangled states is a quite robust effect, which can become accessible in a wide range of experimental devices, even for weak values of electron-vibrational mode coupling. ![Dynamics of populations, concurrence, and fidelity of electrons in quantum dots considering $\delta=\varepsilon_1-\varepsilon_2=0$, $\varepsilon_3=\varepsilon_4=0$, $g=0.5\omega$ and $\Delta_{12}=\Delta_{34}=5\times 10^{-2}\omega$. (a) Populations $P_{1001}$ (black line), $P_{0110}$ (red line) and $P_{1010}=P_{0101}$ (green line), and concurrence (blue line) as functions of $\omega t$. (b) Dynamics of the fidelity between the evolved state and the target state given by Eq. (\[eq:psi\_fase\]), for different values of relative phase $\varphi$: $\varphi=-\pi/2$ (blue line), $\varphi=0$ (green line), $\varphi=+\pi/2$ (black line), and $\varphi=\pi$ (red line).[]{data-label="fig:popconcfid"}](figure4){width="1\linewidth"} ![Concurrence as a function of $\omega t$ for the same values of $\delta$, $\varepsilon_3$, $\varepsilon_4$, $\Delta_{12}$ $\Delta_{34}$ used in Fig. \[fig:popconcfid\], considering several values of electron-vibrational mode coupling $g$: $g=0.5\omega$ (black line), $g=0.1\omega$ (red line), and $g=0.05\omega$ (blue line).[]{data-label="fig:concvsg"}](figure5){width="1\linewidth"} Conclusion {#sec:summary} ========== In this work, we analyze a general model which describes a system where two electrons inside a set of four quantum dots interact with vibrational modes, shown in Fig. \[fig:system\]. The goal is to explore the interplay between tunneling, detuning of the electronic levels inside the quantum dots, and the electron-vibrational mode coupling. The model can describe several experimental scenarios, including electrons inside carbon nanotubes quantum dots or the coupling between electrons and an acoustic cavity. Our findings include the study of the characteristics of the spectrum, the eigenstates, and the quantum dynamics of the system, focusing on the search of electronic maximally entangled states. We find that the electron-vibrational mode coupling is responsible for the apparition of dressed electronic Bell states. Concerning the dynamics, the generation of those entangled states is possible for a wide range of physical parameters. Acknowledgments =============== This work was supported by CNPq (grant 307464/2015-6), and the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ). The matrix representation of hamiltonian on the dressed Bell basis {#ap:dressedbell} ================================================================== The entanglement behavior of the system of interest can be explored by writing down its Hamiltonian in terms of the electronic Bell states. Let us calculate the representation of the original Hamiltonian (\[eq:Hgeneral\]) as a matrix written in the dressed basis ${\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}$, where the electronic part is ordered following $${\mathinner{|{\psi_{\mathrm{Bell}}}\rangle}}_{\mathcal{D}}=\{{\mathinner{|{\Psi_{-}}\rangle}},{\mathinner{|{\Phi_{-}}\rangle}},{\mathinner{|{\Psi_{+}}\rangle}},{\mathinner{|{\Phi_{+}}\rangle}}\}_{\mathcal{D}},$$ where ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1001}\rangle}}\pm{\mathinner{|{0110}\rangle}}\right)$ and ${\mathinner{|{\Phi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1010}\rangle}}\pm{\mathinner{|{0101}\rangle}}\right)$. We choose to keep together the states with the same number of total excitations $N=N_1+N_2$. This choice remarks the fact that the basis for the whole system has an internal structure of coupled subspaces. For each value of $N$, there is an associated family of subspaces $\mathcal{S}_{\mathrm{B},(N_1N_2)}$: $N=0$ has only the subspace $\mathcal{S}_{\mathrm{B},(00)}$ with four inner states, $N=1$ has two subspaces being $\mathcal{S}_{\mathrm{B},(10)}$ and $\mathcal{S}_{\mathrm{B},(01)}$ (eight inner states), $N=3$ has twelve inner states associated with $\mathcal{S}_{\mathrm{B},(11)}$, $\mathcal{S}_{\mathrm{B},(20)}$, and $\mathcal{S}_{\mathrm{B},(02)}$, and so on. Let us write the matrix representation of the Hamiltonian for this first six $4$D subspaces $\mathcal{S}_{\mathrm{B},(N_1N_2)}$, ordered as $\{\mathcal{S}_{B,(00)},\mathcal{S}_{B,(01)},\mathcal{S}_{B,(10)},\mathcal{S}_{B,(11)},\mathcal{S}_{B,(02)},\mathcal{S}_{B,(20)}\}$: $$\label{eq:HmatrixBell} H=\left( \begin{array}{c|cc|ccc} B_{00} & G_{2} & G_{1} & 0 & 0 & 0 \\ \hline G_{2} & B_{01} & 0 & G_{1} & \sqrt{2}G_{2} & 0 \\ G_{1} & 0 & B_{10} & G_{2} & 0 & \sqrt{2}G_{1} \\ \hline 0 & G_{1} & G_{2} & B_{11} & 0 & 0 \\ 0 & \sqrt{2}G_{2} & 0 & 0 & B_{02} & 0 \\ 0 & 0 & \sqrt{2}G_{1} & 0 & 0 & B_{02} \\ \end{array} \right).$$ using the order ${\mathinner{|{\Psi_{-},N_1N_2}\rangle}}$, ${\mathinner{|{\Phi_{-},N_1N_2}\rangle}}$, ${\mathinner{|{\Psi_{+},N_1N_2}\rangle}}$, and ${\mathinner{|{\Phi_{+},N_1N_2}\rangle}}$, the $4$D matrices $B_{N_1N_2}$ and $ G_{v}$ are defined as $$\begin{aligned} \label{eq:4DBellmatrix} B_{N_1N_2}&=&\left( \begin{array}{cccc} E_{N_1N_2} & \Delta_- & \delta_-/2 & 0 \\ \Delta_- & E_{N_1N_2} & 0 & \delta_+/2 \\ \delta_-/2 & 0 & E_{N_1N_2} & \Delta_+ \\ 0 & \delta_+/2 & \Delta_+ & E_{N_1N_2} \\ \end{array} \right),\end{aligned}$$ and $$\label{eq:4DGmatrix} G_{v}=\left( \begin{array}{cccc} g_{v} & 0 & 0 & 0 \\ 0 & g_{v}/2 & 0 & (-1)^{(v-1)}g_{v}/2 \\ 0 & 0 & g_{v} & 0 \\ 0 & (-1)^{(v-1)}g_{v}/2 & 0 & g_{v}/2 \\ \end{array} \right),$$ where $E_{N_1N_2}=\sum_{i=1,2}\sum_{j=3,4}\sum_{v=1,2}\left(\varepsilon_i+\varepsilon_j+\omega_v\right)$ are the energy of the state ${\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}$, the tunneling couplings are defined as $\Delta_{\pm}=\Delta_{34}\pm\Delta_{12}$ and $\delta_{\pm}=\delta_{12}\pm\delta_{34}$, with $\delta_{lm}=\varepsilon_l-\varepsilon_m$ ($l=1,3$ and $m=2,4$). The first matrix resembles the rotated matrix on Bell basis, whose properties discussed in details on Ref. , and the matrices $G_v$ depends on $g_v$ and carry on the effect of electron-vibrational mode coupling, where the factor $\sqrt{N_v}$ appears on the specific elements of the matrix (\[eq:HmatrixBell\]) which depends on the values of $N_v$ of the coupled subspaces. If we consider a full resonance condition between electronic levels $\delta_{\pm}=0$, the matrix $B_{N_1N_2}$ becomes $$\begin{aligned} \label{eq:4DBellmatrixv1} B_{N_1N_2}&=&\left( \begin{array}{cccc} E_{N_1N_2} & 0 & 0 & 0 \\ 0 & E_{N_1N_2} & 0 & 0 \\ 0 & 0 & E_{N_1N_2} & \Delta_+ \\ 0 & 0 & \Delta_+ & E_{N_1N_2} \\ \end{array} \right).\end{aligned}$$ At first sight, it seems that the states with electronic part being ${\mathinner{|{\Psi{-}}\rangle}}_\mathcal{D}$ and ${\mathinner{|{\Phi{-}}\rangle}}_\mathcal{D}$ are decoupled, at the same time that ${\mathinner{|{\Psi{+}}\rangle}}_\mathcal{D}$ and ${\mathinner{|{\Phi{+}}\rangle}}_\mathcal{D}$ are not, in the same way that in Ref. . Nevertheless, if elements for the first two lines on matrix (\[eq:HmatrixBell\]), associated with ${\mathinner{|{\Psi_-,00}\rangle}}$ and ${\mathinner{|{\Psi_-,00}\rangle}}$ respectively, are written using the notation ${\mathinner{|{\;}\rangle}}{\mathinner{\langle{\;}|}}$ we obtain: $$\begin{aligned} \label{eq:HtermsPsi-Phi-} H&=&{\mathinner{|{\Psi_-,00}\rangle}}\big(E_{00}{\mathinner{\langle{\Psi_-,00}|}}+g_2{\mathinner{\langle{\Psi_-,01}|}}+g_1{\mathinner{\langle{\Psi_-10}|}}\big)\nonumber\\ &+&{\mathinner{|{\Phi_-,00}\rangle}}\left(E_{00}{\mathinner{\langle{\Phi_-,00}|}} +\frac{g_2}{2}{\mathinner{\langle{\Phi_-,01}|}}-\frac{g_2}{2}{\mathinner{\langle{\Phi_+,01}|}}\right.\nonumber\\ &&\left.+\frac{g_1}{2}{\mathinner{\langle{\Phi_-,10}|}}+\frac{g_1}{2}{\mathinner{\langle{\Phi_+,10}|}}\right)+...\end{aligned}$$ The term with ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ can be written as $\big({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\big)_{\mathcal{D}}\otimes\big(E_{00}{\mathinner{|{00}\rangle}}{\mathinner{\langle{00}|}}+g_2{\mathinner{|{00}\rangle}}{\mathinner{\langle{01}|}}+g_1{\mathinner{|{00}\rangle}}{\mathinner{\langle{10}|}}\big)$, while the others do not permit the same. Continuing with the calculation, we realize that only the terms on Hamiltonian associated with ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ are decoupled, at least from the electronic point of view, from the rest of the Bell basis. In this way, there is a Bell state, dressed by vibrational modes, becoming an eigenstate of the Hamiltonian (\[eq:Hgeneral\]) for the specific condition of equal tunneling couplings, $\Delta_{34}=\Delta_{12}$ and full resonance between the electronic levels. [37]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} [****,  ()](\doibase 10.1002/qua.20484) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, , ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1038/nnano.2009.71) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{},  ed. (, , ) @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.78.5022) [^1]: Although not shown here, the second-order time-dependent perturbation theory can be used to show this transition

ArXiv

--- abstract: 'There are different categorizations of the definition of a [*ring*]{} such as [*Ann-category*]{} \[6\], [*ring category*]{} \[2\],... The main result of this paper is to prove that every axiom of the definition of a [*ring category*]{}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an [*Ann-category*]{}.' author: - | Nguyen Tien Quang, Nguyen Thu Thuy and Che Thi Kim Phung\ [*Hanoi National University of Education*]{} title: 'THE RELATION BETWEEN ANN-CATEGORIES AND RING CATEGORIES' --- Introduction ============ Categories with monoidal structures $\oplus, \otimes$ (also called [*categories with distributivity constraints*]{}) were presented by Laplaza \[3\]. M. Kapranov and V.Voevodsky \[2\] omitted requirements of the axiomatics of Laplaza which are related to the commutativity constraints of the operation $\otimes$ and presented the name [*ring categories*]{} to indicate these categories. To approach in an other way, monoidal categories can be “smoothed” to become a [*category with group structure,*]{} when they are added the definition of invertible objects (see Laplaza \[4\], Saavedra Rivano \[9\]). Now, if the back ground category is a [*groupoid*]{} (i.e., each morphism is an isomorphism) then we have [*monoidal category group-like*]{} (see A. Frölich and C. T. C. Wall \[1\], or a [*Gr-category*]{} (see H. X. Sinh \[11\]). These categories can be classified by $H^{3}(\Pi, A)$. Each Gr-category $\mathcal G$ is determined by 3 invariants: The group $\Pi$ of classes of congruence objects, $\Pi-$module $A$ of automorphisms of the unit $1$, and an element $\overline{h}\in H^{3}(\Pi, A),$ where $h$ is induced by the associativity constraint of $\mathcal G.$ In 1987, in \[6\], N. T. Quang presented the definition of an [*Ann-category*]{}, as a categorization of the definition of rings, when a symmetric Gr-category (also called Pic-category) is equiped with a monoidal structure $\otimes$. In \[8\], \[7\], Ann-categories and [*regular*]{} Ann-categories, developed from the ring extension problem, have been classified by, respectively, Mac Lane ring cohomology \[5\] and Shukla algebraic cohomology \[10\]. The aim of this paper is to show clearly the relation between the definition of an [*Ann-category*]{} and a [*ring category*]{}. For convenience, let us recall the definitions. Moreover, let us denote $AB$ or $A.B$ instead of $A{\otimes }B.$ Fundamental definitions ======================= [**The axiomatics of an Ann-category**]{}\ An Ann-category consists of:\ i) A groupoid ${\mathcal{A}}$ together with two bifunctors ${\oplus},{\otimes }:{\mathcal{A}}\times{\mathcal{A}}\longrightarrow {\mathcal{A}}.$\ ii) A fixed object $0\in {\mathcal{A}}$ together with naturality constraints $a^+,c,g,d$ such that $({\mathcal{A}},{\oplus},a^+,c,(0,g,d))$ is a Pic-category.\ iii) A fixed object $1\in{\mathcal{A}}$ together with naturality constraints $a,l,r$ such that $({\mathcal{A}},{\otimes },a,(1,l,r))$ is a monoidal $A$-category.\ iv) Natural isomorphisms ${\frak{L}},{\frak{R}}$ $${\frak{L}}_{A,X,Y}:A{\otimes }(X{\oplus}Y)\longrightarrow (A{\otimes }X){\oplus}(A{\otimes }Y)$$ $${\frak{R}}_{X,Y,A}:(X{\oplus}Y){\otimes }A\longrightarrow(X{\otimes }A){\oplus}(Y{\otimes }A)$$ such that the following conditions are satisfied:\ (Ann-1) For each $A\in {\mathcal{A}},$ the pairs $(L^A,\breve{L^A}),(R^A,\breve{R^A})$ determined by relations:\ $$\begin{aligned} &L^A & = &A{\otimes }- \;\;\;\; &R^A&=&-{\otimes }A\\ &\breve{L^A}_{X,Y}& = &{\frak{L}}_{A, X, Y}\;\;\;\; &\breve{R^A}_{X, Y}&=&\mathcal{R}_{X, Y, A} \end{aligned}$$ are ${\oplus}$-functors which are compatible with $a^+$ and $c.$\ (Ann-2) For all $ A,B,X,Y\in {\mathcal{A}},$ the following diagrams: $$\begin{diagram} \node{(AB)(X{\oplus}Y)}\arrow{s,l}{\breve{L}^{AB}} \node{ A(B(X{\oplus}Y))}\arrow{w,t}{\quad a_{A, B, X{\oplus}Y}\quad}\arrow{e,t}{\quad id_A{\otimes }\breve{L}^B\quad}\node{ A(BX{\oplus}BY)}\arrow{s,r}{\breve{L}^A}\\ \node{(AB)X{\oplus}(AB)Y}\node[2]{A(BX){\oplus}A(BY)}\arrow[2]{w,t} {\quad\quad a_{A, B, X}{\oplus}a_{A, B, Y} \quad\quad} \end{diagram}\tag{1.1}$$ $$\begin{diagram} \node{(X{\oplus}Y)(BA)}\arrow{s,l}{\breve{R}^{BA}}\arrow{e,t}{\quad a_{ X{\oplus}Y, B, A}\quad}\node{((X{\oplus}Y)B)A}\arrow{e,t}{\quad \breve{R}^B{\otimes }id_A\quad}\node{ (XB{\oplus}YB)A}\arrow{s,r}{\breve{R}^A}\\ \node{X(BA){\oplus}Y(BA)}\arrow[2]{e,t} {\quad\quad a_{X, B, A}{\oplus}a_{Y, B, A} \quad\quad}\node[2]{(XB)A{\oplus}(YB)A} \end{diagram}\tag{1.1'}$$ $$\begin{diagram} \node{(A(X{\oplus}Y))B}\arrow{s,l}{\breve{L}^{A}{\otimes }id_B} \node{ A((X{\oplus}Y)B)}\arrow{w,t}{\quad a_{A, X{\oplus}Y, B}\quad}\arrow{e,t}{\quad id_A{\otimes }\breve{R}^B\quad}\node{ A(XB{\oplus}YB)}\arrow{s,r}{\breve{L}^A}\\ \node{(AX{\oplus}AY)B}\arrow{e,t}{\quad \breve{R}^B \quad}\node{(AX)B{\oplus}(AY)B}\node{A(XB){\oplus}A(YB)}\arrow{w,t} {\quad a{\oplus}a\quad} \end{diagram}\tag{1.2}$$ $$\begin{diagram} \node{(A{\oplus}B)X{\oplus}(A{\oplus}B)Y}\arrow{s,r}{\breve{R}^{X}{\oplus}\breve{R}^Y} \node{(A{\oplus}B)(X{\oplus}Y)}\arrow{w,t}{\breve{L}^{A{\oplus}B}}\arrow{e,t}{ \breve{R}^{X{\oplus}Y}}\node{A(X{\oplus}Y){\oplus}B(X{\oplus}Y)}\arrow{s,l}{\breve{L}^A{\oplus}\breve{L}^B}\\ \node{(AX{\oplus}BX){\oplus}(AY{\oplus}BY)}\arrow[2]{e,t} {\quad\quad v \quad\quad}\node[2]{(AX{\oplus}AY){\oplus}(BX{\oplus}BY)} \end{diagram}\tag{1.3}$$ commute, where $v=v_{U,V,Z,T}:(U{\oplus}V){\oplus}(Z{\oplus}T)\longrightarrow(U{\oplus}Z){\oplus}(V{\oplus}T)$ is the unique functor built from $a^+,c,id$ in the monoidal symmetric category $({\mathcal{A}},{\oplus}).$\ (Ann-3) For the unity object $1\in {\mathcal{A}}$ of the operation ${\oplus},$ the following diagrams: $$\begin{diagram} \node{1(X\oplus Y)} \arrow[2]{e,t}{\breve{L}^1} \arrow{se,b}{l_{X\oplus Y}} \node[2]{1X\oplus 1Y} \arrow{sw,b}{l_X\oplus l_Y} \\ \node[2]{X\oplus Y} \end{diagram}\tag{1.4}$$ $$\begin{diagram} \node{(X\oplus Y)1} \arrow[2]{e,t}{\breve{R}^1} \arrow{se,b}{r_{X\oplus Y}} \node[2]{X1\oplus Y1} \arrow{sw,b}{r_X\oplus r_Y} \\ \node[2]{X\oplus Y} \end{diagram}\tag{1.4'}$$ commute. [**Remark.**]{} The commutative diagrams (1.1), (1.1’) and (1.2), respectively, mean that:\ $$\begin{aligned} (a_{A, B, -})\;:\;& L^A.L^B &\longrightarrow \;& L^{AB}\\ (a_{-,A,B})\;:\;&R^{AB}&\longrightarrow \;&R^A.R^B\\ (a_{A, - ,B})\;:\;&L^A.R^B&\longrightarrow \;&R^B.L^A \end{aligned}$$ are ${\oplus}$-functors.\ The diagram (1.3) shows that the family $(\breve{L}^Z_{X,Y})_Z=(\mathcal{L}_{-,X,Y})$ is an ${\oplus}$-functor between the ${\oplus}$-functors $Z\mapsto Z(X{\oplus}Y)$ and $Z\mapsto ZX{\oplus}ZY$, and the family $(\breve{R}^C_{A,B})_C=(\mathcal{R}_{A, B,-})$ is an ${\oplus}$-functor between the functors $C\mapsto (A{\oplus}B)C$ and $C\mapsto AC{\oplus}BC.$\ The diagram (1.4) (resp. (1.4’)) shows that $l$ (resp. $r$) is an ${\oplus}$-functor from $L^1$ (resp. $R^1$) to the unitivity functor of the ${\oplus}$-category ${\mathcal{A}}$. [**The axiomatics of a ring category**]{} A [*ring category*]{} is a category ${\mathcal{R}}$ equiped with two monoidal structures ${\oplus},{\otimes }$ (which include corresponding associativity morphisms $a^{{\oplus}}_{A,B,C},a^{{\otimes }}_{A,B,C}$ and unit objects denoted 0, 1) together with natural isomorphisms $$u_{A,B}:A{\oplus}B\to B{\oplus}A,\qquad\qquad v_{A,B,C}:A{\otimes }(B{\oplus}C)\to (A{\otimes }B){\oplus}(A{\otimes }C)$$ $$w_{A,B,C}:(A{\oplus}B){\otimes }C\to (A{\otimes }C){\oplus}(B{\otimes }C),$$ $$x_A:A{\otimes }0\to 0,\qquad y_A:0{\otimes }A\to 0.$$ These isomorphisms are required to satisfy the following conditions. $K1 (\bullet{\oplus}\bullet)$ The isomorphisms $u_{A,B}$ define on ${\mathcal{R}}$ a structure of a symmetric monoidal category, i.e., they form a braiding and $u_{A,B}u_{B,A}=1.$ $K2 (\bullet{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C$ the diagram [$$\begin{diagram} \node{A{\otimes }(B{\oplus}C)}\arrow{s,l}{A{\otimes }u_{B,C}} \arrow{e,t}{v_{A,B,C}}\node{(A{\otimes }B){\oplus}(A{\otimes }C)}\arrow{s,r}{u_{A{\otimes }B,A{\otimes }C}} \\ \node{A{\otimes }(C{\oplus}B)}\arrow{e,t}{v_{A,C,B}}\node{(A{\otimes }C){\oplus}(A{\otimes }B)} \end{diagram}$$]{} is commutative. $K3 ((\bullet{\oplus}\bullet){\otimes }\bullet)$ For any objects $A,B,C$ the diagram [$$\begin{diagram} \node{(A{\oplus}B){\otimes }C}\arrow{s,l}{u_{A,B}{\otimes }C} \arrow{e,t}{w_{A,B,C}}\node{(A{\otimes }C){\oplus}(B{\otimes }C)}\arrow{s,r}{u_{A{\otimes }C,B{\otimes }C}} \\ \node{(B{\oplus}A){\otimes }C}\arrow{e,t}{w_{B,A,C}}\node{(B{\otimes }C){\oplus}(A{\otimes }C)} \end{diagram}$$]{} is commutative. $K4 ((\bullet{\oplus}\bullet{\oplus}\bullet){\otimes }\bullet)$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram} \node{(A{\oplus}(B{\oplus}C)D)}\arrow{s,l}{a^{{\oplus}}_{A,B,C}{\otimes }D} \arrow{e,t}{w_{A,B{\oplus}C,D}}\node{AD{\oplus}((B{\oplus}C)D)}\arrow{e,t}{AD{\oplus}w_{B,C,D}}\node{AD{\oplus}(BD{\oplus}CD)}\arrow{s,r}{a^{{\oplus}}_{AD,BD,CD}} \\ \node{((A{\oplus}B){\oplus}C)D}\arrow{e,t}{w_{A{\oplus}B,C,D}}\node{(A{\oplus}B)D{\oplus}CD}\arrow{e,t}{w_{A,B,D}{\oplus}CD}\node{(AD{\oplus}BD){\oplus}CD} \end{diagram}$$]{} is commutative. $K5 (\bullet{\otimes }(\bullet{\oplus}\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram} \node{A(B{\oplus}(C{\oplus}D))}\arrow{s,l}{A{\otimes }a^{{\oplus}}_{B,C,D}} \arrow{e,t}{v_{A,B,C{\oplus}D}}\node{AB{\oplus}A(C{\oplus}D)}\arrow{e,t}{AB{\oplus}v_{A,C,D}}\node{AB{\oplus}(AC{\oplus}AD)}\arrow{s,r}{a^{{\oplus}}_{AB,AC,AD}} \\ \node{A((B{\oplus}C){\oplus}D)}\arrow{e,t}{v_{A,B{\oplus}C,D}}\node{A(B{\oplus}C){\oplus}AD}\arrow{e,t}{v_{A,B,C}{\oplus}AD}\node{(AB{\oplus}AC){\oplus}AD} \end{diagram}$$]{} is commutative. $K6 (\bullet{\otimes }\bullet{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram} \node{A(B(C{\oplus}D))}\arrow{s,l}{a^{{\otimes }}_{A,B,C{\oplus}D}} \arrow{e,t}{A{\otimes }v_{B,C,D}}\node{A(BC{\oplus}BD)}\arrow{e,t}{v_{A,BC,BD}}\node{A(BC){\oplus}A(BD)}\arrow{s,r}{a^{{\otimes }}_{A,B,C}{\oplus}a^{{\otimes }}_{A,B,D}} \\ \node{(AB)(C{\oplus}D)} \arrow[2]{e,t}{v_{AB,C,D}}\node[2]{(AB)C{\oplus}(AB)D} \end{diagram}$$]{} is commutative. $K7 ((\bullet{\oplus}\bullet){\otimes }\bullet{\otimes }\bullet)$ Similar to the above. $K8 (\bullet{\otimes }(\bullet{\oplus}\bullet){\otimes }\bullet)$ Similar to the above. $K9 ((\bullet{\oplus}\bullet){\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram (12,4.5) (0,0)[$((AC{\oplus}BC){\oplus}AD){\oplus}BD$]{} (0.1,1.5)[$(AC{\oplus}BC){\oplus}(AD{\oplus}BD)$]{} (0.3,3)[$(A{\oplus}B)C{\oplus}(A{\oplus}B)D$]{} (0.8,4.5)[$(A{\oplus}B)(C{\oplus}D)$]{} (10,0)[$(AC{\oplus}(BC{\oplus}AD)){\oplus}BD$]{} (10,1.5)[$(AC{\oplus}(AD{\oplus}BC)){\oplus}BD$]{} (9.8,3)[$((AC{\oplus}AD){\oplus}BC){\oplus}BD$]{} (9.8,4.5)[$(AC{\oplus}AD){\oplus}(BC{\oplus}BD)$]{} (5.2,4.5)[$A(C{\oplus}D){\oplus}B(C{\oplus}D)$]{} (1.8,4.3)[(0,-1)[0.8]{}]{} (1.8,2.8)[(0,-1)[0.8]{}]{} (1.8,1.3)[(0,-1)[0.8]{}]{} (11.6,4.3)[(0,-1)[0.8]{}]{} (11.6,1.9)[(0,1)[0.8]{}]{} (11.6,1.3)[(0,-1)[0.8]{}]{} (3.1,4.6)[(1,0)[1.9]{}]{} (8.3,4.6)[(1,0)[1.3]{}]{} (9.8,0.1)[(-1,0)[6.2]{}]{} is commutative (the notation for arrows have been omitted, they are obvious). $K10 (0{\otimes }0)$ The maps $x_0,y_0:0{\otimes }0\to 0$ coincide. $K11 (0{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B$ the diagram [$$\begin{diagram} \node{0{\otimes }(A{\oplus}B)}\arrow{s,l}{y_{A{\oplus}B}} \arrow{e,t}{v_{0,A,B}}\node{(0{\otimes }A){\oplus}(0{\otimes }B)}\arrow{s,r}{y_a{\oplus}y_B}\\ \node{0}\node{0{\oplus}0}\arrow{w,t}{l^{{\oplus}}_0=r^{{\oplus}}_0} \end{diagram}$$]{} is commutative. $K12 ((\bullet{\oplus}\bullet){\otimes }0)$ Similar to the above. $K13 (0{\otimes }1)$ The maps $y_1,r^{{\otimes }}_0:0{\otimes }1\to 0$ coincide. $K14 (1{\otimes }0)$ Similar to the above. $K15 (0{\otimes }\bullet{\otimes }\bullet)$ For any objects $A,B$ the diagram [$$\begin{diagram} \node{0{\otimes }(A{\otimes }B)}\arrow{s,l}{y_{A{\otimes }B}} \arrow{e,t}{a^{{\otimes }}_{0,A,B}}\node{(0{\otimes }A){\otimes }B}\arrow{s,r}{y_A{\otimes }B}\\ \node{0}\node{0{\otimes }B}\arrow{w,t}{y_B} \end{diagram}$$]{} is commutative. $K16 (\bullet{\otimes }0{\otimes }\bullet),(\bullet{\otimes }\bullet{\otimes }0)$ For any objects $A,B$ the diagrams [$$\begin{diagram} \node{A{\otimes }(0{\otimes }B)}\arrow{s,l}{A{\otimes }y_{B}} \arrow[2]{e,t}{a^{{\otimes }}_{A,0,B}}\node[2]{(A{\otimes }0){\otimes }B}\arrow{s,r}{x_A{\otimes }B}\\ \node{A{\otimes }0}\arrow{e,t}{x_A}\node{0} \node{0{\otimes }B}\arrow{w,t}{y_B} \end{diagram}$$]{} [$$\begin{diagram} \node{A{\otimes }(B{\otimes }0)}\arrow{s,l}{A{\otimes }x_B}\arrow{e,t}{a^{{\otimes }}_{A,B,0}}\node{(A{\otimes }B){\otimes }0}\arrow{s,r}{x_{A{\otimes }B}}\\ \node{A{\otimes }0}\arrow{e,t}{x_A}\node{0} \end{diagram}$$]{} are commutative. $K17 (\bullet(0{\oplus}\bullet))$ For any objects $A,B$ the diagram [$$\begin{diagram} \node{A{\otimes }(0{\oplus}B)}\arrow{s,l}{A{\otimes }l^{{\oplus}}_B} \arrow{e,t}{v_{A,0,B}}\node{(A{\otimes }0){\oplus}(A{\otimes }B)}\arrow{s,r}{x_A{\oplus}(A{\otimes }B)}\\ \node{A{\otimes }B}\node{0{\oplus}(A{\otimes }B)}\arrow{w,t}{l^{{\oplus}}_{A{\otimes }B}} \end{diagram}$$]{} is commutative. $K18 ((0{\oplus}\bullet){\otimes }\bullet),(\bullet{\otimes }(\bullet{\oplus}0)),((\bullet{\oplus}0){\otimes }\bullet)$ Similar to the above. The relation between an Ann-category and a ring category ======================================================== In this section, we will prove that the axiomatics of a ring category, without K10, can be deduced from the one of an Ann-category. First, we can see that, the functor morphisms $a^{\oplus}, a^{\otimes}, u, l^{{\oplus}}, r^{{\oplus}}, v, w,$ in Definiton 2 are, respectively, the functor morphisms $a_{+}, a, c, g, d, {\frak{L}}, {\frak{R}}$ in Definition 1. Isomorphisms $x_A, y_A$ coincide with isomorphisms $\widehat{L}^A, \widehat{R}^A$ referred in Proposition 1. We now prove that diagrams which commute in a ring category also do in an Ann-category. K1 obviously follows from (ii) in the definition of an Ann-category. The commutative diagrams $K2, K3, K4, K5$ are indeed the compatibility of functor isomorphisms $(L^A, \Breve L^A), (R^A, \Breve R^A)$ with the constraints $a_{+}, c$ (the axiom Ann-1). The diagrams $K5-K9,$ respectively, are indeed the ones in (Ann-2). Particularly, K9 is indeed the decomposition of (1.3) where the morphism $v$ is replaced by its definition diagram: [$$\begin{diagram} \node{(P{\oplus}Q){\oplus}(R{\oplus}S)}\arrow{s,l}{v} \arrow{e,t}{a_{+}}\node{((P{\oplus}Q){\oplus}R){\oplus}S}\node{(P{\oplus}(Q{\oplus}R)){\oplus}S}\arrow{w,t}{a_{+}{\oplus}S}\arrow{s,r}{(P{\oplus}c){\oplus}S}\\ \node{(P{\oplus}R){\oplus}(Q{\oplus}S)}\arrow{e,t}{a_{+}}\node{((P{\oplus}R){\oplus}Q){\oplus}S}\node{(P{\oplus}(R{\oplus}Q)){\oplus}S.}\arrow{w,t}{a_{+}{\oplus}S} \end{diagram}$$]{} [**The proof for K17, K18**]{} Let $P,$ $P^{'}$ be Gr-categories, $(a_{+}, (0, g, d)), (a^{'}_{+}, (0^{'}, g^{'}, d^{'}))$ be respective constraints, and $(F, \Breve F):P\rightarrow P^{'}$ be $\oplus$-functor which is compatible with $(a_{+}, a^{'}_{+}).$ Then $(F, \Breve F)$ is compatible with the unitivity constraints $(0, g, d)), (0^{'}, g^{'}, d^{'})).$ First, the isomorphism $\widehat{F}:F0\to 0'$ is determined by the composition $$\begin{diagram} \node{u=F0{\oplus}F0}\node{F(0{\oplus}0)} \arrow {w,t}{\widetilde{F}} \arrow{e,t}{F(g)} \node{F0}\node{0'{\oplus}F0.}\arrow{w,t}{g'} \end{diagram}$$ Since $F0$ is a regular object, there exists uniquely the isomorphism $\widehat{F}:F0\to 0'$ such that $\widehat{F}{\oplus}id_{F0}=u.$ Then, we may prove that $\widehat{F}$ satisfies the diagrams in the definition of the compatibility of the ${\oplus}$-functor $F$ with the unitivity constraints. In an Ann-category ${\mathcal{A}},$ there exist uniquely isomorphisms $$\hat L^A: A{\otimes }0 \longrightarrow 0, \qquad \hat R^A: 0{\otimes }A \longrightarrow 0$$ such that the following diagrams [$$\begin{diagram} \node{AX}\node{A(0{\oplus}X)}\arrow{w,t}{L^A(g)}\arrow{s,r}{\breve L^A\qquad(2.1)} \node{AX}\node{A(X{\oplus}0)}\arrow{w,t}{L^A(d)}\arrow{s,r}{\breve L^A\qquad(2.1')}\\ \node{0{\oplus}AX}\arrow{n,l}{g}\node{A0{\oplus}AX}\arrow{w,t}{\hat L^A{\oplus}id} \node{AX{\oplus}0}\arrow{n,l}{d}\node{AX{\oplus}A0}\arrow{w,t}{id{\oplus}\hat L^A} \end{diagram}$$]{} [$$\begin{diagram} \node{AX}\node{(0{\oplus}X)A}\arrow{w,t}{R^A(g)}\arrow{s,r}{\breve R^A\qquad(2.2)} \node{AX}\node{(X{\oplus}0)A}\arrow{w,t}{R^A(d)}\arrow{s,r}{\breve R^A\qquad(2.2')}\\ \node{0{\oplus}AX}\arrow{n,l}{g}\node{0A{\oplus}XA}\arrow{w,t}{\hat R^A{\oplus}id} \node{AX{\oplus}0}\arrow{n,l}{d}\node{XA{\oplus}0A}\arrow{w,t}{id{\oplus}\hat R^A} \end{diagram}$$]{} commute, i.e., $L^A$ and $R^A$ are U-functors respect to the operation ${\oplus}$. Since $(L^A, \breve L^A)$ are ${\oplus}$-functors which are compatible with the associativity constraint $a^{{\oplus}}$ of the Picard category $({\mathcal{A}},{\oplus}),$ it is also compatible with the unitivity constraint $(0,g,d)$ thanks to Lemma 1. That means there exists uniquely the isomorphism $\hat L^A$ satisfying the diagrams $(2.1)$ and $(2.1')$. The proof for $\hat R^A$ is similar. The diagrams commute in Proposition 1 are indeed K17, K18. [**The proof for 15, K16**]{} Let $(F,\breve F), (G,\breve G)$ be ${\oplus}$-functors between ${\oplus}$-categories ${\mathcal{C}}, {\mathcal{C}}'$ which are compatible with the constraints $(0, g, d), (0', g', d')$ and $\widetilde F: F(0)\longrightarrow 0', \widetilde G: G(0)\longrightarrow 0'$ are respective isomorphisms. If $\alpha: F \longrightarrow G$ in an ${\oplus}$-morphism such that $\alpha_0$ is an isomorphism, then the diagram [$$\begin{diagram} \node{F0}\arrow[2]{r,t}{\alpha_0}\arrow{se,b}{\hat F}\node[2]{G0}\arrow{sw,b}{\hat G}\\ \node[2]{0'} \end{diagram}$$]{} commutes. Let us consider the diagram (9,3.8) (-0.2,0.9)[$F0$]{} (2,0.9)[$F(0{\oplus}0)$]{} (4.8,0.9)[$G(0{\oplus}0)$]{} (7.8,0.9)[$G0$]{} (-0.4,2.5)[$0'{\oplus}F0$]{} (2,2.5)[$F0{\oplus}F0$]{} (4.8,2.5)[$G0{\oplus}G0$]{} (7.6,2.5)[$0'{\oplus}G0$]{} (3.8,0.1)[$u_0$]{} (3.6,3.7)[$id{\oplus}u_0$]{} (0.8,1.1)[$F(g)$]{} (3.6,1.1)[$u_{0{\oplus}0}$]{} (6.5,1.1)[$G(g)$]{} (0.9,2.7)[$\breve{F}{\oplus}id$]{} (3.5,2.7)[$u_0{\oplus}u_0$]{} (6.3,2.7)[$\breve{G}{\oplus}id$]{} (-0.3,1.7)[$g'$]{} (2.3,1.7)[$\widetilde{F}$]{} (5.1,1.7)[$\widetilde{G}$]{} (7.7,1.7)[$g'$]{} (0,0.8)[(0,-1)[0.8]{}]{} (0,0)[(1,0)[8]{}]{} (8,0)[(0,1)[0.8]{}]{} (0,3.6)[(0,-1)[0.7]{}]{} (0,3.6)[(1,0)[8]{}]{} (8,3.6)[(0,-1)[0.7]{}]{} (0,2.3)[(0,-1)[1]{}]{} (8,2.3)[(0,-1)[1]{}]{} (2.6,1.3)[(0,1)[1]{}]{} (5.4,1.3)[(0,1)[1]{}]{} (1.8,2.6)[(-1,0)[1]{}]{} (3.3,2.6)[(1,0)[1.3]{}]{} (6.1,2.6)[(1,0)[1.3]{}]{} (0.3,1)[(1,0)[1.5]{}]{} (3.3,1)[(1,0)[1.3]{}]{} (6.1,1)[(1,0)[1.5]{}]{} (3.8,3.2)[(I)]{} (1,1.7)[(II)]{} (3.7,1.7)[(III)]{} (6.4,1.7)[(IV)]{} (3.8,0.5)[(V)]{} In this diagram, (II) and (IV) commute thanks to the compatibility of ${\oplus}$-functors $(F,\breve F), (G,\breve G)$ with the unitivity constraints; (III) commutes since $u$ is a ${\oplus}$-morphism; (V) commutes thanks to the naturality of $g'.$ Therefore, (I) commutes, i.e., $$\breve{G}\circ u_0{\oplus}u_0=\breve{F}{\oplus}u_0.$$ Since $F0$ is a regular object, $\breve{G}\circ u_0=\breve{F}.$ For any objects $X, Y\in \text{ob}{\mathcal{A}}$ the diagrams $$\begin{aligned} {\scriptsize\begin{diagram} \node{X{\otimes }(Y{\otimes }0)}\arrow{e,t}{id{\otimes }\widehat{L}^Y}\arrow{s,l}{a}\node{X{\otimes }0} \arrow{s,r}{\widehat L^X \qquad(2.3)} \node{0{\otimes }(X{\otimes }Y)}\arrow{e,t}{\widehat R^{XY}}\arrow{s,l}{a}\node{0}\\ \node{(X{\otimes }Y){\otimes }0}\arrow{e,t}{\widehat L^{XY}}\node{0} \node{(0{\otimes }X){\otimes }Y}\arrow{e,t}{\widehat R^X{\otimes }id}\node{0{\otimes }Y}\arrow{n,r}{\widehat R^Y \qquad(2.3')} \end{diagram}\nonumber}\end{aligned}$$ [$$\begin{diagram} \node{X{\otimes }(0{\otimes }Y)}\arrow[2]{e,t}{a}\arrow{s,l}{id{\otimes }\hat R^Y}\node[2]{(X{\otimes }0){\otimes }Y} \arrow{s,r}{\widehat L^X{\otimes }id\qquad(2.4)}\\ \node{X{\otimes }0}\arrow{e,t}{\widehat L^X}\node{0} \node{0{\otimes }Y}\arrow{w,t}{\widehat R^Y} \end{diagram}$$]{} commute. To prove the first diagram commutative, let us consider the diagram [$$\begin{diagram} \node{X{\otimes }(Y{\otimes }0)}\arrow{e,t}{id{\otimes }\hat L^Y}\arrow{s,l}{a}\arrow{se,t}{\widehat{L^X\circ L^Y}} \node{X{\otimes }0}\arrow{s,r}{\hat L^X}\\ \node{(X{\otimes }Y){\otimes }0}\arrow{e,t}{\hat L^{XY}}\node{0} \end{diagram}$$]{} According to the axiom (1.1), $(a_{X, Y, Z})_Z$ is an ${\oplus}$-morphism from the functor $L = L^X\circ L^Y$ to the functor $G = L^{XY}$. Therefore, from Lemma 2, (II) commutes. (I) commutes thanks to the determination of $\hat L$ of the composition $L = L^\circ L^Y$. So the outside commutes. The second diagram is proved similarly, thanks to the axiom (1.1’). To prove that the diagram (2.4) commutes, let us consider the diagram [$$\begin{diagram} \node{X{\otimes }(0{\otimes }Y)}\arrow[2]{e,t}{a}\arrow{s,l}{id{\otimes }\widehat{R^Y}}\arrow{se,t}{\hat H}\node[2]{(X{\otimes }0){\otimes }Y}\arrow{s,r}{\widehat{L^X}{\otimes }id}\arrow{sw,t}{\hat K}\\ \node{X{\otimes }0}\arrow{e,b}{\widehat{L^X}}\node{0}\node{0{\otimes }Y}\arrow{w,b}{\widehat{R^Y}} \end{diagram}$$]{} where $H = L^X\circ R^Y$ and $K = R^Y\circ L^X$. Then (II) and (III) commute thanks to the determination of the isomorphisms $H$ and $K$. From the axiom (1.2), $(a_{X,Y,Z})_Z$ is an ${\oplus}$-morphism from the functor $H$ to the functor $K$. So from Lemma 2, (I) commutes. Therefore, the outside commutes. The diagrams in Proposition 2 are indeed K15, K16. [**Proof for K11**]{} In an Ann-category, the diagram [$$\begin{diagram} \node{0{\oplus}0}\arrow{e,t}{g_0=d_0}\node{0}\\ \node{(0{\otimes }X){\oplus}(0{\otimes }Y)}\arrow{n,l}{\widehat{R}^X{\oplus}\widehat{R}^Y} \node{0{\otimes }(X{\oplus}Y)}\arrow{n,r}{\widehat{R}^{XY}\qquad(2.5)}\arrow{w,t}{\breve{L}^0} \end{diagram}$$]{} commutes. Let us consider the diagram (14,7.3) (1.2,0)[$A(B{\oplus}C){\oplus}0(B{\oplus}C)$]{} (8.1,0)[$A(B{\oplus}C){\oplus}0$]{} (1,1.7)[$(AB{\oplus}AC){\oplus}(0B{\oplus}0C)$]{} (7.9,1.7)[$(AB{\oplus}AC){\oplus}(0{\oplus}0)$]{} (1.1,3.5)[$(AB{\oplus}0B){\oplus}(AC{\oplus}0C)$]{} (8.2,3.5)[$(AB{\oplus}0){\oplus}(AC{\oplus}0)$]{} (1.3,5.2)[$(A{\oplus}0)B{\oplus}(A{\oplus}0)C$]{} (9,5.2)[$AB{\oplus}AC$]{} (1.7,7)[$(A{\oplus}0)(B{\oplus}C)$]{} (8.9,7)[$A(B{\oplus}C)$]{} (2.9,0.8)[$\breve{L}^A{\oplus}\breve{L}^0$]{} (2.9,2.9)[$v$]{} (2.9,4.5)[$\breve{R}^B{\oplus}\breve{R}^C$]{} (2.9,6.2)[$\breve{L}^{A{\oplus}0}$]{} (9.8,0.8)[$\breve{L}^A{\oplus}d_0^{-1}$]{} (9.8,2.9)[$v$]{} (9.8,4.5)[$d_{AB}{\oplus}d_{AC}$]{} (9.8,6.2)[$\breve{L}^A$]{} (0.2,2.6)[$\breve{R}^{B{\oplus}C}$]{} (11.7,2.6)[$d$]{} (5.5,0.3)[$f'_A{\oplus}id$]{} (4.6,2)[$(id{\oplus}id){\oplus}(\widehat{R}^B{\oplus}\widehat{R}^C)$]{} (4.7,3.8)[$(id{\oplus}\widehat{R}^B){\oplus}(id{\oplus}\widehat{R}^C)$]{} (4.9,5.5)[$(d_A{\otimes }id){\oplus}(d_A{\otimes }id)$]{} (5.6,7.3)[$d_A{\otimes }id$]{} (5.5,6.3)[(I)]{} (5.5,4.5)[(II)]{} (5.5,2.7)[(III)]{} (5.5,1)[(IV)]{} (0.5,4.3)[(V)]{} (11.2,4)[(VI)]{} (2.7,0.5)[(0,1)[0.9]{}]{} (2.7,3.3)[(0,-1)[0.9]{}]{} (2.7,5)[(0,-1)[0.9]{}]{} (2.7,6.8)[(0,-1)[1.1]{}]{} (9.6,0.4)[(0,1)[1]{}]{} (9.6,3.2)[(0,-1)[0.9]{}]{} (9.6,4)[(0,1)[1]{}]{} (9.6,6.8)[(0,-1)[1.1]{}]{} (4.2,0.1)[(1,0)[3.7]{}]{} (4.4,1.8)[(1,0)[3.3]{}]{} (4.4,3.6)[(1,0)[3.6]{}]{} (4.2,5.3)[(1,0)[4.6]{}]{} (3.9,7.1)[(1,0)[4.7]{}]{} (0,0.1)[(1,0)[1]{}]{} (12,7.1)[(-1,0)[1.7]{}]{} (0,0.1)[(0,1)[7]{}]{} (0,7.1)[(1,0)[1.5]{}]{} (12,0.1)[(-1,0)[1.9]{}]{} (12,0.1)[(0,1)[7]{}]{} (13,4)[(2.6)]{} In this diagram, (V) commutes thanks to the axiom I(1.3), (I) commutes thanks to the functorial property of ${\frak{L}};$ the outside and (II) commute thanks to the compatibility of the functors $R^{B{\oplus}C},R^B,R^C$ with the unitivity constraint $(0,g,d);$ (III) commutes thanks to the functorial property $v;$ (VI) commutes thanks to the coherence for the ACU-functor $(L^A,\breve{L}^A).$ So (IV) commutes. Note that $A(B{\oplus}C)$ is a regular object respect to the operation ${\oplus},$ so the diagram (2.5) commutes. We have K11. Similarly, we have K12. [**Proof for K13, K14**]{} In an Ann-category, we have $$\widehat{L}^1=l_0,\widehat{R}^1=r_0.$$ We will prove the first equation, the second one is proved similarly. Let us consider the diagram (2.7). In this diagram, the outside commutes thanks to the compatibility of ${\oplus}$-functor $(L^1,\breve{L}^1)$ with the unitivity constraint $(0,g,d)$ respect to the operation ${\oplus};$ (I) commutes thanks to the functorial property of the isomorphism $l;$ (II) commutes thanks to the functorial property of $g;$ (III) obviously commutes; (IV) commutes thanks to the axiom I(1.4). So (V) commutes, i.e., $$\widehat{L}^1{\oplus}id_{1.0}=l_0{\oplus}id_{1.0}$$ Since 1.0 is a regular object respect to the operation ${\oplus},$ $\widehat{L}^1=l_0.$ (5.5,4) (0,0)[$0{\oplus}(1.0)$]{} (4.5,0)[$(1.0){\oplus}(1.0)$]{} (1.5,1.3)[$0{\oplus}0$]{} (3.3,1.3)[$0{\oplus}0$]{} (3.3,2.5)[$0{\oplus}0$]{} (1.8,2.5)[$0$]{} (0.4,3.5)[$1.0$]{} (4.7,3.5)[$1.(0{\oplus}0)$]{} (4.3,0.1)[(-1,0)[3.1]{}]{} (4.5,3.6)[(-1,0)[3.6]{}]{} (0.5,0.3)[(0,1)[3]{}]{} (5.3,3.4)[(0,-1)[3]{}]{} (0.7,3.3)[(3,-2)[1]{}]{} (5,3.3)[(-3,-2)[1]{}]{} (0.7,0.3)[(1,1)[0.9]{}]{} (4.8,0.3)[(-1,1)[0.9]{}]{} (2.3,1.4)[(1,0)[0.8]{}]{} (3.6,1.6)[(0,1)[0.8]{}]{} (1.9,2.4)[(0,-1)[0.8]{}]{} (3.1,2.6)[(-1,0)[1]{}]{} (2.3,-0.3)[$\widehat{L}^1{\oplus}id$]{} (1.3,0.6)[$id{\oplus}l_0$]{} (3.5,0.6)[$l_0{\oplus}l_0$]{} (2.5,0.6)[$(V)$]{} (2.5,1.1)[$id$]{} (-0.1,2)[$g_{1.0}$]{} (0.8,2)[$(II)$]{} (1.6,2)[$g_0$]{} (2.3,2)[$(III)$]{} (3.3,2)[$id$]{} (4.2,2)[$(IV)$]{} (5.4,2)[$\breve{L}^1$]{} (7.5,2)[$(2.7)$]{} (2.5,2.7)[$g_0$]{} (1.2,3.1)[$l_0$]{} (2.5,3.1)[$(I)$]{} (3.9,3.1)[$l_{0{\oplus}0}$]{} (1.6,3.7)[$L^1(g_0)=id{\otimes }g_0$]{} We have K14. Similarly, we have K13. An Ann-category ${\mathcal{A}}$ is [*strong*]{} if $\widehat{L}^0=\widehat{R}^0.$ All the above results can be stated as follows Each strong Ann-category is a ring category. In our opinion, in the axiomatics of a [*ring category,*]{} the compatibility of the distributivity constraint with the unitivity constraint $(1, l, r)$ respect to the operation $\otimes$ is necessary, i.e., the diagrams of (Ann-3) should be added. Moreover, if the symmetric monoidal structure of the operation $\oplus$ is replaced with the symmetric categorical groupoid structure, then each ring category is an Ann-category. An open question: May the equation $\widehat{L}^0=\widehat{R}^0$ be proved to be independent in an Ann-category? [99]{} A. Frölich and C. T. C. Wall, [*Graded monoidal categories*]{}, Compositio Mathematica, tom 28, No 3 (1974), 229-285. M. M. Kapranov and V. A. Voevodsky, *2-Categories and Zamolodchikov tetrahedra equations,* Proc. Symp Pure Math. 56 (1994), par 2,177-259. M. L. Laplaza, *Coherence for distributivity,* Lecture Notes in Math, 281 (1972), 29-65. M. L. Laplaza, *Coherence for categories with group structure: an antenative approach*, J. Algebra, 84 (1983), 305-323. S. Mac Lane, *Homologie des anneaux et des modules,* Collque de Topologie algebrique. Louvain (1956), 55-80. N. T. Quang, *Introduction to Ann-categories,* J. Math. Hanoi, No.15, 4 (1987), 14-24.(arXiv:math. CT/0702588v2 21 Feb 2007) N. T. Quang, *Structure of Ann-categories and Mac Lane-Shukla cohomology of rings,*(Russian) Abelian groups and modues, No. 11,12, Tomsk. Gos. Univ., Tomsk (1994), 166-183. N. T. Quang, *Cohomological classification of Ann-categories,* arXiv:math. 2009. N. Saavedra Rivano, [*Categoryes Tannakiennes,*]{} Lecture Notes in Math.vol. 265, Spriger-Verlag, Berlin and New York, 1972. U.Shukla, *Cohomologie des algebras associatives.* Ann.Sci.Ecole Norm.,Sup.,7 (1961), 163-209. H. X. Sinh, *Gr-categories*, Universite Paris VII, Thèse de doctorat (1975).

ArXiv

--- abstract: 'How to improve the quality of conversations in online communities has attracted considerable attention recently. Having engaged, urbane, and reactive online conversations has a critical effect on the social life of Internet users. In this study, we are particularly interested in identifying a post in a multi-party conversation that is unlikely to be further replied to, which therefore kills that thread of the conversation. For this purpose, we propose a deep learning model called the ConverNet. ConverNet is attractive due to its capability of modeling the internal structure of a long conversation and its appropriate encoding of the contextual information of the conversation, through effective integration of attention mechanisms. Empirical experiments on real-world datasets demonstrate the effectiveness of the proposal model. For the widely concerned topic, our analysis also offers implications for improving the quality and user experience of online conversations.' author: - Yunhao Jiao - Cheng Li - Fei Wu - Qiaozhu Mei title: 'Find The Conversation Killers: A Predictive Study of Thread-ending Posts' ---

ArXiv

--- author: - Christopher Deninger date:   title: Number theory and dynamical systems on foliated spaces --- Introduction ============ In this paper we report on some developments in the search for a dynamical understanding of number theoretical zeta functions that have taken place since my ICM lecture [@D2]. We also point out a number of problems in analysis that will have to be solved in order to make further progress. In section 2 we give a short introduction to foliations and their cohomology. Section 3 is devoted to progress on the dynamical Lefschetz trace formula for one-codimensional foliations mainly due to Álvarez López and Kordyukov. In section 4 we make the comparison with the “explicit formulas” in analytic number theory. Finally in section 5 we generalize the conjectural dynamical Lefschetz trace formula of section 3 to phase spaces which are more general than manifolds. This was suggested by the number theoretical analogies of section 4. This account is written from an elementary point of view as far as arithmetic geometry is concerned, in particular motives are not mentioned. In spirit the present article is therefore a sequel to [@D1]. There is a different approach to number theoretical zeta functions using dynamical systems by A. Connes [@Co]. His phase space is a non-commutative quotient of the adèles. Although superficially related, the two approaches seem to be deeply different. Whereas Connes’ approach generalizes readily to automorphic $L$-functions [@So] but not to motivic $L$-functions, it is exactly the opposite with our picture. One may wonder whether there is some kind of Langlands correspondence between the two approaches. I would like to thank the Belgium and German mathematical societies very much for the opportunity to lecture about this material during the joint BMS–DMV meeting in Liège 2001. Foliations and their cohomology =============================== A $d$-dimensional foliation ${{\mathcal F}}= {{\mathcal F}}_X$ on a smooth manifold $X$ of dimension $a$ is a partition of $X$ into immersed connected $d$-dimensional manifolds $F$, the “leaves”. Locally the induced partition should be trivial: Every point of $X$ should have an open neighborhood $U$ diffeomorphic to an open ball $B$ in ${{\mathbb{R}}}^a$ such that the leaves of the induced partition on $U$ correspond to the submanifolds $B \cap ({{\mathbb{R}}}^d \times \{ y \})$ of $B$ for $y$ in ${{\mathbb{R}}}^{a-d}$. One of the simplest non-trivial examples is the one-dimensional foliation of the two-dimensional torus $T^2 = {{\mathbb{R}}}^2 / {{\mathbb{Z}}}^2$ by lines of irrational slope $\alpha$. These are given by the immersions $${{\mathbb{R}}}\hookrightarrow T^2 \; , \; t \mapsto (x + t \alpha , t) {\;\mathrm{mod}\;}{{\mathbb{Z}}}^2$$ parametrized by $x {\;\mathrm{mod}\;}{{\mathbb{Z}}}+ \alpha {{\mathbb{Z}}}$. In this case every leaf is dense in $T^2$ and the intersection of a global leaf with a small open neighborhood $U$ as above decomposes into countably many connected components. It is the global behaviour which makes foliations complicated. For a comprehensive introduction to foliation theory, the reader may turn to [@Go] for example. To a foliation ${{\mathcal F}}$ on $X$ we may attach its tangent bundle $T {{\mathcal F}}$ whose total space is the union of the tangent spaces to the leaves. By local triviality of the foliation it is a sub vector bundle of the tangent bundle $TX$. It is integrable i.e. the commutator of any two vector fields with values in $T {{\mathcal F}}$ again takes values in $T{{\mathcal F}}$. Conversely a theorem of Frobenius asserts that every integrable sub vector bundle of $TX$ arises in this way. Differential forms of order $n$ along the leaves are defined as the smooth sections of the real vector bundle $\Lambda^n T^* {{\mathcal F}}$, $${{\mathcal A}}^n_{{{\mathcal F}}} (X) = \Gamma (X, \Lambda^n T^* {{\mathcal F}}) \; .$$ The same formulas as in the classical case define exterior derivatives along the leaves: $$d^n_{{{\mathcal F}}} : {{\mathcal A}}^n_{{{\mathcal F}}} (X) \longrightarrow {{\mathcal A}}^{n+1}_{{{\mathcal F}}} (X) \; .$$ They satisfy the relation $d^{n+1}_{{{\mathcal F}}} {\mbox{\scriptsize $\,\circ\,$}}d^n_{{{\mathcal F}}} = 0$ so that we can form the leafwise cohomology of ${{\mathcal F}}$: $$H^n_{{{\mathcal F}}} (X) = {\mathrm{Ker}\,}d^n_{{{\mathcal F}}} / {\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}} \; .$$ For our purposes these invariants are actually too subtle. We therefore consider the reduced leafwise cohomology $${\bar{H}}^n_{{{\mathcal F}}} (X) = {\mathrm{Ker}\,}d^n_{{{\mathcal F}}} / \overline{{\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}} \; .$$ Here the quotient is taken with respect to the topological closure of ${\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}$ in the natural Fréchet topology on ${{\mathcal A}}^n_{{{\mathcal F}}} (X)$. The reduced cohomologies are nuclear Fréchet spaces. Even if the leaves are dense, already ${\bar{H}}^1_{{{\mathcal F}}} (X)$ can be infinite dimensional. The cup product pairing induced by the exterior product of forms along the leaves turns ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ into a graded commutative ${\bar{H}}^0_{{{\mathcal F}}} (X)$-algebra. The Poincare Lemma extends to the foliation context and implies that $$H^n_{{{\mathcal F}}} (X) = H^n (X, {{\mathcal R}}) \; .$$ Here ${{\mathcal R}}$ is the sheaf of smooth real valued functions which are locally constant on the leaves. In particular $${\bar{H}}^0_{{{\mathcal F}}} (X) = H^0_{{{\mathcal F}}} (X) = H^0 (X, {{\mathcal R}})$$ consists only of constant functions if ${{\mathcal F}}$ contains a dense leaf. For the torus foliation above with $\alpha \notin {{\mathbb{Q}}}$ we therefore have ${\bar{H}}^0_{{{\mathcal F}}} (T^2) = {{\mathbb{R}}}$. Some Fourier analysis reveals that ${\bar{H}}^1_{{{\mathcal F}}} (T^2) \cong {{\mathbb{R}}}$. The higher cohomologies vanish since almost by definition we have $$H^n_{{{\mathcal F}}} (X) = 0 \quad \mbox{for all} \; n > d = \dim {{\mathcal F}}\; .$$ For a smooth map $f : X \to Y$ of foliated manifolds which maps leaves into leaves, continuous pullback maps $$f^* : {{\mathcal A}}^n_{{{\mathcal F}}_Y} (Y) \longrightarrow {{\mathcal A}}^n_{{{\mathcal F}}_X} (X)$$ are defined for all $n$. They commute with $d_{{{\mathcal F}}}$ and respect the exterior product of forms. Hence they induce a continuous map of reduced cohomology algebras $$f^* : {\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}_Y} (Y) \longrightarrow {\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}_X} (X) \; .$$ A (complete) flow is a smooth ${{\mathbb{R}}}$-action $\phi : {{\mathbb{R}}}\times X \to X , (t,x) \mapsto \phi^t (x)$. It is called ${{\mathcal F}}$-compatible if every diffeomorphism $\phi^t : X \to X$ maps leaves into leaves. If this is the case we obtain a linear ${{\mathbb{R}}}$-action $t \mapsto \phi^{t*}$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ for every $n$. Let $$\Theta : {\bar{H}}^n_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^n_{{{\mathcal F}}} (X)$$ denote the infinitesimal generator of $\phi^{t*}$: $$\Theta h = \lim_{t\to 0} \frac{1}{t} (\phi^{t*} h -h ) \; .$$ The limit exists and $\Theta$ is continuous in the Fréchet topology. As $\phi^{t*}$ is an algebra endomorphism of the ${{\mathbb{R}}}$-algebra ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ it follows that $\Theta$ is an ${{\mathbb{R}}}$-linear derivation. Thus we have $$\label{eq:1} \Theta (h_1 \cup h_2) = \Theta h_1 \cup h_2 + h_1 \cup \Theta h_2$$ for all $h_1 , h_2$ in ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$. For arbitrary foliations the reduced leafwise cohomology does not seem to have a good structure theory. For Riemannian foliations however the situation is much better. These foliations are characterized by the existence of a “bundle-like” metric $g$. This is a Riemannian metric whose geodesics are perpendicular to all leaves whenever they are perpendicular to one leaf. For example any one-codimensional foliation given by a closed one-form without singularities is Riemannian. The graded Fréchet space ${{\mathcal A}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ carries a canonical inner product: $$(\alpha , \beta) = \int_X \langle \alpha , \beta \rangle_{{{\mathcal F}}} {\mathrm{vol}}\; .$$ Here $\langle , \rangle_{{{\mathcal F}}}$ is the Riemannian metric on $\Lambda^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}} T^* {{\mathcal F}}$ induced by $g$ and ${\mathrm{vol}}$ is the volume form or density on $X$ coming from $g$. Let $$\Delta_{{{\mathcal F}}} = d_{{{\mathcal F}}} d^*_{{{\mathcal F}}} + d^*_{{{\mathcal F}}} d_{{{\mathcal F}}}$$ denote the Laplacian using the formal adjoint of $d_{{{\mathcal F}}}$ on $X$. Since ${{\mathcal F}}$ is Riemannian the restriction of $\Delta_{{{\mathcal F}}}$ to any leaf $F$ is the Laplacian on $F$ with respect to the induced metric [@AK1] Lemma 3.2, i.e. $$(\Delta_{{{\mathcal F}}} \alpha) \, |_F = \Delta_F (\alpha \, |_F) \quad \mbox{for all} \; \alpha \in {{\mathcal A}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X) \; .$$ We now assume that $T {{\mathcal F}}$ is orientable. Via $g$ the choice of an orientation determines a volume form ${\mathrm{vol}}_{{{\mathcal F}}}$ in ${{\mathcal A}}^d_{{{\mathcal F}}} (X)$ and hence a Hodge $*$-operator $$*_{{{\mathcal F}}} : \Lambda^n T^*_x {{\mathcal F}}{\stackrel{\sim}{\longrightarrow}}\Lambda^{d-n} T^*_x {{\mathcal F}}\quad \mbox{for every} \; x \; \mbox{in} \; X \; .$$ It is determined by the condition that $$v \wedge *_{{{\mathcal F}}} w = \langle v,w \rangle_{{{\mathcal F}}} \, {\mathrm{vol}}_{{{\mathcal F}}, x} \quad \mbox{for} \; v,w \; \mbox{in} \; \Lambda^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}} T^*_x {{\mathcal F}}\; .$$ These fibrewise star-operators induce the leafwise $*$-operator on forms: $$*_{{{\mathcal F}}} : {{\mathcal A}}^n_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{{\mathcal A}}^{d-n}_{{{\mathcal F}}} (X) \; .$$ We now list some important properties of leafwise cohomology. [**Properties**]{} Assume that $X$ is compact, ${{\mathcal F}}$ a $d$-dimensional oriented Riemannian foliation and $g$ a bundle-like metric for ${{\mathcal F}}$. Then the natural map $$\label{eq:2} {\mathrm{Ker}\,}\Delta^n_{{{\mathcal F}}} {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^n_{{{\mathcal F}}} (X) \; ,\; \omega \longmapsto \omega {\;\mathrm{mod}\;}\overline{{\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}}$$ is a topological isomorphism of Fréchet spaces. We denote its inverse by ${{\mathcal H}}$. This result is due to Álvarez López and Kordyukov [@AK1]. It is quite deep since $\Delta_{{{\mathcal F}}}$ is only elliptic along the leaves so that the ordinary elliptic regularity theory does not suffice. For non-Riemannian foliations (\[eq:2\]) does not hold in general [@DS1]. All the following results are consquences of this Hodge theorem. The Hodge $*$-operator induces an isomorphism $$*_{{{\mathcal F}}} : {\mathrm{Ker}\,}\Delta^n_{{{\mathcal F}}} {\stackrel{\sim}{\longrightarrow}}{\mathrm{Ker}\,}\Delta^{d-n}_{{{\mathcal F}}}$$ since it commutes with $\Delta_{{{\mathcal F}}}$ up to sign. From (\[eq:2\]) we therefore get isomorphisms for all $n$: $$\label{eq:3} *_{{{\mathcal F}}} : {\bar{H}}^n_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{d-n}_{{{\mathcal F}}} (X) \; .$$ For the next property define the trace map $${\mathrm{tr}}: {\bar{H}}^d_{{{\mathcal F}}} (X) \longrightarrow {{\mathbb{R}}}$$ by the formula $${\mathrm{tr}}(h) = \int_X *_{{{\mathcal F}}} (h) {\mathrm{vol}}:= \int_X *_{{{\mathcal F}}} ({{\mathcal H}}(h)) {\mathrm{vol}}\; .$$ It is an isomorphism if ${{\mathcal F}}$ has a dense leaf. Note that for [*any*]{} representative $\alpha$ in the cohomology class $h$ we have $${\mathrm{tr}}(h) = \int_X *_{{{\mathcal F}}} (\alpha) {\mathrm{vol}}\; .$$ Namely $\alpha - {{\mathcal H}}(h) = d_{{{\mathcal F}}} \beta$ and $$\begin{aligned} \int_X *_{{{\mathcal F}}} (d_{{{\mathcal F}}} \beta) {\mathrm{vol}}& = & \pm \int_X d^*_{{{\mathcal F}}} (*_{{{\mathcal F}}} \beta) {\mathrm{vol}}\\ & = &\pm ( 1 , d^*_{{{\mathcal F}}} (*_{{{\mathcal F}}} \beta)) \\ & = & \pm ( d_{{{\mathcal F}}} (1) , *_{{{\mathcal F}}} \beta)\\ & = & 0 \; .\end{aligned}$$ Alternatively the trace functional is given by $${\mathrm{tr}}(h) = \int_X \alpha \wedge *_{\perp} (1)$$ where $*_{\perp} (1)$ is the transverse volume element for $g$ c.f. [@AK1], §3. It is not difficult to see using (\[eq:2\]) that we get a scalar product on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ for every $n$ by setting: $$\begin{aligned} \label{eq:4} (h,h') & = & {\mathrm{tr}}(h \cup *_{{{\mathcal F}}} h') \\ & = & \int_X \langle {{\mathcal H}}(h) , {{\mathcal H}}(h') \rangle_{{{\mathcal F}}} {\mathrm{vol}}\; . \nonumber\end{aligned}$$ It follows from this that the cup product pairing $$\label{eq:5} \cup : {\bar{H}}^n_{{{\mathcal F}}} (X) \times {\bar{H}}^{d-n}_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^d_{{{\mathcal F}}} (X) \xrightarrow{tr} {{\mathbb{R}}}$$ is non-degenerate. Next we discuss the Künneth formula. Assume that $Y$ is another compact manifold with a Riemannian foliation ${{\mathcal F}}_Y$. Then the canonical map $$H^n_{{{\mathcal F}}_X} (X) \otimes H^m_{{{\mathcal F}}_Y} (Y) \longrightarrow H^{n+m}_{{{\mathcal F}}_X \times {{\mathcal F}}_Y} (X \times Y)$$ induces a topological isomorphism [@M]: $$\label{eq:6} {\bar{H}}^n_{{{\mathcal F}}_X} (X) \hat{\otimes} {\bar{H}}^m_{{{\mathcal F}}_Y} (Y) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{n+m}_{{{\mathcal F}}_X \times {{\mathcal F}}_Y} (X \times Y) \; .$$ Since the reduced cohomology groups are nuclear Fréchet spaces, it does not matter which topological tensor product is chosen in (\[eq:6\]). The proof of this Künneth formula uses (\[eq:2\]) and the spectral theory of the Laplacian $\Delta_{{{\mathcal F}}}$. Before we deal with more specific topics let us mention that also Hodge–Kähler theory can be generalized. A complex structure on a foliation ${{\mathcal F}}$ is an almost complex structure $J$ on $T {{\mathcal F}}$ such that all restrictions $J \, |_F$ to the leaves are integrable. Then the leaves carry holomorphic structures which vary smoothly in the transverse direction. A foliation ${{\mathcal F}}$ with a complex structure $J$ is called Kähler if there is a hermitian metric $h$ on the complex bundle $T_c {{\mathcal F}}= (T {{\mathcal F}}, J)$ such that the Kähler form along the leaves $$\omega_{{{\mathcal F}}} = - {\frac{1}{2}}{\mathrm{im}\,}h \in {{\mathcal A}}^2_{{{\mathcal F}}} (X)$$ is closed. Note that for example any foliation by orientable surfaces can be given a Kählerian structure by choosing a metric on $X$, c.f. [@MS] Lemma A.3.1. Let $$L_{{{\mathcal F}}} : {\bar{H}}^n_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^{n+2}_{{{\mathcal F}}} (X) \; , \; L_{{{\mathcal F}}} (h) = h \cup [\omega_{{{\mathcal F}}}]$$ denote the Lefschetz operator. The following assertions are consequences of (\[eq:2\]) combined with the classical Hodge–Kähler theory. See [@DS3] for details. Let $X$ be a compact orientable manifold and ${{\mathcal F}}$ a Kählerian foliation with respect to the hermitian metric $h$ on $T_c {{\mathcal F}}$. Assume in addition that ${{\mathcal F}}$ is Riemannian. Then we have: $$\label{eq:7} {\bar{H}}^n_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}= \bigoplus_{p+q=n} H^{pq} \; , \quad \mbox{where} \; \overline{H^{pq}} = H^{qp} \; .$$ Here $H^{pq}$ consists of those classes that can be represented by $(p,q)$-forms along the leaves. Moreover there are topological isomorphisms $$H^{pq} \cong {\bar{H}}^q (X , \Omega^p_{{{\mathcal F}}})$$ with the reduced cohomology of the sheaf of holomorphic $p$-forms along the leaves. Furthermore the Lefschetz operator induces isomorphisms $$\label{eq:8} L^i_{{{\mathcal F}}} : {\bar{H}}^{d-i}_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{d+i}_{{{\mathcal F}}} (X) \quad \mbox{for} \; 0 \le i \le d \; .$$ Finally the space of primitive cohomology classes ${\bar{H}}^n_{{{\mathcal F}}} (X)_{{\mathrm{prim}}}$ carries the structure of a polarizable ${\mathrm{ind}}\, {{\mathbb{R}}}$-Hodge structure of weight $n$. After this review of important properties of the reduced leafwise cohomology of Riemannian foliations we turn to a specific result relating flows and cohomology. \[t21\] Let $X$ be a compact $3$-manifold and ${{\mathcal F}}$ a Riemannian foliation by surfaces with a dense leaf. Let $\phi^t$ be an ${{\mathcal F}}$-compatible flow on $X$ which is conformal on $T{{\mathcal F}}$ with respect to a metric $g$ on $T{{\mathcal F}}$ in the sense that for some constant $\alpha$ we have: $$\label{eq:9} g (T_x \phi^t (v) , T_x \phi^t (w)) = e^{\alpha t} g (v,w) \; \mbox{for all} \; v,w \in T_x {{\mathcal F}}, x \in X \; \mbox{and} \; t \in {{\mathbb{R}}}\; .$$ Then we have for the infinitesimal generator of $\phi^{t*}$ that: $$\Theta = 0 \; \mbox{on} \; {\bar{H}}^0_{{{\mathcal F}}} (X) = {{\mathbb{R}}}\quad \mbox{and} \quad \Theta = \alpha \; \mbox{on} \; {\bar{H}}^2_{{{\mathcal F}}} (X) \cong {{\mathbb{R}}}\; .$$ On ${\bar{H}}^1_{{{\mathcal F}}} (X)$ the operator $\Theta$ has the form $$\Theta = \frac{\alpha}{2} + S$$ where $S$ is skew-symmetric with respect to the inner product $( , )$ above. For the bundle-like metric on $X$ required for the construction of $(,)$ we take any extension of the given metric on $T {{\mathcal F}}$ to a bundle-like metric on $TX$. Such extensions exist. [**\[t21\]**]{} Because we have a dense leaf, ${\bar{H}}^0_{{{\mathcal F}}} (X) = H^0_{{{\mathcal F}}} (X)$ consists only of constant functions. On these $\phi^{t*}$ acts trivially so that $\Theta = 0$. Since $$*_{{{\mathcal F}}} : {{\mathbb{R}}}= {\bar{H}}^0_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^2_{{{\mathcal F}}} (X)$$ is an isomorphism and since $$\phi^{t*} (*_{{{\mathcal F}}} (1)) = e^{\alpha t} (*_{{{\mathcal F}}} 1)$$ by conformality, we have $\Theta = \alpha$ on ${\bar{H}}^2_{{{\mathcal F}}} (X)$. For $h_1 , h_2$ in ${\bar{H}}^1_{{{\mathcal F}}} (X)$ we find $$\label{eq:10} \alpha (h_1 \cup h_2) = \Theta (h_1 \cup h_2) \stackrel{\rm (\ref{eq:1})}{=} \Theta h_1 \cup h_2 + h_1 \cup \Theta h_2 \; .$$ By conformality $\phi^{t*}$ commutes with $*_{{{\mathcal F}}}$ on ${\bar{H}}^1_{{{\mathcal F}}} (X)$. Differentiating, it follows that $\Theta$ commutes with $*_{{{\mathcal F}}}$ as well. Since by definition we have $$(h, h') = {\mathrm{tr}}(h \cup *_{{{\mathcal F}}} h') \quad \mbox{for} \; h , h' \in {\bar{H}}^1_{{{\mathcal F}}} (X) \; ,$$ it follows from (\[eq:10\]) that as desired: $$\alpha ( h , h' ) = ( \Theta h , h' ) + ( h , \Theta h' ) \; .$$ Dynamical Lefschetz trace formulas ================================== The formulas we want to consider in this section relate the compact orbits of a flow with the alternating sum of suitable traces on cohomology. A suggestive but non-rigorous argument of Guillemin [@Gu] later rediscovered by Patterson [@P] led to the following conjecture [@D2] §3. Let $X$ be a compact manifold with a one-codimensional foliation ${{\mathcal F}}$ and an ${{\mathcal F}}$-compatible flow $\phi$. Assume that the fixed points and the periodic orbits of the flow are non-degenerate in the following sense: For any fixed point $x$ the tangent map $T_x \phi^t$ should have eigenvalues different from $1$ for all $t > 0$. For any closed orbit $\gamma$ of length $l (\gamma)$ and any $x \in \gamma$ and integer $k \neq 0$ the automorphism $T_x \phi^{kl (\gamma)}$ of $T_x X$ should have the eigenvalue $1$ with algebraic multiplicity one. Observe that the vector field $Y_{\phi}$ generated by the flow provides an eigenvector $Y_{\phi,x}$ for the eigenvalue $1$. Recall that the length $l (\gamma) > 0$ of $\gamma$ is defined by the isomorphism: $${{\mathbb{R}}}/ l (\gamma) {{\mathbb{Z}}}{\stackrel{\sim}{\longrightarrow}}\gamma \; , \; t \longmapsto \phi^t (x) \; .$$ For a fixed point $x$ we set[^1] $$\varepsilon_x = {\mathrm{sgn}\,}\det (1 - T_x \phi^t {\, | \,}T_x {{\mathcal F}}) \; .$$ This is independent of $t > 0$. For a closed orbit $\gamma$ and $k \in {{\mathbb{Z}}}{\setminus}0$ set$^1$ $$\varepsilon_{\gamma} (k) = {\mathrm{sgn}\,}\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x X / {{\mathbb{R}}}Y_{\phi,x}) = {\mathrm{sgn}\,}\det ( 1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \; .$$ It does not depend on the point $x \in \gamma$. Finally let ${{\mathcal D}}' (J)$ denote the space of Schwartz distributions on an open subset $J$ of ${{\mathbb{R}}}$. \[t31\] For $X , {{\mathcal F}}$ and $\phi$ as above there exists a natural definition of a ${{\mathcal D}}' ({{\mathbb{R}}}^{>0})$-valued trace of $\phi^*$ on the reduced leafwise cohomology ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ such that in ${{\mathcal D}}' ({{\mathbb{R}}}^{>0})$ we have: $$\label{eq:11} \sum\limits^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) = \sum\limits_{\gamma} l (\gamma) \sum\limits^{\infty}_{k=1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum\limits_x \varepsilon_x |1 - e^{\kappa_x t}|^{-1} \; .$$ Here $\gamma$ runs over the closed orbits of $\phi$ which are not contained in a leaf and $x$ over the fixed points. For $a \in {{\mathbb{R}}}, \delta_a$ is the Dirac distribution in $a$ and $\kappa_x$ is defined by the action of $T_x \phi^t$ on the $1$-dimensional vector space $T_x X / T_x {{\mathcal F}}$. That action is multiplication by $e^{\kappa_x t}$ for some $\kappa_x \in {{\mathbb{R}}}$ and all $t$. The conjecture is not known (except for $\dim X = 1$) if $\phi$ has fixed points. It may well have to be amended somewhat in that case. The analytic difficulty in the presence of fixed points lies in the fact that in this case $\Delta_{{{\mathcal F}}}$ has no chance to be transversally elliptic to the ${{\mathbb{R}}}$-action by the flow, so that the methods of transverse index theory do not apply directly. In the simpler case when the flow is everywhere transversal to ${{\mathcal F}}$, Álvarez López and Kordyukov have proved a beautiful strengthening of the conjecture. Partial results were obtained by other methods in [@Laz], [@DS2]. We now describe their result in a convenient way for our purposes: \[t32\] Assume $X$ is a compact oriented manifold with a one codimensional foliation ${{\mathcal F}}$. Let $\phi$ be a flow on $X$ which is everywhere transversal to the leaves of ${{\mathcal F}}$. Then ${{\mathcal F}}$ inherits an orientation and it is Riemannian [@Go] III 4.4. Fixing a bundle-like metric $g$ the cohomologies ${\bar{H}}^n_{{{\mathcal F}}} (X)$ acquire pre-Hilbert structures (\[eq:4\]) and we can consider their Hilbert space completions $\hat{H}^n_{{{\mathcal F}}} (X)$. For every $t$ the linear operator $\phi^{t*}$ is bounded on $({\bar{H}}^n_{{{\mathcal F}}} (X) , \| \; \|)$ and hence can be continued uniquely to a bounded operator on $\hat{H}^n_{{{\mathcal F}}} (X)$ c.f. theorem \[t34\]. By transversality the flow has no fixed points. We assume that all periodic orbits are non-degenerate. \[t33\] Under the conditions of (\[t32\]), for every test function $\varphi \in {{\mathcal D}}({{\mathbb{R}}}) = C^{\infty}_0 ({{\mathbb{R}}})$ the operator $$A_{\varphi} = \int_{{{\mathbb{R}}}} \varphi (t) \phi^{t*} \, dt$$ on $\hat{H}^n_{{{\mathcal F}}} (X)$ is of trace class. Setting: $${\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) (\varphi) = {\mathrm{tr}}A_{\varphi}$$ defines a distribution on ${{\mathbb{R}}}$. The following formula holds in ${{\mathcal D}}' ({{\mathbb{R}}})$: $$\label{eq:12} \sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \delta_0 + \sum_{\gamma} l (\gamma) \sum_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} \; .$$ Here $\chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu)$ denotes Connes’ Euler characteristic of the foliation with respect to the transverse measure $\mu$ corresponding to $*_{\perp} (1)$. (See [@MS].) It follows from the theorem that if the right hand side of (\[eq:12\]) is non-zero, at least one of the cohomology groups ${\bar{H}}^n_{{{\mathcal F}}} (X)$ must be infinite dimensional. Otherwise the alternating sum of traces would be a smooth function and hence have empty singular support. By the Hodge isomorphism (\[eq:2\]) one may replace cohomology by the spaces of leafwise harmonic forms. The left hand side of the dynamical Lefschetz trace formula then becomes the ${{\mathcal D}}' ({{\mathbb{R}}})$-valued transverse index of the leafwise de Rham complex. Note that the latter is transversely elliptic for the ${{\mathbb{R}}}$-action $\phi^t$. Transverse index theory with respect to compact group actions was initiated in [@A]. A definition for non-compact groups of a transverse index was later given by Hörmander [@Si] Appendix II. As far as we know the relation of (\[eq:12\]) with transverse index theory in the sense of Connes–Moscovici still needs to be clarified. Let us now make some remarks on the operators $\phi^{t*}$ on $\hat{H}^n_{{{\mathcal F}}} (X)$ in a more general setting: \[t34\] Let ${{\mathcal F}}$ be a Riemannian foliation on a compact manifold $X$ and $g$ a bundle like metric. As above $\hat{H}^n_{{{\mathcal F}}} (X)$ denotes the Hilbert space completion of ${\bar{H}}^n_{{{\mathcal F}}} (X)$ with respect to the scalar product (\[eq:4\]). Let $\phi^t$ be an ${{\mathcal F}}$-compatible flow. Then the linear operators $\phi^{t*}$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ induce a strongly continuous operator group on $\hat{H}^n_{{{\mathcal F}}} (X)$. In particular the infinitesimal generator $\Theta$ exists as a closed densely defined operator. On ${\bar{H}}^n_{{{\mathcal F}}} (X)$ it agrees with the infinitesimal generator in the Fréchet topology defined earlier. There exists $\omega > 0$ such that the spectrum of $\Theta$ lies in $- \omega \le {\mathrm{Re}\,}s \le \omega$. If the operators $\phi^{t*}$ are orthogonal then $T = - i \Theta$ is a self-adjoint operator on $\hat{H}^n_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}$ and we have $$\phi^{t*} = \exp t \Theta = \exp it T$$ in the sense of the functional calculus for (unbounded) self-adjoint operators on Hilbert spaces. [**Sketch of proof**]{} Estimates show that $\| \phi^{t*} \|$ is locally uniformly bounded in $t$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$. Approximating $h \in \hat{H}^n_{{{\mathcal F}}} (X)$ by $h_{\nu} \in {\bar{H}}^n_{{{\mathcal F}}} (X)$ one now shows as in the proof of the Riemann–Lebesgue lemma that the function $t \mapsto \phi^{t*} h$ is continuous at zero, hence everywhere. Thus $\phi^{t*}$ defines a strongly continuous group on $\hat{H}^n_{{{\mathcal F}}} (X)$. The remaining assertions follow from semigroup theory [@DSch], Ch. VIII, XII, and in particular from the theorem of Stone. We now combine theorems \[t21\], \[t33\] and \[t34\] to obtain the following corollary: \[t35\] Let $X$ be a compact $3$-manifold with a foliation ${{\mathcal F}}$ by surfaces having a dense leaf. Let $\phi^t$ be a non-degenerate ${{\mathcal F}}$-compatible flow which is everywhere transversal to ${{\mathcal F}}$. Assume that $\phi^t$ is conformal (\[eq:9\]) with respect to a metric $g$ on $T{{\mathcal F}}$. Then $\Theta$ has pure point spectrum ${\mathrm{Sp}}^1 (\Theta)$ on $\hat{H}^1_{{{\mathcal F}}} (X)$ which is discrete in ${{\mathbb{R}}}$ and we have the following equality of distributions on ${{\mathbb{R}}}$: $$\label{eq:13} 1 - \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)} e^{t\rho} + e^{t\alpha} = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \delta_0 + \sum_{\gamma} l (\gamma) \sum_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \delta_{k l (\gamma)} \; .$$ In the sum the $\rho$’s appear with their geometric multiplicities. All $\rho \in {\mathrm{Sp}}^1 (\Theta)$ have ${\mathrm{Re}\,}\rho = \frac{\alpha}{2}$. [**Remarks**]{} 1) Here $e^{t\rho} , e^{t \alpha}$ are viewed as distributions so that evaluated on a test function $\varphi \in {{\mathcal D}}({{\mathbb{R}}})$ the formula reads: $$\label{eq:14} \Phi (0) - \sum\limits_{\rho \in {\mathrm{Sp}}^1 (\Theta)} \Phi (\rho) + \Phi (\alpha) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \varphi (0) + \sum\limits_{\gamma} l (\gamma) \sum\limits_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \varphi (k l (\gamma)) \; .$$ Here we have put $$\Phi (s) = \int_{{{\mathbb{R}}}} e^{ts} \varphi (t) \, dt \; .$$ 2) Actually the conditions of the corollary force $\alpha = 0$ i.e. the flow must be isometric with respect to $g$. We have chosen to leave the $\alpha$ in the fomulation since there are good reasons to expect the corollary to generalize to more general phase spaces $X$ than manifolds, where $\alpha \neq 0$ becomes possible i.e. to Sullivan’s generalized solenoids. More on this in section 5.\ 3) One can show that the group generated by the lengths of closed orbits is a finitely generated subgroup of ${{\mathbb{R}}}$ under the assumptions of the corollary. In order to achieve an infinitely generated group the flow must have fixed points. [**\[t35\]**]{} By \[t21\], \[t33\] we need only show the equation $$\label{eq:15} {\mathrm{Tr}}(\phi^{t*} {\, | \,}{\bar{H}}^1_{{{\mathcal F}}} (X)) = \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)} e^{t \rho}$$ and the assertions about the spectrum of $\Theta$. As in the proof of \[t21\] one sees that on $\hat{H}^1_{{{\mathcal F}}} (X)$ we have $$( \phi^{t*} h , \phi^{t*} h') = e^{\alpha t} (h , h') \; .$$ Hence $e^{-\frac{\alpha}{2} t} \phi^{t*}$ is orthogonal and by the theorem of Stone $$T = -iS$$ is selfadjoint on $\hat{H}^1_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}$, if $\Theta = \frac{\alpha}{2} + S$. Moreover $$e^{-\frac{\alpha}{2} t} \phi^{t*} = \exp it T \; ,$$ so that $$\label{eq:16} \phi^{t*} = \exp t \Theta \; .$$ In [@DS2] proof of 2.6, for isometric flows the relation $$-\Theta^2 = \Delta^1 \, |_{\ker \Delta^1_{{{\mathcal F}}}}$$ was shown. Using the spectral theory of the ordinary Laplacian $\Delta^1$ on $1$-forms it follows that $\Theta$ has pure point spectrum with finite multiplicities on ${\bar{H}}^1_{{{\mathcal F}}} (X) \cong \ker \Delta^1_{{{\mathcal F}}}$ and that ${\mathrm{Sp}}^1 (\Theta)$ is discrete in ${{\mathbb{R}}}$. Alternatively, without knowing $\alpha = 0$, that proof gives: $$- \left( \Theta - \frac{\alpha}{2} \right)^2 = \Delta^1 \, |_{\ker \Delta^1_{{{\mathcal F}}}} \; .$$ This also implies the assertion on the spectrum of $\Theta$ on $\hat{H}^1_{{{\mathcal F}}} (X)$. Now (\[eq:15\]) follows from (\[eq:16\]) and the fact that the operators $A_{\varphi}$ are of trace class. In more general situations where $\Theta$ may not have a pure point spectrum of $\hat{H}^1_{{{\mathcal F}}} (X)$ but where $e^{-\frac{\alpha}{2}} \phi^{t*}$ is still orthogonal, we obtain: $$\langle {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^1_{{{\mathcal F}}} (X)) , \varphi \rangle = \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)_{\mathrm{point}}} \Phi (\rho) + \int^{\frac{\alpha}{2} + i \infty}_{\frac{\alpha}{2} - i \infty} \Phi (\lambda) m (\lambda) \, d \lambda$$ where $m (\lambda) \ge 0$ is the spectral density function of the continuous part of the spectrum of $\Theta$. Comparison with the “explicit formulas” in analytic number theory ================================================================= Consider a number field $K / {{\mathbb{Q}}}$. The explicit formulas in analytic number theory relate the primes of $K$ to the non-trivial zeroes of the Dedekind zeta function $\zeta_K (s)$ of $K$. \[t41\] For $\varphi \in {{\mathcal D}}({{\mathbb{R}}})$ define $\Phi (s)$ as in the preceeding section. Then the following fomula holds: $$\begin{aligned} \label{eq:17} \lefteqn{\Phi (0) - \sum_{\rho} \Phi (\rho) + \Phi (1) = - \log |d_{K / {{\mathbb{Q}}}}| \varphi (0)} \nonumber \\ & & + \sum_{{\mathfrak{p}}\nmid \infty} \log N {\mathfrak{p}}\left( \sum_{k \ge 1} \varphi (k \log N {\mathfrak{p}}) + \sum_{k \le -1} N {\mathfrak{p}}^k \varphi (k \log N {\mathfrak{p}}) \right) \nonumber \\ & & + \sum_{{\mathfrak{p}}{\, | \,}\infty} W_{{\mathfrak{p}}} (\varphi) \; .\end{aligned}$$ Here $\rho$ runs over the non-trivial zeroes of $\zeta_K (s)$ i.e. those that are contained in the critical strip $0 < {\mathrm{Re}\,}s < 1$. Moreover ${\mathfrak{p}}$ runs over the places of $K$ and $d_{K / {{\mathbb{Q}}}}$ is the discriminant of $K$ over ${{\mathbb{Q}}}$. For ${\mathfrak{p}}{\, | \,}\infty$ the $W_{{\mathfrak{p}}}$ are distributions which are determined by the $\Gamma$-factor at ${\mathfrak{p}}$. If $\varphi$ has support in ${{\mathbb{R}}}^{> 0}$ then $$W_{{\mathfrak{p}}} (\varphi) = \int^{\infty}_{-\infty} \frac{\varphi (t)}{1 - e^{\kappa_{{\mathfrak{p}}} t}} \, dt$$ where $\kappa_{{\mathfrak{p}}} = -1$ if ${\mathfrak{p}}$ is complex and $\kappa_{{\mathfrak{p}}} = -2$ if ${\mathfrak{p}}$ is real. If $\varphi$ has support on ${{\mathbb{R}}}^{< 0}$ then $$W_{{\mathfrak{p}}} (\varphi) = \int^{\infty}_{-\infty} \frac{\varphi (t)}{1 - e^{\kappa_{{\mathfrak{p}}} |t|}} \, e^t \, dt \; .$$ There are different ways to write $W_{{\mathfrak{p}}}$ on all of ${{\mathbb{R}}}$ but we will not discuss this here. See for example [@Ba] which also contains a proof of the theorem for much more general test functions. Formula (\[eq:17\]) implies the following equality of distributions on ${{\mathbb{R}}}^{> 0}$: $$\label{eq:18} 1 - \sum_{\rho} e^{t\rho} + e^t = \sum_{{\mathfrak{p}}\nmid \infty} \log N{\mathfrak{p}}\sum^{\infty}_{k=1} \delta_{k \log N {\mathfrak{p}}} + \sum_{{\mathfrak{p}}{\, | \,}\infty} (1 - e^{\kappa_{{\mathfrak{p}}} t})^{-1} \; .$$ This fits rather nicely with formula (\[eq:11\]) and suggests the following analogies: ------------------------------------------------------------------------------ --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\mathrm{spec}\,}{\mathfrak{o}}_K \cup \{ {\mathfrak{p}}{\, | \,}\infty \}$ ${\;\widehat{=}\;}$ $3$-dimensional dynamical system $(X , \phi^t)$ with a one-codimensional foliation ${{\mathcal F}}$ satisfying the conditions of conjecture \[t31\] finite place ${\mathfrak{p}}$ ${\;\widehat{=}\;}$ closed orbit $\gamma = \gamma_{{\mathfrak{p}}}$ not contained in a leaf and hence transversal to ${{\mathcal F}}$ such that $l (\gamma_{{\mathfrak{p}}}) = \log N {\mathfrak{p}}$ and $\varepsilon_{\gamma_{{\mathfrak{p}}}} (k) = 1$ for all $k \ge 1$. infinite place ${\mathfrak{p}}$ ${\;\widehat{=}\;}$ fixed point $x_{{\mathfrak{p}}}$ such that $\kappa_{x_{{\mathfrak{p}}}} = \kappa_{{\mathfrak{p}}}$ and $\varepsilon_{x_{{\mathfrak{p}}}} = 1$. ------------------------------------------------------------------------------ --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In order to understand number theory more deeply in geometric terms it would be very desirable to find a system $(X , \phi^t , {{\mathcal F}})$ which actually realizes this correspondence. For this the class of compact $3$-manifolds as phase spaces has to be generalized as will become clear from the following discussion. Formula (\[eq:17\]) can be written equivalently as an equality of distributions on ${{\mathbb{R}}}$: $$\begin{aligned} \label{eq:19} \lefteqn{1 - \sum_{\rho} e^{t\rho} + e^t = - \log |d_{K / {{\mathbb{Q}}}}| \delta_0} \nonumber \\ & & + \sum_{{\mathfrak{p}}\nmid \infty} \log N{\mathfrak{p}}\left( \sum_{k \ge 1} \delta_{k \log N {\mathfrak{p}}} + \sum_{k \le 1} N {\mathfrak{p}}^{k} \delta_{k \log N {\mathfrak{p}}} \right) \nonumber \\ & & + \sum_{{\mathfrak{p}}{\, | \,}\infty} W_{{\mathfrak{p}}} \; .\end{aligned}$$ Let us compare this with formula (\[eq:13\]) in Corollary \[t35\]. This corollary is the best result yet on the dynamical side but still only a first step since it does not allow for fixed points which as we have seen must be expected for dynamical systems of relevance for number fields. Ignoring the contributions $W_{{\mathfrak{p}}}$ from the infinite places for the moment we are suggested that $$\label{eq:20} - \log |d_{K/ {{\mathbb{Q}}}}| {\;\widehat{=}\;}\chi_{Co} ({{\mathcal F}}, \mu) \; .$$ There are two nice points about this analogy. Firstly there is the following well known fact due to Connes: \[t42\] Let ${{\mathcal F}}$ be a foliation of a compact $3$-manifold by surfaces such that the union of the compact leaves has $\mu$-measure zero, then $$\chi_{Co} ({{\mathcal F}}, \mu) \le 0 \; .$$ Namely the non-compact leaves are known to be complete in the induced metric. Hence they carry no non-zero harmonic $L^2$-functions, so that Connes’ $0$-th Betti number $\beta_0 ({{\mathcal F}}, \mu) = 0$. Since $\beta_2 ({{\mathcal F}}, \mu) = \beta_0 ({{\mathcal F}}, \mu)$ it follows that $$\begin{aligned} \chi_{Co} ({{\mathcal F}}, \mu) & = & \beta_0 ({{\mathcal F}}, \mu) - \beta_1 ({{\mathcal F}}, \mu) + \beta_2 ({{\mathcal F}}, \mu) \\ & = & - \beta_1 ({{\mathcal F}}, \mu) \le 0 \; .\end{aligned}$$ The reader will have noticed that in accordance with \[t42\] the left hand side of (\[eq:20\]) is negative as well: $$- \log |d_{K / {{\mathbb{Q}}}}| \le 0 \quad \mbox{for all} \; K / {{\mathbb{Q}}}\; .$$ The second nice point about (\[eq:20\]) is this. The bundle-like metric $g$ which we have chosen for the definition of $\Delta_{{{\mathcal F}}}$ and of $\chi_{Co} ({{\mathcal F}}, \mu)$ induces a holomorphic structure of ${{\mathcal F}}$ [@MS], Lemma A3.1. The space $X$ is therefore foliated by Riemann surfaces. Let $\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu)$ denote the holomorphic Connes Euler characteristic of ${{\mathcal F}}$ defined using $\Delta_{\overline{\partial}}$-harmonic forms on the leaves instead of $\Delta$-harmonic ones. According to Connes’ Riemann–Roch Theorem [@MS] Cor. A.2.3, Lemma A3.3 we have: $$\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu) = {\frac{1}{2}}\chi_{Co} ({{\mathcal F}}, \mu) \; .$$ Therefore $\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu)$ corresponds to $-\log \sqrt{|d_{K / {{\mathbb{Q}}}}|}$. For completely different reasons this number is defined in Arakelov theory as the Arakelov Euler characteristic of $\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K} = {\mathrm{spec}\,}{\mathfrak{o}}_K \cup \{ {\mathfrak{p}}{\, | \,}\infty \}$: $$\label{eq:21} \chi_{Ar} ({{\mathcal O}}_{\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}}) = - \log {\textstyle \sqrt{|d_{K / {{\mathbb{Q}}}}|}} \; .$$ See [@N] for example. Thus we see that $$\chi_{Ar} ({{\mathcal O}}_{\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}}) \quad \mbox{corresponds to} \; \chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu) \; .$$ It would be very desirable of course to understand Arakelov Euler characteristics in higher dimensions even conjecturally in terms of Connes’ holomorphic Euler characteristics. Note however that Connes’ Riemann–Roch theorem in higher dimensions does not involve the $R$-genus appearing in the Arakelov Riemann–Roch theorem. The ideas of Bismut [@Bi] may be relevant in this connection. He interpretes the $R$-genus in a natural way via the geometry of loop spaces. Further comparison of formulas (\[eq:13\]) and (\[eq:19\]) shows that in a dynamical system corresponding to number theory we must have $\alpha = 1$. This means that the flow $\phi^{t*}$ would act by multiplication with $e^t$ on the one-dimensional space ${\bar{H}}^2_{{{\mathcal F}}} (X)$. As explained before this would be the case if $\phi^t$ were conformal on $T{{\mathcal F}}$ with factor $e^t$: $$\label{eq:22} g (T_x \phi^t (v) , T_x \phi^t (w)) = e^t g (v,w) \quad \mbox{for all} \; v,w \in T_x {{\mathcal F}}\; .$$ However as mentioned before, this is not possible in the manifold setting of corollary \[t35\] which actually implies $\alpha = 0$.\ An equally important difference between formulas (\[eq:13\]) and (\[eq:19\]) is between the coefficients of $\delta_{kl (\gamma)}$ and of $\delta_{k \log N{\mathfrak{p}}}$ for $k \le -1$. In the first case it is $\pm 1$ whereas in the second it is $N{\mathfrak{p}}^k = e^{k \log N {\mathfrak{p}}}$ which corresponds to $e^{kl (\gamma)}$. Thus it becomes vital to find phase spaces $X$ more general than manifolds for which the analogue of corollary \[t35\] holds and where $\alpha \neq 0$ and in particular $\alpha = 1$ becomes possible. In the new context the term $\varepsilon_{\gamma} (k) \delta_{k l (\gamma)}$ for $k \le -1$ in formula (\[eq:13\]) should become $\varepsilon_{\gamma} (k) e^{\alpha k l (\gamma)} \delta_{kl (\gamma)}$. The next section is devoted to a discussion of certain laminated spaces which we propose as possible candidates for this goal. Remarks on dynamical Lefschetz trace formulas on laminated spaces ================================================================= In this section we extend the previous discussion to more general phase spaces than manifolds. The class of spaces we have in mind are the foliated spaces with totally disconnected transversals in the sense of [@MS]. We will call them laminated spaces for short. They also go by the name of (generalized) solenoids c.f. [@Su]. \[t51\] An $a$-dimensional laminated space is a second countable metrizable topological space $X$ which is locally homeomorphic to the product of a non-empty open subset of ${{\mathbb{R}}}^a$ with a totally disconnected space. Then $a$ is the topological dimension of $X$. Transition functions between local charts $\varphi_1$ and $\varphi_2$ have the following form locally: $$\label{eq:23} \varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x,y) = (F_1 (x,y) , F_2 (y)) \; .$$ Here $x,y$ denote the euklidean resp. totally disconnected components. This is due to the fact, that continuous functions from connected subsets of ${{\mathbb{R}}}^a$ into a totally disconnected space are constant. Because of (\[eq:23\]) the inverse images $\varphi^{-1} (\, , *)$ patch together, to give a partition ${{\mathcal L}}$ of $X$ into $a$-dimensional topological manifolds. These [*leaves*]{} of the laminated space $X$ are exactly the path components of $X$. The classical solenoid $$\label{eq:24} {{\mathbb{S}}}^1_p = {{\mathbb{R}}}\times_{{{\mathbb{Z}}}} {{\mathbb{Z}}}_p = \lim_{\leftarrow} (\ldots \to {{\mathbb{R}}}/ {{\mathbb{Z}}}\xrightarrow{p} {{\mathbb{R}}}/ {{\mathbb{Z}}}\xrightarrow{p} \ldots )$$ is an example of a compact connected one-dimensional laminated space with dense leaves homeomorphic to the real line. A $C^{\infty , 0}$-structure on a laminated space is a maximal atlas of local charts whose transition functions are smooth in the euklidean component and continuous in the totally disconnected one. Furthermore all derivatives in the euklidean directions should be continuous in all components. The leaves then become $a$-dimensional smooth manifolds. $C^{\infty , 0}$-laminated spaces are examples of foliated spaces in the sense of [@MS] Def. 2.1. A stronger structure that may exist on a laminated space was introduced by Sullivan \cite{}. A $C^{\infty , \infty}$- or $TLC$-structure on a laminated space $X$ is given by a maximal atlas whose transition functions are smooth in the euklidean component and uniformly locally constant in the totally disconnected one. That is, locally they have the form: $$\label{eq:25} \varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x,y) = (F_1 (x) , F_2 (y))$$ with $F_1$ smooth and $F_2$ locally constant. Every $C^{\infty , \infty}$-structure gives rise to a $C^{\infty , 0}$-structure. It is clear that ${{\mathbb{S}}}^1_p$ is naturally a $C^{\infty , \infty}$-laminated space. For a $C^{\infty , 0}$-laminated space $X$ let $TX = T {{\mathcal L}}$ denote its tangent bundle in the sense of [@MS] p. 43. For a point $x \in X$, the fibre $T_x X$ is the ordinary tangent space to the leaf through $x$. A Riemannian metric on $X$ is one on $TX$. Morphisms between $C^{\infty , 0}$-laminated spaces are continuous maps which induce smooth maps between the leaves of the lamination. They induce morphisms of tangent bundles. The two most prominent places in mathematics where laminated spaces occur naturally are in number theory e.g. as adelic points of algebraic groups and in the theory of dynamical systems as attractors. \[t52\] We now introduce foliations of laminated spaces. Let $X$ be an $a$-dimensional laminated space. For our purposes, a foliation ${{\mathcal F}}$ of $X$ by laminated spaces is a partition of $X$ into $d$-dimensional laminated spaces. The foliation is supposed to be locally trivial with euclidean transversals. More precisely ${{\mathcal F}}$ is given by a maximal atlas of local charts on $X$ $$\varphi : U {\stackrel{\sim}{\longrightarrow}}V_1 \times V_2 \times Y$$ with $V_1 \subset {{\mathbb{R}}}^d , V_2 \subset {{\mathbb{R}}}^{a-d}$ open and $Y$ totally disconnected, having the following property: The transition maps have the form: $$\varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x_1 , x_2 , y) = (G_1 (x_1 , x_2 , y) , G_2 (x_2 , y) , G_3 (y))$$ where $G_1 , G_2$ are smooth in the $x_1 , x_2$ components and $G_3$ and all $\partial^{\alpha_1 , \alpha_2}_{x_1 , x_2} G_1$ and $\partial^{\alpha_2}_{x_2} G_2$ are continuous. Setting $x = (x_1 , x_2)$, $$F_1 (x,y) = (G_1 (x_1 , x_2 , y_2) , G_2 (x_2 , y))$$ and $F_2 = G_2$ this induces a $C^{\infty , 0}$-structure on $X$ which is supposed to agree with the given one. The leaves of ${{\mathcal F}}$ are obtained by patching together the sets $\varphi^{-1} (V_1 \times \{ * \} \times Y )$. They are $d$-dimensional $C^{\infty , 0}$-laminated spaces with local transition functions given by: $$(x_1 , y) \longmapsto (G_1 (x_1 , * , y) , G_3 (y)) \; .$$ Their leaves are $d$-dimensional manifolds which foliate the $a$-dimensional manifolds which occur as leaves of the lamination on $X$. Thus $X$ is partitioned into $a$-dimensional manifolds each of which carries a $d$-dimensional foliation in the usual sense. Besides ${{\mathcal L}}$ and ${{\mathcal F}}$ there is a third foliated structure denoted ${{\mathcal F}}{{\mathcal L}}$ on $X$. The space $X$ is foliated with leaves the $d$-dimensional manifolds that occur as path components of the ${{\mathcal F}}$-leaves. Here the transverse space is of the form open subspace of ${{\mathbb{R}}}^{a-d} \times$ totally disconnected. The local transition maps are given by: $$(x_1 , z) \longmapsto (H_1 (x_1 , z) , H_2 (z))$$ with $z = (x_2 , y) , H_1 (x_1 , z) = G_1 (x_1 , x_2 , y)$ and $H_2 (z) = (G_2 (x_2 , y) , G_3 (y))$. Of the three foliated structures ${{\mathcal L}}, {{\mathcal F}}$ and ${{\mathcal F}}{{\mathcal L}}$ on $X$ the first and the last fit into the context of [@MS] but not the second. \[t53\] We now turn to cohomology. The foliation ${{\mathcal F}}{{\mathcal L}}$ gives rise to the integrable rank $d$ subbundle $T{{\mathcal F}}{{\mathcal L}}$ of $TX = T {{\mathcal L}}$. These bundles are tangentially smooth in the sense of [@MS], p. 43 with respect to ${{\mathcal L}}$. The leafwise cohomology of $X$ along ${{\mathcal F}}{{\mathcal L}}$ is by definition the cohomology of the sheaf ${{\mathcal R}}_{{{\mathcal F}}{{\mathcal L}}}$ of real valued smooth functions on $X$ wich are locally constant along the ${{\mathcal F}}{{\mathcal L}}$ leaves. Here a continuous function or form on an open subset of $X$ is called smooth if its restrictions to the ${{\mathcal L}}$-leaves are smooth. The sheaf ${{\mathcal R}}_{{{\mathcal F}}{{\mathcal L}}}$ is resolved by the de Rham complex of smooth forms along ${{\mathcal F}}{{\mathcal L}}$. Using the natural Fréchet topology on the spaces of global differential forms, one defines the reduced version ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ of leafwise cohomology $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ as its maximal Hausdorff quotient. By $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ we denote the cohomology of the sheaf ${{\mathcal R}}_{{{\mathcal F}}}$ of real valued smooth functions on $X$ which are locally constant along the ${{\mathcal F}}$-leaves. \[t54\] A flow $\phi$ on $X$ is a continuous ${{\mathbb{R}}}$-action such that the induced ${{\mathbb{R}}}$-actions on the leaves of ${{\mathcal L}}$ are smooth. It respects ${{\mathcal F}}$ if every $\phi^t$ maps leaves of ${{\mathcal F}}$ into leaves of ${{\mathcal F}}$. It follows that $\phi^t$ maps ${{\mathcal F}}{{\mathcal L}}$-leaves into ${{\mathcal F}}{{\mathcal L}}$-leaves. Thus $(X , {{\mathcal F}}, \phi^t)$ is partitioned into the foliated dynamical systems $(L, {{\mathcal F}}_L , \phi^t \, |_L)$ for $L \in {{\mathcal L}}$. Here $${{\mathcal F}}_L = {{\mathcal F}}{{\mathcal L}}\, |_L = \{ S \in {{\mathcal F}}{{\mathcal L}}{\, | \,}S \subset L \} \; .$$ Any ${{\mathcal F}}$-compatible flow $\phi^t$ induces pullback actions $\phi^{t*}$ on $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ and ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$. \[t55\] We now state as a working hypotheses a generalization of the conjectured dynamical trace formula \[t31\]. We allow the phase space to be a laminated space. Moreover we extend the formula to an equality of distributions on ${{\mathbb{R}}}^*$ instead of ${{\mathbb{R}}}^{>0}$. After checking various compatibilities we state a case where our working hypotheses can be proved and give a number theoretical example. [**Working hypotheses:**]{} \[t56\] Let $X$ be a compact $C^{\infty , 0}$-laminated space with a one-codimensional foliation ${{\mathcal F}}$ and an ${{\mathcal F}}$-compatible flow $\phi$. Assume that the fixed points and the periodic orbits of the flow are non-degenerate. Then there exists a natural definition of a ${{\mathcal D}}' ({{\mathbb{R}}}^*)$-valued trace of $\phi^{t*}$ on ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ such that in ${{\mathcal D}}' ({{\mathbb{R}}}^*)$ we have: $$\begin{aligned} \label{eq:26} \hspace*{0.5cm} \lefteqn{\sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X)) = }\\ & & \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{k l (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (|k|) \det (-T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \delta_{kl (\gamma)} \right) \nonumber \\ & & + \sum_x W_x \; . \nonumber\end{aligned}$$ Here $\gamma$ runs over the closed orbits not contained in a leaf and in the sums over $k$’s any point $x \in \gamma$ can be chosen. The second sum runs over the fixed points $x$ of the flow. The distributions $W_x$ on ${{\mathbb{R}}}^*$ are given by: $$W_x \, |_{{{\mathbb{R}}}^{> 0}} = \varepsilon_x \, |1 - e^{\kappa_x t}|^{-1}$$ and $$W_x \, |_{{{\mathbb{R}}}^{< 0}} = \varepsilon_x \det (-T_x \phi^t {\, | \,}T_x {{\mathcal F}}) \, |1 - e^{\kappa_x |t|}|^{-1} \; .$$ \[t57\] [**0)**]{} It may actually be better to use a version of foliation cohomology where transversally forms are only supposed to be locally $L^2$ instead of being continuous.\ [**1)**]{} In the situation described in \[t58\] below the working hypotheses can be proved if ${\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X)$ is replaced by $H^n_{{{\mathcal F}}} (X)$, Theorem \[t59\]. In those cases there are no fixed points, only closed orbits. Thus Theorem \[t59\] dictated only the coefficients of $\delta_{kl (\gamma)}$ for $k \in {{\mathbb{Z}}}{\setminus}0$, but not the contributions $W_x$ from the fixed points.\ [**2)**]{} The coefficients of $\delta_{kl (\gamma)}$ for $k \in {{\mathbb{Z}}}{\setminus}0$ can be written in a uniform way as follows. They are equal to: $$\label{eq:27} \frac{\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})}{|\det (1 - T_x \phi^{|k| l (\gamma)} {\, | \,}T_x X / {{\mathbb{R}}}Y_{\phi , x})|} = \frac{\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})}{| \det (1 - T_x \phi^{|k| l (\gamma)} {\, | \,}T_x {{\mathcal F}})|} \; .$$ Here $x$ is any point on $\gamma$. Namely, for $k \ge 1$ this equals $\varepsilon_{\gamma} (k)$ whereas for $k \le -1$ we obtain $$\label{eq:28} \varepsilon_{\gamma} (|k|) \det (- T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) = \varepsilon_{\gamma} (k) \, |\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| \; .$$ The expression on the left hand side of (\[eq:27\]) motivated our conjecture about the contributions on ${{\mathbb{R}}}^*$ from the fixed points $x$. Since $Y_{\phi , x} = 0$, they should be given by: $$\frac{\det (1 - T_x \phi^t {\, | \,}T_x {{\mathcal F}})}{| \det (1 - T_x \phi^{|t|} {\, | \,}T_x X)|} \overset{!}{=} W_x \; .$$ [**3)**]{} One can prove that in the manifold setting of theorem \[t33\] we have $$|\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| = 1 \; .$$ By (\[eq:26\]), our working hypotheses \[t56\] is therefore compatible with formula (\[eq:12\]). Compatibility with conjecture \[t31\] is clear.\ [**4)**]{} We will see below that in our new context metrics $g$ on $T{{\mathcal F}}$ can exist for which the flow has the conformal behaviour (\[eq:22\]). Assuming we are in such a situation and that ${{\mathcal F}}$ is $2$-dimensional, we have: $$|\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| = e^{kl (\gamma)} \quad \mbox{for} \; x \in \gamma , k \in {{\mathbb{Z}}}$$ and $$|\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}})| = e^t \quad \mbox{for a fixed point} \; x \; .$$ In the latter case, we even have by continuity: $$\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}}) = e^t \; ,$$ the determinant being positive for $t = 0$. Hence by (\[eq:28\]) the conjectured formula (\[eq:26\]) reads as follows in this case: $$\begin{aligned} \label{eq:29} \lefteqn{\sum^2_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X))} \\ & = & \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (k) e^{kl (\gamma)} \delta_{kl (\gamma)} \right) \nonumber \\ & & + \sum_x W_x \; . \nonumber\end{aligned}$$ Here: $$W_x \, |_{{{\mathbb{R}}}^{> 0}} = \varepsilon_x \, |1 - e^{\kappa_x t}|^{-1}$$ and $$W_x \, |_{{{\mathbb{R}}}^{< 0}} = \varepsilon_x e^t \, |1 - e^{\kappa_x |t|}|^{-1} \; .$$ This fits perfectly with the explicit formula (\[eq:19\]) if all $\varepsilon_{\gamma_{{\mathfrak{p}}}} (k) = 1$ and $\varepsilon_{x_{{\mathfrak{p}}}} = 1$. Namely if $l (\gamma_{{\mathfrak{p}}}) = \log N {\mathfrak{p}}$ for ${\mathfrak{p}}\nmid \infty$ and $\kappa_{x_{{\mathfrak{p}}}} = \kappa_{{\mathfrak{p}}}$ for ${\mathfrak{p}}{\, | \,}\infty$, then we have: $$e^{kl (\gamma_{{\mathfrak{p}}})} = e^{k \log N{\mathfrak{p}}} = N {\mathfrak{p}}^k \quad \mbox{for finite places} \; {\mathfrak{p}}$$ and $$W_{x_{{\mathfrak{p}}}} = W_{{\mathfrak{p}}} \quad \mbox{on} \; {{\mathbb{R}}}^* \; \mbox{for the infinite places} \; {\mathfrak{p}}\; .$$ [**5)**]{} In the setting of the preceeding remark the automorphisms $$e^{-\frac{k}{2} l (\gamma)} T_x \phi^{kl (\gamma)} \quad \mbox{of} \; T_x {{\mathcal F}}\quad \mbox{for} \; x \in \gamma$$ respectively $$e^{-\frac{t}{2}} T_x \phi^t \quad \mbox{of} \; T_x {{\mathcal F}}\quad \mbox{for a fixed point} \; x$$ are orthogonal automorphisms. For a real $2 \times 2$ orthogonal determinant $O$ with $\det O = -1$ we have: $$\det (1 - uO) = 1 - u^2 \; .$$ The condition $\varepsilon_{\gamma} (k) = +1$ therefore implies that $\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})$ is positive for $k \ge 1$ and hence for all $k \in {{\mathbb{Z}}}$. The converse is also true. For a fixed point we have already seen directly that $\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}})$ is positive for all $t \in {{\mathbb{R}}}$. Hence we have the following information. [**Fact**]{} In the situation of the preceeding remark, $\varepsilon_k (\gamma) = +1$ for all $k \in {{\mathbb{Z}}}{\setminus}0$ if and only if on $T_x {{\mathcal F}}$ we have: $$T_x \phi^{kl (\gamma)} = e^{\frac{k}{2} l (\gamma)} \cdot O_k \quad \mbox{for} \; O_k \in {\mathrm{SO}}(T_x {{\mathcal F}}) \; .$$ For fixed points, $\varepsilon_x = 1$ is automatic and we have: $$T_x \phi^t = e^{\frac{t}{2}} O_t \quad \mbox{for} \; O_t \in {\mathrm{SO}}(T_x {{\mathcal F}}) \; .$$ In the number theoretical case the eigenvalues of $T_x \phi^{\log N{\mathfrak{p}}}$ on $T_x {{\mathcal F}}$ for $x \in \gamma_{{\mathfrak{p}}}$ would therefore be complex conjugate numbers of absolute value $N{\mathfrak{p}}^{1/2}$. If they are real then $T_x \phi^{\log N{\mathfrak{p}}}$ would simply be mutliplication by $\pm N{\mathfrak{p}}^{1/2}$. If not, the situation would be more interesting. Are the eigenvalues Weil numbers (of weight $1$)? If yes there would be some elliptic curve over ${\mathfrak{o}}_K / {\mathfrak{p}}$ involved by Tate–Honda theory.\ [**6)**]{} It would of course be very desirable to extend the hypotheses \[t56\] to a conjectured equality of distributions on all of ${{\mathbb{R}}}$. By theorem \[t33\] we expect one contribution of the form $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) \cdot \delta_0 \; .$$ The analogy with number theory suggests that there will also be somewhat complicated contributions from the fixed points in terms of principal values which are hard to guess at the moment. After all, even the simpler conjecture \[t31\] has not yet been verified in the presence of fixed points!\ [**7)**]{} If there does exist a foliated dynamical system attached to $\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}$ with the properties dictated by our considerations we would expect in particular that for a preferred transverse measure $\mu$ we have: $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) = - \log |d_{K / {{\mathbb{Q}}}}| \; .$$ This gives some information on the space $X$ with its ${{\mathcal F}}{{\mathcal L}}$-foliation. If $K / {{\mathbb{Q}}}$ is ramified at some finite place i.e. if $d_{K / {{\mathbb{Q}}}} \neq \pm 1$ then $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) < 0$. Now, since ${\bar{H}}^2_{{{\mathcal F}}{{\mathcal L}}}$ must be one-dimensional, it follows that $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \nu) < 0 \quad \mbox{for all non-trivial transverse measures} \; \nu \; .$$ Hence by a result of Candel [@Ca] there is a Riemannian metric on $T {{\mathcal F}}{{\mathcal L}}$, such that every ${{\mathcal F}}{{\mathcal L}}$-leaf has constant curvature $-1$. Moreover $(X , {{\mathcal F}}{{\mathcal L}})$ is isomorphic to $${{\mathcal O}}(H , X) / {\mathrm{PSO}}(2) \; .$$ Here ${{\mathcal O}}(H , X)$ is the space of conformal covering maps $u : H \to N$ as $N$ runs through the leaves of ${{\mathcal F}}{{\mathcal L}}$ with the compact open topology. See [@Ca] for details. In the unramified case, $|d_{K / {{\mathbb{Q}}}}| = 1$ we must have $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \nu) = 0$ for all transverse measures by the above argument. Hence there is an ${{\mathcal F}}{{\mathcal L}}$-leaf which is either a plane, a torus or a cylinder c.f. [@Ca]. \[t58\] In this final section we describe a simple case where the working hypothesis \[t56\] can be proved. Consider an unramified covering $f : M \to M$ of a compact connected orientable $d$-dimensional manifold $M$. We set $$\bar{M} = \lim_{\leftarrow} ( \ldots \xrightarrow{f} M \xrightarrow{f} M \to \ldots) \; .$$ Then $\bar{M}$ is a compact topological space equipped with the shift automorphism $\bar{f}$ induced by $f$. It can be given the structure of a $C^{\infty ,\infty}$-laminated space as follows. Let $\tilde{M}$ be the universal covering of $M$. For $i \in {{\mathbb{Z}}}$ there exists a Galois covering $$p_i : \tilde{M} \longrightarrow M$$ with Galois group $\Gamma_i$ such that $p_i = p_{i+1} {\mbox{\scriptsize $\,\circ\,$}}f$ for all $i$. Hence we have inclusions: $$\ldots \subset \Gamma_{i+1} \subset \Gamma_i \subset \ldots \subset \Gamma_0 =: \Gamma \cong \pi_1 (M , x_0) \; .$$ Writing the operation of $\Gamma$ on $\tilde{X}$ from the right, we get commutative diagrams for $i \ge 0$: $$\begin{CD} \tilde{M} \times_{\Gamma} (\Gamma / \Gamma_{i+1}) @= \tilde{M} / \Gamma_{i+1} @>{\overset{p_{i+1}}{\sim}}>> M \\ @VV{{\mathrm{id}}\times {\mathrm{proj}}}V @VV{{\mathrm{proj}}}V @VV{f}V \\ \tilde{M} \times_{\Gamma} (\Gamma / \Gamma_i) @= \tilde{M} / \Gamma_i @>{\overset{p_i}{\sim}}>> M \end{CD}$$ It follows that $$\label{eq:30} \tilde{M} \times_{\Gamma} \bar{\Gamma} {\stackrel{\sim}{\longrightarrow}}\bar{M}$$ where $\bar{\Gamma}$ is the pro-finite set with $\Gamma$-operation: $$\bar{\Gamma} = \lim_{\leftarrow} \Gamma / \Gamma_i \; .$$ The isomorphism (\[eq:30\]) induces on $\bar{M}$ the structure of a $C^{\infty , \infty}$-laminated space with respect to which $\bar{f}§$ becomes leafwise smooth. Fix a positive number $l > 0$ and let $\Lambda = l {{\mathbb{Z}}}\subset {{\mathbb{R}}}$ act on $\bar{M}$ as follows: $\lambda = l \nu$ acts by $\bar{f}^{\nu}$. Define a right action of $\Lambda$ on $\bar{M} \times {{\mathbb{R}}}$ by the formula $$(m,t) \cdot \lambda = (- \lambda \cdot m , t + \lambda) = (\bar{f}^{-\lambda / l} (m) , t + \lambda) \; .$$ The suspension: $$X = \bar{M} \times_{\Lambda} {{\mathbb{R}}}$$ is an $a = d+1$-dimensional $C^{\infty , \infty}$-laminated space with a one-codimensional foliation ${{\mathcal F}}$ as in \[t52\]. The leaves of ${{\mathcal F}}$ are the fibres of the natural fibration of $X$ over the circle ${{\mathbb{R}}}/ \Lambda$: $$X \longrightarrow {{\mathbb{R}}}/ \Lambda \; .$$ The leaves are also the images of $\bar{M} \times \{ t \}$ for $t \in {{\mathbb{R}}}$ under the natural projection. Translation in the ${{\mathbb{R}}}$-variable $$\phi^t [m,t'] = [m,t+t']$$ defines an ${{\mathcal F}}$-compatible flow $\phi$ on $X$ which is everywhere transverse to the leaves of ${{\mathcal F}}$ and in particular has no fixed points. The map $$\gamma \longmapsto \gamma_M = \gamma \cap (\bar{M} \times_{\Lambda} \Lambda)$$ gives a bijection between the closed orbits $\gamma$ of the flow on $X$ and the finite orbits $\gamma_M$ of the $\bar{f}$- or $\Lambda$-action. These in turn are in bijection with the finite orbits of the original $f$-action on $M$. We have: $$l (\gamma) = |\gamma_M| l \; .$$ \[t59\] In the situation of \[t58\] assume that all periodic orbits of $\phi$ are non-degenerate. Let ${\mathrm{Sp}}^n (\Theta)$ denote the set of eigenvalues with their algebraic multiplicities of the infinitesimal generator $\Theta$ of $\phi^{t*}$ on $H^n_{{{\mathcal F}}} (X)$. Then the trace $${\mathrm{Tr}}(\phi^* {\, | \,}H^n_{{{\mathcal F}}} (X)) := \sum_{\lambda \in {\mathrm{Sp}}^n (\Theta)} e^{t\Theta}$$ defines a distribution on ${{\mathbb{R}}}$ and the following formula holds true in ${{\mathcal D}}' ({{\mathbb{R}}})$: $$\begin{aligned} \lefteqn{\sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}H^n_{{{\mathcal F}}} (X)) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) \cdot \delta_0 \; + }\\ && \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (|k|) \det (-T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \delta_{kl (\gamma)} \right) \; .\end{aligned}$$ Here $\gamma$ runs over the closed orbits of $\phi$ and in the sum over $k$’s any point $x \in \gamma$ can be chosen. Moreover $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu)$ is the Connes’ Euler characteristic of ${{\mathcal F}}{{\mathcal L}}$ with respect to a certain canonical transverse measure $\mu$. Finally we have the formula: $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) = \chi (M) \cdot l \; .$$ The definition of $\mu$ and the proof of the theorem will be given elsewhere. Let $E / {{\mathbb{F}}}_p$ be an ordinary elliptic curve over ${{\mathbb{F}}}_p$ and let ${{\mathbb{C}}}/ \Gamma$ be a lift of $E$ to a complex elliptic curve with $CM$ by the ring of integers ${\mathfrak{o}}_K$ in an imaginary quadratic field $K$. Assume that the Frobenius endomorphism of $E$ corresponds to the prime element $\pi$ in ${\mathfrak{o}}_K$. Then $\pi$ is split, $\pi \bar{\pi} = p$ and for any embedding ${{\mathbb{Q}}}_l \subset {{\mathbb{C}}}, l \neq p$ the pairs $$(H^*_{{\mathrm{\acute{e}t}}} (E \otimes \bar{{{\mathbb{F}}}}_p , {{\mathbb{Q}}}_l) \otimes {{\mathbb{C}}}, {\mathrm{Frob}}^*) \quad \mbox{and} \quad (H^* ({{\mathbb{C}}}/ \Gamma , {{\mathbb{C}}}) , \pi^*)$$ are isomorphic. Setting $M = {{\mathbb{C}}}/ \Gamma , f = \pi$ we are in the situation of \[t58\] and we find: $$X = ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{\Lambda} {{\mathbb{R}}}\; .$$ Here $$T_{\pi} \Gamma = \lim_{\leftarrow} \Gamma / \pi^i \Gamma \cong {{\mathbb{Z}}}_p$$ is the $\pi$-adic Tate module of ${{\mathbb{C}}}/ \Gamma$. It is isomorphic to the $p$-adic Tate module of $E$. Setting $l = \log p$, so that $\Lambda = (\log p) {{\mathbb{Z}}}$ and passing to multiplicative time, $X$ becomes isomorphic to $$X \cong ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{p^{{{\mathbb{Z}}}}} {{\mathbb{R}}}^*_+$$ which may be a more natural way to write $X$. Note that $p^{\nu}$ acts on ${{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma$ by diagonal multiplication with $\pi^{\nu}$. It turns out that the right hand side of the dynamical Lefschetz trace formula established in theorem \[t59\] equals the right hand side in the explicit formulas for $\zeta_E (s)$. Moreover the metric $g$ on $T {{\mathcal F}}$ given by $$g_{[z,y,t]} (\xi, \eta) = e^t {\mathrm{Re}\,}(\xi \bar{\eta}) \quad \mbox{for} \; [z,y,t] \; \mbox{in}\; ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{\Lambda} {{\mathbb{R}}}$$ satisfies the conformality condition (\[eq:9\]) for $\alpha = 1$. The proof of Theorem \[t21\] can be easily adapted to $X$ above and shows that $\Theta = {\frac{1}{2}}+ S$ on ${\bar{H}}^1_{{{\mathcal F}}{{\mathcal L}}} (X)$ where $S$ is skew symmetric. This gives a dynamical proof for the Riemann hypotheses for $\zeta_E (s)$ along the lines that we hope for in the case of $\zeta (s)$. The construction of $(X , \phi^t)$ that we made for ordinary elliptic curves is misleading however, since it almost never happens that a variety in characteristic $p$ can be lifted to characteristic zero [*together with its Frobenius endomorphism*]{}. Moreover for ordinary elliptic curves the Riemann hypotheses can already be proved using Hodge cohomology of the lifted curve. This was essentially Hasse’s proof. Our present dream for the general situation is this: To an algebraic sum ${{\mathcal X}}/ {{\mathbb{Z}}}$ one should first attach an infinite dimensional dissipative dynamical system, possibly using ${\mathrm{GL}\,}_{\infty}$ in some way. The desired dynamical system should then be obtained by passing to the finite dimensional compact global attractor, c.f. [@La] Part I. [9999]{} J.A. Álvarez López, Y. Kordyukov, Long time behaviour of leafwise heat flow for Riemannian foliations. Compositio Math. [**125**]{} (2001), 129–153 J. Álvarez López, Y. Kordyukov, Distributional Betti numbers of transitive foliations of codimension one. Preprint 2000 M.F. Atiyah, Elliptic operators and compact groups. Springer LNM [**401**]{}, 1974 K. Barner, On A. Weil’s explicit formula. J. Reine Angew. Math. [**323**]{} (1981), 139–152 J.-M. Bismut, Complex equivariant intersection, excess normal bundles and Bott–Chern currents. Comm. Math. Phys. [**148**]{} (1992), 1–55 A. Candel, Uniformization of surface laminations. Ann. scient. Éc. Norm. Sup. 4$^e$ série [**26**]{} (1993), 489–516 A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Sel. math., New ser. 5 (1999), 29–106 C. Deninger, Evidence for a cohomological approach to analytic number theory. in: Joseph, Rentschler (eds.): Proceedings of the EMS conference 1992, 491–510, Birkhäuser 1994 C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces. Doc. Math. J. DMV. Extra Volume ICM I (1998), 23–46 C. Deninger, W. Singhof, A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas. Ann. Int. Fourier, Grenoble, [**51**]{}, 1 (2001), 209–219 C. Deninger, W. Singhof, A note on dynamical trace formulas. To appear in a proceedings volume on dynamical systems edited by M.L. Lapidus, M. van Frankenhuysen C. Deninger, W. Singhof, Real polarizable Hodge structures arising from foliations, Preprint 2001 N. Dumford, J.T. Schwartz, Linear operators, Parts I and II. Wiley Classics Library 1988 C. Godbillon, Feuilletages. Progress in Math. [**98**]{}, Birkhäuser 1991 V. Guillemin, Lectures on spectral theory of elliptic operators. Duke Math. J. [**44**]{} (1977), 485–517 O. Ladyzhenskaya, Attractors for semigroups and evolution equations. Cambridge university press 1991 C. Lazarov, Transverse index and periodic orbits. GAFA [**10**]{} (2000), 124–159 C.C. Moore, C. Schochet, Global analysis on foliated spaces. MSRI Publications [**9**]{}, Springer 1988 B. Mümken, Thesis in preparation J. Neukirch, Algebraic number theory. Springer Grundlehren [**322**]{}, 1999 S.J. Patterson, On Ruelle’s zeta-function. Israel Math. Conf. Proc. [**3**]{} (1990), 163–184 C. Soulé, Sur les zéros des fonctions $L$ automorphes. CR Acad. Sci. Paris Sér. I Math. [**328**]{} (1999), 955–958 I.M. Singer, Index theory for elliptic operators, Proc. Symp. Pure Math. [**28**]{} (1973), 11–31 D. Sullivan, Linking the universalities of Milnor–Thurston Feigenbaum and Ahlfors-Bers. In: Topological methods in modern mathematics. Publish or Perish 1993, pp. 543–564. [^1]: This is different from the normalization in [@D2] §3.

ArXiv

--- abstract: 'Moderated by a director, laymen and students are encouraged to assume the role of quanta and enact a quantum cryptographic protocol. The performance is based on a generalized urn model capable of reproducing complementarity even for classical chocolate balls.' author: - Karl Svozil title: 'Staging quantum cryptography with chocolate balls[^1]' --- > [Dedicated to Antonin Artaud,\ > author of [*Le th[é]{}[â]{}tre et son double*]{} [@Arthaud]. ]{} Background ========== Quantum cryptography is a relatively recent and extremely active field of research within quantum physics. Its main characteristic is the use of (at least ideally) individual particles for encrypted information transmission. Its objective is to encrypt messages, or to create and enlarge a set of secret equal random numbers, between two spatially separated agents by means of elementary particles, such as single photons, which are transmitted through a quantum channel. The history of quantum cryptography dates back to around 1970, to the manuscript by Wiesner [@wiesner] and a protocol by Bennett [&]{} Brassard in 1998 [@benn-82; @benn-84; @ekert91; @benn-92; @gisin-qc-rmp] henceforth called “BB84”. Since then, experimental prototyping has advanced rapidly. Without going into too much detail and just to name a few examples, the work ranges from the very first experiments carried out in the IBM Yorktown Heights Laboratory by Bennett and co-workers in 1989 [@benn-92], to signal transmissions across Lake Geneva in 1993 [@gisin-qc-rmp], and the network in the Boston Metropolitan Area which has been sponsored by DARPA since 2003 [@ell-co-05]. In a much publicized, spectacular demonstration, a quantum cryptographic aided bank transfer took place via optical fibers installed in the sewers of Vienna in the presence of some local politicians and bank representatives [@pflmubpskwjz]. Quantum cryptography forms an important link between quantum theory and experimental technology, and possibly even industrial applications. The public is highly interested in quantum physics and quantum cryptography, but the protocols used are rarely made available to the layman or student in any detail. For an outsider these subjects seem to be shrouded in some kind of “mystic veil” and are very difficult to understand, although great interest in the subject prevails. In what follows, we shall use a simple but effective generalized urn model introduced by Wright [@wright; @wright:pent; @svozil-2001-eua] to mimic complementarity. A generalized urn model is characterized by an ensemble of balls with black background color. Printed on these balls are some color symbols from a symbolic alphabet. The colors are elements of a set of colors. A particular ball type is associated with a unique combination of mono-spectrally (no mixture of wavelength) colored symbols printed on the black ball background. Every ball contains just one single symbol per color. Assume further some mono-spectral filters or eyeglasses which are “perfect” by totally absorbing light of all other colors but a particular single one. In that way, every color can be associated with a particular eyeglass and vice versa. When a spectator looks at a particular ball through such an eyeglass, the only operationally recognizable symbol will be the one in the particular color which is transmitted through the eyeglass. All other colors are absorbed, and the symbols printed in them will appear black and therefore cannot be differentiated from the black background. Hence the ball appears to carry a different “message” or symbol, depending on the color at which it is viewed. We will present an explicit example featuring complementarity, in very similar ways as quantum complementarity. The difference between the chocolate balls and the quanta is the possibility to view all the different symbols on the chocolate balls in all different colors by taking off the eyeglasses. Quantum mechanics does not provide us with such a possibility. On the contrary, there are strong formal arguments suggesting that the assumption of a simultaneous physical existence [@epr] of such complementary observables yields a complete contradiction [@kochen1]. Principles of conduct ===================== In order to make it a real-life experience, we have aimed at dramatizing quantum cryptography. The quantum world is turned into a kind of drama, in which actors and a moderator present a quantum cryptographic protocol on stage. The audience is actively involved and invited to participate in the dramatic presentation. If at all possible, the event should be moderated by a well-known comedian, or by a physics teacher. The entire process is principally analogous to an experiment in a slightly surreal sense: just like humans, single quanta are never completely predictable. Among other things, they are in fact determined by random events, and marked by a certain “noise” similar to the chaos that will certainly accompany the public presentation of the quantum cryptographic protocols. Therefore, the interference of individual participants is even encouraged and not a deficiency of the model. Throughout the performance, everybody should have fun, relax, and try to feel and act like an elementary particle – rather in the spirit of the meditative Zen koan “Mu.” The participants might manage to feel like Schrödinger’s cat [@schrodinger], or like a particle simultaneously passing through two spatially separated slits. In idle times, one may even contemplate how conscious minds could experience a coherent quantum superposition between two states of consciousness. However, this kind of sophistication is neither necessary, nor particularly important for dramatizing quantum cryptographic protocols. Our entire empirical knowledge of the world is based on the occurrence of elementary (binary) events, such as the reactions caused by quanta in particle detectors yielding either a “click” or none. Therefore, the following simple syntactic rules should not be dismissed as mere cooking recipes, for quantum mechanics itself can actually be applied merely as a sophisticated set of laws with a possibly superfluous [@fuchs-peres] semantic superstructure. Instructions for staging the protocol ===================================== Our objective is to generate a secret sequence of random numbers only known by two agents called Alice and Bob. In order to do so, the following utensils depicted in Figure \[2005-ln1e-utensils\] will be required: ![Utensils required for staging the BB84 protocoll.[]{data-label="2005-ln1e-utensils"}](2005-ln1e-utensils){width="8.2cm"} - Two sets each of fully saturated glasses in red and green (complementary colors) - An urn or bucket - A large number of foil-wrapped chocolate balls (in Austria called “Mozartkugeln”) or similar – each with a black background, imprinted with one red and one green symbol (either 0 or 1) – to be placed inside the urn. According to all possible combinations, there are four types altogether, which can be found in Table \[2005-nl1-t1\]. There needs to be an equal number of each type in the urn. Balltyp [red]{} [green]{} --------- --------- ----------- Typ 1 [0]{} [0]{} Typ 2 [0]{} [1]{} Typ 3 [1]{} [0]{} Typ 4 [1]{} [1]{} : Schema of imprinting of the chocolate balls.\[2005-nl1-t1\] - Small red and green flags, two of each - Two blackboards and chalk (or two secret notebooks) - Two coins The following acting persons are involved: - A moderator who makes comments and ensures that the participants more or less adhere to the protocol as described below. The moderator has many liberties and may even choose to stage cryptographic attacks. - Alice and Bob, two spatially separated parties - Ideally, but not necessary are some actors who know the protocol and introduce new visitors to the roles of Alice, Bob and the quanta. - A large number of people assuming the roles of the quanta. They are in charge of transmitting the chocolates and may eat them in the course of events or afterwards. In our model, chocolates marked with the symbols 0 and 1 in red, correspond to what in quantum optics correspond to horizontally ($\leftrightarrow$) and vertically ($\updownarrow$) polarized photons, respectively. Accordingly, chocolates marked with the symbols 0 and 1 in green, correspond to left ($\circlearrowleft$) and right ($\circlearrowright$) circularly polarized photons, or alternatively to linearly polarized photons with polarization directions ($\ddarrow$) and ($\cddarrow$) rotated by 45$°$ ($\pi / 4$) from the horizontal and the vertical, respectively. The protocol is to be carried out as follows: - Alice flips a coin in order to chose one of two pairs of glasses: heads is for the green glasses, tails for the red ones. She puts them on and randomly draws one chocolate from the urn. She can only read the symbol in the color of her glasses (due to subtractive color the other symbol in the complementary color appears black and cannot be differentiated from the black background). This situation is illustrated in Figure \[f-gum-w\]. She writes the symbol she could read, as well as the color used, either on the blackboard or into her notebook. Should she attempt to take off her glasses or look at the symbols with the other pair, the player in the role of the quantum is required to eat the chocolate at once. 0.6mm (118.33,55.67) (35.00,35.00)[(0,-1)[25.00]{}]{} (55.00,9.67)[(40.00,9.33)\[b\]]{} (75.00,35.00)[(0,-1)[25.00]{}]{} (40.62,10.00) (40.62,10.00)[(0,0)\[cc\][**[1]{} [0]{}**]{}]{} (50.29,10.00) (50.29,10.00)[(0,0)\[cc\][**[1]{} [1]{}**]{}]{} (60.33,10.00) (69.91,10.00) (60.33,10.00)[(0,0)\[cc\][**[0]{} [1]{}**]{}]{} (69.91,10.00)[(0,0)\[cc\][**[0]{} [0]{}**]{}]{} (46.33,19.00) (46.33,19.00)[(0,0)\[cc\][**[0]{} [1]{}**]{}]{} (56.00,18.67) (56.00,18.67)[(0,0)\[cc\][**[1]{} [1]{}**]{}]{} (65.81,19.00) (65.81,19.00)[(0,0)\[cc\][**[1]{} [0]{}**]{}]{} (40.15,26.33) (40.15,26.33)[(0,0)\[cc\][**[0]{} [1]{}**]{}]{} (51.76,27.57) (51.76,27.57)[(0,0)\[cc\][**[0]{} [0]{}**]{}]{} (69.66,27.91) (69.66,27.91)[(0,0)\[cc\][**[1]{} [0]{}**]{}]{} (5.00,41.00) (13.67,41.00) (9.33,42.17)[(3.33,2.33)\[t\]]{} (16.33,41.00)(0.11,0.16)[16]{}[(0,1)[0.16]{}]{} (18.16,43.55)(0.12,0.15)[14]{}[(0,1)[0.15]{}]{} (19.79,45.67)(0.11,0.13)[13]{}[(0,1)[0.13]{}]{} (21.24,47.38)(0.11,0.12)[11]{}[(0,1)[0.12]{}]{} (22.49,48.67)(0.13,0.11)[8]{}[(1,0)[0.13]{}]{} (23.56,49.54)(0.22,0.11)[4]{}[(1,0)[0.22]{}]{} (24.43,49.99)[(1,0)[0.68]{}]{} (25.11,50.02)(0.12,-0.10)[4]{}[(1,0)[0.12]{}]{} (25.60,49.64)(0.10,-0.27)[3]{}[(0,-1)[0.27]{}]{} (25.90,48.83)[(0,-1)[1.83]{}]{} (2.33,41.00)(0.11,0.16)[16]{}[(0,1)[0.16]{}]{} (4.16,43.55)(0.12,0.15)[14]{}[(0,1)[0.15]{}]{} (5.79,45.67)(0.11,0.13)[13]{}[(0,1)[0.13]{}]{} (7.24,47.38)(0.11,0.12)[11]{}[(0,1)[0.12]{}]{} (8.49,48.67)(0.13,0.11)[8]{}[(1,0)[0.13]{}]{} (9.56,49.54)(0.22,0.11)[4]{}[(1,0)[0.22]{}]{} (10.43,49.99)[(1,0)[0.68]{}]{} (11.11,50.02)(0.12,-0.10)[4]{}[(1,0)[0.12]{}]{} (11.60,49.64)(0.10,-0.27)[3]{}[(0,-1)[0.27]{}]{} (11.90,48.83)[(0,-1)[1.83]{}]{} (97.00,41.33) (105.67,41.33) (101.33,42.50)[(3.33,2.33)\[t\]]{} (108.33,41.33)(0.11,0.16)[16]{}[(0,1)[0.16]{}]{} (110.16,43.88)(0.12,0.15)[14]{}[(0,1)[0.15]{}]{} (111.79,46.01)(0.11,0.13)[13]{}[(0,1)[0.13]{}]{} (113.24,47.72)(0.11,0.12)[11]{}[(0,1)[0.12]{}]{} (114.49,49.01)(0.13,0.11)[8]{}[(1,0)[0.13]{}]{} (115.56,49.88)(0.22,0.11)[4]{}[(1,0)[0.22]{}]{} (116.43,50.33)[(1,0)[0.68]{}]{} (117.11,50.36)(0.12,-0.10)[4]{}[(1,0)[0.12]{}]{} (117.60,49.97)(0.10,-0.27)[3]{}[(0,-1)[0.27]{}]{} (117.90,49.16)[(0,-1)[1.83]{}]{} (94.33,41.33)(0.11,0.16)[16]{}[(0,1)[0.16]{}]{} (96.16,43.88)(0.12,0.15)[14]{}[(0,1)[0.15]{}]{} (97.79,46.01)(0.11,0.13)[13]{}[(0,1)[0.13]{}]{} (99.24,47.72)(0.11,0.12)[11]{}[(0,1)[0.12]{}]{} (100.49,49.01)(0.13,0.11)[8]{}[(1,0)[0.13]{}]{} (101.56,49.88)(0.22,0.11)[4]{}[(1,0)[0.22]{}]{} (102.43,50.33)[(1,0)[0.68]{}]{} (103.11,50.36)(0.12,-0.10)[4]{}[(1,0)[0.12]{}]{} (103.60,49.97)(0.10,-0.27)[3]{}[(0,-1)[0.27]{}]{} (103.90,49.16)[(0,-1)[1.83]{}]{} (7.67,23.67) (7.67,23.67)[(0,0)\[cc\][**?**]{}]{} (101.33,24.33) (101.33,24.33)[(0,0)\[cc\][**?**]{}]{} (101.33,0.00)[(0,0)\[cc\][green eyeglass]{}]{} (7.33,-0.33)[(0,0)\[cc\][red eyeclass]{}]{} (54.00,0.00)[(0,0)\[cc\][urn]{}]{} - After writing down the symbol, Alice hands the chocolate to the quantum, who in turn carries it to the recipient Bob. During this process, it could, however, get lost and for some reason never reach its destination (those with a sweet tooth might for example not be able to wait and eat their chocolate immediately). - Before Bob may take the chocolate and look at it, he, too, needs to flip a coin in order to choose one pair of glasses. Again, heads is for the green, and tails for the red ones. He puts them on and takes a look at the chocolate ball he has just received. He, too, will only be able to read one of the symbols, as the other one is imprinted in the complementary color and appears black to him. Then he makes a note of the symbol he has read, as well as of the color used. As before, should he attempt to take off his glasses or look at the symbols with the other pair, the quantum is required to eat the chocolate at once. After the legal transmission has taken place, the “quantum” may eat the chocolate ball just transfered from Alice to Bob, or give it away, anyway. - Now Bob uses one of the two flags (red or green) to tell Alice whether he has received anything at all, and what color his glasses are. He does not, however, communicate the symbol itself. At the same time, Alice uses one of her flags to inform Bob of the color of her glasses. Again, she does not tell Bob the symbol she identified. - Alice and Bob only keep the symbol if they both received the corresponding chocolate, and if the color of their glasses (i.e. their flags) matched. Otherwise, they dismiss the entry. The whole process (1-5) is then repeated several times. As a result, Alice and Bob obtain an identical random sequence of the symbols 0 and 1. They compare some of the symbols directly to make sure that there has been no attack by an eavesdropper. The random key can be used in many cryptographic applications, for instance as one-time pad (like TANs in online banking). A more amusing application is to let Alice communicate to Bob secretly whether (1) or not (0) she would consider giving him her mobile phone number. For this task merely a single bit of the sequence they have created is required. Alice forms the sum $i\oplus j= i+j \,{\rm mod}\, 2$ of her decision and the secret bit and cries it out loudly over to Bob. Bob can decode Alice’s message to plain text by simply forming the sum $s\oplus t$ of Alice’s encrypted message $s$ and the secret bit $t=j$ shared with Alice, for $j\oplus t=0$. Indeed, this seems to be a very romantic and easily communicable way of employing one-time pads generated by quantum cryptography. (And seems not too far away from the phantasies of its original inventors ;-) Alternative protocol versions ============================= There exist numerous possible variants of the dramatization of the BB84 protocol. A great simplification can be the total abandonment of the black background of the chocolate balls, as well as the colored eyeglasses. In this case, both Alice and Bob simply decide by themselves which color to take, and record the symbols in the color cosen. In the following, we will present yet another BB84-type protocol with the context translation principle [@svozil-2003-garda]. First of all, we define one of two possible contexts (either red or green). Then we randomly measure another context, which is independent of this choice. If the two contexts do not match (red-green or green-red), a context translation [@svozil-2003-garda] is carried out by flipping a coin. In this case, there is no correlation between the two symbols. If, however, the two contexts match (red-red or green-green), the results, i.e. the symbols, are identical. In this protocol, we use sets of two chocolate figures shaped like 0 and 1, and uniformly colored in red and green, as shown in Table \[2005-nl1-t1a\]. An equal amount of each type of figures is placed inside an urn. No colored glasses are necessary to carry out this protocol. Balltyp [red]{} [green]{} --------- --------- ----------- Typ 1 — [0]{} Typ 2 — [1]{} Typ 3 [0]{} — Typ 4 [1]{} — : Coloring and geometry of the four chocolate figures.\[2005-nl1-t1a\] The protocol is to be carried out as follows: - First of all, Alice randomly draws one figure from the urn and makes a note of its value (0 or 1) and of its color. Then she gives the figure to one of the quanta. - The quantum carries the figure to Bob. - Bob flips a coin and thus chooses one of two colors. Heads is for green, tails for red. If the color corresponds to that of the figure drawn by Alice and presented by the quantum (red-red or green-green), the symbol of the figure counts. If it does not correspond (red-green or green-red), Bob takes the result of the coin he has just flipped and assigns heads to 0 and tails to 1. If he wants to, he may flip it again and use the new result instead. In any case Bob writes down the resulting symbol. - The rest corresponds to the protocol presented previously. Further dramaturgical aspects, attacks and realization ====================================================== It is possible to scramble the protocol in its simplest form and thus the encryption by drawing two or more chocolate balls, with or without identical symbols on them, from the urn at once; or by breaking the time order of events. It is allowed to carry out peaceful attacks in order to to eavesdrop on the encrypted messages. In the case of the first protocol, every potential attacker needs to wear colored glasses herself. Note that no one (not even the quanta) may take additional chocolates or chocolate figures from the urn, which are identical to the one originally drawn by Alice. In a sense, this rule implements the no-cloning theorem stating that it is not possible to copy an arbitrary quantum if it is in a coherent superposition of the two classical states. The most promising eavesdropping strategy is the so-called man-in-the-middle attack, which is often used in GSM networks. The attacker manages to impersonate Bob when communicating with Alice and vice versa. What basically happens is that two different quantum cryptographic protocols are connected in series, or carried out independently from each other. Quantum cryptography is not immune to this kind of attack. The first performance of the quantum drama sketched above took place in Vienna at the University of Technology as a parallel part of an event called “Lange Nacht der Forschung” (“long night of science”). Figure \[2005-ln1e-pics\]a) depicts the “quantum channel,” a “catwalk” constructed from yellow painted form liners lifted on the sides with planed wood planks, through which the individual “quanta” had to pass from Alice to Bob. In the middle of the catwalk, the path forked into two passes, which joined again – some allusion to quantum interference. The photographs in Figure \[2005-ln1e-pics\]b)-f) depict some stages of the performance. Experience showed that a considerable fraction of the audience obtained some understanding of the protocol; in particular the players acting as Alice and Bob. Most people from the audience got the feeling that quantum cryptography is not so cryptic after all, if they are capable of performing the protocol and even have fun experiencing it. For the student of physics probably the most important questions are those related to the differences and similarities between chocolate balls and quanta. This quasi-classical analogy may serve as a good motivation and starting point to consider the type of complementarity encountered in quantum physics, and the type of experience presented by single-quantum experiments. Acknowledments {#acknowledments .unnumbered} ============== The idea was born over a coffee conversation with Günther Krenn. The first public performance was sponsored by “Lange Nacht der Forschung” [http://www.langenachtderforschung.at]{}. The chocoate balls “Mozartkugeln” were donated by [*Manner*]{} [http://www.manner.com]{}. The black foils covering the balls were donated by [*Constantia Packaging*]{} [http://www.constantia-packaging.com]{}. Thanks go to Karin Peter and the public relation office of the TU Vienna for providing the infrastructure, to the [*Impro Theater*]{} for the stage performance, and to Martin Puntigam moderating part of the performances. [10]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{} A. Artaud, *Le th[é]{}[â]{}tre et son double* (Gallimard, Paris, 1938). S. Wiesner, “Conjugate coding,” Sigact News **15**, 78–88 (1983). Manuscript written [*circa*]{} 1970 [@benn-92 Ref. 27]. C. H. Bennett, G. Brassard, S. Breidbart, and S. Wiesner, “Quantum cryptography, or unforgable subway tokens,” in *Advances in Cryptography: Proceedings of Crypto ’82*, pp. 78–82 (Plenum Press, New York, 1982). C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” in *Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India*, pp. 175–179 (IEEE Computer Society Press, 1984). A. Ekert, “Quantum cryptography based on [B]{}ell’s theorem,” Physical Review Letters **67**, 661–663 (1991). C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental Quantum Cryptography,” Journal of Cryptology **5**, 3–28 (1992). N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Review of Modern Physics **74**, 145–195 (2002). <http://dx.doi.org/10.1103/RevModPhys.74.145>. C. Elliott, A. Colvin, D. Pearson, O. Pikalo, J. Schlafer, and H. Yeh, “Current status of the [DARPA]{} Quantum Network,” (2005). . A. Poppe, A. Fedrizzi, T. Loruenser, O. Maurhardt, R. Ursin, H. R. Boehm, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical Quantum Key Distribution with Polarization-Entangled Photons,” Optics Express **12**, 3865–3871 (2004). , <http://www.opticsinfobase.org/ViewMedia.cfm?id=80796&seq=0>. R. Wright, “Generalized urn models,” Foundations of Physics **20**, 881–903 (1990). R. Wright, “The state of the pentagon. [A]{} nonclassical example,” in *Mathematical Foundations of Quantum Theory*, A. R. Marlow, ed., pp. 255–274 (Academic Press, New York, 1978). K. Svozil, “Logical equivalence between generalized urn models and finite automata,” International Journal of Theoretical Physics p. in print (2005). . A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Physical Review **47**, 777–780 (1935). <http://dx.doi.org/10.1103/PhysRev.47.777>. S. Kochen and E. P. Specker, “The Problem of Hidden Variables in Quantum Mechanics,” Journal of Mathematics and Mechanics **17**(1), 59–87 (1967). Reprinted in [@specker-ges pp. 235–263]. E. Schr[ö]{}dinger, “Die gegenw[ä]{}rtige [S]{}ituation in der [Q]{}uantenmechanik,” Naturwissenschaften **23**, 807–812, 823–828, 844–849 (1935). [E]{}nglish translation in [@trimmer] and [@wheeler-Zurek:83 pp. 152-167]; http://www.emr.hibu.no/lars/eng/cat/, <http://www.emr.hibu.no/lars/eng/cat/>. C. A. Fuchs and A. Peres, “Quantum theory needs no ‘Interpretation´,” Physics Today **53**(4), 70–71 (2000). Further discussions of and reactions to the article can be found in the September issue of Physics Today, [*53*]{}, 11-14 (2000), [http://www.aip.org/web2/aiphome/pt/vol-53/iss-9/p11.html and http://www.aip.org/web2/aiphome/pt/vol-53/iss-9/p14.html](http://www.aip.org/web2/aiphome/pt/vol-53/iss-9/p11.html and http://www.aip.org/web2/aiphome/pt/vol-53/iss-9/p14.html). K. Svozil, “Quantum information via state partitions and the context translation principle,” Journal of Modern Optics **51**, 811–819 (2004). . E. Specker, *Selecta* (Birkh[ä]{}user Verlag, Basel, 1990). J. D. Trimmer, “The present situation in quantum mechanics: a translation of [S]{}chr[ö]{}dinger’s “cat paradox”,” Proc. Am. Phil. Soc. **124**, 323–338 (1980). Reprinted in [@wheeler-Zurek:83 pp. 152-167]. J. A. Wheeler and W. H. Zurek, *Quantum Theory and Measurement* (Princeton University Press, Princeton, 1983). -------------------------------------------- ------------------------------------------- {width="7.2cm"} {width="7.2cm"} $\;$ -------------------------------------------- ------------------------------------------- [lr]{}\ \ \ \ [lr]{} $\;$\ \ \ \ \ [^1]: The author reserves the copyright for all public performances. Performance lincenses are granted for educational institutions and other not-for-profit performances for free; these institutions are kindly asked to send a small note about the performance to the author.

ArXiv

--- abstract: Correct quantization of free electromagnetic field is proposed author: - 'D.Yearchuck' - 'Y.Yerchak' - 'A.Alexandrov' title: To Quantization of Free Electromagnetic Field --- In 1873 “A Treatise on Electricity and Magnetism” by Maxwell [@Maxwell] was published, in which the discovery of the system of electrodynamics equations was reported. The equations are in fact the symmetry expressions for experimental laws, established by Faraday, and, consequently, they are mathematical mapping of experimentally founded symmetry of EM-field. It means in its turn that if some new experimental data will indicate, that symmetry of EM-field is higher, then Maxwell equations have to be generalized. That is the reason why the symmetry study of Maxwell equations is the subject of many research in field theory up to now. Heaviside [@Heaviside] in twenty years after Maxwell discovery was the first, who payed attention to the symmetry between electrical and magnnetic quantities in Maxwell equations. Mathematical formulation of given symmetry, consisting in invariance of Maxwell equations for free EM-field under the duality transformations $$\label{eq1d} \vec {E} \rightarrow \pm\vec {H}, \vec {H} \rightarrow \mp\vec {E},$$ gave Larmor [@Larmor]. Duality transformations (\[eq1d\]) are private case of the more general dual transformations, established by Rainich [@Rainich]. Dual transformations produce oneparametric abelian group $U_1$ of chiral transformations and they are $$\label{eq2d} \begin{split} \raisetag{40pt} \vec {E} \rightarrow \vec {E} cos\theta + \vec {H} sin\theta\\ \vec {H} \rightarrow \vec {H} cos\theta - \vec {E} sin\theta. \end{split}$$ Given symmetry indicates, that both constituents $\vec {E}$ and $\vec {H}$ of EM-field are possessing equal rights, in particular they both have to consist of component with different parity. Subsequent extension of dual symmetry for the EM-field with sources leads to requirement of two type of charges. Examples of the dual symmetry display are for instance the equality of magnetic and electric energy values in LC-tank or in free electromagnetic wave. Recently concrete experimental results have been obtained concerning dual symmetry of EM-field in the matter. Two new physical phenomena - ferroelectric [@Yearchuck_Yerchak] and antiferroelectric [@Yearchuck_PL] spin wave resonances have been observed. They were predicted on the base of the model [@Yearchuck_Doklady] for the chain of electrical “spin” moments, that is intrinsic electrical moments of (quasi)particles. Especially interesting, that in [@Yearchuck_PL] was experimentally proved, that really purely imaginary electrical “spin” moment, in full correspondence with Dirac prediction [@Dirac], is responsible for the phenomenon observed. Earlier on the same samples has been registered ferromagnetic spin wave resonance, [@Ertchak_J_Physics_Condensed_Matter]. The values of splitting parameters $\mathfrak{A}^E$ and $\mathfrak{A}^H$ in ferroelectric and ferromagnetic spin wave resonance spectra allowed to find the ratio $J_{E }/J_{H}$ of exchange constants in the range of $(1.2 - 1.6)10^{4}$. Given result seems to be direct proof, that the charge, that is function, which is invariant under gauge transformations is two component function. The ratio of imagine $e_{H} \equiv g$ to real $e_{E}\equiv e $ components of complex charge is $\frac{g}{e} \sim \sqrt{J_{E }/J_{H}} \approx (1.1 - 1.3)10^{2}$. At the same time in classical and in quantum theory dual symmetry of Maxwell eqations does not take into consideration. Moreover the known solutions of Maxwell eqations do not reveal given symmetry even for free EM-field, see for instance [@Scully], although it is understandable, that the general solutions have to posseess by the same symmetry. The aim of given work is to find the cause of symmetry difference of Maxwell eqations and their solutions and to propose correct field functions for classical and quantized EM-field. Suppose EM-field in volume rectangular cavity. Suppose also, that the field polarization is linear in z-direction. Then the vector of electrical component can be represented in the form $$E_x(z,t) = \sum_{\alpha=1}^{\infty}A_{\alpha}q_{\alpha}(t)\sin(k_{\alpha}z),$$ where $q_{\alpha}(t)$ is amplitude of $\alpha$-th normal mode of the cavity, $\alpha \in N$, $k_{\alpha} = \alpha\pi/L$, $A_{\alpha}=\sqrt{2 \nu_{\alpha}^2m_{\alpha}/(V\epsilon_0)}$, $\nu_{\alpha} = \alpha\pi c/L$, $L$ is cavity length along z-axis, $V$ is cavity volume, $m_{\alpha}$ is parameter, which is introduced to obtain the analogy with mechanical harmonic oscillator. Using the equation $$\epsilon_0\partial_t \vec{E}(z,t) = \left[ \nabla\times\vec{H}(z,t)\right]$$ we obtain in suggestion of transversal EM-field the expression for magnetic field $${H}_y(z,t) = \sum_{\alpha=1}^{\infty}\epsilon_0\frac{A_{\alpha}}{k_{\alpha}}\frac{dq_{\alpha}}{dt}\cos(k_{\alpha}z) + H_{y0}(t),$$ where $H_{y0} = \sum_{\alpha=1}^{\infty} f_{\alpha}(t)$, $\{f_{\alpha}(t)\}$, $\alpha \in N$, is the set of arbitrary functions of the time. The partial solution is usually used, in which the function $H_{y0}(t)$ is identically zero. The field Hamiltonian $\mathcal{H}^{[1]}(t)$, corresponding given partial solution, is $$\begin{split} &\mathcal{H}^{[1]}(t) = \frac{1}{2}\iiint\limits_{(V)}\left[\epsilon_0E_x^2(z,t)+\mu_0H_y^2(z,t)\right]dxdydz\\ &= \frac{1}{2}\sum_{\alpha=1}^{\infty}\left[m_{\alpha}\nu_{\alpha}^2q_{\alpha}^2(t) + \frac{p_{\alpha}^2(t)}{m_{\alpha}} \right], \end{split}$$ where $$p_{\alpha} = m_{\alpha} \frac{dq_{\alpha}(t)}{dt}.$$ Then, using the equation $$\left[ \nabla\times\vec{E}\right] = -\frac{\partial \vec{B}}{\partial t} = -\mu_0 \frac{\partial \vec{H}}{\partial t}$$ it is easily to find the field functions $\{q_{\alpha}(t)\}$. They will satisfy to differential equation $$\frac{d^2q_{\alpha}(t)}{dt^2}+\frac{k_{\alpha}^2}{\mu_0\epsilon_0}q_{\alpha}(t)=0.$$ Consequently, taking into account $\mu_0\epsilon_0 = 1/c^2$, we have $$q_{\alpha}(t) = C_1e^{i\nu_{\alpha}t}+C_2e^{-i\nu_{\alpha}t}$$ Thus, real free Maxwell field equations result in well known in the theory of differential equations situation - the solutions are complex-valued functions. It means, that generally the field function for free Maxwell field produce complex space. From general expression for the field $\vec{H}(\vec{r},t)$ $$\vec{H}(\vec{r},t) = \left[\sum_{\alpha=1}^{\infty}A_{\alpha}\frac{\epsilon_0}{k_{\alpha}}\frac{dq_{\alpha}(t)}{dt}\cos(k_{\alpha}z) + f_{\alpha}(t)\right]\vec{e}_y$$ it is easily to obtain differential equation for $f_{\alpha}(t)$ $$\begin{split} &\frac{d f_{\alpha}(t)}{dt} + A_{\alpha}\frac{\epsilon_0}{k_{\alpha}}\frac{\partial^2q_{\alpha}(t)}{\partial t^2}\cos(k_{\alpha}z) \\ &- \frac {1}{\mu_0} A_{\alpha}k_{\alpha}q_{\alpha}(t)cos(k_{\alpha}z) = 0. \end{split}$$ Its solution in general case is $$f_{\alpha}(t) = \int A_{\alpha} \cos(k_{\alpha}z)\left[q_{\alpha}(t)\frac{k_{\alpha}}{\mu_0}-\frac{d^2q_{\alpha}(t)}{dt^2}\frac{\epsilon_0}{k_{\alpha}}\right] dt +C_{\alpha}$$ Then we have another solution of Maxwell equations $$\vec{H}(\vec{r},t) = \frac{1}{\mu_0}\left\{\sum_{\alpha=1}^{\infty}k_{\alpha}A_{\alpha} \cos(k_{\alpha}z) q_{\alpha}'(t)\right\}\vec{e}_y,$$ $$\vec{E}(\vec{r},t) = \left\{\sum_{\alpha=1}^{\infty}A_{\alpha}\frac{dq_{\alpha}'(t)}{dt}\sin(k_{\alpha}z)\right\}\vec{e}_x,$$ where $q_{\alpha}'(t) = \int q_{\alpha}(t) dt + C_{\alpha}'$ Then Hamiltonian $\mathcal{H}^{[2]}(t)$ is $$\mathcal{H}^{[2]}(t) = \frac{1}{2}\sum_{\alpha=1}^{\infty}\left[m_{\alpha} \nu_{\alpha}^4 q{'}_{\alpha}^2(t) + {m_{\alpha}\nu_{\alpha}^2 (\frac{dq_{\alpha}'(t)}{dt})^2} \right].$$ Let us introduce new variables $$\begin{split} &q{''}_{\alpha}(t) = \nu_{\alpha}q{'}_{\alpha}(t) \\ &p{''}_{\alpha}(t) = m_{\alpha}\nu_{\alpha}\frac{dq_{\alpha}'(t)}{dt} \end{split}$$ Then $$\mathcal{H}^{[2]}(t) = \frac{1}{2}\sum_{\alpha=1}^{\infty}\left[m_{\alpha}\nu_{\alpha}^2q{''}_{\alpha}^2(t) + \frac{p{''}_{\alpha}^2(t)}{m_{\alpha}} \right].$$ We use further the standard procedure of field quantization. So for the first partial solution we have $$\begin{split} &\left[\hat {p}_{\alpha}(t) , \hat {q}_{\beta}(t)\right] = i\hbar\delta_{{\alpha}\beta}\\ &\left[\hat {q}_{\alpha}(t) , \hat {q}_{\beta}(t)\right] = \left[\hat {p}_{\alpha}(t) , \hat {p}_{\beta}(t)\right] = 0, \end{split}$$ where $\alpha, \beta \in N$. Introducing the operators $\hat{a}_{\alpha}(t)$ and $ \hat{a}^{+}_{\alpha}(t)$ $$\begin{split} &\hat{a}_{\alpha}(t) = \frac{1}{ \sqrt{ \frac{1}{2} \hbar m_{\alpha} \nu_{\alpha}}} \left[ m_{\alpha} \nu_{\alpha}\hat {q}_{\alpha}(t) + i \hat {p}_{\alpha}(t)\right]\\ &\hat{a}^{+}_{\alpha}(t) = \frac{1}{ \sqrt{ \frac{1}{2} \hbar m_{\alpha} \nu_{\alpha}}} \left[ m_{\alpha} \nu_{\alpha}\hat {q}_{\alpha}(t) - i \hat {p}_{\alpha}(t)\right], \end{split}$$ we have for the operators of canonical variables $$\begin{split} &\hat {q}_{\alpha}(t) = \sqrt{\frac{\hbar}{2 m_{\alpha} \nu_{\alpha}}} \left[\hat{a}^{+}_{\alpha}(t) + \hat{a}_{\alpha}(t)\right]\\ &\hat {p}_{\alpha}(t) = i \sqrt{\frac{\hbar m_{\alpha} \nu_{\alpha}}{2}} \left[\hat{a}^{+}_{\alpha}(t) - \hat{a}_{\alpha}(t)\right]. \end{split}$$ Then field function operators are $$\hat{\vec{E}}(\vec{r},t) = \{\sum_{\alpha=1}^{\infty} \sqrt{\frac{\hbar \nu_{\alpha}}{V\epsilon_0}} \left[\hat{a}^{+}_{\alpha}(t) + \hat{a}_{\alpha}(t)\right] sin(k_{\alpha} z)\} \vec{e}_x,$$ $$\hat{\vec{H}}(\vec{r},t) = ic\epsilon_0 \{\sum_{\alpha=1}^{\infty} \sqrt{\frac{\hbar \nu_{\alpha}}{V\epsilon_0}} \left[\hat{a}^{+}_{\alpha}(t) - \hat{a}_{\alpha}(t)\right] cos(k_{\alpha} z)\} \vec{e}_y,$$ For the second partial solution, corresponding to Hamiltonian $\mathcal{H}^{[2]}(t)$ we have $$\begin{split} &\left[\hat{p}{''}_{\alpha}(t) , \hat {q}{''}_{\beta}(t)\right] = i\hbar\delta_{{\alpha}\beta}\\ &\left[\hat {q}{''}_{\alpha}(t) , \hat {q}{''}_{\beta}(t)\right] = \left[\hat {p}{''}_{\alpha}(t) , \hat {p}{''}_{\beta}(t)\right] = 0, \end{split}$$ $\alpha, \beta \in N$. The operators $\hat{a}{''}_{\alpha}(t)$, $\hat{a}{''}^{+}_{\alpha}(t)$ are introduced analogously $$\begin{split} &\hat{a}{''}_{\alpha}(t) = \frac{1}{ \sqrt{ \frac{1}{2} \hbar m_{\alpha} \nu_{\alpha}}} \left[ m_{\alpha} \nu_{\alpha}\hat {q}{''}_{\alpha}(t) + i \hat {p}{''}_{\alpha}(t)\right]\\ &\hat{a}{''}^{+}_{\alpha}(t) = \frac{1}{ \sqrt{ \frac{1}{2} \hbar m_{\alpha} \nu_{\alpha}}} \left[ m_{\alpha} \nu_{\alpha}\hat {q}{''}_{\alpha}(t) - i \hat {p}{''}_{\alpha}(t)\right] \end{split}$$ Relationships for canonical variables are $$\begin{split} &\hat {q}{''}_{\alpha}(t) = \sqrt{\frac{\hbar}{2 m_{\alpha} \nu_{\alpha}}} \left[\hat{a}{''}^{+}_{\alpha}(t) + \hat{a}{''}_{\alpha}(t)\right]\\ &\hat {p}{''}_{\alpha}(t) = i \sqrt{\frac{\hbar m_{\alpha} \nu_{\alpha}}{2}} \left[\hat{a}{''}^{+}_{\alpha}(t) - \hat{a}{''}_{\alpha}(t)\right] \end{split}$$ For the field function operators we obtain $$\hat{\vec{E}}^{[2]}(\vec{r},t) = i \{\sum_{\alpha=1}^{\infty} \sqrt{\frac{\hbar \nu_{\alpha}}{V\epsilon_0}} \left[\hat{a}{''}^{+}_{\alpha}(t) - \hat{a}{''}_{\alpha}(t)\right] sin(k_{\alpha} z)\} \vec{e}_x,$$ $$\begin{split} \hat{\vec{H}}^{[2]}(\vec{r},t) = \frac{1}{\mu_0 c} \{\sum_{\alpha=1}^{\infty} \sqrt{\frac{\hbar \nu_{\alpha}}{V\epsilon_0}} \left[\hat{a}{''}^{+}_{\alpha}(t) + \hat{a}{''}_{\alpha}(t)\right] cos(k_{\alpha} z)\} \vec{e}_y, \end{split}$$ Let us designate $\sqrt{\frac{\hbar \nu_{\alpha}}{V\epsilon_0}} = E_0$. In accordance with definition of complex quantities we have $$(\vec{E}(\vec{r},t), \vec{E}^{[2]}(\vec{r},t)) \rightarrow \vec{E}(\vec{r},t) + i \vec{E}^{[2]}(\vec{r},t) = \vec{E}(\vec{r},t).$$ Consequently, correct field operators for quantized EM-field are $$\begin{split} \hat{\vec{E}}(\vec{r},t) = \{\sum_{\alpha=1}^{\infty} E_0 \{\left[\hat{a}^{+}_{\alpha}(t) + \hat{a}_{\alpha}(t)\right]\\ + \left[\hat{a}{''}_{\alpha}(t) - \hat{a}{''}^{+}_{\alpha}(t)\right]\} sin(k_{\alpha} z)\} \vec{e}_x, \end{split}$$ and $$(\vec{H}^{[2]}(\vec{r},t), \vec{H}(\vec{r},t)) \rightarrow \vec{H}^{[2]}(\vec{r},t) + i \vec(\vec{r},t) = \vec{H}(\vec{r},t)$$ $$\begin{split} \hat{\vec{H}}(\vec{r},t) = \{\sum_{\alpha=1}^{\infty} E_0 \{\frac{1}{\mu_0 c} \left[\hat{a}{''}_{\alpha}(t) + \hat{a}{''}^{+}_{\alpha}(t)\right]\\ + c \epsilon_0 \left[\hat{a}_{\alpha}(t) - \hat{a}^{+}_{\alpha}(t)\right]\} cos(k_{\alpha} z) + C_{\alpha}\hat{e}\} \vec{e}_y, \end{split}$$ Maxwell J C, A Treatise on Electricity and Magnetism, Oxford, Clarendon Press, V.1, 1873, 438, V.2 1873, 464 Heaviside O, Phil.Trans.Roy.Soc.A,**183** (1893) 423-430 Larmor J, Collected papers , London, 1928 Rainich G Y, Trans.Am.Math.Soc.,**27** (1925) 106 Dirac P.A M, Proceedigs of the Royal Society **117A** (1928) 610 - 624 Yearchuck D, Yerchak Y, Red’kov V, Doklady NANB **51**, N 5 (2007) 57 - 64 Yearchuck D, Yerchak Y, Alexandrov A, Phys.Lett.A, **373**, N 4 (2009) 489 - 495 Yearchuck D, Yerchak Y, Kirilenko A, Popechits V, Doklady NANB **52**, N 1 (2008) 48 - 53 Ertchak D P, Kudryavtsev Yu P, Guseva M B, Alexandrov A F et al, J.Physics: Condensed Matter, **11**, N3 (1999) 855 -870 Scully M O, Zubairy M S, Quantum Optics, Cambridge University Press, 1997, 650

ArXiv

--- abstract: 'The union of an ascending chain of prime ideals is not always prime. The union of an ascending chain of semi-prime ideals is not always semi-prime. We show that these two properties are independent. We also show that the number of non-prime unions of subchains in a chain of primes in a PI-algebra does not exceed the PI-class minus one, and this bound is tight.' address: 'Department of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel' author: - 'Be’eri Greenfeld' - 'Louis H. Rowen' - Uzi Vishne title: Unions of chains of primes --- Introduction {#sec:intro} ============ In a commutative ring, the union of a chain of prime ideals is prime, and the union of a chain of semiprime ideals is semiprime. This paper demonstrates and measures the failure of these chain conditions in general. A ring has the [**[(semi)prime chain property]{}**]{} (denoted  and , respectively) if the union of any countable chain of (semi)prime ideals is always (semi)prime.[^1] The property  was recognized by Fisher and Snider [@FS] as the missing hypothesis for Kaplansky’s conjecture on regular rings, and they gave an example of a ring without . Our focus is on . The class of rings satisfying  is quite large. An easy exercise shows that every commutative ring satisfies , and the same argument yields that the union of strongly prime ideals is strongly prime ($P \normali R$ is strongly prime if $R/P$ is a domain). In fact, we have the following result: \[first\] Every ring $R$ which is a finite module over a central subring, satisfies . Write $R = \sum_{i=1}^t Cr_i$ where $C \sub \operatorname{Cent}(R)$. Suppose $P_1 \subset P_2 \subset \cdots$ is a chain of prime ideals, with $P = \cup P_i$. If $a,b \in R$ with $$\sum C ar_i b = \sum aCr_i b = aRb \subseteq P,$$ then there is $n$ such that $ar_i b \in P_n$ for $1 \le i \le t,$ implying $aRb = \sum Car_i b \subseteq P_n$, and thus $a \in P_n$ or $b \in P_n$. (For a recent treatment of the correspondence of infinite chains of primes between a ring $R$ and a central subring, see [@Shai]). The class of rings satisfying  also contains every ring that satisfies ACC (ascending chain condition) on primes, and is closed under homomorphic images and central localizations. This led some mathematicians to believe that it holds in general. On the other hand, Bergman produced an example lacking  (see [[Example \[Ex1\]]{}]{} below), implying that the free algebra on two generators does not have . Obviously, the property  follows from the maximum property on families of primes. On the other hand,  implies (by Zorn’s lemma) the following maximum property: for every prime $Q$ contained in any ideal $I$, there is a prime $P$ maximal with respect to $Q \sub P \sub I$. In [[Section \[sec:mat\]]{}]{} we show that  and  are independent, by presenting an example (due to Kaplansky and Lanski) of a ring satisfying  and not , and an example of a ring satisfying  but not . We say that an ideal is [****]{} if it is a union of a chain of primes, but is not itself prime. (If ${{\{P_\lam\}}}$ is an ascending chain of primes, then $R/\bigcup P_{\lambda} = \lim_{\rightarrow} R/P_{\lam}$ is a direct limit of prime rings). The [**[$\PP$-index]{}**]{} of the ring $R$ is the maximal number of non-prime unions of subchains of a chain of prime ideals in $R$ (or infinity if the number is unbounded, see [[Proposition \[PPindex\]]{}]{}). [[Section \[sec:mon\]]{}]{} extends Bergman’s example by showing that the -index of the free (countable) algebra is infinity. A variation of this construction, based on free products, is presented in [[Section \[sec:example2.2\]]{}]{}. After defining the -index in [[Section \[sec:PP\]]{}]{}, in [[Section \[sec:PI\]]{}]{} we discuss PI-rings, showing that the -index does not exceed the PI-class minus one, and this bound is tight. We thank the anonymous referee for careful comments on a previous version of this paper. Monomial algebras {#sec:mon} ================= Fix a field $F$. We show that  and  fail in the free algebra (over $F$) by constructing an (ascending) chain of primitive ideals whose union is not semiprime. Let us start with a simpler theme, whose variations have extra properties. \[Ex1\] Let $R$ be the free algebra in the (noncommuting) variables $x,y$. For each $n$, let $$P_n = {{\left<xx,xyx, xy^2x, \dots, xy^{n-1}x\right>}}.$$ As a monomial ideal, it is enough to check primality on monomials. If $uRu' \sub P_n$ for some words $u,u'$, then in particular $uy^{n}u' \in P_n$, which forces a subword of the form $xy^ix$ (with $i<n$) in $u$ or in $u'$; hence either $u\in P_n$ or $u'\in P_n$. On the other hand $\bigcup P_n = (RxR)^2$ which is not semiprime. This example, due to G. Bergman, appears in [@P Exmpl. 4.2]. Interestingly, primeness is always maintained in the following sense ([@P Lem. 4.1], also due to Bergman): for every countable chain of primes $P_1 \subset P_2 \subset \cdots$ in a ring $R$, the union $\bigcup (P_n[[\zeta]])$ is a prime ideal of the power series ring $R[[\zeta]]$. Since in [[Example \[Ex1\]]{}]{} $\bigcup P_n = (RxR)^2$, if $Q \normali R$ is a prime containing the union then $x \in Q$ so $R/Q$ is commutative. In particular, a chain of prime ideals starting from the chain $P_1 \subset P_2 \subset \cdots$ has only one . Let us exhibit a (countable) chain providing infinitely many s. Let $R$ be the free algebra generated by $x,y,z$. For a monomial $w$ we denote by $\deg_yw$ the degree of $w$ with respect to $y$. For $i,n \geq 1$, consider the monomial ideals $$I_{i,n} = RxxR+RxzxR+\cdots+Rxz^{i-1}xR+\sum_{\deg_y w < n} R xz^ixwxz^ix R,$$ which form an ascending chain with respect to the lexicographic order on the indices $(i,n)$, since $xz^ix \in I_{i',n}$ for every $i'>i$. To show that $I_{i,n}$ are prime, suppose that $u,u'$ are monomials such that $u,u' \not \in I_{i,n}$ but $u R u' \sub I_{i,n}$. Then $u z^i y^n z^i u' \in I_{i,n}$. Since none of the monomials $xz^{i'}x$ ($i'<i$) is a subword of $u$ or $u'$, they are not subwords of $u z^i y^n z^i u'$, forcing $u z^i y^n z^i u'$ to have a subword of the form $xz^ixwxz^ix$ where $\deg_yw < n$. It follows that $z^iy^nz^i$ is a subword of $z^ixwxz^i$, contrary to the degree assumption. Now, for every $i$, $$\bigcup_{n} I_{i,n} = RxxR+RxzxR+\cdots+Rxz^{i-1}xR+(Rxz^ixR)^2,$$ which contains $(Rxz^ixR)^2$ but not $Rxz^ixR$, so it is not semiprime. In particular the -index of $R$ (see [[Proposition \[PPindex\]]{}]{}) is infinity. In Section \[sec:PI\] we show that this phenomenon is impossible in PI algebras: there, the number of s in a prime chain is bounded by the PI-class. The ideals $P_n$ in [[Example \[Ex1\]]{}]{} are in fact primitive. Indeed, Bell and Colak [@Bell] proved that any finitely presented prime monomial algebra is either primitive or PI (also see [@Ok]), and $R/P_n$ contains a free subalgebra, e.g. $k{{\left<xy^n,xy^{n+1},\dots\right>}}$. The same effect can be achieved by using idempotents. Let $R$ be a binomial algebra, namely a quotient of a free algebra with respect to relations of the form $w_1 = w_2$ (or $w_1 = 0$) where $w_1,w_2$ are monomials. We say that an ideal $I$ in $R$ is monomially prime if $uRu' \sub I$ forces $u \in I$ or $u' \in I$ for every two monomials $u,u'$. Let $R$ be the free algebra generated by $e,y$ subject to the relation $e^2 = e$. Then every monomial ideal which is monomially prime is prime. Let $I$ be a monomial ideal which is monomially prime. Let $f,g \in R$ be any two elements such that $fRg \sub I$. \[Ex3\] Let $R$ be the free algebra in the variables $e,y$, modulo the relation $e^2 = e$. Every monomial has a unique shortest presentation as a word (replacing $e^2$ by $e$ throughout). Ordering monomials first by length and then lexicographically, every element $f$ has an upper monomial $\bar{f}$. Notice that $\overline{f y^n g} = \bar{f}y^n \bar{g}$. For each $n$, let $$P_n = {{\left<eye, ey^2e, \dots, ey^{n-1}e\right>}}.$$ To show that $P_n$ is a prime ideal, assume that $f y^n g \in P_n$. Then $\bar{f}y^n \bar{g} = \overline{f y^n g} \in P_n$, forcing $\bar{f} \in P_n$ or $\bar{g} \in P_n$ as in [[Example \[Ex1\]]{}]{}. The claim follows by induction on the number of monomials. To show that the ideal $P_n$ is primitive, it is enough by [@LRS] to prove that $e(R/P_n)e$ is a primitive ring. We construct an isomorphism between $e(R/P_n)e$ and the countably generated free algebra $F{{\left<z_0,z_1,\dots\right>}}$ by sending $ey^me$ for $m \geq n$ (which clearly generate a free algebra) to $z_{m-n}$. But the free algebra is primitive (see [@Lam Prop. 11.23]). On the other hand $\bigcup P_n = ReyReR = ReRyeR$, which contains $(ReyR)^2$ but not $ey$, so is not semiprime. We say that a ring is [**[uniquely-]{}**]{} if it has a unique minimal prime over every chain of prime ideals. Since the intersection of a descending chain of primes is prime, Zorn’s lemma shows that there are minimal primes over every ideal, in particular over every . In the topology of the spectrum, a net ${{\{P_{\lambda}\}}}_{\lambda \in \Lambda}$ of primes converges to a prime $Q$ if and only if $\bigcap_{\lambda \in \Lambda}\bigcup_{\lambda' \geq \lambda} P_{\lambda'} \subseteq Q$; in particular when ${{\{P_{\lambda}\}}}$ is an ascending chain, $\lim P_\lambda = Q$ if and only if $\bigcup P_\lambda \sub Q$. Therefore, the spectrum can identify minimal primes over s. It seems that the spectrum cannot distinguish  from uniquely-. In the examples of this section, there is a unique minimal prime over every . In [[Example \[Ex4\]]{}]{} below the situation is different: the  ideal constructed there is the intersection of two primes containing it. Prime ideals in free products {#sec:example2.2} ============================= In [[Example \[Ex1\]]{}]{} there are infinitely many (incomparable) prime ideals lying over the chain. We modify this example, in order to obtain a chain over which there is unique prime. In [[Example \[Ex1\]]{}]{} we considered ideals of the free algebra, which can be written as a free product $F[x] *_F F[y]$. The quotient over the radical of the union over the chain is the “second” component $F[y]$, which we would like to replace by the field $F(y)$. The proof that the ideals are prime is somewhat delicate; we thank the referee for pointing this out. \[Ex1div\] Let $D$ be the quotient division ring of the free algebra $F{{\left<x,y\right>}}$. Let $R$ be the subalgebra generated by $x$ and the subfield $F(y)$. Extend $\deg_y \co F[y] \ra \N$ to $\deg_y \co {{{F(y)}^{\times}}} \ra \Z$ in the obvious manner. Similarly to the previous example, take the prime ideals $$P_n = {{\left<xax {{\,:\ }}a \in F(y), \deg_y(a) < n\right>}}.$$ Again $P = \bigcup P_n = (RxR)^2$, which is not semiprime. But now $R/\sqrt P = R/{{\left<x\right>}} \cong F(y)$. Let $F$ be a field and let $A,B$ be $F$-algebras, with given vector space decompositions $A = F \oplus A_0$ and $B = F \oplus B_0$. The free product $A *_F B$ can be viewed as the tensor algebra $T(A_0 \oplus B_0) = F \oplus \bigoplus_{n\geq 1} (A_0\oplus B_0)^{{{\otimes_{}}}n}$, modulo the relations $a {{\otimes_{}}}a' = aa'$ and $b {{\otimes_{}}}b' = bb'$ for every $a,a' \in A$ and $b,b' \in B$. We will omit the tensor symbol. Fixing the decomposition $F[x] = F \oplus xF[x]$ and an arbitrary decomposition $B = F \oplus B_0$, we consider ideals of the free product $R = F[x] *_F B$. The tensor algebra is graded by $x$, once we declare that $\deg(b) = 0$ for every $b \in B_0$, and this grading induces a grading on $R$. Let $W \sub B$ be a vector space containing $F$. We say that $W$ is [**[restricted]{}**]{} if for every finite dimensional subspace $V \sub B$ there is an element $b \in B$ such that $Vb \sub B_0$ and $Vb \not \sub W$; and an element $b' \in B$ such that $b'V \sub B_0$ and $b'V \not \sub W$. \[main\] The ideal $P = RxWxR$ of $R$ is prime whenever $W \sub B$ is a restricted subspace. Write $W = F \oplus W_0$ where $W_0 = W \cap B_0$. Let $L'$ be the ideal of $R$ generated by $x^2$. For $n \geq 0$ let us denote the vector spaces $$L_n = B x B_0 x B_0 \cdots x B_0 x B,$$ where the degree with respect to $x$ is $n$; so that $L_0 = B$ and $L_1 = B x B$. Setting $L = \sum_{n\geq 0} L_n$, we have that $R = L' \oplus L$. Let $P_n = L_n \cap P$; so $P_0 = P_1 = 0$, and for $n \geq 2$, $$P_n = \sum B x B_0 x \cdots x B_0 x W_0 x B_0 x \cdots x B_0 x B$$ where in each summand one of the intermediate entries is $W_0$ and all the others are equal to $B_0$. For example $P_2 = B x W_0 x B$ and $P_3 = B x W_0 x B_0 x B + B x B_0 x W_0 x B$. Now, since $F \sub W$, we have that $L' = RxxR = RxFxR \sub RxWxR = P$, and we can compute: $$\begin{aligned} P & = & L' + P \\ & = & L' + (L'+L)x W x(L'+L) \\ & = & L' + Lx W x L \\ & = & L' + Lx W_0 x L \\ & = & L ' + \sum_{d,d' \geq 0} L_d x W_0 x L_{d'} \\ & = & L ' + \sum_{n \geq 0} \(\sum_{d+d' = n} L_d x W_0 x L_{d'}\) \\ & = & L' + \sum_{n \geq 2} P_n,\end{aligned}$$ since modulo $L'$, $xBx \equiv xB_0x$ and $xWx \equiv xW_0 x$. Let $m \geq 1$. As a vector space, $L_m {{\,\cong\,}}B {{\otimes_{}}}B_0 {{\otimes_{}}}\cdots {{\otimes_{}}}B_0 {{\otimes_{}}}B$ with $m+1$ factors. This isomorphism carries $P_m$ to $\sum B {{\otimes_{}}}B_0 {{\otimes_{}}}\cdots {{\otimes_{}}}B_0 {{\otimes_{}}}W_0 {{\otimes_{}}}B_0 {{\otimes_{}}}\cdots {{\otimes_{}}}B_0 {{\otimes_{}}}B$ as above, and there is an isomorphism $$\psi_m \co L_m / P_m \,\longrightarrow\, B {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}\cdots {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}B$$ with $m-1$ factors of the form $\overline{B_0} = B_0/W_0$. The image of $g \in L_m$ in $\overline{L_m} = L_m/P_m$ will be denoted by $\overline{g}$, hoping that no confusion is incurred by the double usage of the over-line. We need to show that $P$ is prime. Since $L' \sub P$, it suffices to show that if $f,f' \in L$ and $f,f' \not \in P$, then $fBf' \not \sub P$. Furthermore since $R$ is graded with respect to $x$, and $P$ is a homogeneous ideal with respect to this grading, we may assume that $f,f'$ are homogeneous with respect to $x$, so we can write $$f = \sum_i a_{0,i} x a_{1,i} x \cdots x a_{n,i} \in L_n$$ and $$f' = \sum_j a'_{0,j} x a'_{1,j} x \cdots x a'_{n',j} \in L_{n'}$$ where $a_{0,i},a_{n,i}, a'_{0,j}, a'_{n',j} \in B$ and $a_{t,i}, a'_{t',j} \in B_0$ for $0 < t < n$ and $0 < t' < n'$. Let $V$ be the vector space spanned by all the $a_{n,i}$ and $V'$ the vector space spanned by all the $a'_{0,j}$. We say that $f$ “ends in $V$” and $f'$ “begins in $V'$”. By assumption, there are elements $b,b' \in B$ such that $Vb, b'V' \sub B_0$, while $Vb \not \sub W_0$ and $b'V' \not \sub W_0$. Since $Vb, b'V' \sub B_0$, we have that $f b x b' f' \in L_{n+n'+1}$. Consider the commutative diagram $$\xymatrix@C=32pt{L_n {{\otimes_{}}}L_{n'} \ar@{->}[d]^{m} \ar@{->}[r]^{\theta} & \overline{L_n} {{\otimes_{}}}\overline{L_{n'}} \ar@{->}[r]^(0.24){\psi_n {{\otimes_{}}}\psi_{n'}} \ar@{->}[d]^{\bar{m}} & (B {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}\cdots {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}B) {{\otimes_{}}}(B {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}\cdots {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}B) \ar@{->}[d] \\ L_{n+n'+1} \ar@{->}[r] & \overline{L_{n+n'+1}} \ar@{->}[r]^(0.4){\psi_{n+n'+1}} & B {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}\cdots {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}B }$$ $$\xymatrix{L_n {{\otimes_{}}}L_{n'} \ar@{->}[r]^{m} \ar@{->}[d]^{\theta} & L_{n+n'+1} \ar@{->}[d] \\ \overline{L_n} {{\otimes_{}}}\overline{L_{n'}} \ar@{->}[d]^{\psi_n {{\otimes_{}}}\psi_{n'}} \ar@{->}[r]^{\bar{m}} & \overline{L_{n+n'+1}} \ar@{->}[d]^{\psi_{n+n'+1}} \\ (B {{\otimes_{}}}\cdots {{\otimes_{}}}B) {{\otimes_{}}}(B {{\otimes_{}}}\cdots {{\otimes_{}}}B) \ar@{->}[r] & B {{\otimes_{}}}\cdots {{\otimes_{}}}B }$$ where the domain of definition of the top-to-bottom maps is the elements in $L_n {{\otimes_{}}}L_{n'}$ such that the left factor ends in $B_0$ and the right factor begins in $B_0$ (and not merely in $B$), and their image. Here, $m(g {{\otimes_{}}}g') = g x g'$ and $\bar{m}(\overline{g} {{\otimes_{}}}\overline{g'}) = \overline{g x g'}$ which is easily checked to be well-defined. The right-most arrow is reduction of the two intermediate factors along $B_0 \ra B_0/W_0$. $$\xymatrix{L_n {{\otimes_{}}}L_{n'} \ar@{->}[r]^(0.4){m} \ar@{->}[d]^{\theta} & (L'+L_{n+n'+1})/L' \ar@{->}[d]\ar@{->}[r]^(0.61)\cong & L_{n+n'+1} \ar@{->}[d] \\ \overline{L_n} {{\otimes_{}}}\overline{L_{n'}} \ar@{->}[d]^{\psi_n {{\otimes_{}}}\psi_{n'}} \ar@{->}[r]^(0.3){\bar{m}} & (L'+L_{n+n'+1})/(L'+P_{n+n'+1}) \ar@{->}[r]^(0.7)\cong & \overline{L_{n+n'+1}} \ar@{->}[d]^{\psi_{n+n'+1}} \\ (B {{\otimes_{}}}\cdots {{\otimes_{}}}B) {{\otimes_{}}}(B {{\otimes_{}}}\cdots {{\otimes_{}}}B) \ar@{-->}[rr] & & B {{\otimes_{}}}\cdots {{\otimes_{}}}B }$$ where: $m(g {{\otimes_{}}}g') = g x g' + L'$; $\bar{m}(\overline{g} {{\otimes_{}}}\overline{g'}) = g x g' + P_{n+n'+1}+L'$ which is easily checked to be well-defined. The isomorphisms to the right are the natural ones, induced by the fact that $L' \cap L_{n+n'+1} = 0$; the bottom arrow is reduction of the two intermediate factors along $B_0 \ra B_0/W_0$, and is defined on the subspace of elements whose two intermediate factors are indeed in $B_0$. Now consider the element $f b{{\otimes_{}}}b'f' \in L_n {{\otimes_{}}}L_{n'}$, for the given $f \in L_n$ and $f' \in L_{n'}$. By assumption $\theta (f b{{\otimes_{}}}b'f') = \overline{fb} {{\otimes_{}}}\overline{b'f'}$ is non-zero, because $\overline{fb}, \overline{bf'} \neq 0$. Furthermore $\psi_n(\overline{fb})$ ends in $B_0$ and $\psi_{n'}(\overline{b'f'})$ begins in $B_0$, so $\psi_n(\overline{fb}) {{\otimes_{}}}\psi_{n'}(\overline{b'f'})$ is in the domain of definition of the right-most arrow, which takes this element to $\psi_{n+n'+1}(\overline{fbxbf'})$. It remains to show that this element is nonzero. But $\psi_n(\overline{fb})$ does not end in $W_0$, and $\psi_{n'}(\overline{b'f'})$ does not begin in $W_0$; hence their images in $B {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}\cdots {{\otimes_{}}}\overline{B_0} {{\otimes_{}}}B$ are nonzero, and their tensor product, equal to $\psi_{n+n'+1}(\overline{fbxbf'})$, is nonzero as well. \[3.2\] Let $(K,\nu)$ be a valued field containing $F$ as a field of scalars. For any $m \geq 0$, $W = {{\{k \in K {{\,:\ }}\nu(k) \geq -m\}}}$ is a restricted subspace of $K$ (where the decomposition $K = F \oplus K_0$ is arbitrary). We first claim that if $U \sub K$ is an $F$-vector subspace of finite codimension, then ${{\{\nu(u) {{\,:\ }}u \in U\}}}$ is unbounded from below. Indeed, choose any finite dimensional complement $U'$, and notice that ${{\{\nu(u'){{\,:\ }}u' \in U'\}}}$ is bounded from below; so if $\nu(u)$ were bounded for $u \in U$, then $\nu(k)$ would be bounded over the set of $k \in K$. Let $V\neq 0$ be a finite dimensional space. The space of elements $y$ such that $Vy \sub K_0$ has finite codimension, so by the previous argument contains elements of arbitrarily small value, for which $Vy \not \sub W$. \[almostexample\] Let $(K,\nu)$ be a valued field containing $F$ as a field of scalars. For fixed $m \geq 0$, let $W_m = {{\{k \in K {{\,:\ }}\nu(k) \geq -m\}}}$. Then the ideal generated by $xW_mx$ in $R = F[x] *_F K$ is prime. With the notation of [[Corollary \[almostexample\]]{}]{}, we now formulate the promised counterexample: \[6.4\] Let $F$, $K$, $R$ and the $W_m$ be as above. By definition $\bigcup_{m\geq 0} W_m = K$. Let $P_m$ be the (prime) ideal generated by $xW_mx$. Then $P_1 \sub P_2 \sub \cdots$ is a chain of prime ideals in $R$, and $\bigcup_{m\geq 0} RxW_mxR = RxKxR = (RxR)^2$. The radical $RxR$ is thus maximal, as $R/RxR \,{{\,\cong\,}}\, K$. Matrix constructions {#sec:mat} ==================== This section shows that  and  are independent: the algebra in [[Example \[ExKL\]]{}]{} satisfies  but not , and the algebra in [[Example \[Ex4\]]{}]{} satisfies  but not .  does not imply ---------------- As mentioned in the introduction, Kaplansky conjectured that a semiprime ring all of whose prime quotients are von Neumann regular, is regular. Fisher and Snider  [@FS] proved that this is the case if the ring satisfies  (also see [@G Thm. 1.17]), and gave a counterexample which lacks this property, due to Kaplansky and Lanski [@G Example 1.19]. We repeat the example and exhibit, in this ring, an explicit ascending chain of semiprime ideals whose union is not semiprime. \[ExKL\] (A ring whose prime ideals are maximal, but without ). Let $R$ be the ring of sequences of $2$-by-$2$ matrices which eventually have the form $\begin{pmatrix} \alpha & \beta_n \\ 0 & \alpha \end{pmatrix}$ in the $n$th place, clearly a semiprime ring. Let $I_n$ be the set of sequences in $R$, which are zero from the $n$th place onward. Clearly $R/I_n {{\,\cong\,}}R$, so the ideals are semiprime. However $\bigcup I_n$ is composed of sequences of matrices which are eventually zero, and $aRa$ is eventually zero for $a = \left(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \dots \right)$; hence $R/\bigcup I_n$ is not semiprime. On the other hand by the argument in [@FS], every prime ideal of $R$ is maximal, so there are no infinite chains of primes and  holds trivially.  does not imply ---------------- In the rest of this section we investigate  and  for rings of the form $\hat{A} = {{\left(\begin{array}{cc} {A} & {M} \\ {M} & {A} \end{array}\right)}}$ where $A$ is an integral domain and $M \normali A$ is a nonzero ideal. We show that they always satisfy , and give an example which does not have . Clearly $\hat{A}$ is a prime ring. Let us describe the ideals of this ring. \[whoissp\] 1. The ideals of $\hat{A}$ have the form $\hat{I} = {{\left(\begin{array}{cc} {I_{11}} & {I_{12}} \\ {I_{21}} & {I_{22}} \end{array}\right)}}$, where for $1 \le i,j\le 2$, $I_{ij} \normal A$ (not necessarily proper), $I_{ii'} \sub M$, and $MI_{ij} \sub I_{i'j} \cap I_{ij'}$ (where $1' = 2$ and $2' = 1$). 2. The semiprime ideals of $\hat{A}$ are of the form $${{\left(\begin{array}{cc} {I} & {M\cap I} \\ {M \cap I'} & {I'} \end{array}\right)}},$$ where $I,I'$ are semiprime ideals of $A$, and $M \cap I' = M \cap I$. <!-- --> 1. This is well known and easy. 2. Write $A_{ij} = A$ if $i = j$ and $A_{ij} = M$ otherwise. We are given an ideal $\hat{I} \normali \hat{A}$ which thus can be written as $\hat{I} = {{\left(\begin{array}{cc} {I_{11}} & {I_{12}} \\ {I_{21}} & {I_{22}} \end{array}\right)}}$, satisfying the conditions of (1). Clearly $\hat{I}$ is semiprime if and only if for every $a_{ij} \in A_{ij}$, ( $\sum_{j,k} A_{jk} a_{ij}a_{k\ell} \sub I_{i\ell}$ for every $i,\ell$) implies ( $a_{i\ell} \in I_{i\ell}$ for every $i,\ell$). Assuming that this condition holds, fix $i,j$ and choose $a_{k\ell} = 0$ for every $(k,\ell) \neq (i,j)$; then $\COND$ for $a_{ij} \in A_{ij}$, $A_{ji} a_{ij}^2 \sub I_{ij}$ implies $a_{ij} \in I_{ij}$. On the other hand if Condition $\COND$ holds and $\sum_{j,k} A_{jk} a_{ij}a_{k\ell} \sub I_{i\ell}$ for every $i,\ell$, then in particular $A_{ji} a_{ij}^2 \sub I_{ij}$, so each $a_{i\ell} \in I_{i\ell}$. We conclude that $\hat{I}$ is semiprime if and only if $\COND$ holds for every $i,j$. We claim that $\COND$ is equivalent to $I_{ii}$ being semiprime in $A$ with $$M \cap I_{11} \sub I_{ij},\qquad \forall i\neq j.$$ Indeed, for $i = j$, condition $\COND$ requires that the $I_{ii}$ are semiprime in $A$. Assuming this, the condition is “for $a_{ij} \in A_{ij}$, $Ma_{ij}^2 \sub I_{ij}$ implies $a_{ij} \in I_{ij}$ for $i \neq j$.” In light of the standing assumption that $a_{ij} \in A_{ij} = M$, we claim that this is equivalent to $M \cap I_{11} \sub I_{ij}$. Indeed, for every $b \in M$, $Mb^2 \sub I_{ij}$ iff $b \in I_{11}$ (Proof: If $b^2 M \sub I_{ij}$ then $b^4 \in (bM)^2 = b^2M\cdot M \sub I_{ij}M \sub I_{11}$, so $b \in I_{11}$. On the other hand if $b \in M \cap I_{11}$ then $b^2 M \sub MI_{11} \sub I_{ij}$), so the condition becomes “for $b \in M$, $b \in I_{11}$ implies $b \in I_{ij}$ for $i\neq j$”, as claimed. We have shown that $\hat{I}$ is semiprime if and only if $I_{11},I_{22}$ are semiprime in $A$ and $M \cap I_{11} \sub I_{12} \cap I_{21}$. Now assume that $I_{ii}$ are semiprime, and that $M \cap I_{11} \sub I_{12} \cap I_{21}$. Denote the idealizer of an ideal $I$ by $(I:M) = {{\{x \in A {{\,:\ }}xM \sub I\}}}$, and notice that $M \cap (I:M) = M \cap I$ when $I$ is semiprime. But $I_{12} M \sub I_{11}$, implying $$I_{12}\sub M \cap (I_{11}:M) = M \cap I_{11} \sub I_{12},$$ so $I_{12} = M \cap I_{11}$ and likewise $I_{21} = M\cap I_{11}$. By symmetry $I_{12} = M \cap I_{22}$ as well, so $M \cap I_{11} = M \cap I_{22}$. Recall that for ideals $I,M$ in a commutative ring $A$, $(I:M) = {{\{a \in A{{\,:\ }}aM \sub I\}}}$. The properties of $(I:M)$ are well known, and we review the ones that we need. \[compactness\] Let $A$ be a commutative ring, with an ideal $M \normali A$. For any semiprime ideal $I \normali A$, $$M \cap (I:M) = M \cap I.$$ The inclusion $I \sub (I:M)$ is trivial. In the other direction let $x \in M \cap (I:M)$, then $x^2 \in xM \sub I$, so $x \in I$ by assumption. \[prep\] Let $A$ be a commutative ring with an ideal $M \normali A$. Let $I,J \sub M$ be ideals of $A$ such that $I$ is semiprime, $M I\sub J$ and $M J\sub I$. 1. \[x1\] For $b \in M$, $b^2 M \sub J$ iff $b \in I$. 2. \[x4\] The condition “For every $b \in M$, if $b^2M \sub J$, then $b \in I$" is equivalent to $M \cap I \sub J$. 3. \[x3\] Let $I'$ be another semiprime ideal such that $MI' \sub J$ and $MJ \sub I'$. Then $M \cap I = M \cap I'$. <!-- --> 1. If $b^2 M \sub J$ then $(bM)^2 \sub JM \sub I$ so $bM \sub I$ and $b \in M \cap (I:M) \sub I$ by [[Proposition \[compactness\]]{}]{}. On the other hand if $b \in M \cap I$ then $b^2 \in MI \sub J$ and $b^2M \sub J$. 2. This is [[(\[x1\])]{}]{}. 3. Indeed, by [[Proposition \[compactness\]]{}]{} and [[(\[x1\])]{}]{}, $b \in M \cap I$ iff $b \in M \cap (I:M)$ iff ($b \in M$ and $b^2M \sub J$) iff $b \in M \cap (I':M)$ iff $b \in M \cap I'$. \[main0\] The ring $\hat{A}$ satisfies . By [[Proposition \[whoissp\]]{}]{} every chain of semiprime ideals $T_1 \sub T_2 \sub \cdots$ in $\hat{A}$ has the form $T_n = {{\left(\begin{array}{cc} {I_n} & {J_n} \\ {J_n} & {I'_n} \end{array}\right)}}$, $I_n$ and $I'_n$ are ascending chains of semiprime ideals of $A$, and $J_n = M \cap I_n = M \cap I_n'$. The union of this chain is ${{\left(\begin{array}{cc} {\bigcup I_n} & {L} \\ {L} & {\bigcup I_n'} \end{array}\right)}}$ where $L = M \cap \bigcup I_n = M \cap \bigcup I_n'$, which is semiprime. Using the description of the semiprime ideals, it is not difficult to obtain the following. \[main0.2\] 1. \[Y5\] The prime ideals of $\hat{A}$ are ${{\left(\begin{array}{cc} {J} & {M} \\ {M} & {A} \end{array}\right)}}$ and ${{\left(\begin{array}{cc} {A} & {M} \\ {M} & {J} \end{array}\right)}}$ for prime ideals $J \normali A$ containing $M$, and $I^{0} = {{\left(\begin{array}{cc} {I} & {M \cap I} \\ {M \cap I} & {I} \end{array}\right)}}$ for prime ideals $I \normali A$ not containing $M$. 2. The  ideals of $\hat{A}$ are of the form ${{\left(\begin{array}{cc} {M'} & {M} \\ {M} & {M'} \end{array}\right)}}$ where $M' \normali A$ is a prime ideal containing $M$, which can be written as a union of an ascending chain of primes not containing $M$. Notation as in [[Proposition \[whoissp\]]{}]{}(1), we are done by [[Proposition \[whoissp\]]{}]{}(2) unless some ${I_{11}} = A$, in which case we must have (1). If the chain of primes includes an ideal with $A$ in one of the corners, then every higher term has the same form, and the union is determined by the union of the ideals in the other corner, which is prime since $A$ is commutative. We thus assume that the chain has the form $I_1^0 \sub I_2^0 \sub \cdots$ where $I_1 \sub I_2 \sub \cdots$ are primes in $A$, not containing $M$. The union is clearly $\hat I^0$ where $\hat I = \bigcup I_n$ is prime, and $\hat I^0$ is not a prime iff $M \sub \hat I$. Suppose $A$ is a prime PI-ring, integral over its center $C$. In [@BV] it is shown that $A$ satisfies the properties Lying Over and Going Up over $C$, which gives a correspondence of chains of primes between the two rings. The next example shows that the union of chains in not preserved. \[Ex4\] Let $F$ be a field. Let $\hat{A} = {{\left(\begin{array}{cc} {A} & {M} \\ {M} & {A} \end{array}\right)}}$, where $A = F[\lam_1,\dots]$ is the ring of polynomials in countably many variables $\lam_1,\lam_2,\dots$, and $M = {{\left<\lam_1,\dots\right>}}$. Clearly $\hat{A} \subset M_2(A)$ is integral over $A$. Choose $I_n = {{\left<\lam_1,\dots,\lam_n\right>}}$. Then $T_n = {{\left(\begin{array}{cc} {I_n} & {I_n} \\ {I_n} & {I_n} \end{array}\right)}} \normali \hat{A}$ form an ascending chain of primes by [[Proposition \[main0.2\]]{}]{}.[[(\[Y5\])]{}]{}, but their union $\bigcup T_n = {{\left(\begin{array}{cc} {M} & {M} \\ {M} & {M} \end{array}\right)}}$ is obviously not prime. Therefore $\hat{A}$ satisfies  ([[Proposition \[main0\]]{}]{}) but not . Furthermore $\hat{A}/\bigcup T_n\cong A/M\times A/M$ which has two minimal ideals, so uniquely- also fails. The $\PP$-index {#sec:PP} =============== Let $\tilde P = {{\{P_{\alpha}\}}}$ be a chain of prime ideals in a ring $R$. The number of non-prime unions of subchains of $\tilde P$ is called the [**[index]{}**]{} of $\tilde P$ (either finite or infinite). The [**[-index]{}**]{} of $R$, denoted by $\PP(R)$, is the supremum of the indices of all chains of primes in $R$. For any ring $R$, $\PP(R) = \sup \PP(R/P)$ where $P\normali R$ ranges over the prime ideals. By definition $\PP(R/P)$ is the supremum of indices of chains of primes containing $P$. Therefore, the supremum on the right-hand side is the supremum of indices of chains of primes containing some prime $P$, but any chain contains its own intersection, so this supremum is by definition $\PP(R)$. If $n = \PP(R)$ then $R$ has a chain of $n$ . It is not clear if the converse holds. For example if ${{\{P_\lam\}}}$ and ${{\{P_{\lam}'\}}}$ are ascending chains of primes such that $\bigcup P_{\lam} \subset \bigcup P_{\lam}'$ are not primes, does it follow that there is a chain of primes with at least two non-prime unions? We claim that: \[PPindex\] For any ring $R$, $$\PPind(R) = \left\{ \begin{array}{cl} 0 & i\!f\mbox{\, $R$\, has the property\, \PP} \\ \sup_I \PPind(R/I)+1 & \mbox{otherwise} \end{array} \right.$$ where the supremum is taken over the  ideals of $R$ (when they exist). Indeed, $\PPind(R) = 0$ if and only if there are no s (which are non-prime by definition), if and only if $R$ satisfies $\PP$. Now assume $R$ does not satisfy . Consider the set ${{\{\PPind(R/I)\}}}$ ranging over the s $I$. If this set is unbounded, then clearly $\PPind(R) = \infty$. Otherwise, take a $I$ such that $n = \PPind(R/I)$ is maximal among the -indices of the quotients. If $J_1/I \subset \cdots \subset J_n/I$ are s in a chain of primes in $R/I$, then $I \subset J_1 \subset \cdots \subset J_n$ are in a chain in $R$. On the other hand if $J_0 \subset J_1 \subset \cdots \subset J_n$ are in a chain in $R$, then $\PPind(R/J_0) \geq n$. For example, $\PPind(R) = 1$ if and only if the union of an ascending chain of primes starting from a  ideal is necessarily prime. The property $\PP$ in PI-rings {#sec:PI} ============================== In this section we show that for PI-rings, the $\PP$-index is bounded by the PI-class. \[Azumaya\_CP\] Any Azumaya algebra satisfies (and ). Let $A$ be an Azumaya algebra over a commutative ring $C$. There is a 1:1 correspondence between ideals of $A$ and the ideals of $C$, preserving inclusion, primality and semiprimality. The claim follows since the center satisfies  (and ). Recall that by Posner’s theorem ([@RowenPI]), a prime PI-ring $R$ is [*representable*]{}, namely embeddable in a matrix algebra ${{\operatorname{M}_{n}}}(C)$ over a commutative ring $C$. The minimal such $n$ is the PI-class of $R$, denoted $\PIdeg(R)$. Although PI-rings do not necessarily satisfy the property , we show that the PI-class bounds the extent in which  may fail. We are now ready for our main positive result about PI-rings. \[mainPI\] Let $R$ be a (prime) PI-ring. Then $\PPind(R) < \PIdeg(R)$. Let $R$ be a prime PI-ring of PI-class $n$. If the PI-class is $1$ then $R$ is commutative, and has $\PPind(R) = 0$. We continue by induction on $n$. Let $$0 = P_0 \subset P_1 \subset \cdots$$ be an ascending chain of primes, and assume that $\bigcup P_n$ is not a prime ideal. Let $Q \supset \bigcup P_n$ be a prime ideal. We want to prove that the PI-class of $R/Q$ is smaller than that of $R$. Assume otherwise. Let $g_n$ be a central polynomial for $n \times n$ matrices (see [@RowenPI p. 26]). Since $\PIdeg(R/Q) = n$, there is a value $\gamma \neq 0$ of $g_n$ in the center of $R$, which is not in $Q$. Since the center is a domain we can consider the localization $A[\gamma^{-1}]$ (see [@R-RT Section 2.12]), which is Azumaya by Rowen’s version of the Artin-Procesi Theorem [@RowenPI Theorem 1.8.48], since $1$ is a value of $g_n$ on this algebra. But then the union of $$0 \subset P_1[\gamma^{-1}] \subset P_2[\gamma^{-1}] \subset \cdots$$ is prime by [[Proposition \[Azumaya\_CP\]]{}]{}, so $\bigcup P_n$ is prime as well, contrary to assumption. We now show the bound is tight. Notice that the ring constructed in [[Example \[Ex4\]]{}]{} has PI-class $2$ and is not (and thus has $\PPind(R) = 1$). Let us generalize this. \[PI\_non\_CP\] Let $A_{(n)}=F\left[\lambda_i^{(j)} : 1 \leq j < n,\, i = 1,2,\dots\right]$. Let $M_n = 0$ and, for $j = n-1,n-2,\dots,1$, take $M_j=M_{j+1}+\left<\lambda_1^{(j)},\lambda_2^{(j)},\cdots\right>$, so that $0 = M_n \subset M_{n-1} \subset \cdots \subset M_1 \normali A_{(n)}$. Let $e_{ij}$ denote the matrix units of the matrix algebra over $A_{(n)}$. Let $J_{(n)} = \sum_{i,j} e_{ij}M_{\max(i,j)-1}$, $S_{(n)} = \sum e_{ii} A_{(n)}$. Let $$R_{(n)} = J_{(n)}+S_{(n)} =\begin{pmatrix} A_{(n)} & M_1 & M_2 & \cdots & M_{n-1} \\ M_1 & A_{(n)} & M_2 & \cdots & M_{n-1} \\ M_2 & M_2 & A_{(n)} & \cdots & M_{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ M_{n-1} & M_{n-1} & M_{n-1} & \cdots & A_{(n)} \\ \end{pmatrix}.$$ Clearly $S_{(n)} J_{(n)}, \, J_{(n)} S_{(n)} \sub J_{(n)}$, and $S_{(n)}S_{(n)}= S_{(n)}$. Moreover $M_{k}M_{\ell} \sub M_{\max{{\{k,\ell\}}}}$ for every $k,\ell$, so that $J_{(n)}J_{(n)} \sub J_{(n)}$. It follows that $R_{(n)} = J_{(n)}+S_{(n)}$ is a ring. The ring of central fractions of $R_{(n)}$ is the simple ring ${{\operatorname{M}_{n}}}(\operatorname{q}(A_{(n)}))$, so $R_{(n)}$ is prime, of PI-class $n$. When $n = 2$ we obtain the ring of [[Example \[Ex4\]]{}]{}, so $\PPind(R_{(2)}) = 1$. For arbitrary $n$, consider the chain $I_1 \subset I_2 \subset \cdots$ of ideals of $A$ defined by $I_i = {{\left<\lam_1^{(n-1)},\dots,\lam_i^{(n-1)}\right>}}$; thus $\bigcup I_i = M_{n-1}$. Let $\tilde{I}_i = {{\operatorname{M}_{n}}}(I_i)$. Each ideal $\tilde{I}_i$ is prime (again by central fractions), and their union is the set of matrices over $M_{n-1}$. The quotient ring is therefore $R_{(n)}/\bigcup_i \tilde{I}_i \,{{\,\cong\,}}\, R_{(n-1)} \times A_{(n-1)}$, which is not prime, and $\PPind(R_{(n-1)}) = n-2$ by induction. We conclude that $\PPind(R_{(n)}) = n-1$. \[end\] 1. \[AffinePIhasACCsp\] Although PI-rings do not necessarily satisfy (see [[Example \[Ex4\]]{}]{}), affine PI-rings over commutative Noetherian rings do satisfy ACC on semiprime ideals, and in particular are  and (by Schelter’s theorem, [@RowenPI Thm 4.4.16]). 2. PI-rings of finite Gel’fand-Kirillov dimension satisfy ACC on primes, since a prime PI-ring is Goldie, and then every prime ideal contains a regular element which reduces the dimension. 3. On the other hand, we have examples of a (non-affine) locally nilpotent monomial algebra, which does not satisfy , and of affine algebras of $\GK=2$ which do not satisfy (details will appear elsewhere.) In both cases the -index is uncountable. 4. Graded affine algebras of quadratic growth and graded affine domains of cubic growth have finite Krull dimension and so satisfy , [@BS; @GLSZ]. Further ideas ============= (This section will be omitted from the submitted paper). Normalizing extensions ---------------------- The most general version: If $S$ is a subring of $R$ and there is a finite $S$-module $M$ such that $aR \subseteq Ma$ for all $a\in R$, then the P-index of $R$ does not the P-index of exceed $S$. This should cover all the cases that we know. If what I wrote is true, I suggest that Beeri write down a short exposition of this, together with the examples about PI-rings and submit it on the ArXiv in parallel to our article, so that nobody from the outside jumps in with the result (which is not difficult to discover given the thrust of the article). \[first.1\] Every ring $R$ which is a finite normalizing extension satisfies . Write $R = \sum_{i=1}^t Cr_i$ where $R$ normalizes $C$ (namely $aC = Ca$ for every $a \in R$). Suppose $P_1 \subset P_2 \subset \cdots$ is a chain of prime ideals, with $P = \cup P_i$. If $a,b \in R$ with $$\sum C ar_i b = \sum aCr_i b = aRb \subseteq P$$ then there is $n$ such that $ar_i b \in P_n$ for $1 \le i \le t,$ implying $aRb = \sum aCr_i b \subseteq P_n$, and thus $a \in P_n$ or $b \in P_n$. \[first+\] Let $R$ be a normalizing extension of $C$, and $Q \normali R$ prime. Then $Q \cap C$ is prime. Indeed, assume $aCb \sub Q \cap C$. Then $aRb = aCRb = aCbR \sub Q$ so $a \in Q$ or $b \in Q$. Assume $R$ is a finite normalizing extension of a ring $C$. Assume $R$ is PI. Then $C$ (and $R$) satisfy . The claim for $R$ is [[Proposition \[first.1\]]{}]{}. Let $P_n$ be an ascending chain of ideals in $C$. By the LO for PI rings, there is a chain $Q_n$ in $R$ such that each $P_n$ is minimal prime over $Q_n \cap C$, and since $Q_n \cap C$ is prime by the remark, $P_n = Q_n \cap C$. But $\bigcup Q_n$ is prime since $R$ is a finite normalizing extension ([[Proposition \[first.1\]]{}]{}), so $\bigcup P_n = \bigcup (Q_n \cap C) = (\bigcup Q_n) \cap C$ is prime by the [[Remark \[first+\]]{}]{}. Comments on [[Section \[sec:example2.2\]]{}]{} ---------------------------------------------- ((This is only needed if more details are required above)) - In \[Hungerford, 1967\], he defines the free product for augmented algebras. So the definition depends on the augmentation, and one needs to be extra careful about this, because $F(y)$ has no natural augmentation. It seems best to take $F[x] = F \oplus xF[x]$ and $F(y) = F \oplus {\operatorname{span}}_F {{\{\frac{f}{g} {{\,:\ }}(f,g) = 1, fg \not \in {{{F}^{\times}}}\}}}$. Finally $F[x] *_F F(y)$ is well defined. - We could extend $\nu$ to $F[x] *_F F(y)$ by taking $\nu()$ of a monomial to be the maximal $\nu(a)$ for $xax$ within the monomial; and $\nu()$ of a sum of monomials to be the minimal $\nu()$ of a monomial. Since $\nu(xa_1xa_2\cdots a_kx) = \max(\nu(a_1),\dots,\nu(a_k))$, we have that $\nu(xa_1xa_2\cdots a_kx) \geq -n$ iff for some $a_i$ we have $\nu(a_i) \geq -n$, iff some factor $xa_i x \in P_n$. Which is fine. In this definition of $\nu()$ on monomials we ignore initial or terminal coefficients from $F(y)$. So for example $\nu(axbx) = \nu(b)$, regardless of $a$. Likewise $\nu(\sum w_i) = \min(\nu(w_i))$ where $w_i$ are monomials; so $\nu(\sum w_i) \geq -n$ iff $\nu(w_i) \geq -n$ for each $i$, iff $w_i \in P_n$ for each $i$; which is fine again, because we want $f \in P_n$ iff $\nu(f) \geq -n$ for every $f \in R$. Of course there is still a problem of defining everything; what are monomials, etc. - Write $F[x] = F \oplus A$ and $F(y) = F\oplus B$. As in \[Hungerford, 1967\], define $T_n$ and $T_n'$ to be the tensor products $A {{\otimes_{}}}B {{\otimes_{}}}\cdots$ and $B {{\otimes_{}}}A {{\otimes_{}}}\cdots$ of length $n$, respectively. The free product is, by definition, $F[x]*_FF(y) = F\oplus \bigoplus(T_n \oplus T_n')$. We suppress the tensor notating for elements. Since $A = \bigoplus Fx^i$, every simple tensor can be written as a sum of [**[monomials]{}**]{}, which have the form $x^{i_0}a_1x^{i_1}a_2x^{i_3}a_3\cdots x^{i_{t-1}}a_{t}x^{i_t}$, where $t \geq 0$, $i_0, i_t \geq 0$, $i_1,\dots,i_{t-1} > 0$ and $a_1,\dots,a_t \in B$. The components $a_i$ of a monomial are uniquely determined up to (balanced) multiplication by a nonzero scalar. The presentation of an element as a sum of monomials is not unique, but the sum of monomials of any given [**[signature]{}**]{} $(t;i_0,\dots,i_t)$ – is. Every monomial can be uniquely written in the form $w = x^{j_0}a_1 x a_2 x a_3\cdots x a_{\ell}x^{j_\ell}$, where $\ell \geq 0$, $j_0, j_t \in {{\{0,1\}}}$, and $a_1,\dots,a_\ell \in B \cup F$ (the number of appearances of $x$ in $w$ is $\ell+j_0+j_{\ell}-1$ if $\ell > 0$, or $j_0$ for $\ell = 0$). Let us now compute $P_n$. By definition, $$P_n = \begin{cases} \sum_{a \in B, \nu(a) \geq -n} R xax R & n < 0 \\ P_n = \sum_{a \in B, \nu(a) \geq -n} R xax R + RxxR & n \geq 0\end{cases}.$$ $P_n$ is monomial (in the sense that if it contains an element, it contains all of its monomials). A monomial is in $P_n$ iff it has $xax$ as a subword where $\nu(a) \geq -n$. We extend the definition of $\nu \co F(y) \ra \Z$ to $F[x]*_F F(y)$ as follows. For a monomial, we set $$\nu(x^{j_0}a_1 x a_2 x a_3\cdots x a_{\ell}x^{j_\ell}) = \begin{cases}\max{{\{\nu(a_2),\dots,\nu(a_{\ell-1})\}}} & j_0 = 0, j_{\ell} = 0 \\ \max{{\{\nu(a_2),\dots,\nu(a_{\ell})\}}} & j_0 = 0, j_{\ell} = 1 \\ \max{{\{\nu(a_1),\dots,\nu(a_{\ell-1})\}}} & j_0 = 1, j_{\ell} = 0 \\ \max{{\{\nu(a_1),\dots,\nu(a_{\ell})\}}} & j_0 = 1, j_{\ell} = 1 \end{cases}$$ Proofs for Section \[sec:mat\] ------------------------------ Recall that for ideals $I,M$ in a commutative ring, $(I:M) = {{\{x \in A{{\,:\ }}xM \sub I\}}}$. \[compactness\] Let $A$ be a commutative ring, with an ideal $M \normali A$. For any semiprime $I \normali A$, $$M \cap (I:M) = M \cap I.$$ The inclusion $I \sub (I:M)$ is trivial. In the other direction let $x \in M \cap (I:M)$, then $x^2 \in xM \sub I$, so $x \in I$ by assumption. \[prep\] Let $A$ be a commutative ring with an ideal $M \normali A$. Let $I,J \sub M$ be ideals of $A$ such that $I$ is semiprime, $M I\sub J$ and $M J\sub I$. 1. \[x1\] For $b \in M$, $b^2 M \sub J$ iff $b \in I$. 2. \[x4\] The condition “For every $b \in M$, if $b^2M \sub J$, then $b \in J$" is equivalent to $M \cap I \sub J$. 3. \[x3\] Let $I'$ be another semiprime ideal such that $MI' \sub J$ and $MJ \sub I'$. Then $M \cap I = M \cap I'$. <!-- --> 1. If $b^2 M \sub J$ then $(bM)^2 \sub JM \sub I$ so $bM \sub I$ and $b \in M \cap (I:M) \sub I$ by [[Proposition \[compactness\]]{}]{}. On the other hand if $b \in M \cap I$ then $b^2 \in MI \sub J$ and $b^2M \sub J$. 2. This is [[(\[x1\])]{}]{}. 3. Indeed, by [[Proposition \[compactness\]]{}]{} and [[(\[x1\])]{}]{}, $b \in M \cap I$ iff $b \in M \cap (I:M)$ iff ($b \in M$ and $b^2M \sub J$) iff $b \in M \cap (I':M)$ iff $b \in M \cap I'$. \[main0FI\] 1. \[Y1\] The ideals of $\hat{A}$ are the subsets $\hat{I} = {{\left(\begin{array}{cc} {I_{11}} & {I_{12}} \\ {I_{21}} & {I_{22}} \end{array}\right)}}$, where for every $i,j$, $I_{ij} \normal A$ (not necessarily proper), $I_{ii'} \sub M$, and $MI_{ij} \sub I_{i'j} \cap I_{ij'}$ (where $1' = 2$ and $2' = 1$). 2. \[Y3\] $\hat{I}$ is semiprime iff - $I_{11}$ and $I_{22}$ are semiprime, and - $M \cap I_{11} \sub I_{12} \cap I_{21}$; iff - $I_{11}$ and $I_{22}$ are semiprime, and - $I_{12} = I_{21} = M \cap I_{11} = M \cap I_{22}$. 3. \[Y33\] The semiprime ideals of $\hat{A}$ are of the form ${{\left(\begin{array}{cc} {I} & {M\cap I} \\ {M \cap I} & {I'} \end{array}\right)}}$ where $I,I'$ are semiprime, and $M \cap I' = M \cap I$. 4. \[Y7\] The ring $\hat{A}$ satisfies . <!-- --> 1. Follows by computing the principal ideals generated by monomial matrices. 2. Write $A_{ij} = A$ if $i = j$ and $A_{ij} = M$ otherwise. Clearly $\hat{I}$ is semiprime if for every $a_{11} \in A_{11}$,..., $a_{22} \in A_{22}$, if $(\sum a_{ij}e_{ij})(\sum A_{rs}e_{rs})(\sum a_{k\ell}e_{k\ell}) \sub \sum I_{i\ell}e_{i\ell}$ then $a_{ij} \in I_{ij}$ for each $i,j$. In other words, if (for every $i,\ell$, $\sum_{j,k} A_{jk} a_{ij}a_{k\ell} \sub I_{i\ell}$) implies (for every $i,\ell$, $a_{i\ell} \in I_{i\ell}$). Assuming this is the case, fix $i,j$ and choose $a_{k\ell} = 0$ for every $(k,\ell) \neq (i,j)$; then $\COND \quad A_{ji} a_{ij}^2 \sub I_{ij}$ implies $a_{ij} \in I_{ij}$. On the other hand if Condition $\COND$ holds and for every $i,\ell$, $\sum_{j,k} A_{jk} a_{ij}a_{k\ell} \sub I_{i\ell}$, then in particular $A_{ji} a_{ij}^2 \sub I_{ij}$ so each $a_{i\ell} \in I_{i\ell}$. Therefore, $\hat{I}$ is semiprime iff $\COND$ holds for every $i,j$. Let us interpret Condition $\COND$. For $i = j$ it requires that $I_{ii}$ are semiprime. Assuming this is the case, for $i \neq j$ the condition is “$Ma_{ij}^2 \in I_{ij}$ implies $a_{ij} \in I_{ij}$”, which in light of the standing assumption that $a_{ij} \in A_{ij}$, is equivalent by [[Lemma \[prep\]]{}]{}.[[(\[x4\])]{}]{} to $M \cap I_{11} \sub I_{ij}$. Now assume that $I_{ii}$ are semiprime, and that $M \cap I_{11} \sub I_{12} \cap I_{21}$. Since $I_{12} M \sub I_{11}$, we have that $I_{12}\sub M \cap (I_{11}:M) = M \cap I_{11} \sub I_{12}$ so $I_{12} = M \cap I_{11}$ and likewise $I_{21} = M\cap I_{11}$. Finally the equality $M \cap I_{11} = M \cap I_{22}$ is [[Lemma \[prep\]]{}]{}.[[(\[x3\])]{}]{}. 3. Clear from [[(\[Y3\])]{}]{}, noting that $(M \cap I)M \sub I,I'$ and $MI,MI' \sub M\cap I$ by [[(\[Y1\])]{}]{}. 4. By [[(\[Y3\])]{}]{} every chain of semiprimes $T_1 \sub T_2 \sub \cdots$ in $\hat{A}$ has the form $T_n = {{\left(\begin{array}{cc} {I_n} & {J_n} \\ {J_n} & {I'_n} \end{array}\right)}}$, $I_n$ and $I'_n$ are ascending chains of semiprimes, and $J_n = M \cap I_n = M \cap I_n'$. The union of this chain is ${{\left(\begin{array}{cc} {\bigcup I_n} & {L} \\ {L} & {\bigcup I_n'} \end{array}\right)}}$ where $L = M \cap \bigcup I_n = M \cap \bigcup I_n'$. Again by [[(\[Y3\])]{}]{} the union is semiprime. \[main0.1\] 1. \[Y4\] An ideal $\hat{I}$ (as in [[Proposition \[main0\]]{}]{}.[[(\[Y1\])]{}]{}) is prime iff - $I_{11}$ and $I_{22}$ are prime, - $I_{12} = I_{21} = M \cap I_{11} = M \cap I_{22}$. - If $M \sub I_{ii}$ for some $i$, then $I_{jj} = A$ for some $j$. - $I_{11} \sub I_{22}$ or vise versa. 2. \[Y5FI\] The prime ideals of $\hat{A}$ are ${{\left(\begin{array}{cc} {J} & {M} \\ {M} & {A} \end{array}\right)}}$ and ${{\left(\begin{array}{cc} {A} & {M} \\ {M} & {J} \end{array}\right)}}$ for prime ideals $J \normali A$ containing $M$, and $I^{0} = {{\left(\begin{array}{cc} {I} & {M \cap I} \\ {M \cap I} & {I} \end{array}\right)}}$ for prime ideals $I \normali A$ not containing $M$. 3. $\hat{A}$ is prime. 4. The almost prime ideals of $\hat{A}$ are of the form ${{\left(\begin{array}{cc} {M'} & {M} \\ {M} & {M'} \end{array}\right)}}$ where $M' \normali A$ is a prime containing $M$, which can be presented as a union over an ascending chain of primes not containing $M$. <!-- --> 1. As in [[Proposition \[main0\]]{}]{}.[[(\[Y3\])]{}]{}, $\hat{I}$ is prime if for every $a_{11},a_{11}' \in A_{11}$,..., $a_{22},a_{22}' \in A_{22}$, the following condition holds: ($\forall i,j,k,\ell$, $A_{jk} a_{ij}a'_{k\ell} \sub I_{i\ell}$) implies ($\forall i,\ell$, $a_{i\ell} \in I_{i\ell}$ or $\forall i,\ell$, $a_{i\ell}' \in I_{i\ell}$). Assuming this is the case, fix $i,j$ and choose $a_{k\ell} = a'_{k\ell} = 0$ for every $(k,\ell) \neq (i,j)$; then $\COND' \quad A_{ji} a_{ij}a_{ij}' \sub I_{ij}$ implies $a_{ij} \in I_{ij}$ or $a'_{ij} \in I_{ij}$, and in particular $I_{ii}$ is prime (for each $i$). Since $\hat{I}$ is semiprime, we also have that $I_{12} = I_{21} = M \cap I_{ii}$, as claimed. If, moreover, $M \sub I_{ii}$ for some $i$, then $I_{12} = I_{21} = M$ and $\hat{A}/\hat{I} = (A/I_{11}) \times (A/I_{22})$ so one of the components is zero. Finally suppose there are $a \in I_{22}$ and $a' \in I_{11}$ such that $a \not \in I_{11}$ and $a' \not \in I_{22}$. Then $(ae_{11})\hat{A}(a'e_{22}) = aa'Me_{12} \sub aI_{11}Me_{12}\sub aI_{12}e_{12} \sub \hat{I}$, whereas $ae_{11}, a'e_{22} \not \in \hat{I}$, a contradiction. This proves the fourth condition. On the other hand, assume the four conditions hold. If $M \sub I_{ii}$ then we may assume $I_{ii} = A$ and $\hat{A}/\hat{I} = A/I_{i'i'}$ is prime. So we assume $M \nsub I_{ii}$ for $i = 1,2$. Suppose $\forall i,j,k,\ell$, $A_{jk} a_{ij}a'_{k\ell} \sub I_{i\ell}$, but for some (fixed) $i,j,k,\ell$, $a_{ij} \not \in I_{ij}$ and $a'_{k\ell} \not \in I_{k\ell}$. We have that $A_{jk} a_{ij}a'_{k\ell} \sub I_{i\ell}$, contained in the prime ideals $I_{ii}, I_{\ell\ell}$ (whether or not $i = \ell$). But since $M \nsub I_{ii}, I_{\ell\ell}$, we must have $a_{ij}a'_{k\ell} \in I_{ii}, I_{\ell\ell}$. We claim that $a_{ij} \not \in I_{ii}$. Indeed if $j = i$ this is the assumption, while if $j \neq i$ the claim follows since $a_{ij} \not \in I_{ij} = M \cap I_{ii}$. Likewise $a'_{k\ell} \not \in I_{\ell\ell}$. Therefore $a_{ij} \in I_{\ell\ell}$ and $a_{k\ell}' \in I_{ii}$. Now, if $j \neq i$, then $a_{ij} \in M$, so $a_{ij} \in M \cap I_{\ell\ell} = I_{ij}$, a contradiction, so we may assume $i = j$, and likewise $k = \ell$. Since $a_{ij} \in I_{k\ell}$ but $a_{ij} \not \in I_{ij}$, we must have $i \neq k$, but $a_{ij} \in I_{k \ell}\setminus I_{ii}$ and $a_{k\ell}' \in I_{ij} \setminus I_{k\ell}$, contradicting the fourth condition. 2. Following the conditions given in [[(\[Y4\])]{}]{}, assume $I_{11} \sub I_{22}$. Then $I_{22}M \sub I_{22} \cap M = I_{11} \cap M \sub I_{11}$, so since $I_{11}$ is prime there are two options: either $I_{22} \sub I_{11}$, in which case $I_{22} = I_{11}$; or $M \sub I_{11}$, in which case $I_{22} = A$ by the third condition. 3. Zero is a prime ideal of $\hat{A}$ by [[(\[Y4\])]{}]{}. 4. If the chain of primes includes an ideal with $A$ in one of the corners, then every higher term has the same form, and the union is determined by the union of entries in the other corner, which is prime since $A$ is commutative. We thus assume the chain has the form $I_1^0 \sub I_2^0 \sub \cdots$ where $I_1 \sub I_2 \sub \cdots$ are primes in $A$, not containing $M$. The union is clearly $M'^0$ where $M' = \bigcup I_n$ is prime, and by [[(\[Y5FI\])]{}]{}, $M'^0$ is not a prime iff $M \sub M'$. Chain of primitive ideals ------------------------- It would be nice to find an example of a chain of primitive ideals with prime non primitive union. This one does not work (the ideals $P_{ij}$ are not prime): In $F{{\left<e,y\right>}}$ with $e$ idempotent define $I_{i,j} = {{\left<y^iey^j-y^jey^i\right>}}$. Then, taking lexicographic order we can make a chain $P_{ij} = \sum_{(i',j')\leq (i,j)} I_{i'j'}$. I didn’t check details but the idea is that the corner is primitive (is it?) modulo every ideal in the chain but isomorphic to a polynomial ring (hence non primitive) modulo the union. P-index of a module ------------------- (BTW, what about P-index of a module?) up and down ----------- We might also want to consider how the property lifts and descends. GU -- If $R \subset T$ satisfies GU and every prime ideal of $T$ intersects $R$ nontrivially, then descends from $T$ to $R$. I think this holds for finite normalizing extensions, maybe finite extensions in general by work of Letzter. Compactness ----------- (One could define ‘compactness’ with respect to semiprime ideals: $M$ is compact if whenever it is contained in a union of a chain of semiprime ideals, it is contained in one of the ideals. Perhaps our ring $\hat{A}$ has non-semiprime chains iff $M$ is not compact?) From the section on matrices ---------------------------- \[main0.2FI\] Let $A$ be an integral domain, and $M$ an ideal. Let $\hat{A} = {{\left(\begin{array}{cc} {A} & {M} \\ {M} & {A} \end{array}\right)}}$. 1. \[z1\] The left ideals of $\hat{A}$ are the subsets $\hat{I} = {{\left(\begin{array}{cc} {I_{11}} & {I_{12}} \\ {I_{21}} & {I_{22}} \end{array}\right)}}$, where for every $i,j$, $I_{ij} \normal A$ (not necessarily proper), $I_{ii'} \sub M$, and $MI_{ij} \sub I_{i'j}$. 2. The maximal left ideals of $\hat{A}$ are of the form ${{\left(\begin{array}{cc} {I} & {M} \\ {M \cap I} & {A} \end{array}\right)}}$ and ${{\left(\begin{array}{cc} {A} & {M \cap I} \\ {M} & {I} \end{array}\right)}}$ for maximal ideals $I \normali A$. 3. In light of [[(\[z1\])]{}]{}, a maximal left ideal has the form ${{\left(\begin{array}{cc} {I} & {M} \\ {J} & {A} \end{array}\right)}}$ for $I \normali A$ and $J \sub M$; or have the symmetric form. If $I$ is not maximal and $I \subset I'$, this is contained in ${{\left(\begin{array}{cc} {I'} & {M} \\ {J+MI'} & {A} \end{array}\right)}}$, so we assume $I$ is maximal. But then $J^2 \sub MJ \sub I$ implies $J \sub I$ and since $J \sub M$, ${{\left(\begin{array}{cc} {I} & {M} \\ {J} & {A} \end{array}\right)}} \sub {{\left(\begin{array}{cc} {I} & {M} \\ {I \cap M} & {A} \end{array}\right)}}$, so we have an equality. Since ideals of the form $ {{\left(\begin{array}{cc} {I} & {M} \\ {I \cap M} & {A} \end{array}\right)}}$ do not contain each other, and every maximal has this form, they are all maximal. 4. The Jacobson radical of $\hat{A}$ is ${{\left(\begin{array}{cc} {J(A)} & {J(A) \cap M} \\ {J(A) \cap M} & {J(A)} \end{array}\right)}}$. 5. $\hat{A}$ is semiprimitive iff $A$ is semiprimitive. 6. \[z5\] $\hat{A}$ is not primitive. 7. Every maximal left ideal contains a (nonzero) two-sided ideal: for ${{\left(\begin{array}{cc} {I} & {M \cap I} \\ {M\cap I} & {I} \end{array}\right)}} \sub {{\left(\begin{array}{cc} {I} & {M} \\ {M\cap I} & {A} \end{array}\right)}}$ is an ideal. 8. The only primitive ideals of $\hat{A}$ are the maximal ones, which are of the forms: ${{\left(\begin{array}{cc} {J} & {M} \\ {M} & {A} \end{array}\right)}}$ and ${{\left(\begin{array}{cc} {A} & {M} \\ {M} & {J} \end{array}\right)}}$ for $J \normali A$ maximal; and ${{\left(\begin{array}{cc} {I} & {M \cap I} \\ {M\cap I} & {I} \end{array}\right)}}$ where $I+M = A$. 9. Primitive PI-rings are simple. But here is a direct proof: For prime ideals of the form ${{\left(\begin{array}{cc} {J} & {M} \\ {M} & {A} \end{array}\right)}}$ or ${{\left(\begin{array}{cc} {A} & {M} \\ {M} & {J} \end{array}\right)}}$, the quotient is commutative, so the ideal is primitive if and only if it is maximal. For the prime ideals of the other form, ${{\left(\begin{array}{cc} {I} & {M \cap I} \\ {M \cap I} & {I} \end{array}\right)}}$ where $I$ is a prime not containing $M$, the quotient is isomorphic to ${{\left(\begin{array}{cc} {A/I} & {(M+I)/I} \\ {(M+I)/I} & {A/I} \end{array}\right)}}$. If $M+I = A$ this is a simple ring so the ideal is maximal; otherwise, it is of the same structure as $\hat{A}$ (since $0 \neq (M+I)/I \normali A/I$), so it is not primitive by [[(\[z5\])]{}]{}. Questions: - How do semiprime quotients look like? - Which semiprime ideals are semiprimitive? - Is the union over a semiprimitive chain necessarily semiprimitive? Stability --------- Question: assuming $A$ is , does $A[\lam]$ have ? \*Noncomm. Hilbert basis thm? A is CP (or ACC(primes)). Is the same true for A\[X\]? Proof that commutative has --------------------------- Given an ascending chain of prime ideals in a commutatuve ring, the union is prime too (indeed, an ideal in a commutative ring is prime if and only if $ab\in I$ implies either $a\in I$ or $b\in I$, so if $ab\in \bigcup{Q_i}$ then for some $i$ we have $ab\in Q_i$, which is prime, hence we may assume $a\in Q_i\subseteq \bigcup{Q_i}$). Finite GK-dim ------------- We pose the following question. Is there an affine ring of finite  which does not have ? Note that all quotient rings $R/P_n$ considered in our examples \[Ex1\],\[Ex3\],\[PI\_non\_CP\] are large - in the sense that they have infinite . On the other hand, many division rings (obviously ) contain a free subalgebra. It is reasonable therefore to ask whether there are natural restrictions on a ring in order to consist of a chain of prime ideals with non-prime union. \*An example of an affine algebra with GKdim=2 which has infinite P-index (based on an earlier construction in locally finite nil algebras. This is not \*that\* short, so might not suite; on the other hand, this could not hold an independent paper but it would be a shame to burry it.) locally primeness ----------------- Here are some properties of ideals $I$ ($\exists a_i$ marks finitely many, $P$ is always a prime). The rightmost one is called ‘locally prime’ in [@TK]. (I didn’t give the reverse implications much thought). $$\xymatrix{ {} & \mbox{prime} \ar@{=>}[d] & {} & {} & {} \\ {} & \bigcup_{\mbox{chain}}\mbox{primes} \ar@{=>}[dr] & {} & {} & {} \\ {} & {} & {{\mbox{$\begin{array}{c} \forall a_i \in I\\ \exists P \sub I: a_i \in P \end{array}$}}} \ar@{=>}[dl] \ar@{=>}[dr] & {} & {} \\ {} & \bigcup \mbox{primes} = {{\mbox{$\begin{array}{c} \forall a \in I\\ \exists P \sub I: a \in P \end{array}$}}} \ar@{=>}[dr] & {} & {{\mbox{$\begin{array}{c} \forall a_i \in I, b_j \not \in I\\ \exists P: a_i \in P, b_j \not \in P \end{array}$}}} \ar@{=>}[dl] \ar@/_25pt/@{..>}[lluuu]|{quasi-commutative} & {} \\ {} & {} & {{\mbox{$\begin{array}{c} \forall a \in I, b \not \in I\\ \exists P: a \in P, b \not \in P \end{array}$}}} & {} & {} \\ }$$ In [@TK] they give examples of rings which are not quasi-commutative. We should check their examples for our ‘union of primes is prime’. A ring is ‘spectral’ if $\operatorname{spec}R$ is a spectral topological space (a property characterized by being the spectrum of some commutative ring, but which also has topological characterization). The space of locally prime ideals is spectral. So quasi-commutative implies spectral. Again, how does  fit in? By [@TK Rem. 15], $R$ is quasy-commutative iff: QC: For every $x,y$ there are $r_1,\dots,r_n$ such that if $x,y \not \in P$ then some $xr_iy \not \in P$. Equivalently, QC: $\forall x,y$ there are $r_i$ such that if ${{\left<xr_1y,\dots,xr_ny\right>}} \sub P$ then $x\in P$ or $y \in P$. Weaker version of quasi-commutative ----------------------------------- What about the following \*weaker\* condition: QCF: $\forall x,y$ $\exists$ f.g. ideal $I$ such that for every $P$, $xIy \sub P$ implies $x\in P$ or $y \in P$ What does it imply? Is our chain condition the only way to violate quasi-commutativity? ------------------------------------------------------------------- Let us then phrase ‘not quasi-commutative’: $\sim{}QC$: For some $x,y$, for every finite set $r_1,\dots,r_n$, there is a prime $P$ such that all $xr_iy \in P$ but $x,y \not \in P$. Reformulation: $\sim{}QC$: For some $x,y$, in every quotient of the form $\bar{R} = R/{{\left<xr_1y,\dots,xr_ny\right>}}$, there is a prime $\bar{P}$ such that $x,y \not \in \bar{P}$. The challenge: prove that if $\sim{}QC$, then there is a chain of primes whose union is not prime. Notice that if $x,y \not \in P$ then there is some $r$ such that $xry \not \in P$. Let’s see what the condition gives, for some $x,y$. - There is $P_1$ such that $x,y \not \in P_1$. So there is some $r_1$ such that $xr_1y \not \in P_1$. - There is $P_2$ such that $xr_1y \in P_2$ but $x,y \not \in P_2$. \[So $P_2 \not \sub P_1$\] So there is some $r_2$ such that $xr_2y \not \in P_2$. - There is $P_3$ such that $xr_2y \in P_3$ but $x,y \not \in P_3$. \[So $P_3 \not \sub P_1 \cup P_2$\] So there is some $r_3$ such that $xr_3y \not \in P_3$. - There is $P_4$ such that $xr_3y \in P_4$ but $x,y \not \in P_4$. \[So $P_4 \not \sub P_1 \cup P_2 \cup P_3$\] So there is some $r_4$ such that $xr_4y \not \in P_4$. - ... The key issue is the following property: (\*\*) For every prime $P$ and $a\not \in P$, $x,y \not \in {{\left<P,a\right>}}$, there is a prime $Q \supset P$ such that $a \in Q$ and $x,y \not \in Q$. If this is granted, we can construct our chain. Note that passing to the quotient, what we ask is this (in a prime ring): (\*\*) For every $a \neq 0$, $x,y \not \in {{\left<a\right>}}$, there is a prime $Q$ such that $a \in Q$ and $x,y \not \in Q$. Passing to the quotient modulo ${{\left<a\right>}}$, what we ask is this (in an arbitrary ring): (\*\*) For every $x,y \neq 0$, there is a prime $Q$ such that $x,y \not \in Q$. (Take $Q$ maximal with respect to not containing $x,y$? the monoid generated by $x,y$?) Compare and consider the condition: (\*) For every $x \neq 0$, there is a prime $Q$ such that $x \not \in Q$. (obviously, even in a commutative ring this would require $x$ to be non-nil. So are we asking too much?) Primes are f.g. --------------- \* Consider the class of rings where primes are f.g. Neocommutativity ---------------- In [@FS] it is mentioned that ACC(ideals) implies ‘neocommutativity’ (the product of f.g. ideals is f.g.; the notion is due to Kaplansky, unpublished). LO, GU etc. ----------- \* Consider comparison properties such as LO, GU, GD, INC with respect to . -index ------ \* Concerning the “CP dimension”: we want $\PPind(A) \leq \PPind(A/I) + \PPind(I)$. What about $\PPind(A\otimes B)$? \* We can ask questions like catenarity on the -index. regular rings ------------- (Remark: it seems that ${{\operatorname{End}}}(V)$ (vN-regular) have all ideals prime and principal, so in particular  holds. ) Finite module over center ------------------------- \* Question: suppose $A$ is a finite module over its center. Does it satisfy ? Strongly primes --------------- \* The union of a chain of strongly prime is always strongly prime. An example to check ------------------- (This example does not hold). \*An example (of something) in finite GK. Take the ring spanned - as a module - by all subwords of $xy^2xy^4xy^8\cdots$ and obtain a chain of ideals by ’shaving’ the word from the left to the right. These ideals are \*not\* prime, but modulo each one of them the ideal generated by $x$ is not nilpotent, whereas modulo the union, $xRx=0$. \*An attempt in finite GKdim: Define ’Agata numbers’ - a sequence $\{a_1,a_2,...\}$ with incredible growth. Let $R$ be the quotient of the free algebra obtained by setting $x^2=0$ and also $xy^mx=0$ for all $m$ not an Agata number. What about GKdimR? Let $I_n$ be the ideal of $R$ generated by all $xy^{a_i}x$ for $1 \leq i \leq n$. They are prime but their union is not even semiprime. ACC(primes) ----------- Does an affine  ring necessarily satisfy ACC(primes)? Affine algebras --------------- \*Is it true that any PI ring finitely generated as a ring over some commutative ring is ? If not, there exists a nice, quite surprising example of a ring which is a finite module both over a commutative ring and over a non-CP ring. This is obtained by a combination of a theorem of Schelter and Noether normalization lemma (which does not exist yet). This also means that if we prove that it is possible to raise and lower CP between finite modules then we prove that PI f.g. over comm. ring is CP. Crossed products ---------------- Use properties of normalizing extensions. Generic flatness ---------------- ? Further ideas for another article ================================= Chains of rings --------------- It is also possible to discuss chains of rings. It is well known that the union of a chain of prime (semiprime, simple) rings is prime (semiprime, simple respectively). The case of primitive rings is slightly different, however. Consider the Leavitt path algebras $L(E_{\alpha})$ of the graphs $E_{\alpha}$, where $E_{\alpha}$, for $\alpha$ an ordinal, is the graph whose vertices are the elements of $\alpha$, with an edge from $\beta$ to $\gamma$ iff $\beta<\gamma$. As in [@Leavitt], for all countable ordinals $\alpha$, the algebra $L(E_\alpha)$ is primitive but the algebra $E_{\aleph_1}$ is not primitive (as it does not satisfy the countable separation property), resulting in a chain of primitive rings with non-primitive union. However, the chain is uncountable and the rings are not finitely generated. Is there a countable chain of affine primitive rings with non-primitive union? A general theory for monomials ------------------------------ Let $F$ be a field, $S$ an algebra over $F$, and $R = F[x] * S$ the amalgamated product (which is the co-product in the category of algebras). The notion of a monomial is clearly defined. It is easy to see that if $I$ is generated by monomials then if $f \in I$, every monomial of $f$ is in $I$. There is also the notion of a leading monomial, but we might need some assumption on $S$ for this to satisfy the conditions below. We denote by $\overline{g}$ the leading monomial. Assume for any monomial $w$ and elements $f,g \in R$, we have that $\overline{fwg} = \overline{f}w \overline{g}$. We claim that if $P$ is ‘monomially prime’ (if $uRv \sub P$ then $u\in P$ or $v \in P$, for any two monomials $u,v$), then $P$ is prime. Indeed, assume $f R g \sub P$. Then for every monomial $w$ we have that $fwg \in P$ so $\overline{f}w\overline{g} = \overline{fwg} \in P$, which proves $\overline{f}R\overline{g} \sub P$, so by assumption $\overline{f} \in P$ or $\overline{g} \in P$, and the claim on $f,g$ follows by induction on the number of monomials. ((In order to produce the required technique, we need to define a leading monomial in a way that $\overline{fwg} = \overline{f}w \overline{g}$.)) [99]{} J. Bell and P. Colak, [*[Primitivity of finitely presented monomial algebras]{}*]{}, Journal of Pure and Applied Algebra [**[213]{}**]{}, (2009), 1299–1305. J. Bell, A. Smoktunowicz, [*[The prime spectrum of algebras of quadratic growth]{}*]{}, J. Algebra [**[319]{}**]{}(1) (2008), 414–431. A. Braun and N. Vonessen, [*[Integrality for PI-rings]{}*]{}, [**[151]{}**]{}(1), (1992), 39–79. J.W. Fisher and R.L. Snider, [*[On the von Neumann regularity of rings with regular prime factor rings]{}*]{}, Pacific J. Math. [**[54]{}**]{}(1), (1974), 135–144. K.R. Goodearl, “Von Neumann regular rings”, Monographs and Studies in Math. [**[4]{}**]{}, Pitman, London, 1979. Abrams, Bell and Rangswami on Leavitt path algebras. The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings, Igor Klep and Marcus Tressl, (2007). \[http://personalpages.manchester.ac.uk/staff/Marcus.Tressl/papers/Klep,Tressl-ThePrimeSpectrumandtheExtendedPrimeSpectrumofNoncommutativeRings.pdf\], T.Y. Lam, “A First Course in Noncommutative Rings”, Springer, 2001. C. Lanski, R. Resco and L. Small, [*[On the primitivity of prime rings]{}*]{}, J. Algebra [**[59]{}**]{}(2), (1979), 395-398. J. Okniński, [*[Structure of prime finitely presented monomial algebras]{}*]{}, J. Algebra [**[320]{}**]{}, (2008), 3199–3205. D. Passman, [*[Prime Ideals in Normalizing Extensions]{}*]{}, J. Algebra [**[73]{}**]{} (1981), 556–572. L.H. Rowen, “Polynomial Identities in Ring Theory”, 1980. L.H. Rowen, “Ring Theory”, Acedemic Press, 1988. S. Sarussi, [*[Maximal covers of chains of prime ideals]{}*]{}, preprint, [[`arXiv:1301.4340`]{}]{}. B. Greenfeld, A. Leroy, A. Smoktunowicz and M. Ziembowski, [*[Chains of Prime Ideals and Primitivity of $\mathbb Z$-Graded Algebras]{}*]{}, Alg. Rep. Theory [**[18]{}**]{}(3), 777–800, (2015). [^1]: For simplicity we deal only with countable chains throughout the paper, but the arguments are general.

ArXiv

--- abstract: | Let $\varphi: D\rightarrow \Omega$ be a homeomorphism from a circle domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$. We obtain necessary and sufficient conditions (1) for $\varphi$ to have a continuous extension to the closure $\overline{D}$ and (2) for such an extension to be injective. Further assume that $\varphi$ is conformal and that $\partial\Omega$ has at most countably many non-degenerate components $\{P_n\}$ whose diameters have a finite sum $\displaystyle\sum_n{\rm diam}(P_n)<\infty$. When the point components of $\partial D$ or those of $\partial \Omega$ form a set of $\sigma$-finite linear measure, we can show that $\varphi$ continuously extends to $\overline{D}$ if and only if all the components of $\partial\Omega$ are locally connected. This generalizes Carathéodory’s Continuity Theorem, that concerns the case when $D$ is the open unit disk $\left\{z\in\hat{{\mathbb{C}}}: |z|<1\right\}$, and allows us to derive a new generalization of the Osgood-Taylor-Caratheodry Theorem. 0.5cm **Keywords.** *Carathéodory’s Continuity Theorem, Peano Compactum, generalized Jordan domain* **MSC 2010: Primary 30A72, 30D40, Secondary 54C20, 54F25.** author: - Jun Luo - 'Xiao-Ting Yao' title: 'To Generalize Carathéodory’s Continuity Theorem[^1]' --- Introduction and What We Study ============================== There are two questions that are of particular interest from a topological viewpoint. In the first, we want to decide whether two spaces $X$ and $Y$ are topologically equivalent or homeomorphic, in the sense that there is a homeomorphism $h_1:X\rightarrow Y$. In the second, the spaces $X$ and $Y$ are respectively embedded in two larger spaces, say $\hat{X}$ and $\hat{Y}$, and we wonder whether a continuous map $h_2:X\rightarrow Y$ allows a continuous extension $\hat{h}_2: \hat{X}\rightarrow\hat{Y}$. Our study concerns a special case of the second question, when $X$ is a circle domain and $h_2$ a conformal homeomorphism sending $X$ onto a domain $Y\subset\hat{{\mathbb{C}}}$. In such a case $X$ and $Y$ are said to be [**conformally equivalent**]{}. Our major aim in this paper is to generalize Carathëodory’s Continuity Theorem [@Caratheodory13-a]. See also [@Arsove68-a Theorem 3] or [@Pom92 p.18]. A conformal homeomorphism $\varphi:{\mathbb{D}}\rightarrow \Omega\subset\hat{{\mathbb{C}}}$ of the unit disk ${\mathbb{D}}=\{z: |z|<1\}$ has a continuous extension $\overline{\varphi}: \overline{{\mathbb{D}}}\rightarrow\overline{\Omega}$ if and only if the boundary $\partial\Omega$ is a Peano continuum, [*i.e.*]{} a continuous image of the interval $[0,1]$. If $\Omega$ in the above theorem is a [**Jordan domain**]{}, so that its boundary is a [**Jordan curve**]{}, the extension $\overline{\varphi}: \overline{{\mathbb{D}}}\rightarrow\overline{\Omega}$ is actually injective. This has been obtained earlier by Osgood and Taylor [@Osgood-Taylor1913 Corollary 1] and independently by Carathéodory [@Caratheodory13-b]. It will be referred to as the Osgood-Taylor-Carathéodory Theorem. See for instance [@Arsove68-a Theorem 4]. Here we also call it shortly the OTC Theorem. A conformal homeomorphism $\varphi:{\mathbb{D}}\rightarrow \Omega\subset\hat{{\mathbb{C}}}$ has a continuous and injective extension to $\overline{{\mathbb{D}}}$ if and only if the boundary $\partial\Omega$ is a simple closed curve. There are very recent generalizations of the above OTC Theorem. See [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], and [@Ntalampekos-Younsi19 Theorem 6.1]. Those generalizations are closely connected with a very famous example of the first question, proposed in 1909 by Koebe [@Koebe09]. Is every domain $\Omega\subset\hat{{\mathbb{C}}}$ conformally equivalent to a circle domain ? When $\Omega$ is finitely connected, in the sense that its boundary has finitely many components, the above question is resolved by Koebe [@Koebe18]. See the following theorem. The special case when $\Omega$ is simply connected is discussed in the well known Riemann Mapping Theorem. Each finitely connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to a circle domain $D$, unique up to Möbius transformations. When $\Omega$ is at most countably connected, He and Schramm [@He-Schramm93] obtained the same result. Each countably connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to a circle domain, unique up to Möbius transformations. This covers some earlier and more resticted results that partially solve [**Koebe’s Question**]{}, when additional conditions on a countably connected domain $\Omega$ are assumed. Among others, one may see [@Strebel51] for such a result. A slightly more general version of the above theorem, on almost circle domains, is given by He and Schramm in [@He-Schramm95a]. Here $\Omega\subset A$ is a relative circle domain in $\Omega$ provided that each component of $A\setminus\Omega$ is either a point or a closed geometric disk. An equivalent statement, pointed out by He and Schramm in [@He-Schramm95a], reads as follows. Given a countably connected domain $A\subset\hat{{\mathbb{C}}}$, every relative circle domain $\Omega\subset A$ is conformally equivalent to a circle domain $D$, unique up to Möbius transformations. The uniqueness part of the above extended versions of [**Koebe’s Theorem**]{} comes from the conformal rigidity of specific circle domains. For circle domains that are at most countably connected and even for those that have a boundary with $\sigma$-finite linear measure, the conformal rigidity is known. See [@He-Schramm93 Theorem 3.1] and [@He-Schramm94]. To obtain the conformal rigidity of the underlying circle domains, He and Schramm actually employ some extended version of the OTC Theorem. See [@He-Schramm93 Theorem 3.2] for the case of countably connected domains. See [@He-Schramm95a Lemma 5.3] and [@He-Schramm95a Theorem 6.1] for the case of almost circle domains. Before addressing on what we study, we recall that in Carathéodory’s Continuity Theorem, the “only if” part follows from very basic observations. On the other hand, the “if” part may be obtained by using the prime ends of $\Omega$, or equivalently, the cluster sets of $\varphi$. See [@Caratheodory13-a] and [@CL66] for the theory of prime ends and for that of cluster sets. Moreover, by the Hahn-Mazurkewicz-Sierpiński Theorem [@Kuratowski68 p,256, $\S50$, II, Theorem 2], a compact connected metric space is a Peano continuum if and only if it is locally connected. Therefore, in Carathéodory’s Continuity Theorem one may replace the property of being a Peano continuum with that of being locally connected. In such a form, the same result still holds, if we change ${\mathbb{D}}$ into a circle domain that is finitely connected, [*i.e.*]{}, having finitely many boundary components. We will characterize all homeomorphisms $\varphi: D\rightarrow\Omega$ of an arbitrary circle domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ that allow a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ to the closure $\overline{D}$. We also analyse the restriction of $\overline{\varphi}$ to any boundary component of $D$, trying to find conditions for such a restriction to be injective. More importantly, we will find answers to the following. Under what conditions does $\varphi$ extend continuously to $\overline{D}$, if it is further assumed to be a conformal map ? What We Obtain and What Are Known ================================= In the first theorem we find a topological counterpart for Carathéodory’s Continuity Theorem. \[topological-cct\] Any homeomorphism $\varphi$ of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if the conditions below are both satisfied. - The boundary $\partial\Omega$ is a Peano compactum. - The oscillations of $\varphi$ satisfy $\displaystyle\underline{\lim}_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$. A [**Peano compactum**]{} means a compact metrisable space whose components are each a Peano continuum such that for any $C>0$ at most finitely many of the components are of diameter $>C$. A [**generalized Jordan domain**]{} is defined to be a domain $\Omega\subset\hat{{\mathbb{C}}}$ whose boundary $\partial\Omega$ is a Peano compactum, such that all the components of $\partial\Omega$ are each a point or a Jordan curve. And, for any $r>0$ and any point $z_0\in\partial D$, the oscillation of $\varphi$ at $C_r(z_0)\cap D$ is $\sigma_r(z_0)=\sup\{|\varphi(x)-\varphi(y)|: x,y\in D, |x-z|=|y-z|=r\}$. Here $C_r(z_0)=\{z: |z-z_0|=r\}$. The same philosophy has been employed by Arsove [@Arsove68-a]. Indeed, the result of Theorem \[topological-cct\] for simply connected $D$ is known [@Arsove68-a Theorem 1]. In the same work, Arsove also gives a topological counterpart for the OTC Theorem [@Arsove68-a Theorem 2]. In the next theorem,, we continue to obtain a topological counterpart for generalized Jordan domains in the second theorem. \[topological-otc\] Any homeomorphism $\varphi$ of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$ has a continuous injective extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if the conditions below are satisfied: - The domain $\Omega$ is a generalized Jordan domain. - The oscillations of $\varphi$ satisfy $\displaystyle\underline{\lim}_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$, - No arc on $\partial D$ of positive length is sent by $\overline{\varphi}$ to a single point of $\partial\Omega$. In the above theorems the homeomorphism $\varphi$ is not required to be conformal. When this is assumed and $D$ is a circle domain, three special cases are already known in which $\varphi$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$. See [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], and [@Ntalampekos-Younsi19 Theorem 6.1]. In each of these cases, the circle domain $D$ is required to have a boundary with $\sigma$-finite linear measure or to satisfy a quasi-hyperbolic condition, while $\Omega$ is either a circle domain or a generalized Jordan domain that is [*cofat*]{} in Schramm’s sense, so that all its complementary components are each a single point or closed Jordan domain that is not far from a geometric disk. When both $D$ and $\Omega$ are required to be generalized Jordan domains that are countably connected and cofat, any conformal homeomorphism $\varphi: D\rightarrow\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$ provided that the boundary map $\varphi^B$ gives a bijection between the point components of $\partial D$ and those of $\partial\Omega$. See [@Schramm95 Theorem 6.2]. Removing the requirement of cofatness, we will find new conditions for an arbitrary conformal homeomorphism $\varphi: D\rightarrow\Omega$ to extend continuously to the closure $\overline{D}$. This extends Carathéodory’s Continuity Theorem to infinitely connected circle domains and leads us to a new generalization of the OTC Theorem. Such a generalization has overlaps with but is not covered by any of the known extended versions of the OTC Theorem, that have been obtained in [@He-Schramm93; @He-Schramm94; @Schramm95; @Ntalampekos-Younsi19]. Recall that, by Theorem \[topological-cct\](1), we may confine ourselves to the case that the boundary $\partial\Omega$ is a Peano compactum. Therefore, in the third theorem we characterize all domains $\Omega\subset\hat{{\mathbb{C}}}$ such that the boundary $\partial\Omega$ is a Peano compactum. \[topology\_metric\] Each of the following is necessary and sufficient for an arbitrary domain $\Omega\subset\hat{{\mathbb{C}}}$ to have its boundary being a Peano compactum: \(1) $\Omega$ has property S, \(2) every point of $\partial \Omega$ is locally accessible, \(3) every point of $\partial \Omega$ is locally sequentially accessible, \(4) $\Omega$ is finitely connected at the boundary, and \(5) the completion of $\Omega$ under the diameter distance is compact. On the one hand, Theorem \[topology\_metric\] demonstrates an interplay between the topology of $\Omega$, that of the boundary $\partial\Omega$, and the completion of the metric space $(\Omega,d)$. Here $d$ denotes the diameter distance, which is also called the Mazurkiewicz distance. See [@Herron12] for a special sub-case of the above Theorem \[topology\_metric\], when $\Omega$ is assumed to be simply connected. On the other, Theorem \[topology\_metric\] is also motivated by and actually provides a generalization for a fundamental characterization of planar domains that have property $S$. See for instance [@Whyburn42 p.112, Theorem (4.2)], which will be cited wholly in this paper and is to appear as Theorem \[whyburn\_112\] (in Section \[proof\_1/2\] of this paper). Note that the completion of $(\Omega,d)$ is compact if and only if $\Omega$ is finitely connected at the boundary [@BBS16 Theorem 1.1]. The authors of [@BBS16] also obtain the equivalences between (2), (4) and (5) for countably connected domains $\Omega\subset\hat{{\mathbb{C}}}$ [@BBS16 Theorem 1.2] or slightly more general choices of $\Omega$ [@BBS16 Theorem 4.4]. The above Theorem \[topology\_metric\] improves these earlier results, by obtaining all these equivalences for an arbitrary planar domain $\Omega$ and relating them to the property of having a boundary that is a Peano compactum. Now, we are ready to present on two approaches, that are new, to generalize Carathéodory’s Continuity Theorem. To do that, we further suppose that the domain $\Omega$ has at most countably many non-degenerate boundary components $P_n$ whose diameters satisfy $\sum_n{\rm diam}(P_n)<\infty$. For the sake of convenience, a domain $\Omega$ satisfying the above inequality $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ concerning the diameters of its non-degenerate boundary components will be called a domain with [**diameter control**]{}. By the first approach, we obtain the following. \[arsove\] Let $\Omega\subset\hat{{\mathbb{C}}}$ be a domain with countably many non-degenerate boundary components $P_n$ such that the sum of diameters $\sum_n{\rm diam}(P_n)$ is finite. Suppose that the linear measure of $\displaystyle\partial\Omega\setminus\bigcup_nP_n$ is $\sigma$-finite. Then any conformal homeomorphism $\varphi:D\rightarrow \Omega$ from a circle domain $D$ onto $\Omega$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if $\partial\Omega$ is a Peano compactum. In the second approach, we require instead that the point components of $\partial D$ form a set of $\sigma$-finite linear measure. This happens if and only if the whole boundary $\partial D$ has a $\sigma$-finite linear measure. In other words, we have the following. \[arsove\_sigma\] Let $\Omega\subset\hat{{\mathbb{C}}}$ be a domain with diameter control, so that $\partial\Omega$ has at most countably many non-degenerate boundary components $P_n$ satisfying $\sum_n{\rm diam}(P_n)<\infty$. Let $D$ be a circle domain whose boundary has $\sigma$-finite linear measure. Then any conformal homeomorphism $\varphi:D\rightarrow \Omega$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ if and only if $\partial\Omega$ is a Peano compactum. Note that, in Theorems \[arsove\] and \[arsove\_sigma\], the continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ exists if and only if one of the five conditions given in Theorem \[topology\_metric\] is satisfied. Among others, Theorem \[arsove\_sigma\] has motivations from a recent work by He and Schramm [@He-Schramm94]. This works centers around the conformal rigidity of circle domains that have a boundary with $\sigma$-finite linear measure. Particularly, in the proof for [@He-Schramm94 Theorem 2.1] we find detailed techniques that are very useful in our study. He and Schramm [@He-Schramm94] consider conformal homeomorphisms between circle domains, while in Theorem \[arsove\_sigma\] we study conformal homeomorphisms from a circle domain $D$ onto a general planar domain $\Omega$. Note that the inequalities obtained in [@He-Schramm94 Lemmas 1.1 and 1.2 and 1.4] are among the crucial elements that constitute the proof for [@He-Schramm94 Theorem 2.1]. In order to obtain these inequalities, one needs to assume at least that the complementary components of $\Omega$ are $L$-nondegenerate for some constant $L>0$. Such domains are also called [**cofat domains**]{} in [@Ntalampekos-Younsi19] and in [@Schramm95]. Instead of assuming the property of being cofat, we focus on domains $\Omega$ with diameter control. This is the major difference between Theorem \[arsove\_sigma\] and the earlier results obtained in [@He-Schramm94; @Ntalampekos-Younsi19; @Schramm95]. For this flexibility, to choose $\Omega$ more freely among a large family of planar domains, we pay a price by assuming in addition the [**diameter control**]{}, so that $\Omega$ has at most countably many components whose diameters have a finite sum $\sum_n{\rm diam}(P_n)<\infty$. Note that in the cofat situation, there is a natural inequality $\sum_n\left({\rm diam}(P_n)\right)^2<\infty$, ensured by the fact that every domain on the sphere has a finite area. Theorems \[arsove\] and \[arsove\_sigma\] may be slightly improved by replacing $D$ with a generalized circle domain. See Theorems \[arsove-new\] and \[arsove\_sigma-new\]. From this we can infer a new generalization of the OTC Theorem. Such a generalization has overlaps with and is not covered by any of the earlier ones obtained in [@He-Schramm93 Theorem 3.2], [@He-Schramm94 Theorem 2.1], [@Ntalampekos-Younsi19 Theorem 1.6], and [@Schramm95 Theorem 6.2]. The original form of the OTC Theorem is about a conformal homeomorphism between two Jordan domains. In the next theorem, we extend the OTC Theorem to conformal homeomorphisms between two generalized Jordan domains with [**diameter control**]{}. \[OTC-b\] Given a conformal map $h: D\rightarrow \Omega$ between two generalized Jordan domains, such that both $\partial D$ and $\partial\Omega$ have at most countably many non-degenerate components, say $\{Q_n\}$ and $\{P_n\}$, whose diameters have a finite sum $\sum_n{\rm diam}P_n+\sum_n{\rm diam}(Q_n)<\infty$. Suppose that the point components of $\partial D$ or those of $\partial\Omega$ form a set of $\sigma$-finite linear measure. Then $\varphi$ extends to be a homeomorphism from $\overline{D}$ onto $\overline{\Omega}$. The other parts of our paper are arranged as follows. In section \[proof\_1/2\] we prove Theorems \[topological-cct\] and \[topological-otc\]. To do that, we firstly establish in subsection \[s-pc\] a connection between the topology of a planar domain $\Omega$ and that of its boundary $\partial\Omega$, showing that $\Omega$ has property $S$ if and only if $\partial\Omega$ is a Peano compactum. See Theorem \[property\_s\]. Then we discuss in subsection \[cluster\_set\] continuous function of a generalized Jordan domain and show that all the cluster sets of such a function are connected. See Theorem \[connected\_cluster\]. In this subsection, we also provide a non-trivial characterization of generalized Jordan domain. See Theorem \[jordan\]. Then, in subsection \[outline\] and in subsection \[outline-otc\], we respectively prove Theorems \[topological-cct\] and \[topological-otc\]. In section \[topology\], we prove Theorems \[topology\_metric\]. In section \[outline-1\] we firstly discuss a special case of Theorem \[arsove\], when the point components of $\partial\Omega$ form a set of zero linear measure. See Theorem \[sufficient\]. Then we use very similar arguments, with necessary adjustments and more complicated details, to construct a proof for Theorem \[arsove\]. In section \[outline-2\] we will prove Theorem \[arsove\_sigma\], when the point components of $\partial D$ form a set of zero linear measure. The proofs for this theorem and Theorem \[arsove\] are both based on an estimate of the oscillations for some conformal homeomorphism $\varphi: D\rightarrow\Omega$ of a circle domain $D$, so that Theorem \[topological-cct\] may be applied. Note that the results for Theorems \[arsove\] and \[arsove\_sigma\] still hold, even if the circle domain $D$ is replaced by a generalized Jordan domain. See Theorems \[arsove-new\] and \[arsove\_sigma-new\]. Finally, in section \[final\] we will prove Theorem \[OTC-b\]. Here we also recall earlier results that provide generalized versions of the classical OTC Theorem. See Theorems \[OTC-countable\] to \[OTC-3\]. These results arise very recent studies that provide the latest partial solutions to Koebe’s Question. They are comparable with Theorem \[OTC-b\], especially Theorem \[OTC-3\]. To Extend Homeomorphisms on a Circle Domain {#proof_1/2} =========================================== The target of this section is to prove Theorems \[topological-cct\] and \[topological-otc\]. To do that, we need a result that connects the topology of a planar domain $\Omega\subset\hat{{\mathbb{C}}}$ to that of its boundary, stating that $\Omega$ has property $S$ if and only if $\partial\Omega$ is a Peano compactum. We also need to analyze the cluster sets of a homeomorphism $h$, possibly not conformal, that sends a generalized Jordan domain $D$ onto a planar domain $\Omega$. Then we will be ready to construct the proofs for Theorems \[topological-cct\] and \[topological-otc\]. All these materials are presented separately in the following four subsections. Property $S$ and the property of being a Peano Compactum {#s-pc} -------------------------------------------------------- The property $S$ for planar domains and the property of being a Peano compactum, for compact planar sets, are closely connected. Such a connection is motivated by and provides a partial generalization for [@Whyburn42 p.112, Theorem (4.2)], which reads as follows. \[whyburn\_112\] If $\Omega\subset{\mathbb{C}}$ is a region whose boundary is a continuum the following are equivalent: - that $\Omega$ have Property $S$, - that every point of $\partial\Omega$ be regularly accessible from $\Omega$, - that every point of $\partial\Omega$ be accessible from all sides from $\Omega$, - that $\partial\Omega$ be locally connected, or equivalently, a Peano continuum. Here a region is a synonym of a domain and a metric space $X$ is said to have Property $S$ provided that for each $\epsilon>0$ the set $X$ is the union of finitely many connected sets of diameter less than $\epsilon$ [@Whyburn42 p.20]. Also, note that a point $p\in\partial\Omega$ is said to be [*regularly accessible from $\Omega$*]{} provided that for any $\epsilon>0$ there is a number $\delta>0$ such that for any $x\in\Omega$ with $|x-p|<\delta$ one can find a simple arc $\overline{xp}\subset \Omega\cup\{p\}$ that joins $x$ to $p$ and has a diameter $<\epsilon$ [@Whyburn42 p.111]. Note that a point $x\in\partial\Omega$ regularly accessible is also said to be [*locally accessible*]{} [@Arsove67]. The above theorem provides another motivation for Theorem \[property\_s\] that is of its own interest. We find a partial generalization for it, keeping items (ii) and (iii) untouched for the moment. \[property\_s\] A domain $\Omega\subset\hat{{\mathbb{C}}}$ has Property $S$ if and only if $\partial\Omega$ is a Peano compactum. When proving Theorem \[property\_s\] we will use two notions introduced in [@LLY-2019], the [**Schönflies condition**]{} and the [**Schönflies relation**]{} for planar compacta. \[Schonflies\_condition\] A compactum $K\subset{\mathbb{C}}$ satisfies the Schönflies condition provided that for the strip $W=W(L_1,L_2)$ bounded by two arbitrary parallel lines $L_1$ and $L_2$, the [**difference**]{} $\overline{W}\setminus K$ has at most finitely many components intersecting $L_1$ and $L_2$ at the same time. \[Schonflies\_relation\] Given a compact set $K\subset{\mathbb{C}}$. The Schönflies relation on $K$, denoted as $R_K$, is a reflexive relation such that two points $x_1\ne x_2\in K$ are related under $R_K$ if and only if there are two disjoint simple closed curves $J_i\ni x_i$ such that $\overline{U}\cap K$ has infinitely many components intersecting $J_1, J_2$ both. Here $U$ is the component of  $\hat{{\mathbb{C}}}\setminus(J_1\cup J_2)$ with $\partial U=J_1 \cup J_2$. By [@LLY-2019 Theorem 3], a compact $K\subset{\mathbb{C}}$ is a Peano compactum if and only if it satisfies the Schönflies condition. On the other hand, by [@LLY-2019 Theorem 7], a compact $K\subset{\mathbb{C}}$ is a Peano compactum if and only if $R_K$ is [**trivial**]{}, so that $(x,y)\in R_K$ indicates $x=y$. These results have motivations from recently developed topological models that are very helpful in the study of polynomial Julia sets. See for instance [@BCO11; @BCO13; @Curry10; @Kiwi04]. It is noteworthy that these models also date back to the 1980’s, when Thurston and Douady and their colleagues started applying [**Carathéodory’s Continuity Theorem**]{} to the study of polynomial Julia sets, which are assumed to be connected and locally connected. See for instance [@Douady93] and [@Thurston09]. We start from a proof by contradiction for the “only if” part. Suppose on the contrary that $\Omega$ has Property $S$ but $\partial\Omega$ is not a Peano compactum. There would exist two parallel lines $L_1,L_2$ such that for the unbounded strip $W=W(L_1,L_2)$ lying between $L_1$ and $L_2$, the [**difference**]{} $\overline{W}\setminus \partial\Omega$ has infinitely many components intersecting both $L_1$ and $L_2$. Denote those components as $W_1,W_2,\ldots$. Since every $W_i$ is arcwise connected, we may choose simple open arcs $\alpha_i\subset W_i$ joining a point $a_n$ on $\overline{W_i}\cap L_1$ to a point $b_n$ on $\overline{W_i}\cap L_2$. Renaming the arcs $\alpha_n$ if necessary, we may assume that for any $n>1$, the two arcs $\alpha_{n-1}$ and $\alpha_{n+1}$ lie in different components of $W\setminus\alpha_n$. Thus the arcs $\alpha_n$ may be arranged inside $W$ linearly from left to right. See the following figure for a simplified depiction of this arrangement. -0.25cm iin [1,...,50]{} [ (3+3\*i/50,5) – (2+3\*i/50,3) – (3.5+3\*i/50,1.5) – (3+3\*i/50,0); (11+3\*i/50,5) – (10+3\*i/50,3) – (11.5+3\*i/50,1.5) – (11+3\*i/50,0); ]{} (-2,0)–(25,0); (-2,5)–(25,5); (3,5) – (2,3) – (3.5,1.5) – (3,0); (6,5) – (5,3) – (6.5,1.5) – (6,0); (11,5) – (10,3) – (11.5,1.5) – (11,0); (14,5) – (13,3) – (14.5,1.5) – (14,0); at (25.75,5) [$L_1$]{}; at (25.75,0) [$L_2$]{}; at (1.4,2.9) [$\alpha_1$]{}; at (3.75,2.9) [$D_1$]{}; at (11.75,2.9) [$D_n$]{}; at (6.0,2.9) [$\alpha_2$]{}; at (9.4,2.9) [$\alpha_n$]{}; at (14.5,2.9) [$\alpha_{n+1}$]{}; at (11,5.5) [$a_n$]{}; at (11,-0.5) [$b_n$]{}; at (9.0,1.4) [$\cdots\cdots$]{}; at (16.5,1.4) [$\cdots\cdots$]{}; (11,0) circle (0.1); (11,5) circle (0.1); (14,0) circle (0.1); (14,5) circle (0.1); (6,0) circle (0.1); (6,5) circle (0.1); (3,0) circle (0.1); (3,5) circle (0.1); -0.75cm -0.5cm Let $D_n (n\ge1)$ be the unique bounded component of ${\mathbb{C}}\setminus(L_1\cup L_2\cup \alpha_n\cup\alpha_{n+1})$. Then each $D_n$ is a Jordan domain; moreover, the closed disk $\overline{D_n}$ contains a continuum $M_n\subset\partial\Omega$ that separates $\alpha_n$ from $\alpha_{n+1}$ in $\overline{D_n}$. Such a continuum $M_n$ must intersect both $L_1$ and $L_2$. Therefore, we can choose $x_n\in M_{2n-1}$ for all $n\ge1$ with $${\rm dist}(x_n,L_1)={\rm dist}(x_n,L_2):=\min\left\{\left|x_n-z\right|:\ z\in L_2\right\}.$$ Let $\epsilon>0$ be a number smaller than $\frac14\text{\rm dist}(L_1,L_2)$. Since $x_n\in M_{2n-1}\subset\partial\Omega$ we may find a point $y_n\in \Omega\cap D_{2n-1}$ such that $|x_n-y_n|<\epsilon$. Clearly, for any $m,n\ge1$ the two points $y_n, y_{n+m}\in \Omega$ are separated in $\overline{W}$ by $M_{2n}$. In other words, we have obtained an infinite set $\{y_n\}$ of points in $\Omega$, no two of which may be contained in a single connected subset of $\Omega$ that are of diameter less than $\epsilon$. This leads to a contradiction to the assumption that $\Omega$ has Property $S$. Then we continue to prove the “if” part. Again we will construct a proof by contradiction. Suppose on the contrary that $\partial\Omega$ is a Peano compactum but $\Omega$ does not have Property $S$. Then we could find a number $\epsilon>0$ and an infinite set $\{x_i\}$ of points $\Omega$ no two of which lie together in a single connected subset of $\Omega$ having diameter less than $3\epsilon$. By compactness of $\overline{\Omega}$, we may assume that $\lim\limits_{i\rightarrow\infty} x_i=x$. The way we choose the points $x_i$ then implies that $x\in\partial\Omega$. In the following, let $D_r(z)=\{w\in{\mathbb{C}}:\ |z-w|<r\}$ for $r>0$. Given a number $r\in(0,\epsilon)$, there exists an integer $i_0\ge1$ such that $x_i\in D_r(x)$ for all $i\ge i_0$. Fix a point $x_0\in \Omega$ with $|x-x_0|>\epsilon$ and choose arcs $\alpha_i\subset \Omega$ starting from $x_0$ and ending at $x_i$. Now for any $i\ge i_0$ let $a_i\in\alpha_i$ be the last point at which $\alpha_i$ leaves $\partial D_\epsilon(x)$; let $b_i\in\alpha_i$ be the first point after $a_i$ at which $\alpha_i$ encounters $\partial D_r(x)$. Let $\beta_i$ be the sub-arc of $\alpha_i$ between $a_i$ and $b_i$. Let $\gamma_i$ be the sub-arc of $\alpha_i$ between $b_i$ and $x_i$. (0,0) circle (5); (0,0) circle (2.5); at (6.25,0) [$\partial D_\epsilon(x)$]{}; at (0.75,-3.00) [$\partial D_r(x)$]{}; at (0.75,0) [$x$]{}; (0,0)circle (0.2); at (-6.75,0) [$x_0$]{}; (-6,0)circle (0.2); at (0.8,-1.2) [$x_i$]{}; (0,-1.2)circle (0.2); (-1.5,-2.0)circle (0.2); (-4,-3.0)circle (0.2); at (-4.5,-3.5) [$a_i$]{}; at (-1.35,-1.1) [$b_i$]{}; (-4,-3) – (-2.75,-3) – (-1.5,-2); at (-2.25,-3.5) [$\beta_i$]{}; -0.75cm Since no two of the points $\{x_i\}$ are contained by a single connected subset of $\Omega$ that is of diameter less than $3\epsilon$, we see that all those arcs $\{\beta_i: \ i\ge i_0\}$ are disjoint. Moreover, we can further infer that no two of them may be contained in the same component of $A\setminus\partial\Omega$, where $A$ denotes the closed annulus with boundary circles $\partial D_r(x)$ and $\partial D_\epsilon(x)$. Indeed, if this happens for $\beta_i, \beta_j$ with $k\ne j\ge i_0$ then $\beta_k\cup\beta_j$ lies in a component $P$ of $A\setminus \partial\Omega$, which is necessarily a subset of $\Omega$. In such a case the union $\gamma_k\cup\beta_k\cup P\cup\beta_j\cup\gamma_j$ would be a connected subset of $\Omega$ that contains $x_k, x_j$ both and is of diameter $<2\epsilon$. This is prohibited, by the choices of $\{x_i\}$. Therefore, if we denote by $P_i (i\ge i_0)$ the component of $A\setminus \partial\Omega$ that contains $\beta_i$ then $P_i\cap P_j=\emptyset$ for all $i\ne j\ge i_0$, indicating that $A\setminus\partial\Omega$ has infinitely many components that intersect the two circles $\partial D_r(x)$ and $\partial D_\epsilon(x)$ both. By [@LLY-2019 Definition 4], we see that the Schönflies relation on $\partial\Omega$ is not trivial. Thus, by [@LLY-2019 Theorem 7] we can infer that $\partial\Omega$ is not a Peano compactum. This is absurd, since we assume $\partial\Omega$ to be a Peano compactum. Theory of Cluster Sets for Generalized Jordan Domains {#cluster_set} ----------------------------------------------------- In this subsection we recall from [@CL66] some elements of cluster sets and characterize generalized Jordan domains as those that are simply connected at the boundary. For the sake of convenience, we will focus on continuous maps $h$ defined on generalized Jordan domains $U\subset\hat{{\mathbb{C}}}$. Since a Jordan curve separates $\hat{{\mathbb{C}}}$ into two domains, we see that $\partial U$ contains at most countably many components that are Jordan curves. Denote these boundary components of $U$ as $\{\Gamma_n\}$. Moreover, denote by $W_n$ the components of $\hat{{\mathbb{C}}}\setminus\Gamma_n$ that is disjoint from $U$. Here we are mostly interested in the case when $U$ is a circle domain and when $h$ is conformal. Given a continuous map $h: U\rightarrow V\subset\hat{{\mathbb{C}}}$. The cluster set $C(h,z_0)$ for $z_0\in\partial U$ is defined as $$\bigcap_{r>0}\overline{h(D_r(z_0)\cap U)},$$ where $D_r(z_0)=\{z: |z-z_0|<r\}$. This is a nonempty compact set, since these closures $\overline{h\left(D_r(z_0)\cap U\right)}$ with $r>0$ are considered as subsets of $\hat{{\mathbb{C}}}$. In the following, we will obtain the connectivity of all of them, by showing that [*every neighborhood of an arbitrary point $x\in\partial U$ contains a smaller neighborhood $N_x$ (in $\hat{{\mathbb{C}}}$) with $N_x\cap U$ connected.*]{} A domain with this property will be said to be [**simply connected at the boundary**]{}. This is a special sub-case for the property of being finitely connected at the boundary. \[connected\_cluster\] Each generalized Jordan domain is simply connected at the boundary. Consequently, if $h: U\rightarrow\hat{{\mathbb{C}}}$ is a continuous map every cluster set $C(h,z_0)$ with $z_0\in\partial U$ is a continuum. In particular, if $h$ is a homeomorphism its cluster sets are sub-continua of $\partial h(U)$. We need [**Zoretti Theorem**]{} [@Whyburn64 p.35,Corollary 3.11], which reads as follows. \[Zoretti\] If $K$ is a component of a compact set $M$ (in the plane) and $\epsilon$ is any positive number, then there exists a simple closed curve $J$ which encloses $K$ and is such that $J\cap M=\emptyset$, and every point of $J$ is at a distance less than $\epsilon$ from some point of $K$. By [**Zoretti Theorem**]{}, We only consider the case that $z_0$ lies on a non-degenerate boundary component $\Gamma_p$ for some $p\ge1$, which is a Jordan curve. By the well known Schönflies Theorem [@Moise p.72,Theorem 4], we may assume that $\Gamma_p=\{|z|=1\}$ and $U\subset {\mathbb{D}}^*:=\{|z|>1\}\subset\hat{{\mathbb{C}}}$. Given an open subset $V_0$ of $\hat{{\mathbb{C}}}$ that contains $z_0$, we may fix a closed geometric disk $D$ on $\hat{{\mathbb{C}}}$ that is centered at $z_0$ and is such that $(D\cap U)\subset V_0$. Denote by $\rho$ the distance between $D$ and $\hat{{\mathbb{C}}}\setminus V_0$. Since $U$ has property $S$, we may find finitely many regions that are of diameter less than $\rho$, say $M_n(1\le n\le N)$, so that $\bigcup_nM_n=U$ and that every $M_n$ has property $S$. See for instance [@Whyburn42 p.21, Theorem (15.41)]. Let $W$ be the union of all those $M_n$ with $z_0\in\overline{M_n}$. Renaming the regions $M_n$, we may assume that $z_0\in\overline{M_n}$ if and only if $1\le n\le N_0$ for some integer $N_0<N$. Using [**Zoretti Theorem**]{} repeatedly, we may choose a sequence of Jordan curves $\gamma_k\subset U$ that converge to $\Gamma_p$ under Hausdorff distance. Fix a point $z_k\in\gamma_k$ that is not contained in $D$, so that $z_\infty=\lim\limits_{k\rightarrow\infty}z_k\in\Gamma_p$. Assume that every $\gamma_k$ is parameterized as $g_k: [0,1]\rightarrow U$, with $g_k(0)=g_k(1)=x_k$, so that $g_k(t)$ traverses along $\gamma_k$ counter clockwise as $t$ runs through $[0,1]$. Fix a point $w_0\in U$ that lies in ${\mathbb{D}}^*\cap\partial D$, an open arc that is separated by $w_0$ into two open arcs, say $a$ and $b$. Going to an appropriate sub-sequence, if necessary, we may assume that every $\gamma_k$ separates $w_0$ from $z_0$ thus intersects both $a$ and $b$. Let $x_k\in\gamma_k$ be the last point at which $\gamma_k$ leaves $a$. Let $y_k\in\gamma_k$ be the first point, after $x_k$, that lies on $b$. Denote by $\alpha_k$ the sub-arc of $\gamma_k$ lying in $D$ that connects $x_k$ to $y_k$. Then $\alpha_k$ converges to the arc $D\cap\Gamma_p$ under Hausdorff distance. Now, let $W_k$ be the union of all these $M_n(1\le n\le N)$ that intersects $\alpha_k$. Then $W_k$ is connected hence is a region, that contains the whole arc $\alpha_k$. Since there are finitely many choices for the regions $M_n$, we can find an infinite subsequence, say $\{k_i: i\ge1\}$, such that these regions $W_{k_i}$ coincide with each other. We claim that each of these regions $W_{k_i}$ contains $W$. With this we see that for any open disk $D_r(z_0)\subset D$ with $r$ small enough (say, smaller than the distance from $z_0$ to $U\setminus W$), the union $V_1=W_{k_1}\cup D_r(z_0)$ is an open subset of $\hat{{\mathbb{C}}}$ we are searching for. This $V_1$ contains $z_0$, lies in $V_0$, and is such that $V_1\cap U$ is connected. To verify the above mentioned claim, we connect $z_0$ to a point $w_n\in M_n$ by an open arc $\beta_n\subset M_n$ for $1\le n\le N_0$. Since $\lim\limits_{k\rightarrow\infty}\alpha_k=D\cap\Gamma_p$ under Hasudorff distance and since $z_0$ is the center of $D$, we see that $\beta_n$ and hence $M_n$ intersects $\alpha_{k_i}$ for infinitely many $i$. From this we can obtain $M_n\subset W_{k_1}$ for $1\le n\le N_0$, indicating that $W\subset W_{k_i}$ for all $k_i$. In [@Ntalampekos-Younsi19 Proposition 3.5], Ntalampekos and Younsi obtain the result of Theorem \[connected\_cluster\], assuming in addition that $f$ be a homeomorphism of a generalized Jordan domain $D$ onto another planar domain. In Theorem \[connected\_cluster\], we only require that $f$ be a continuous map and the codomain may not be the complex plane or the extended complex plane. Our arguments are more direct and the whole proof is shorter. Moreover, we do not use Moore’s decomposition theorem [@Moore25]; actually we can not refer to this famous theorem, since $h$ may send $D$ into an arbitrary space. We refer to [@Ntalampekos-Younsi19 Theorem 3.6] and [@Ntalampekos-Younsi19 Lemma 3.7] for details concerning the roles that Moore’s decomposition theorem plays in the proof for [@Ntalampekos-Younsi19 Proposition 3.5]. There is another merit of Theorem \[connected\_cluster\] that is noteworthy, if one wants to characterize all planar domains that are simply connected at the boundary. By Theorem \[connected\_cluster\], a generalized Jordan domain is such a region. On the other hand, Theorem \[topology\_metric\] ensures that a region simply connected at the boundary necessarily has property $S$. For such a region $U$, all of its boundary components are Peano continua. Moreover, the assumption of simple connectedness at the boundary implies that none of them has a cut point. This means that the region $U$ is necessarily a generalized Jordan domain. From this we can infer a nontrivial criterion for generalized Jordan domain, in terms of simple connectedness at the boundary. This provides another justification for the introduction of generalized Jordan domain as a new term. \[jordan\] A planar domain is simply connected at the boundary if and only if it is a generalized Jordan domain. A Topological Counterpart for Generalized Continuity Theorem {#outline} ------------------------------------------------------------ This subsection proves [**Theorem \[topological-cct\]**]{}, a topological counterpart for [**Theorems \[arsove\] and \[arsove\_sigma\]**]{}. To begin with, let us recall a recent result by He and Schramm: [*each countably connected domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally homeomorphic to a circle domain $D$, unique up to Möbius equivalence*]{} [@He-Schramm93 Theorem 0.1]. Slightly later, they even prove that any domain $\Omega\subset\hat{{\mathbb{C}}}$ is conformally equivalent to some circle domain (1) if $\partial\Omega$ has at most countably many components that are not geometric circles or single points and (2) if the collection of those components has a countable closure in the space formed by all the components of $\partial\Omega$ [@He-Schramm95b; @He-Schramm95a]. However, Koebe’s conjecture is still open if $\partial\Omega$ has a complicated part like a cantor set of segments. Therefore, we may focus on domains $\Omega$ such that the boundary $\partial\Omega$ is “simple” in some sense, say from a topological point of view. In other words, we would like to limit our discussions to the case when $\partial\Omega$ does not possess a difficult topology. To this end, we examine the necessary conditions for $\varphi: D\rightarrow \Omega$ to have a continuous extension to the closure $\overline{D}$. At this point, we even do not assume the homeomorphism $\varphi: D\rightarrow \Omega$ to be conformal. \[necessary\] If a homeomorphism $\varphi: D\rightarrow \Omega$ of a generalized Jordan domain $D$ admits a continuous extension to $\overline{D}$ then $\partial\Omega$ is a Peano compactum and $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for $z_0\in\partial D$. Here $\displaystyle\sigma_r(z_0)=\sup_{D\cap C_r(z_0)}|\varphi(z_1)-\varphi(z_2)|$, with $C_r(z_0)=\{z: \ |z-z_0|=r\}$. This quantity is often called the [**oscillation**]{} of $\varphi$ on $C_r(z_0)\cap D$. Clearly, the uniform continuity of $\overline{\varphi}:\ \overline{D}\rightarrow\overline{\Omega}$ indicates that $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$. So the only thing to be verified is that the boundary $\partial\Omega$ is a Peano compactum. Assume that $\varphi$ has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Since $D$ is a generalized Jordan domain, it has Property $S$. Then the uniform continuity of $\overline{\varphi}$ ensures that $\Omega$ also has Property $S$, which then indicates that $\partial\Omega$ is a Peano compactum. The “only if” part of Theorem \[topological-cct\] is given in Theorem \[necessary\]. Before we continue to prove the “if” part, we want to mention some basic observations that are noteworthy. Firstly, the union of finitely many Peano continua is a Peano compactum. Secondly, if $\varphi$ is conformal then we always have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ provided that the following are satisfied: - $D$ has finitely many boundary components and each of them is locally connected, - $\varphi: D\rightarrow\Omega$ is a conformal homeomorphism. Therefore, Theorem \[topological-cct\] includes a simple case that extends the [**Continuity Theorem**]{} to the case of finitely connected circle domains $D$. Finally, the proof for [@Arsove68-a Theorem 1] already contains the necessary elements that will lead us to the result of Theorem \[topological-cct\], which includes Arsove’s theorem [@Arsove68-a Theorem 1] as a special subcase. In order to provide a self-contained argument and to make concrete clarifications, that become necessary when we involve infinitely connected domains, we also provide a proof for Theorem \[topological-cct\] that comes from a slight modification of Arsove’s proof for [@Arsove68-a Theorem 1]. Exactly the same argument is used in [@Arsove67 Lemma 2] which, as well as that used in [@Arsove68-a Theorem 1], employs the property of being locally sequentially connected. Here we follow the same line of arguments, as those adopted in [@Arsove68-a Theorem 1]. The only difference is that we use [**Property $S$**]{}, instead of the property of [**being locally sequentially accessible**]{}. Let $\varphi$ be a homeomorphism of a generalized Jordan domain $D$ onto a domain $\Omega\subset\hat{{\mathbb{C}}}$. Suppose that $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial D$ and that $\partial\Omega$ is a Peano compactum. It will suffice if we can show that each cluster set $C(\varphi,z_0)$ is a singleton. Suppose on the contrary that the cluster set $C(\varphi,z_0)$ at $z_0\in\partial D$ contains two points, say $w_1\ne w_2$. Then we can find an infinite sequence $z_n\rightarrow z_0$ of distinct points satisfying $\varphi\left(z_{2n-1}\right)\rightarrow w_1$ and $\varphi\left(z_{2n}\right)\rightarrow w_2$. Since $\partial\Omega$ is a Peano compactum, by Theorem \[property\_s\] we see that $\Omega$ has Property $S$. That is to say, for any number $\varepsilon>0$ we can find finitely many connected subsets of $\Omega$, say $N_1,\ldots,N_k$, satisfying $\displaystyle \bigcup_iN_i=\Omega$ and $\displaystyle\max_{1\le i\le k}{\rm diam}(N_i)<\varepsilon$. Choose a positive number $\varepsilon<\frac13|w_1-w_2|$. Then, there exist two of those connected sets $N_i$, say $N_1$ and $N_2$, such that - $N_1$ contains infinitely many points in $\{\varphi\left(z_{2n-1}\right)\}$, - $N_2$ contains infinitely many points in $\{\varphi\left(z_{2n}\right)\}$. Since $z_n\rightarrow z_0$ and since $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$, we can further choose a small enough number $r>0$ such that $\sigma_r(z_0)<\varepsilon$ and that the intersections $\displaystyle N_1\cap\{\varphi\left(z_{2n-1}\right)\}$ and $\displaystyle N_2\cap \{\varphi\left(z_{2n}\right)\}$ each contains at least one point outside $\varphi(D_r(z_0)\cap D)$ and at least one point inside. Therefore, we have $N_i\cap\varphi(C_r(z_0))\ne\emptyset$ for $i=1,2$. Let $M$ be the union of $\{w_1\}\cup N_1, \ \varphi(C_r(z_0))$, and $\{w_2\}\cup N_2$. As $\sigma_r(z_0)$ is defined to be the diameter of $\varphi(C_r(z_0))$, we have $|w_1-w_2|\le{\rm diam}(M)<3\varepsilon$. This is absurd, since we have chosen $\varepsilon<\frac13|w_1-w_2|$. A Topological Counterpart for Generalized OTC Theorem {#outline-otc} ----------------------------------------------------- This subsection proves Theorem \[topological-otc\]. To this end, we firstly investigate into the boundary behaviour of an arbitrary homeomorphism $\varphi: D\rightarrow\Omega$ of a generalized Jordan domain $D$, which has a continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$ to the whole closure $\overline{D}$. Here we recall that a generalized Jordan domain is a planar domain that satisfies the following two properties: 1. $\partial U$ is a Peano compactum, 2. each component of $\partial U$ is either a point or a Jordan curve. We have the following result, from which the “if” part of Theorem \[topological-otc\] is easily inferred. \[boundary\_property\] The restriction map $\overline{\varphi}_Q: Q\rightarrow P=\overline{\varphi}(Q)$ to any component $Q$ of $\partial D$ is non-alternating. Moreover, the whole extension $\overline{\varphi}: \overline{D}\rightarrow \overline{\Omega}$ is a monotone map if and only if $\Omega$ is also a generalized Jordan domain. Under the assumption in Theorem \[boundary\_property\], the boundary $\partial\Omega$ is a Peano compactum. Therefore, by Torhorst Theorem [@Kuratowski68 p.512, $\S61$, II, Theorem 4], we can infer that $\Omega$ is a generalized Jordan domain if and only if no component of its boundary $\partial\Omega$ has a cut point. Therefore, the “only if” part is indicated by Theorem \[necessary\]. Together with the above Theorem \[boundary\_property\], we have provided a complete proof for Theorem \[topological-otc\]. We firstly obtain the first half of the above theorem, showing that $\left.\overline{\varphi}\right|_{Q}$ is non-alternating for any component $Q$ of $\partial D$. Recall that a continuous map $f: A\rightarrow B$ is called a non-alternating transformation provided that for no two points $x,y\in B$ does there exist a separation $A\setminus f^{-1}(x)=A_1\cup A_2$ such that $y$ lies in $f(A_1)\cap f(A_2)$ [@Whyburn42 p.127, (4.2)]. From this one can infer that $f:A\rightarrow B$ is non-alternating if and only if $f(A_1)\cap f(A_2)=\emptyset$ for any $x\in B$ and for any separation $A\setminus f^{-1}(x)=A_1\cup A_2$. By [**Zoretti Theorem**]{}, the image $P=\overline{\varphi}(Q)$ is a component of $\partial\Omega$. By definition of non-alternating transformation, we only need to show that $\overline{\varphi}(A_1)\cap \overline{\varphi}(A_2)=\emptyset$ for any $x\in P$ and for any separation $Q\setminus\left(\overline{\varphi}\right)^{-1}\!(x)=A_1\cup A_2$. Assume on the contrary that there were a point $x\in P$ and a separation $Q\setminus\left(\overline{\varphi}\right)^{-1}\!(x)=A_1\cup A_2$ such that $\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$ for $z_i\in A_i (i=1,2)$. Set $x'=\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$. Since $D$ is a generalized Jordan domain, the component $Q$ of $\partial D$ must be a simple closed curve. Thus the point inverse $\left(\overline{\varphi}\right)^{-1}\!(x)$ contains two points $y_1\ne y_2$ such that $\{y_1,y_2\}$ separates $z_1$ from $z_2$ in $Q$. Since $D$ has property $S$, all boundary points of $D$ are accessible from $D$. Thus we can find an open arc $\alpha\subset D$ that connects $y_1$ to $y_2$. From this we see that $Q\cup\alpha$ is a $\theta$-curve and that $D\setminus\alpha$ consists of two domains. Let $U_i (i=1,2)$ be the one whose boundary contains $z_i$. Clearly, $J=\varphi(\alpha)\cup\{x\}$ is a Jordan curve and $\Omega\setminus J=\varphi(U_1)\cup \varphi(U_2)$. See the left part of Figure \[5.1\], -0.25cm [cccc]{} (0,0) circle (5); (3,4) – (11,4) –(11,-4)– (3,-4); at (8,-4.5) [$\alpha$]{}; (3,4)circle (0.2); at (2.2,3.7) [$y_1$]{}; (3,-4)circle (0.2); at (2.2,-3.7) [$y_2$]{}; (5,0)circle (0.2); at (4.0,0) [$z_2$]{}; (-5,-0)circle (0.2); at (-4.0,0) [$z_1$]{}; at (8,5) [$U_1$]{}; at (8,0) [$U_2$]{}; &&& (0,0) circle (5); (3,4) – (11,4) –(11,-4)– (3,-4); at (8,-4.5) [$\alpha$]{}; (-5,0) – (-5,5.4) – (8,5.4)– (8,4); (8,4)– (8,2)– (14,2) – (14,0)–(11,0); (11,0)–(5,0); at (15,1) [$\beta$]{}; (-5,0) – (-5,5.4) – (8,5.4)– (8,4); at (2,6.2) [$\beta_1$]{}; (8,4) – (11,4) –(11,0); (8,4)–(11,4)–(11,0); at (12,4) [$\beta_3$]{}; (11,0)–(5,0); at (8,-1) [$\beta_2$]{}; (3,4)circle (0.2); at (2.2,3.7) [$y_1$]{}; (3,-4)circle (0.2); at (2.2,-3.7) [$y_2$]{}; (5,0)circle (0.2); at (4.0,0) [$z_2$]{}; (-5,-0)circle (0.2); at (-4.0,0) [$z_1$]{}; (8,4)circle (0.2); at (8.7,4.7) [$b_1$]{}; (11,0)circle (0.2); at (11.7,-0.7) [$b_2$]{}; -0.25cm for relative locations of the arc $\alpha$, the domains $U_i$ and the points $y_i,z_i$. Now, fix an arc $\beta\subset D$ that connects $z_1$ to $z_2$ and denote by $\beta_i\subset(U_i\cap\beta)$ the maximal open sub-arc of $\beta$ that has $z_i$ as one of its ends. Denote by $b_i$ the other end point of $\beta_i$ for $i=1,2$. Obviously, we have $b_1,b_2\in\alpha$. Let $\beta_3$ be the closed sub-arc of $\alpha$ with ends $b_1,b_2$. Then we have an arc $\beta'=\beta_1\cup\beta_2\cup\beta_3$, lying in $D$ and intersecting $\alpha$ at $\beta_3$. See right part of Figure \[5.1\]. Since $\varphi: D\rightarrow\Omega$ is a homeomorphism, we know that $\varphi(\beta')=\varphi(\beta_1)\cup\varphi(\beta_2)\cup\varphi(\beta_3)$ is an arc contained in $\Omega$ such that (1) $\varphi(\beta')\cap\varphi(\alpha)=\varphi(\beta_3)$ and (2) $\varphi(\beta_i)\subset \varphi(U_i)$ for $i=1,2$. Since the simple closed curve $J=\{x\}\cup\varphi(\alpha)$ does not contain the point $x'=\overline{\varphi}(z_1)=\overline{\varphi}(z_2)$ and since each of $\varphi(\beta_i)$ has $x'$ as one of its ends, we can infer that $\varphi(\beta_1)$ and $\varphi(\beta_2)$ are both contained in a single component of $\hat{{\mathbb{C}}}\setminus J$, thus are both contained in a single component of $\Omega\setminus J$, which is either $\varphi(U_1)$ or $\varphi(U_2)$. This is absurd, since we have chosen $\beta_i\subset U_i(i=1,2)$ so that $\varphi(\beta_i)\subset\varphi(U_i)$. Then we go on to consider the latter half of Theorem \[boundary\_property\]. Since the“only if" part of which is obvious, we just discuss the “if” part. To this end, we recall that a special type of non-alternating maps come from the family of [*monotone maps*]{}. If we confine ourselves to continuous maps between compacta then, under a monotone map $f: X\rightarrow Y$, the pre-image of any point $y\subset Y$ is a sub-continuum of $X$. Therefore, if $P$ is a component of $\partial\Omega$ with $\varphi^B(Q)=P$ and if $P$ is a single point or is a Jordan curve then it has no cut point and hence the inverse $\overline{\varphi}^{-1}(x)$ for any $x\in P$ is a sub-continuum of $Q$. This means that the restriction $\left.\overline{\varphi}\right|_Q$ is monotone. Therefore, the whole extension $\overline{\varphi}$ is monotone provided that $\Omega$ is a generalized Jordan domain, too. On Domains $\Omega\subset\hat{{\mathbb{C}}}$ Whose Boundary is a Peano Compactum {#topology} ================================================================================= In this section we will provide a complete proof for Theorem \[topology\_metric\]. Namely, we shall prove that the following six conditions are equivalent for all domains $\Omega\subset\hat{{\mathbb{C}}}$: 1. $\partial\Omega$ is a Peano compactum. 2. $\Omega$ has property S. 3. All points of $\partial\Omega$ are locally accessible. 4. All points of $\partial\Omega$ are locally sequentially accessible. 5. $\Omega$ is finitely connected at the boundary. 6. The completion $\overline{\Omega}_d$ of the metric space $(\Omega,d)$ is compact. Our arguments will center around two groups of implications: $(1)\Leftrightarrow(2)\Leftrightarrow(5)\Leftrightarrow(6)$ and $(2)\Rightarrow(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$. The equivalence $(5)\Leftrightarrow(6)$ has been given in [@BBS16 Theorem 1.1]. The equivalence $(1)\Leftrightarrow(2)$ is obtained by Theorem \[property\_s\] in the previous section. The equivalence $(1)\Leftrightarrow(5)$ is to be established in Theorem \[S\_finitely\_connected\]. The implication $(2)\Rightarrow(3)$ is already known [@Whyburn42 p.111, (a)] and the implications $(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$ will be discussed in Theorem \[3-1-4-1\]. There are three issues we want to mention. Firstly, the notion of [**local accessibility**]{} coincides with that of [**regular accessibility**]{} in [@Whyburn42 p.112, Theorem (4.2)]. Here a point $x\in\partial\Omega$ is locally accessible from $\Omega$ if for any $\epsilon>0$ there is a number $\delta>0$ such that all points $z\in\Omega$ with $|z-x|<\delta$ may be connected to $x$ by a simple arc inside $\Omega\cup\{x\}$, whose diameter is smaller than $\epsilon$. Secondly, a point $\xi\in\partial\Omega$ is called [**locally sequentially accessible**]{} if for each $r>0$ and for each sequence $\{\xi_n\}$ of points in $\Omega$ that converge to $\xi$ the common part $\Omega\cap D_r(\xi)$, of $\Omega$ and the open disk $D_r(\xi)$ centered at $\xi$ with radius $r$, is an open set such that one of its components contains infinitely many $\xi_n$. Lastly, a domain $\Omega\subset\hat{{\mathbb{C}}}$ is [**finitely connected at the boundary point $x\in\partial\Omega$**]{} provided that for any number $r>0$ there is an open subset $U_x$ of $\hat{{\mathbb{C}}}$, lying in $D_r(x)$, such that $U_x\cap\Omega$ has finitely many components. In particular, if we further require that $U_x\cap\Omega$ be connected, we say that $\Omega$ is [**simply connected at $x$**]{}. If $\Omega$ is finitely connected at every of its boundary points, we say that $\Omega$ is [**finitely connected at the boundary**]{}. Similarly, if $\Omega$ is simply connected at every of its boundary points, we say that $\Omega$ is [**simply connected at the boundary**]{}. See Theorem \[jordan\] for a nontrivial characterization generalized Jordan domain, as planar domains that are simply connected at the boundary. \[S\_finitely\_connected\] $\Omega$ has property S if and only if it is finitely connected at the boundary. Suppose that $\Omega$ is finitely connected at the boundary. Given an arbitrary number $r>0$, we can find for any $x\in\partial\Omega$ an open set $G_x\subset\{z: |z-x|<\frac{r}{2}\}$ such that $G_x\cap\Omega$ has finitely many components [@BBS16 Definition 2.2]. Clearly, the collection $\{G_x: x\in\partial\Omega\}$ gives an open cover of the boundary $\partial\Omega$. So we can find a finite sub-cover of $\partial \Omega$, denoted as $\{G_1,\ldots, G_n\}$. Since $\Omega\setminus\left(\bigcup G_i\right)$ is a compact subset of $\Omega$, we can cover it with finitely many small disks contained in $\Omega$, with radius $<\frac{r}{2}$. For $1\le i\le n$ the intersection $G_i\cap\Omega$ has finitely many components. These components and the above-mentioned small disks, that cover $\Omega\setminus\left(\bigcup G_i\right)$, form a finite cover of $\Omega$ by sub-domains of $\Omega$ having a diameter $<r$. This shows that $\Omega$ has property S. On the other hand, assuming that $\Omega$ has property S. Given an arbitrary point $x\in\partial\Omega$ and any positive number $r$, we can cover $\Omega$ by finitely many domains $W_1,\ldots, W_N\subset\Omega$ of arbitrarily small diameter, say $\varepsilon\in(0,\frac{r}{3})$. Denote by $U_x$ the union of all those $W_i$ whose closure contains $x$ and by $E_x$ the union of all those $W_i$ whose closure does not contain $x$. Then $\overline{E_x}$ is a compact set, whose distance to $x$ is a positive number $r_x>0$. Let $$G_x=U_x\cup\{x\}\cup\left\{z\notin\Omega: |z-x|<\min\left\{\frac{r}{3},r_x\right\}\right\}.$$ Then $G_x\subset\left\{z: |z-x|<\frac{r}{2}\right\}$ is an open set with $G_x\cap\Omega=U_x$, which is the union of some of the domains $W_1,\ldots, W_N$ and hence has finitely many components. This verifies that $\Omega$ is finitely connected at $x$. Since $x$ and $r>0$ are both flexible we see that $\Omega$ is finitely connected at the whole boundary. \[3-1-4-1\] The implications $(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(1)$ hold. Thus Theorem \[topology\_metric\] is true. Without losing generality, we may assume that $\infty\in\Omega$. Under this context $\partial\Omega$ may be considered as a compactum on ${\mathbb{C}}$. Let us start from the implications $(3)\Rightarrow(1)$ and $(4)\Rightarrow(1)$, which will be obtained by a contrapositive proof. Suppose on the contrary that $\partial\Omega$ were not a Peano compactum. Then it would not satisfy the Schönflies condition [@LLY-2019 Theorem 3]. In other words, there would exist an unbounded closed strip $W$, whose boundary consists of two parallel lines $L_1\ne L_2$, such that $W\cap\partial\Omega$ has infinitely many components, say $W_n$ for $n\ge1$, each of which intersects both $L_1$ and $L_2$. See for instance [@LLY-2019 Lemma 3.8]. Let $L$ be the line parallel to $L_1$ with $${\rm dist}(L,L_1)={\rm dist}(L,L_2).$$ Then $L$ intersects $W_n$ for all $n\ge1$. Pick an infinite sequence of points $z_n\in (W_n\cap L)$ which converge to a limit point $z_0\in\partial\Omega$. Pick a point $\xi_n\in \Omega$ such that $\lim\limits_{n\rightarrow\infty}|\xi_n-z_n|=0$. Clearly, for infinitely many choices of $n\ge1$, no arc connecting $\xi_n$ to $z_0$ is disjoint from $L_1\cup L_2$. Thus $z_0$ is not locally accessible from $\Omega$. This verifies the implication $(3)\Rightarrow(1)$. On the other hand, if we fix a neighborhood $V_0$ of $z_0$, which entirely lies in the interior of $W$, then there are infinitely many $\xi_n$ that belong to distinct components of $V_0\cap \Omega$. This indicates that $z_0$ is not locally sequentially accessible from $\Omega$ and verifies the implication $(4)\Rightarrow(1)$. The rest of our proof is to verify the implication $(1)\Rightarrow(4)$. And we will follow the ideas used in the proof for [@Arsove67 Lemma 1]. Indeed, if we suppose on the contrary that some point $z_0\in\partial\Omega$ were not locally sequentially accessible from $\Omega$, then for some $\rho>0$ there would exist infinitely many components of $\Omega\cap D_\rho(z_0)$, with $D_\rho(z_0)=\{z: |z-z_0|\le\rho\}$, that intersect the smaller disk $D_{\rho/2}(z_0)$. Denote these components by $Q_n (n\ge1)$. Since each $Q_n$ intersects $C_\rho(z_0)=\{z: |z-z_0|=\rho\}$ and since each of them is path connected, we can find paths $\gamma_n\subset Q_n$, lying in $A_\rho(z_0)=\{z: \frac{\rho}{2}\le |z-z_0|\le\rho\}$, that connects a point on $C_\rho(z_0)$ to a point on $C_{\rho/2}(z_0)$. Let $P_n$ be the component of $Q_n\cap A_{\rho}(z_0)$ that contains $\gamma_n$. Clearly, all these $P_n(n\ge1)$ are each a component of $\Omega\cap A_\rho(z_0)$. From this we may conclude that the Schönflies relation $R_{\partial\Omega}$ contains a pair $(z_1,z_2)$ for some $z_1\in C_\rho(z_0)$ and some $z_2\in C_{\rho/2}(z_0)$. See [@LLY-2019 Lemma 3.8] and [@LLY-2019 Remark 3.9] for this conclusion. Thus $\partial\Omega$ is not a Peano compactum, since a compact $K\subset\hat{{\mathbb{C}}}$ is a Peano compactum if and only if $R_K$ is a trivial relation. To Generalize Continuity Theorem [— the first approach]{} {#outline-1} ========================================================= Our target of this section is to give a complete proof for Theorem \[arsove\]. Since Theorem \[topological-cct\] provides the “only if” part, we just discuss the “if” part. And the only problem is that, for domains $\Omega\subset\hat{{\mathbb{C}}}$ whose boundary $\partial\Omega$ is a Peano compactum having countably many non-degenerate components $\{P_n\}_{n=1}^\infty$, it is not known whether $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ holds for all $z_0\in\partial D$. We will obtain the following special case for Theorem \[arsove\]. \[sufficient\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ and all its point components form a set of [**zero**]{} linear measure. If $\partial\Omega$ is a Peano compactum then $\varphi$ has a continuous extension to $\overline{D}$. Theorem \[sufficient\] is benefited from ideas used in the main theorem of [@Arsove67], which reads as follows. Each of the following is necessary and sufficient for a bounded simply connected plane region $\Omega$ to have its boundary parametrizable as a closed curve [**(equivalently, being a Peano continuum)**]{}: 1. all points of $\partial\Omega$ are locally accessible, 2. all points of $\partial\Omega$ are locally sequentially accessible, 3. some (equivalently, any) Riemann mapping function $\varphi:{\mathbb{D}}\rightarrow \Omega$ for $\Omega$ can be extended to a continuous mapping of $\overline{{\mathbb{D}}}$ onto $\overline{\Omega}$. Here we use Property $S$ instead of the property of being locally sequentially accessible. As in earlier works, such as [@Arsove67; @Arsove68-a], we also need to estimate from above the oscillations of the homeomorphism $\varphi: D\rightarrow \Omega$. To do that, we assume in addition some control on the diameters of the non-degenerate components of $\partial\Omega$. On the other hand, we also need to deal with the point components of $\partial\Omega$, by assuming that they form a set that is small in terms of linear measure. In order to prove Theorem \[sufficient\], we only need to obtain the following Theorem \[oscillation\]. Our proof for Theorem \[oscillation\] uses a bijection between the boundary components of $D$ and those of $\Omega$. This bijection associates to any component $Q$ of $\partial D$ a component $P$ of $\partial\Omega$, which actually consists of all the cluster sets $C(\varphi,z_0)$ with $z_0\in Q$. In deed, by [**Zoretti Theorem**]{}, we can choose inductively an infinite sequence of simple closed curves $\Gamma_n\subset D$ such that for all $n\ge1$ we have: (1) every point of $\Gamma_n$ is at a distance less than $\frac{1}{n}$ from a point of $Q$; and (2) $\Gamma_{n+1}$ separates $Q$ from $\Gamma_n$. Let $U_n$ be the component of $\hat{{\mathbb{C}}}\setminus\varphi(\Gamma_n)$ that contains $\varphi(\Gamma_{n+1})$. Then $\{U_n\}$ is a decreasing sequence of Jordan domains with $\overline{U_{n+1}}\subset U_n$ for all $n\ge1$. Therefore, we know that $M=\cap_nU_n=\cap_n\overline{U_n}$ is a sub-continuum of $\hat{{\mathbb{C}}}\setminus U$, whose complement is connected. Consequently, $P=\partial M$ is a sub-continuum of $\partial\Omega$ and is a component of $\partial\Omega$, which consists of all the cluster sets $C(\varphi,z_0)$ with $z_0\in D$. Following He and Schramm [@He-Schramm93], we set $\varphi^B(Q)=P$. This gives a well defined bijection between boundary components of $D$ and those of $\Omega$. We can infer Theorem \[sufficient\] by combining Theorem \[topological-cct\] and the theorem below, in which we do not require that $\partial\Omega$ be a Peano compactum. The only assumptions are about the diameters of $P_n$ and about the linear measure of the difference $\partial\Omega\setminus\left(\bigcup_nP_n\right)$, the set consisting of all the point components of $\partial\Omega$. Therefore, the result we obtain here is just the oscillation convergence $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$, without mentioning the cluster sets $C(\varphi,z_0)$ for $z_0\in\partial Q_n$. \[oscillation\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ and all its point components form a set of zero linear measure. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$. Let $\Lambda_r(z_0)$ be the [**arc length**]{} of $\varphi(C_r(z_0)\cap D)$, with $C_r(z_0)=\{|z-z_0|=r\}$. Then we have $\displaystyle\inf\limits_{\rho<r<\sqrt{\rho}}\Lambda_r(z_0)\le\frac{2\pi R}{\sqrt{\log1/\rho}}$ for $0<\rho<1$. This result is often referred to as Wolff’s Lemma. See [@Pom92 p.20, Proposition 2.2] for instance. Therefore, $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$. This is however different from what we need to verify, which is $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$; since the oscillation $\sigma_r(z_0)$ is defined to be the [**diameter**]{} of $\varphi(C_r(z_0)\cap D)$. Let $\{k_i: i\ge1\}$ be the collection of all those integers $k_i$ with $P_{k_i}\subset U_n$, arranged so that $k_1<k_2<\cdots$. Recall that $Q_{k_i}$ denotes the component of $\partial D$ with $P_{k_i}=\varphi^B\left(Q_{k_i}\right)$. Given a point $z_0\in\partial Q_n$ and an arbitrary number $\epsilon>0$, we shall find a positive number $r<\epsilon$ such that $\sigma_r(z_0)<\epsilon$, which then completes our proof. To this end, we firstly fix a point $w_0\in \Omega$ and then use [**Zoretti Theorem**]{} to find a simple closed curve $\Gamma_i$ for each $P_{k_i}$ such that [**$\Gamma_i$ separates $w_0$ from $P_{k_i}$**]{} and that [**every point of $\Gamma_{k_i}$ is at a distance less than $2^{-i}\epsilon$ from some point of $P_{k_i}$**]{}. Clearly, we have $\displaystyle\sum_{i}{\rm diam}(\Gamma_i)<\infty$. For $i\ge1$, let $W_i^*$ denote the component of $\hat{{\mathbb{C}}}\setminus \Gamma_i$ that contains $P_{k_i}$; moreover, let $W_i$ denote the component of $\hat{{\mathbb{C}}}\setminus \varphi^{-1}(\Gamma_i)$ that contains $Q_{k_i}$. Then, fixing an integer $N\ge1$ with $\displaystyle \sum_{i=N+1}^\infty{\rm diam}(\Gamma_i)<\frac12\epsilon$, we continue to choose $r>0$ small enough, with $\varphi\left(C_r(z_0)\cap D\right)\subset U_n$, such that $C_r(z_0)\setminus Q_n$ intersects none of the boundary components $Q_{k_1},\ldots, Q_{k_N}$ of $D$. By Wollf’s Lemma, we have $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$. Thus we may further require that the above number $r$ is chosen so that $\Lambda_r(z_0)<\frac14\epsilon$. \[transboundary\_bridge\] Let $F_r$ consist of all the points $q$ in $C_{r}(z_0)\cap \partial D$ such that $\{q\}$ is a component of $\partial D$. Let $F_r^*$ consist of all the points $q^*\in\partial\Omega$ such that $\{q^*\}=\varphi^B(Q)$ for some component $Q$ of $\partial D$ that intersects $C_{r}(z_0)\cap \partial D$. Then the linear measure of $F_r^*$ is zero. Therefore, for the above $\epsilon>0$, we can find a countable cover of $F_r^*$ by open sets of diameter smaller than any constant $\delta>0$, say $\{V_k^*: k\ge1\}$, such that $\sum_j{\rm diam}\left(V_k^*\right)<\frac14\epsilon$. Since $\partial\Omega$ has at most countably many non-degenerate components and since its point components form a set of zero linear measure, the result of this lemma is immediate. Now, by flexibility of $\epsilon>0$, we see that the following lemma completes our proof. \[estimate\_of\_oscillation\] For the above mentioned $r$, the inequality $|\varphi(z_1)-\varphi(z_2)|<\epsilon$ holds for any fixed points $z_1\ne z_2$ lying on $C_{r}(z_0)\cap D$. To prove this lemma, we may consider the closed sub-arc of $C_{r}(z_0)\setminus Q_n$ from $z_1$ to $z_2$. Denote this arc as $\alpha$. Clearly, it is a compact set disjoint from each of $Q_n, Q_{k_1},\ldots, Q_{k_N}$. Moreover, denote by $M_\alpha$ the union of $\varphi(\alpha\cap D)$ with all the boundary components $\varphi^B(Q)$ of $\Omega$ with $Q$ running through the boundary components of $D$ that intersect $\alpha$. Then, we only need to verify that the diameter of $M_\alpha$ is less than $\epsilon$. Let us now consider the components $Q$ of $\partial D$, with $Q\cap\alpha\ne\emptyset$, such that $\varphi^B(Q)\subset U_n$ is a non-degenerate component of $\partial\Omega$. These components may be denoted as $Q_j$ for $j$ belonging to an index set ${\mathcal{J}}\subset\{k_1<k_2<\cdots\}$. Clearly, we have ${\mathcal{J}}\subset\{k_i: i\ge N+1\}$. Let $\left\{V_k^*: k\in{\mathcal{K}}\right\}$ be the cover of $F_r^*$ given in Lemma \[transboundary\_bridge\], so that $\sum_k{\rm diam}(V_k^*)<\frac14\epsilon$. Since all these sets $V_k^*$ are open in $\hat{{\mathbb{C}}}$, we can choose for each point $w\in F_r^*$ a Jordan curve $J_{w}\subset \Omega$ that lies in some $V_k^*$ and separates $w_0$ from the point component $\{w\}$ of $\partial\Omega$. Let $V_w^*$ be the component of $\hat{{\mathbb{C}}}\setminus J_w$ that contains $w$. Let $V_w$ be the component of $\hat{{\mathbb{C}}}\setminus\varphi^{-1}(J_w)$ that contains $\left(\varphi^B\right)^{-1}(\{w\})$, which is the component of $\partial D$ corresponding to $\{w\}$ under $\varphi^B$. On the other hand, the components of $\alpha\cap D$ form a countable family $\{\alpha_t: t\in{\mathcal{I}}\}$. All these $\alpha_t$ are open arcs or semi-closed arcs on the circle $C_r(z_0)$. In deed, exactly two of them are semi-closed. Now it is easy to see that $$\left\{W_i: i\in{\mathcal{J}}\right\}\ \bigcup\ \left\{V_w: w\in F_m\right\}\ \bigcup\ \left\{\alpha_t: t\in{\mathcal{I}}\right\}$$ is a cover of $\alpha$. Since each $\alpha_t$ is open in $\alpha$, we may choose finite index sets ${\mathcal{J}}_0\subset{\mathcal{J}}$, $F_0\subset F_m$ and ${\mathcal{I}}_0\subset{\mathcal{I}}$, such that $$\left\{W_i: i\in{\mathcal{J}}_0\right\}\ \bigcup\ \left\{V_w: w\in F_0\right\}\ \bigcup\ \left\{\alpha_t: t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $\alpha$. This indicates that $$\left\{W_i^*: i\in{\mathcal{J}}_0\right\}\ \bigcup\ \left\{V_w^*: w\in F_0\right\}\ \bigcup\ \left\{\varphi(\alpha_t): t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $M_\alpha$. Therefore, we can choose a finite subset ${\mathcal{K}}_0\subset{\mathbb{Z}}$ such that $$\left\{W_i^*: i\in{\mathcal{J}}_0\right\}\ \bigcup\ \left\{V_k^*: k\in {\mathcal{K}}_0\right\}\ \bigcup\ \left\{\varphi(\alpha_t): t\in{\mathcal{I}}_0\right\}$$ is a finite cover of $M_\alpha$, too. From this we can infer that, for the above mentioned points $z_1\ne z_2$ lying on $C_r(z_0)\cap D$, the inequality $$|\varphi(z_1)-\varphi(z_2)|<\sum_{j\in{\mathcal{J}}_0}{\rm diam}(\Gamma_j)+\sum_{k}{\rm diam}(V_k^*)+\sum_{t\in{\mathcal{I}}_0}{\rm diam}(\varphi(\alpha_t))<\frac12\epsilon+\frac14\epsilon+\frac14\epsilon=\epsilon$$ always holds. By flexibility of $z_1,z_2\in \alpha\cap D$, this leads to the result of Lemma \[estimate\_of\_oscillation\]. Now we have all the ingredients to construct a proof for Theorem \[arsove\]. To do that, we only need to obtain the result given in Theorem \[oscillation\] under a weaker assumption, saying that the point components of $\partial\Omega$ forms a set of $\sigma$-finite linear measure. Note that, in Theorem \[oscillation\], this set is assumed to be of zero linear measure. \[oscillation-a\] Given a circle domain $D$ and a conformal homeomorphism $\varphi: D\rightarrow \Omega$, where the boundary $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$ and all its point components form a set of $\sigma$-finite linear measure. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$. We shall follow the same ideas in proving Theorem \[oscillation\], except for a couple of minor adjustments. The first one is to infer a slightly more general version of Wolff’s lemma [@Pom92 p.20, Proposition 2.2]. \[Wolff\_lemma\_2\] Let $\varphi$ map a domain $D\subset{\mathbb{C}}$ conformally into a bounded domain $D_R(0)$. Let $C_r(z_0)=\{|z-z_0|=r\}$ and $\Lambda_r(z_0)$ the arc length of $\varphi(C_r(z_0)\cap D)$. Then for any $\epsilon>0$ and any number $\rho\in(0,1)$, there exists $N>0$ such that for all $n>N$, the interval $[\rho^{2^{n+1}},\rho^{2^{n}}]$ has a subset $E_n$ with positive measure such that $\displaystyle\sup_{r\in E_n}\Lambda_r(z_0)<\frac14\epsilon$. Denote $l(r)=\Lambda_r(z_0)$. Suppose on the contrary that there exists $\epsilon_0>0$ and an increasing sequence $\{n_k: k\ge1\}$ of integers such that $l(r){\geqslant}\epsilon_0$ for almost all $r\in A_n=[\rho^{2^{n+1}},\rho^{2^{n}}]$. Then a simple calculation would lead us to the following inequality $$\int_{A_{n_k}}l^2(r)\frac{dr}{r}{\geqslant}\epsilon^2_0\int_{A_{n_k}}\frac{dr}{r}= \epsilon^2_0\log\frac{1}{\rho^{2^{n_k}}}$$ for all $k\ge1$. Thus we have $$\displaystyle \int_0^\infty l^2(r)\frac{dr}{r}{\geqslant}\sum_k\int_{A_{n_k}}l^2(r)\frac{dr}{r}=\infty.$$ This is impossible, since Wolff’s lemma states that $\displaystyle \int_0^\infty l^2(r)\frac{dr}{r}{\leqslant}2\pi^2 R^2$. Therefore, the Generalized Wolff’s Lemma holds. The second adjustment is needed when we prove the result of Lemma \[transboundary\_bridge\]. The aim here is to obtain a number $r$ in the set $E_n$, as defined in the above Lemma \[Wolff\_lemma\_2\], such that $F_r^*$ has zero linear measure. Here we only assume that the point components of $\partial\Omega$ form a set of $\sigma$-finite linear measure. \[zero linear measure\] Let $F_r, F_r^*$ be defined as in Lemma \[transboundary\_bridge\]. The linear measure of $F_r^*$ is zero for all but countably many of $r\in E_n$. In this lemma, we only need to consider the case that the point components of $\partial\Omega$ form a set of finite linear measure. Since $\{F_r^*: r\in E_n\}$ are essentially pairwise disjoint Borel sets, in the sense that every two of them has at most countably many common points, one can directly infer the result of Lemma \[zero linear measure\]. Now, we can copy the result and the proof for Lemma \[estimate\_of\_oscillation\], and then infer Theorem \[oscillation-a\]. Combining this with Theorem \[topological-cct\], we readily have Theorem \[arsove\]. The result of Theorem \[oscillation\] still holds, if $D$ is only required to be a generalized Jordan domain. Actually, if $U_0$ denotes the component of $\hat{{\mathbb{C}}}\setminus Q_n$ containing $D$ then we can find a homeomorphism $H:\hat{{\mathbb{C}}}\rightarrow \hat{{\mathbb{C}}}$, sending $Q_n$ onto the unit circle, such that $\left.H\right|_{U_0}$ is conformal map between $U_0$ and $\{ z\in\hat{{\mathbb{C}}}: |z|>1 \}$. In such a way, we see that all the arguments in the proof for Theorem \[oscillation\] still work. Similarly, all the arguments in the proof for Theorem \[oscillation-a\] are valid, even if the circle domain $D$ is changed into a generalized Jordan domain. Combining this observation with Theorem \[topological-cct\], we can further extend the result of Theorem \[sufficient\] and obtain the following. \[arsove-new\] Let $\Omega_1$ be a generalized Jordan domain. Let $\varphi: \Omega_1\rightarrow \Omega_2$ be a conformal homeomorphism, where the boundary $\partial\Omega_2$ has at most countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ and all its point components form a set of $\sigma$-finite linear measure. Then $\varphi$ extends continuously to the closure $\overline{\Omega_1}$ if and only if $\partial\Omega_2$ is a Peano compactum. To Generalize Continuity Theorem [— the second approach]{} {#outline-2} ========================================================== Our target of this section is to prove Theorem \[arsove\_sigma\]. Let us start from a result that can be inferred as a direct corollary of [@He-Schramm94 Lemma 1.3], which reads as follows. \[HS-1994\_1.3\] Let $Z\subset{\mathbb{R}}^2$ be a Borel set of $\sigma$-finite linear measure, and let $X\subset{\mathbb{R}}$ be the set of points $x$ such that the section $\left(\{x\}\times{\mathbb{R}}\right)\cap Z$ is uncountable. Then $X$ has zero Lebesgue measure. In the above lemma, we may consider ${\mathbb{R}}^2$ as the complex plane ${\mathbb{C}}$, consisting of $re^{{\bf i}\theta}$ with $r>0$ and $0\le\theta<2\pi$. Then, we study the set $R_0$ of numbers $r>0$ such that the circle $\left\{re^{{\bf i}\theta}: 0\le\theta<2\pi\right\}$ intersects $Z$ at uncountably many points. For any $r_2>r_1>0$, we see that the part of $Z$ in the annulus $\{z\in{\mathbb{C}}: r_1\le|z|\le r_2\}$ is sent onto the rectangle $[r_1,r_2]\times[0,2\pi]$ by the map $re^{{\bf i}\theta} \mapsto (r,\theta)$. If we define the distance between $r_1e^{{\bf i}\theta_1}$ and $r_2e^{{\bf i}\theta_2}$ to be $|r_1-r_2|+|\theta_1-\theta_2|$, the previous map is actually bi-Lipschitz. Therefore, by Lemma \[HS-1994\_1.3\], we have \[small\_level\_sets\] Given a domain $D$ and a point $z_0\in \partial D$. Let $R_0$ denote the set of all $r>0$ such that $C_r(z_0)=\{z: |z-z_0|=r\}$ contains uncountably many point components of $\partial D$. If $\partial D$ has $\sigma$-finite linear measure then $R_0$ has zero Lebesgue measure. A combination of Theorem \[topological-cct\] with the following result will lead us to Theorem \[arsove\_sigma\]. \[oscillation2\] Given a conformal homeomorphism $\varphi: D\rightarrow \Omega$ of a circle domain $D$, where $\partial D$ has $\sigma$-finite linear measure and $\partial\Omega$ has countably many non-degenerate components $\{P_n\}$. Let $Q_n$ be the component of $\partial D$ with $\varphi^B(Q_n)=P_n$ for all $n\ge1$. If there exists an open set $U_n\supset P_n$ satisfying $\displaystyle\sum_{P_k\subset U_n}{\rm diam}(P_k)<\infty$ we have $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q_n$. In Theorem \[oscillation2\], we do not require that $\partial\Omega$ be a Peano compactum. The only assumptions are about the linear measure of $\partial D$ and about the diameters of $P_n$. Therefore, the result we obtain here is just the oscillation convergence $\liminf\limits_{r\rightarrow 0}\sigma_r(z_0)=0$ for all $z_0\in\partial Q$. Again, we say nothing about the cluster sets $C(\varphi,z_0)$ for $z_0\in\partial Q$. For any $r>0$ and $z_0\in Q$, let $C_r(z_0)=\{|z-z_0|=r\}$. By Lemma \[small\_level\_sets\], the boundary components of $D$ that intersect $C_r(z_0)$ forms a countable set for all $r$ except those lying in a set $R_0$ of zero Lebesgue measure. Let $\Lambda_r(z_0)$ be the [**arc length**]{} of $\varphi(C_r(z_0)\cap D)$. After a slight modification of the proof for Wolff’s Lemma in [@Pom92 p.20, Proposition 2.2], we can show that $$\displaystyle\inf\limits_{\rho<r<\sqrt{\rho},r\notin R_0}\Lambda_r(z_0)\le\frac{2\pi R}{\sqrt{\log1/\rho}}$$ holds for $0<\rho<1$. Therefore, $\liminf\limits_{r\rightarrow 0}\Lambda_r(z_0)=0$ and we can choose for any $\epsilon>0$ a decreasing sequence of numbers outside $R_0$, say $r_1>r_2>\cdots>r_m>\cdots$, such that $\lim\limits_{m\rightarrow 0}r_m=0$ and $\Lambda_{r_m}(z_0)<\frac12\epsilon$ for all $m{\geqslant}1$. The components of $\partial D$ intersecting $C_{r_m}(z_0)\setminus Q$ for any given $r_m$ form a countable set. Thus we denote them as $\{Q_{k_i}, i=1,2,\cdots\}$. We may assume that every $P_{k_i}=\varphi^B( Q_{k_i})$ lies in the open neighborhood $U_n$ of $P_n=\varphi^B(Q_n)$. This is possible by choosing a sufficiently small $r_1$. Moreover, we may rename $k_i$, if necessary, so that we have $k_1<k_2<\cdots$. Fix a point $w_0\in \Omega$ and then use [**Zoretti Theorem**]{} to find a simple closed curve $\Gamma_i$ for each $P_{k_i}$ such that [**$\Gamma_i$ separates $w_0$ from $P_{k_i}$**]{} and that [**every point of $\Gamma_i$ is at a distance less than $2^{-i}\epsilon$ from some point of $P_{k_i}$**]{}. Clearly, we have $\displaystyle\sum_{i}{\rm diam}(\Gamma_i)<\infty$. For $i\ge1$, let $W_i^*$ denote the component of $\hat{{\mathbb{C}}}\setminus \Gamma_i$ that contains $P_{k_i}$; moreover, let $W_i$ denote the component of $\hat{{\mathbb{C}}}\setminus \varphi^{-1}(\Gamma_i)$ that contains $Q_{k_i}$. Here $Q_{k_i}$ is the boundary component of $D$ with $\varphi^B\left(Q_{k_i}\right)=P_{k_i}$. Then, fixing an integer $N\ge1$ with $\displaystyle \sum_{i=N+1}^\infty{\rm diam}(\Gamma_i)<\frac12\epsilon$, we continue to choose a positive number $r\in \{r_m:m\ge1\}$ that is small enough so that $C_{r}(z_0)\setminus Q$ intersects none of the boundary components $Q_{k_1},\ldots, Q_{k_N}$ of $D$. Moreover, if we let $F_r^*$ be defined as in Lemma \[transboundary\_bridge\], then $F_r^*$ is a countable set and hence we can find a countable open cover $\{V_k^*\}$ of $F_r^*$ such that $\sum_k{\rm diam}(V_k^*)<\frac{\epsilon}{4}$. Consequently, we can follow a similar but simpler argument, as used in Lemma \[estimate\_of\_oscillation\], and verify that for the above $r$, the inequality $|\varphi(z_1)-\varphi(z_2)|<\epsilon$ holds for any fixed points $z_1\ne z_2$ lying on $C_{r}(z_0)\cap D$. This shall complete our proof. The above proof also works, even if the circle domain $D$ in Theorem \[oscillation2\] is changed into a generalized Jordan domain. Combining this observation with Theorem \[topological-cct\], we actually have the following. \[arsove\_sigma-new\] Let $\Omega_1$ be a generalized Jordan domain. Let $\varphi: \Omega_1\rightarrow \Omega_2$ be a conformal homeomorphism, where the boundary $\partial\Omega_2$ has at most countably many non-degenerate components $\{P_n\}$ with $\displaystyle\sum_n{\rm diam}(P_n)<\infty$ while all the point components of $\partial\Omega_1$ form a set of $\sigma$-finite linear measure. Then $\varphi$ extends continuously to the closure $\overline{\Omega_1}$ if and only if $\partial\Omega_2$ is a Peano compactum. To Generalize Osgood-Taylor-Carathéodory Theorem {#final} ================================================ This section addresses on a new generalization of the OTC Theorem, as given in Theorem \[OTC-b\]. We firstly recall some earlier results of a similar nature, which focus on domains that are not far from a circle domain in their metric structure. Then, we give a proof for Theorem \[OTC-b\]. Let us start from four earlier works of a very similar nature. The first comes from an extension theorem by He and Schramm. Let $\Omega,\Omega^*$ be open connected sets in the Riemann sphere and let $f: \Omega\rightarrow\Omega^*$ be a conformal homeomorphism between them. Let $W$ be an open subset of $B(\Omega)$, which is at most countable. Suppose that the boundary components of $\Omega$ corresponding to elements of $W$ are all circles and points and that the corresponding (under $f$) boundary components of $\Omega^*$ are also circles and points. Then $f$ extends continuously to the boundary components in $W$ and extends to be a homeomorphism between $\bigcup\{K: K\in W\}\cup\Omega$ and $\bigcup\{K^*: K^*\in f^B(W)\}\cup\Omega^*$. In the above theorem $B(\Omega)$ denotes the space of boundary components of $\Omega$. As a direct corollary we can obtain the following generalization of OTC Theorem. \[OTC-countable\] Every conformal homeomorphism $\varphi: D\rightarrow\Omega$ of a countably connected circle domain $D$ onto a circle domain $\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$. In the second, the circle domain $D$ is just required to have a boundary with $\sigma$-finite linear measure. Therefore, it will include as a special case the above Theorem \[OTC-countable\]. \[OTC-1\] Let $D$ be a circle domain in $\hat{{\mathbb{C}}}$ whose boundary has $\sigma$-finite linear measure. Let $\Omega$ be another circle domain and let $\varphi: D\rightarrow\Omega$ be a conformal homeomorphism. Then $\varphi$ extends to be a homeomorphism $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. In the third one, the circle domain $D$ is assumed to satisfy the so-called quasihyperbolic condition while $\Omega$ is only required to be a domain whose complement consists of points and a family of uniformly fat closed Jordan domains. Such a domain is just a cofat generalized Jordan domain. \[OTC-2\] Let $D$ be a circle domain with $\infty\in D$ and let $h$ be conformal map from $D$ onto another domain $\Omega$ with $\infty=h(\infty)\in\Omega$. Suppose that $D$ satisfies the quasihyperbolic condition and that the complementary components of $\Omega$ are uniformly fat closed Jordan domains and points. Then $h$ extends to be a homeomorphism from $\overline{D}$ onto $\overline{\Omega}$. The last one may be inferred from [@Schramm95 Theorem 6.2], in which $D$ and $\Omega$ are both allowed to be generalized Jordan domains that are cofat. \[OTC-3\] Let $\varphi: D\rightarrow \Omega$ be a conformal homeomorphism between generalized Jordan domains that are countably connected and cofat. Suppose that for any component $Q$ of $\partial D$ the corresponding component $P=\varphi^B(Q)$ of $\partial\Omega$ is a singleton if and only if $Q$ is a singleton. Then every conformal homeomorphism $\varphi: D\rightarrow\Omega$ of $D$ onto $\Omega$ extends to be a homeomorphism between $\overline{D}$ and $\overline{\Omega}$. Theorem \[OTC-b\] is comparable with Theorem \[OTC-3\]. There are two major differences. Firstly, we do not require the domains $D,\Omega$ to be countably connected. Secondly, the property of being cofat is replaced by two properties: (1) for one of them the point boundary components form a set of $\sigma$-finite linear measure and (2) for both of them the diameters of the non-degenerate boundary components have a finite sum. Therefore, Theorem \[OTC-b\] is an OTC Theorem for generalized Jordan domains that [**may not be cofat**]{}. Its proof is given as below. If the point components of $\partial\Omega$ form a set of $\sigma$-finite linear measure we apply Theorem \[arsove-new\] to the map $\varphi: D\rightarrow\Omega$ and obtain a well-defined continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Then, applying Theorem \[arsove\_sigma-new\], we see that the inverse map $\psi=\varphi^{-1}: \Omega\rightarrow D$ also extends to be acontinuous map $\overline{\psi}: \overline{\Omega}\rightarrow\overline{D}$. Consequently, we can check that $\overline{\varphi}\circ \overline{\psi}=id_{\overline{\Omega}}$ and $\overline{\psi}\circ \overline{\varphi}=id_{\overline{D}}$. This indicates that $\overline{\varphi}$ and $\overline{\psi}$ are both injective. If the point components of $\partial D$ form a set of $\sigma$-finite linear measure we apply Theorem \[arsove\_sigma-new\] to the map $\varphi: D\rightarrow\Omega$ and obtain a well-defined continuous extension $\overline{\varphi}: \overline{D}\rightarrow\overline{\Omega}$. Then, applying Theorem \[arsove-new\] to the inverse map $\psi=\varphi^{-1}: \Omega\rightarrow D$, we obtain another continuous map $\overline{\psi}: \overline{\Omega}\rightarrow\overline{D}$ that extends $\psi$. Similarly, we can infer that $\overline{\varphi}$ and $\overline{\psi}$ are both injective. [**Acknowledgement**]{}. The authors are grateful to Christopher Bishop at SUNY, to Malik Younsi at University of Hawaii Hanoa, and to Xiaoguang Wang at Zhejiang University for private communications that are of great help, including important references and valuable ideas with concrete mathematical details. [10]{} Maynard G. Arsove. Intrinsic characterization of regions bounded by closed curves. , 34:425–429, 1967. Maynard G. Arsove. The [O]{}sgood-[T]{}aylor-[C]{}arathéodory theorem. , 19:38–44, 1968. Anders Björn, Jana Björn, and Nageswari Shanmugalingam. The [M]{}azurkiewicz distance and sets that are finitely connected at the boundary. , 26(2):873–897, 2016. Alexander M. Blokh, Clinton P. Curry, and Lex G. Oversteegen. Locally connected models for [J]{}ulia sets. , 226(2):1621–1661, 2011. Alexander M. Blokh, Clinton P. Curry, and Lex G. Oversteegen. Finitely [S]{}uslinian models for planar compacta with applications to [J]{}ulia sets. , 141(4):1437–1449, 2013. Constantin Carath[é]{}odory. Über die [B]{}egrenzung einfach zusammenhängender [G]{}ebiete. , 73(3):323–370, 1913. Constantin Carath[é]{}odory. Über die gegenseitige [B]{}eziehung der [R]{}änder bei der konformen [A]{}bbildung des [I]{}nneren einer [J]{}ordanschen [K]{}urve auf einen [K]{}reis. , 73(2):305–320, 1913. E. F. Collingwood and A. J. Lohwater. . Cambridge Tracts in Mathematics and Mathematical Physics, No. 56. Cambridge University Press, Cambridge, 1966. Clinton P. Curry. Irreducible [J]{}ulia sets of rational functions. , 16(5-6):443–450, 2010. Adrien Douady. Descriptions of compact sets in [${\bf C}$]{}. In [*Topological methods in modern mathematics ([S]{}tony [B]{}rook, [NY]{}, 1991)*]{}, pages 429–465. Publish or Perish, Houston, TX, 1993. R. W. FitzGerald and P. M. Swingle. Core decomposition of continua. , 61:33–50, 1967. Zheng-Xu He and Oded Schramm. Fixed points, [K]{}oebe uniformization and circle packings. , 137(2):369–406, 1993. Zheng-Xu He and Oded Schramm. Rigidity of circle domains whose boundary has [$\sigma$]{}-finite linear measure. , 115(2):297–310, 1994. Zheng-Xu He and Oded Schramm. The inverse [R]{}iemann mapping theorem for relative circle domains. , 171(1):157–165, 1995. Zheng-Xu He and Oded Schramm. Koebe uniformization for “almost circle domains”. , 117(3):653–667, 1995. David A. Herron. Riemann maps and diameter distance. , 119(2):140–147, 2012. Jan Kiwi. eal laminations and the topological dynamics of complex polynomials. , 184(2):207–267, 2004. Paul Koebe. Über die [U]{}niformisierung der algebraischen [K]{}urven. [I]{}. , 67(2):145–224, 1909. Paul Koebe. Abhandlungen zur [T]{}heorie der konformen [A]{}bbildung. , 2(1-2):198–236, 1918. K. Kuratowski. . New edition, revised and augmented. Translated from the French by A. Kirkor. Academic Press, New York, 1968. Benoît Loridant, Jun Luo, and Yi Yang. A core decomposition of compact sets in the plane. , 343:219–244, 2019. Jun Luo, Yi Yang, and Xiao-Ting Yao. The peano model for rational julia sets. , 2019. Edwin E. Moise. . Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, Vol. 47. R. L. Moore. Concerning upper semi-continuous collections of continua. , 27(4):416–428, 1925. Dimitrios Ntalampekos and Malik Younsi. Rigidity theorems for circle domains. , online, 2019. William F. Osgood and Edson H. Taylor. Conformal transformations on the boundaries of their regions of definitions. , 14(2):277–298, 1913. Christian Pommerenke. , volume 299 of [ *Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, Berlin, 1992. Oded Schramm. Transboundary extremal length. , 66:307–329, 1995. Kurt Strebel. Über das [K]{}reisnormierungsproblem der konformen [A]{}bbildung. , 1951(101):22, 1951. William P. Thurston. On the geometry and dynamics of iterated rational maps. In [*Complex dynamics*]{}, pages 3–137. A K Peters, Wellesley, MA, 2009. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. Gordon Thomas Whyburn. . American Mathematical Society Colloquium Publications, v. 28. American Mathematical Society, New York, 1942. Gordon Thomas Whyburn. . Second, revised edition. Princeton Mathematical Series, No. 23. Princeton University Press, Princeton, N.J., 1964. [^1]: Supported by Chinese National Natural Science Foundation Projects \# 11871483 and 11771391.

ArXiv

--- abstract: | The asymptotic behaviour of the solutions of Poincaré’s functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions $W$ of the complex plain. It is known [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] that $f(z)\sim\exp(z^\rho F(\log_\lambda z))$, if $f(z)\to\infty$ for $z\to\infty$ and $z\in W$, where $F$ denotes a periodic function of period $1$ and $\rho=\log_\lambda\deg(p)$. In the present paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function $F$ is characterised in terms of geometric properties of the Julia set of $p$. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of $f$ is related to the harmonic measure on the Julia set of $p$. author: - | GREGORY DERFEL\ Department of Mathematics and Computer Science,\ Ben Gurion University of the Negev, Beer Sheva 84105, Israel\ e-mail PETER J. GRABNER[^1]\ Institut für Analysis und Computational Number Theory (Math A),\ Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria\ e-mail\ - | FRITZ VOGL\ Institut für Analysis und Scientific Computing, Technische Universität Wien,\ Wiedner Hauptstraße 8–10, 1040 Wien, Austria\ e-mail title: Complex asymptotics of Poincaré functions and properties of Julia sets --- *Dedicated to Robert F. Tichy on the occasion of his ^th^ birthday.* Introduction {#sec:introduction} ============ Historical remarks {#sec:historical-remarks} ------------------ In his seminal papers [@Poincare1886:une_classe_etendue; @Poincare1890:une_classe_nouvelle] H. Poincaré has studied the equation $$\label{Eq 1} f(\lambda z)= R(f(z)),\quad z \in {\mathbb{C}},$$ where $R(z)$ is a rational function and $\lambda\in{\mathbb{C}}$. He proved that, if $R(0)=0$, $R'(0)=\lambda$, and $|\lambda|>1$, then there exists a meromorphic or entire solution of (\[Eq 1\]). After Poincaré, (\[Eq 1\]) is called [*the Poincaré equation*]{} and solutions of (\[Eq 1\]) are called [*the Poincaré functions* ]{}. The next important step was made by G. Valiron [@Valiron1923:lectures_on_general; @Valiron1954:fonctions_analytiques], who investigated the case, where $R(z)=p(z)$ is a polynomial, i.e. $$\label{eq:poincare} f(\lambda z)=p(f(z)),\quad z \in {\mathbb{C}},$$ and obtained conditions for the existence of an entire solution $f(z)$. Furthermore, he derived the following asymptotic formula for $M(r)=\max_{|z|\leq r}|f(z)|$: $$\label{Eq 3} \log M(r)\sim r^{\rho}F\left(\frac{\log r}{\log |\lambda|}\right), \quad r\rightarrow \infty.$$ Here $F(z)$ is a $1$-periodic function bounded between two positive constants, $\rho=\frac{\log d}{\log |\lambda|}$ and $d=\deg p(z)$. Different aspects of the Poincaré functions have been studied in the papers [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Eremenko_Levin1989:periodic_points_polynomials; @Eremenko_Sodin1990:iterations_rational_functions; @Ishizaki_Yanagihara2005:borel_and_julia; @Romanenko_Sharkovsky2000:long_time_properties]. In particular in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions], in addition to (\[Eq 3\]), asymptotics of entire solutions $f(z)$ on various rays $re^{i \vartheta}$ of the complex plane have been found. It turns out that this asymptotic behaviour heavily depends on the arithmetic nature of $\lambda$. For instance, if $\operatorname{\mathrm{arg}}\lambda=2\pi\beta$, and $\beta$ is irrational, then $f(z)$ is unbounded along any ray $\operatorname{\mathrm{arg}}z={\vartheta}$ (cf. [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions]). Assumptions {#sec:assumptions} ----------- In the present paper we concentrate on the simplest, but maybe most important case for applications, namely, when $\lambda$ is real and $p(z)$ is a real polynomial (i. e. all coefficients of $p(z)$ are real). It is known from [@Valiron1954:fonctions_analytiques] and [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] that, if $f(z)$ is an entire solution of , then the only admissible values for $f_0=f(0)$ are the fixed points of $p(z)$ (i. e. $p(f_0)=f_0$). Moreover, entire solutions exist, if and only if there exists an $n_0\in{\mathbb{N}}$ such that $$\lambda^{n_0}=p'(f_0).$$ It was proved in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Propositions 2.1–2.3] that the general case may be reduced to the simplest case $$f(0)=p(0)=0\text{ and }p'(0)=\lambda>1$$ by a change of variables. In the same vein, we can assume without loss of generality that $f'(0)=1$ and the polynomial $p$ is monic (i. e. the leading coefficient is $1$) $$\label{eq:poly} p(z)=z^d+p_{d-1}z^{d-1}+\cdots+p_1z.$$ Poincaré and Schröder equations {#sec:poinc-schr-equat} ------------------------------- The functional equation with the additional (natural) conditions $f(0)=0$ and $f'(0)=1$ is closely related to Schröder’s functional equation (cf. [@Schroeder1871:uber_iterierte_funktionen]) $$\label{eq:schroeder} g(p(z))=\lambda g(z),\quad g(0)=0\text{ and }g'(0)=1$$ which was used by G. Koenigs [@Koenigs1884:recherches_sur_integrales; @Koenigs1885:nouvelles_recherches_sur] to study the behaviour of $p$ under iteration around the repelling fixed point $z=0$. By definition, $g$ is the local inverse of $f$ around $z=0$. Both functions together provide a linearisation of $p$ around its repelling fixed point $z=0$ $$g(p(f(z)))=\lambda z\text{ and }g(p^{ (n)}(f(z)))=\lambda^n z,$$ where $p^{(n)}(z)$ denotes the $n$-th iterate of $p$ given by $p^{(0)}(z)=z$ and $p^{(n+1)}(z)=p(p^{(n)}(z))$. We note here that and are also called Schröder equation by some authors. For instance, the value distribution of solutions of the Poincaré (alias Schröder) equation has been investigated in [@Ishizaki_Yanagihara2005:borel_and_julia]. Branching processes and diffusion on fractals {#sec:branch-proc-diff} --------------------------------------------- Iterative functional equations occur in the context of branching processes (cf. [@Harris1963:theory_branching_processes]). Here a probability generating function $$q(z)=\sum_{n=0}^\infty p_nz^n$$ encodes the offspring distribution, where with $p_n\geq0$ is the probability that an individual has $n$ offspring in the next generation (note that $q(1)=1$). The growth rate $\lambda=q'(1)$ decides whether the population is increasing ($\lambda>1$) or dying out $\lambda\leq1$. In the first case the branching process is called *super-critical*. The probability generating function $q^{(n)}(z)$ ($n$-th iterate of $q$) encodes the distribution of the size $X_n$ of the $n$-th generation under the offspring distribution $q$. In the case of a super-critical branching process it is known that the random variables $\lambda^{-n}X_n$ tend to a limiting random variable $X_\infty$. The moment generating function of this random variable $$f(z)=\mathbb{E}e^{-zX_\infty}$$ satisfies the functional equation (cf. [@Harris1963:theory_branching_processes]) $$f(\lambda z)=q(f(z)),$$ which is , if $q$ is a polynomial. Furthermore, this equation can be transformed into , if $q$ is conjugate to a polynomial by a Möbius transformation, especially $q(z)=\frac1{p(1/z)}$, where $p$ is a polynomial. Branching processes have been used in [@Barlow1998:diffusions_on_fractals; @Barlow_Perkins1988:brownian_motion_sierpinski; @Lindstroem1990:brownian_motion_nested] to model time for the Brownian motion on certain types of self-similar structures such as the Sierpiński gasket. In this context the zeros of the solution of are the eigenvalues of the infinitesimal generator of the diffusion (“Laplacian”), if the generating function of the offspring distribution is conjugate to a polynomial (cf. [@Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Grabner1997:functional_iterations_stopping; @Malozemov_Teplyaev2003:self_similarity_operators; @Teplyaev2004:spectral_zeta_function; @Teplyaev2007:spectral_zeta_functions]). In this case the zeros of $f$ have to be real, since they are eigenvalues of a self-adjoint operator. This motivates the investigation of real Julia sets in Section \[sec:real-julia-set\]. Contents {#sec:contents} -------- The paper is organised as follows. In Section \[sec:asympt-infin-fatou\] we study the asymptotic behaviour of $f(z)$ in those sectors $W$ of the complex plane, where $$\label{eq:infty} f(z)\to\infty \text{ for } z\to\infty,\quad z\in W.$$ It was proved in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] that implies $$f(z)\sim\exp\left(z^\rho F\left(\frac{\log z}{\log\lambda}\right)\right) \text{ for }z\to\infty,\quad z\in W,$$ where $F(z)$ is a periodic function of period $1$. In Section \[sec:asympt-infin-fatou\] we will refine this result to a full asymptotic expansion of $f(z)$, which takes the form $$\label{eq:f-asymp-1} f(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)+ \sum_{n=0}^\infty c_n\exp\left(-nz^\rho F\left(\log_\lambda z\right)\right),$$ where $F$ is a periodic function of period $1$ holomorphic in some strip depending on $W$ and $\rho=\log_\lambda d$. The proof is based on an application of the Böttcher function at $\infty$ of $p(z)$. We note here that E. Romanenko and A. Sharkovsky [@Romanenko_Sharkovsky2000:long_time_properties] have studied equation on ${\mathbb{R}}$ (rather than ${\mathbb{C}}$) and obtained a full asymptotic expansion of this type by Sharkovsky’s method of “first integrals” or “invariant curves”. Further analysis of the periodic function $F$ occurring in is presented in Section \[sec:furth-analys-peri\], where the Fourier coefficients of $F$ are related to the Böttcher function at $\infty$ of $p(z)$ and the harmonic measure on the Julia set of $p$. In Section \[sec:asympt-finite-fatou\] the asymptotic behaviour of $f(z)$ is studied in sectors that are related to basins of attraction of finite attracting fixed points. In Section \[sec:zeros-poinc-funct\] we relate geometric properties of the Julia set to the location of the zeros of $f$. Section \[sec:real-julia-set\] is devoted to the special case of real Julia sets ${\mathcal{J}}(p)$. Here we prove, in particular, the following inequalities of Pommerenke-Levin-Yoccoz type for multipliers of fixed points $\xi$: $$\label{eq:pommerenke} p(\xi)=\xi\Rightarrow \begin{cases} |p'(\xi)|\geq d&\text{ for }\min{\mathcal{J}}(p)<\xi<\max{\mathcal{J}}(p)\\ |p'(\xi)|\geq d^2&\text{ for }\xi=\min{\mathcal{J}}(p)\text{ or }\xi=\max{\mathcal{J}}(p). \end{cases}$$ Furthermore, equality can hold only, if $p$ is linearly conjugate to a Chebyshev polynomial of the first kind. In Section \[sec:zeta-funct-poinc\] we continue the study of Dirichlet generating functions of zeros of Poincaré functions that we started in [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] in the context of spectral zeta functions on certain fractals. We relate the poles and residues of the zeta function of $f$ to the Mellin transform of the harmonic measure $\mu$ on the Julia set of $p$. Furthermore, we show a connection between the zero counting function of $f$ and the harmonic measure $\mu$ of circles around the origin. Relation of complex asymptotics and the Fatou set {#sec:relat-compl-asympt} ================================================= Throughout the rest of the paper we will use the following notations and assumptions. Let $p$ be a real polynomial of degree $d$ as in . We always assume that $p(0)=0$ and $p'(0)=a_1=\lambda$ with $|\lambda|>1$. We refer to [@Beardon1991:iteration_rational_functions; @Milnor2006:dynamics_complex] as general references for complex dynamics. We denote the Riemann sphere by ${\mathbb{C}}_\infty$ and consider $p$ as a map on ${\mathbb{C}}_\infty$. We recall that the Fatou set ${\mathcal{F}}(p)$ is the set of all $z\in{\mathbb{C}}_\infty$ which have an open neighbourhood $U$ such that the sequence $(p^{(n)})_{n\in{\mathbb{N}}}$ is equicontinuous on $U$ in the chordal metric on ${\mathbb{C}}_\infty$. By definition ${\mathcal{F}}(p)$ is open. We will especially need the component of $\infty$ of ${\mathcal{F}}(p)$ given by $$\label{eq:Fatou-infty} {\mathcal{F}}_\infty(p)=\left\{z\in{\mathbb{C}}\mid \lim_{n\to\infty}p^{(n)}(z)=\infty\right\},$$ as well as the basins of attraction of a finite attracting fixed point $w_0$ ($p(w_0)=w_0$, $|p'(w_0)|<1$) $$\label{eq:Fatou-w0} {\mathcal{F}}_{w_0}(p)=\left\{z\in{\mathbb{C}}\mid \lim_{n\to\infty}p^{(n)}(z)=w_0\right\}.$$ The complement of the Fatou set is the Julia set ${\mathcal{J}}(p)={\mathbb{C}}_\infty\setminus{\mathcal{F}}(p)$. The filled Julia set is given by $$\label{eq:filled-Julia} {\mathcal{K}}(p)=\left\{z\in{\mathbb{C}}\mid (p^{(n)}(z))_{n\in{\mathbb{N}}}\text{ is bounded}\right\}= {\mathbb{C}}\setminus{\mathcal{F}}_\infty(p).$$ Furthermore, it is known that (cf. [@Falconer2003:fractal_geometry]) $$\label{eq:boundary} \partial{\mathcal{K}}(p)=\partial{\mathcal{F}}_\infty(p)={\mathcal{J}}(p).$$ In the case of polynomials this can be used as an equivalent definition of the Julia set. We will also use the notations $$\label{eq:W_alpha_beta} W_{\alpha,\beta}=\left\{z\in{\mathbb{C}}\setminus\{0\}\mid \alpha<\arg z<\beta\right\}$$ and $$B(z,r)=\left\{w\in{\mathbb{C}}\mid |z-w|<r\right\}.$$ Asymptotics in the infinite Fatou component {#sec:asympt-infin-fatou} ------------------------------------------- In [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] the asymptotics of the solution of the Poincaré equation was given. We want to present a different approach here, which gives a full asymptotic expansion. \[thm:poincare-asymp\] Let $f$ be the entire solution of the Poincaré equation for a real polynomial $p$ with $\lambda=p'(0)>1$. Assume further that the Fatou component of $\infty$, ${\mathcal{F}}_\infty(p)$ contains an angular region $W_{\alpha,\beta}$. A : Then the following asymptotic expansion for $f$ is valid for all $z\in W_{\alpha,\beta}$ large enough $$\label{eq:f-asymp} f(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)+ \sum_{n=0}^\infty c_n\exp\left(-nz^\rho F\left(\log_\lambda z\right)\right),$$ where $F$ is a periodic function of period $1$ holomorphic in the strip $$\left\{z\in{\mathbb{C}}\mid \frac{\alpha}{\log\lambda}<\Im z<\frac{\beta}{\log\lambda} \right\}$$ and $\rho=\log_\lambda d$. Furthermore, $$\label{eq:Re>0} \forall z\in W_{\alpha,\beta}:\Re z^\rho F(\log_\lambda z)>0$$ holds. B : Let $g$ denote the Böttcher function associated with $p$, *i. e.* $$\label{eq:boettcher} (g(z))^d=g(p(z))$$ in some neighbourhood of $\infty$. Its inverse function is given by the Laurent series around $\infty$ $$\label{eq:Boettcher-inverse-cn} g^{(-1)}\left(w\right)=w+\sum_{n=0}^\infty\frac{c_n}{w^n}.$$ Then we have $$f(z)=g^{(-1)}\left(\exp\left(z^\rho F\left(\log_\lambda z\right)\right)\right)$$ and $c_n$ can be determined from the coefficients of $p$. We recall that $p$ has a super-attracting fixed point of order $d=\deg p$ at infinity. We consider the Böttcher function $g$ associated with this fixed point (cf. [@Beardon1991:iteration_rational_functions; @Blanchard1984:complex_analytic_dynamics; @Boettcher1905:beitraege_zur_theorie; @Kuczma_Choczewski_Ger1990:iterative_functional_equations]), which satisfies the functional equation in some neighbourhood of infinity. The Böttcher function has a Laurent expansion around infinity given by $$\label{eq:Boettcher-Laurent} g(z)=z+\sum_{n=0}^\infty\frac{b_n}{z^n},$$ which converges for $|z|>R$ for some $R>0$. The coefficients $(b_n)_{n\in{\mathbb{N}}_0}$ can be determined uniquely from the coefficients of the polynomial $p$. Using the Böttcher function we can rewrite the Poincaré equation assuming that $|f(z)|>R$ $$\label{eq:Poincare-Boettcher} (g(f(z)))^d=g(p(f(z)))=g(f(\lambda z)).$$ From this we derive that $h(z)=g(f(z))$ satisfies the much simpler functional equation $$(h(z))^d=h(\lambda z),$$ which only holds for those values $z$ for which $|f(z)|>R$. This equation has solutions $$\label{eq:h(z)} h(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)$$ with $\rho=\log_\lambda d$ and $F$ a periodic function of period $1$ holomorphic in some strip parallel to the real axis. Since $|h(z)|>1$ for all $z$ with $|f(z)|>R$ by the properties of the function $g$, we have . By $g$ is invertible in some neighbourhood of $\infty$ and we can write where the coefficients $c_n$ depend only on the coefficients of the polynomial $p$. This function satisfies the functional equation $$\label{eq:Boettcher-inverse} g^{(-1)}(w^d)=p(g^{(-1)}(w))$$ for $w$ in some neighbourhood of $\infty$. Inserting into yields giving an exact and asymptotic expression for $f(z)$. \[rem8\] E. Romanenko and A. Sharkovsky have studied equation on ${\mathbb{R}}$ (rather than on ${\mathbb{C}}$) in [@Romanenko_Sharkovsky2000:long_time_properties]. Applying Sharkovsky’s method of “first integrals” (“invariant graphs”) they obtained a full asymptotic formula of type for all solutions $f(x)$, such that $f(x)\to\infty$ for $x\to\infty$. Böttcher functions, Green functions, and constancy of the periodic function $F$ {#sec:bottch-funct-green} ------------------------------------------------------------------------------- We will make frequent use of the integral representation of the Böttcher function $$\label{eq:Boettcher-int} g(z)=\exp\left(\int_{{\mathcal{J}}(p)}\log(z-x)\,d\mu(x)\right),$$ where $\mu$ denotes the harmonic measure on the Julia set ${\mathcal{J}}(p)$ (cf. [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated; @Brolin1965:invariant_sets_under; @Ransford1995:potential_theory_complex_plane]). This shows that $g$ is holomorphic on any simply connected subset of ${\mathcal{F}}_\infty(p)$. The measure $\mu$ can be given as the weak limit of the measures $$\label{eq:mu_n} \mu_n=\frac1{d^n}\sum_{p^{(n)}(x)=\xi}\delta_x,$$ where $\xi$ can be chosen arbitrarily (not exceptional) and $\delta_x$ denotes the unit point mass at $x$ (cf. [@Brolin1965:invariant_sets_under; @Ransford1995:potential_theory_complex_plane]). The function $g(z)$ can be continued to any simply connected subset $U$ of ${\mathbb{C}}_\infty\setminus{\mathcal{K}}(p)$ (this follows for instance from the integral representation ). Furthermore, it follows from [@Beardon1991:iteration_rational_functions Lemma 9.5.5] and that $$g(U)\subset\{z\in{\mathbb{C}}_\infty\mid |z|>1\}.$$ The function $\log|g(z)|$ is the Green function for the logarithmic potential on ${\mathcal{F}}_\infty(p)$ (cf. [@Beardon1991:iteration_rational_functions Section 9]). Combining classical potential theory with polynomial iteration theory we get $$\label{eq:Julia-condition} \lim_{\substack{z\to z_0\\ z\in {\mathcal{F}}_\infty(p)}}|g(z)|=1\Leftrightarrow z_0\in{\mathcal{J}}(p),$$ where the implication $\Leftarrow$ is [@Beardon1991:iteration_rational_functions Lemma 9.5.5]. The opposite implication is a general property of the Green function (cf. [@Garnett_Marshall2005:harmonic_measure Chapter III], and [@Ransford1995:potential_theory_complex_plane Section 6.5]) combined with the fact that $\partial{\mathcal{F}}_\infty(p)={\mathcal{J}}(p)$ for polynomial $p$. \[thm:constant\] The periodic function $F$ occurring in the asymptotic expression for $f$ is constant, if and only if the polynomial $p$ is either linearly conjugate to $z^d$ or to the Chebyshev polynomial of the first kind $T_d(z)$. The periodic function $F$ is constant, if and only if the function $h(z)=g(f(z))$ introduced above satisfies $$\label{eq:h-exact} h(z)=\exp\left(Cz^\rho\right)$$ for some constant $C\neq0$. This implies that for any $w_0\in{\mathcal{J}}(p)\setminus\{0\}$ the function $g$ has an analytic continuation to some open neighbourhood of $w_0$. Thus can be replaced by $$|g(w_0)|=1 \Leftrightarrow w_0\in{\mathcal{J}}(p)$$ in our case. By this is equivalent to $w_0=f(z_0)$ for $Cz_0^\rho\in i{\mathbb{R}}$. Since $Cz^\rho\in i{\mathbb{R}}$ describes an analytic curve (with a possible cusp at $z=0$), the Julia set of $p$ is the image of this curve under the entire function $f$, thus itself an analytic arc. By [@Hamilton1995:length_julia_curves Theorem 1] ${\mathcal{J}}(p)$ can only be an analytic arc, if the Julia set of $p$ is either a line segment or a circle. The Julia set is a line segment, if and only if $p$ is linearly conjugate to the Chebyshev polynomial $T_d$ (cf. [@Beardon1991:iteration_rational_functions Theorem 1.4.1]); the Julia set is a circle, if and only if $p$ is linearly conjugate to $z^d$ (cf. [@Beardon1991:iteration_rational_functions Theorem 1.3.1]). Suppose that the periodic function $F$ is constant. If $p$ is linearly conjugate to a monomial, then the Böttcher function $g$ and therefore its inverse are linear functions. In this case $\rho=1$. (We recall that we generally assume that $f'(0)=1$.) If $p$ is linearly conjugate to a Chebyshev polynomial, $g^{(-1)}$ is linearly conjugate to the Joukowski function $z+\frac1z$. In this case $\rho=1$, if $0$ is an inner point of the line segment ${\mathcal{J}}(p)$, and $\rho=\frac12$, if $0$ is an end point of the line segment ${\mathcal{J}}(p)$ (cf. Sections \[sec:negative-julia-set\] and \[sec:julia-set-has\]). Furthermore, the asymptotic series is finite, if the periodic function $F$ is constant. Further analysis of the periodic function {#sec:furth-analys-peri} ----------------------------------------- In this section we relate the periodic function $F$ occurring in to the local behaviour of the Böttcher function at the fixed point $f(0)=0$. This will allow to express the Fourier coefficients of $F$ in terms of residues of the Mellin transform (cf. [@Doetsch1971:handbuch_der_laplace; @Oberhettinger1974:tables_mellin_transforms]) of the harmonic measure $\mu$ given by . This Mellin transform was introduced and studied in [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated]. A similar relation was also used in [@Grabner1997:functional_iterations_stopping] to derive an asymptotic expression for $f$ in a special case. We will use the relation $$\label{eq:G(w)} G(w)=\log g(w)=\int_{{\mathcal{J}}(p)}\log(w-x)\,d\mu(x)$$ between the (complex) “Green function” $G$ and the Böttcher function $g$. Assume that the Fatou component ${\mathcal{F}}_\infty(p)$ contains an angular region centred at the fixed point $0$. Furthermore, assume that $\lim_{w\to 0}g(w)=1$. Then holds in this angular region. This fact can be used to analyse the local behaviour of $\log g(w)$ around $w=0$: $$\label{eq:logg} \log g(w)=\left(f^{(-1)}(w)\right)^\rho F\left(\log_\lambda f^{(-1)}(w)\right)= w^\rho F\left(\log_\lambda w\right)+{\mathcal{O}}(w^{\rho+1}).$$ Thus the behaviour of the Green function $G$ at the point $0$ exhibits the same periodic function $F$ as the asymptotic expansion of $\log f$ around $\infty$. ![Paths of integration.[]{data-label="fig:paths"}](path.eps){width="0.8\hsize"} We now relate the Green function $G(w)$ to the Mellin transform of $\mu$ $$\label{eq:Mellin} M_\mu(s)=\int_{{\mathcal{J}}(p)}(-x)^s\,d\mu(x),$$ where the branch cut for the function $(-x)^s$ is chosen to connect $0$ with $\infty$ without any further intersection with ${\mathcal{J}}(p)$. Following the computations in [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated Section 5] we obtain $$M_\mu(s)=\frac1{2\pi i}\oint_\Gamma(-z)^s\,dG(z)= \frac1{2\pi i}\oint_{\Gamma_R}(-z)^s\,dG(z).$$ For $\Re s<0$ we have for the circle of radius $R$ $$\left|\frac1{2\pi i}\int_{|z|=R}(-z)^s\,dG(z)\right|\ll R^{\Re s},$$ which allows to let $R\to\infty$ in this case. This gives $$\begin{gathered} M_\mu(s)=\frac1{2\pi i}\left(\int_{\Lambda_+}(-z)^s\,dG(z)- \int_{\Lambda_-}(-z)^s\,dG(z)\right)\\ =\frac{e^{-i\pi s}-e^{i\pi s}}{2\pi i}\int_0^\infty x^sG'(x)\,dx= s\frac{\sin\pi s}\pi\int_0^\infty x^{s-1}G(x)\,dx,\end{gathered}$$ which relates the Mellin transform of the measure $\mu$ to the Mellin transform of the function $G(z)$ $$\label{eq:Mellin-G} {\mathcal{M}}G(s)=\int_0^\infty x^{s-1}G(x)\,dx=\frac\pi{s\sin\pi s}M_\mu(s)\text{ for } -\rho<\Re s<0.$$ The function $M_\mu(s)$ (and therefore ${\mathcal{M}}G(s)$ by ) has an analytic continuation by the following observation $$\label{eq:continuation} M_\mu(s)=\frac1d\sum_{k=1}^d\int_{{\mathcal{J}}(p)}(-p_k^{(-1)}(x))^s\,d\mu(x),$$ where $p_k^{(-1)}$ ($k=1,\ldots,d$) denote the $d$ branches of the inverse function of $p$; we choose the numbering so that $p_1^{(-1)}(0)=0$. The summands for $k=2,\ldots,d$ are clearly entire functions in $s$, since the integrand is bounded away from $0$ and $\infty$. For the summand with $k=1$ we observe that $$\label{eq:approx} p_1^{(-1)}(x)=\frac1\lambda x+{\mathcal{O}}(x^2)\text{ for }x\to0.$$ Inserting this into gives $$\begin{gathered} M_\mu(s)=\frac1d\lambda^{-s}\int_{{\mathcal{J}}(p)}(-x)^s\,d\mu(x)+ \frac1d\lambda^{-s}\int_{{\mathcal{J}}(p)}(-x)^s{\mathcal{O}}(x)\,d\mu(x)\\+ \frac1d\sum_{k=2}^d\int_{{\mathcal{J}}(p)}(-p_k^{(-1)}(x))^s\,d\mu(x),\end{gathered}$$ where the second term on the right-hand-side originates from inserting the holomorphic function ${\mathcal{O}}(x^2)$ from into the integrand, which gives a function holomorphic in a larger domain. Thus we obtain $$\label{eq:Mellin-continuation} M_\mu(s)=\frac1{d\lambda^s-1}H(s)$$ for some function $H(s)$ holomorphic for $\Re s>-\rho-1$ ($\rho=\log_\lambda d$). The numerator $d\lambda^s-1$ has zeros at $s=-\rho+\frac{2k\pi i}{\log\lambda}$ ($k\in{\mathbb{Z}}$), which give possible poles for the function $M_\mu(s)$. Using the full Taylor expansion of $p_1^{(-1)}(x)$ instead of the ${\mathcal{O}}$-term in would yield the existence of a meromorphic continuation of $M_\mu(s)$ to the whole complex plane. Taking and together gives the analytic continuation of ${\mathcal{M}}G(s)$ to $-\rho-1<\Re s<0$. Then the Mellin inversion formula (cf. [@Doetsch1971:handbuch_der_laplace]) gives (for $-\rho<c<0$) $$\begin{gathered} \label{eq:Mellin-inv-G} G(x)=\frac1{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}{\mathcal{M}}G(s)x^{-s}\,ds= \frac1{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}\frac\pi{s\sin\pi s} \frac1{d\lambda^s-1}H(s)x^{-s}\,ds\\ =\frac1{2\pi i}\int\limits_{-\rho-\frac12-i\infty}^{-\rho-\frac12+i\infty} \frac\pi{s\sin\pi s} \frac1{d\lambda^s-1}H(s)x^{-s}\,ds+ \sum_{k\in{\mathbb{Z}}}\operatorname*{\mathrm{Res}}_{s=-\rho+\frac{2k\pi i}{\log\lambda}} {\mathcal{M}}G(s)x^{-s}.\end{gathered}$$ The integral in the second line is ${\mathcal{O}}(x^{\rho+\frac12})$, the sum of residues can be evaluated further to give the Fourier expansion of the periodic function $F$ $$\label{eq:Fourier-F} \sum_{k\in{\mathbb{Z}}}\operatorname*{\mathrm{Res}}_{s=-\rho+\frac{2k\pi i}{\log\lambda}} {\mathcal{M}}G(s)x^{-s}=x^\rho\sum_{k\in{\mathbb{Z}}}f_k e^{2k\pi i\log_\lambda x}= x^\rho F(\log_\lambda x).$$ The Fourier coefficients $f_k$ are given by $$\begin{gathered} \label{eq:fk} f_k=\operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}{\mathcal{M}}G(s)= \frac\pi{\left(-\rho-\frac{2k\pi i}{\log\lambda}\right) \sin\pi\left(-\rho-\frac{2k\pi i}{\log\lambda}\right)} \operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}M_\mu(s)\\ =\frac\pi{\left(-\log d-2k\pi i\right) \sin\pi\left(-\rho-\frac{2k\pi i}{\log\lambda}\right)} H\left(-\rho-\frac{2k\pi i}{\log\lambda}\right).\end{gathered}$$ Asymptotics in a finite Fatou component – analysis of asymptotic values {#sec:asympt-finite-fatou} ----------------------------------------------------------------------- It is clear from the functional equation for $f$ that any asymptotic value of $f$ has to be an attracting fixed point of the polynomial $p$ (including $\infty$). Thus the analysis in Section \[sec:asympt-infin-fatou\] can be interpreted as the behaviour of $f$ when approaching the asymptotic value $\infty$. In the present section we extend this analysis to all asymptotic values. First we study the case of a finite attracting, but not super-attracting fixed point. Let $w_0$ be an attracting fixed point of $p$ and denote $\eta=p'(w_0)\neq0$ ($|\eta|<1$). Then there exists a solution $\Psi$ of the Schröder equation $$\label{eq:schroeder-Psi} \eta\Psi(z)=\Psi(p(z)),\quad \Psi(w_0)=0,\text{ and }\Psi'(w_0)=1,$$ which is holomorphic in ${\mathcal{F}}_{w_0}(p)$ (for instance, the sequence $(\eta^{-n}(p^{(n)}(z)-w_0))_{n\in{\mathbb{N}}}$ converges to $\Psi$ on any compact subset of ${\mathcal{F}}_{w_0}(p)$). Assume now that ${\mathcal{F}}_{w_0}(p)$ contains an angular region $W_{\alpha,\beta}\cap B(0,r)$ for some $r>0$. Then by conformity of $f$ some angular region at the origin is mapped into $W_{\alpha,\beta}\cap B(0,r)$. We consider the function $$j(z)=\Psi(f(z)),$$ which satisfies the functional equation $$\label{eq:Psif} j(\lambda z)=\Psi(f(\lambda z))=\Psi(p(f(z)))=\eta\Psi(f(z))=\eta j(z).$$ This equation has the solution $$\label{eq:jz} j(z)=z^{\log_\lambda\eta}H(\log_\lambda z)$$ with some periodic function of period $1$, holomorphic in some strip. This periodic function can never be constant, since otherwise $j(z)$ would have an analytic continuation to the slit complex plane. From this it would follow that $f$ is bounded in the slit complex plane, a contradiction. The function $\Psi$ has a holomorphic inverse around $0$ $$\Psi^{(-1)}(z)=w_0+z+\sum_{n=2}^\infty \psi_nz^n$$ which allows us to write $$\label{eq:fw0} f(z)=\Psi^{(-1)}\left(z^{\log_\lambda\eta}H(\log_\lambda z)\right)= w_0+z^{\log_\lambda\eta}H(\log_\lambda z)+ \sum_{n=2}^\infty \psi_nz^{n\log_\lambda\eta}(H(\log_\lambda z))^n,$$ which is valid in the angular region $W_{\alpha,\beta}$ for $z$ large enough. This gives an exact and asymptotic expression for $f$ in an angular region. In the case of a super-attracting fixed point $w_0$ we have $p'(w_0)=0$. Assume that the first $k-1$ derivatives of $p$ vanish in $w_0$, but the $k$-th derivative is non-zero. Then $p(z)=(z-w_0)^kP(z)$ with $P(w_0)=A\neq0$. We use the solution $g$ of the corresponding Böttcher equation $$\label{eq:boettcher-w0} g(p(z))=A(g(z))^k\quad g(w_0)=0,\quad g'(w_0)=1$$ to linearise $$g(f(\lambda z))=g(p(f(z)))=A(g(f(z)))^k.$$ Thus the function $h(z)=g(f(z))$ satisfies $$h(\lambda z)=A(h(z))^k.$$ This equation has solutions $$h(z)=A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)$$ for a periodic function $L$ of period $1$ and a suitable choice of the $(k-1)$-th root. Furthermore, by the fact that $\lim_{z\to\infty}h(z)=0$ we have $$\Re\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)<0\text{ for } f(z)\in{\mathcal{F}}_{w_0}(p).$$ using the local inverse of $g$ around $0$ we get $$\begin{gathered} \label{eq:f-asymp-w0} f(z)=g^{(-1)}\left(A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)\right)\\ =w_0+A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)(1+o(1)).\end{gathered}$$ Summing up, we have proved \[thm10\] Let $w_0$ be an attracting fixed point of $p$ such that the Fatou component ${\mathcal{F}}_{w_0}(p)$ contains an angular region $W_{\alpha,\beta}\cap B(0,r)$ for some $r>0$. Then the asymptotic behaviour of $f$ for $z\to\infty$ and $z\in W_{\alpha,\beta}$ is given by , if $\eta=p'(w_0)\neq0$, and by , if $p(z)-w_0$ has a zero of order $k$ in $w_0$. The periodic function $H$ in cannot be constant, because otherwise $f(z)$ would be bounded. The periodic function $L$ in can only be constant, if $p$ is linearly conjugate to $z^k$, by the same arguments as in the proof of Theorem \[thm:constant\] (the case of Chebyshev polynomials does not occur, because they only have repelling finite fixed points). As a consequence of Ahlfors’ theorem on asymptotic values (cf. [@Goluzin1969:geometric_theory_functions]) and Valiron’s theorem on the growth of $f$ (cf. [@Valiron1923:lectures_on_general; @Valiron1954:fonctions_analytiques]) we get an upper bound for the number of attracting fixed points of a polynomial. Let $p$ be a real polynomial of degree $d>1$ and let $$\gamma=\max\left\{|p'(z)|\mid p(z)=z\right\}.$$ Then the number of (finite) attracting fixed points of $p$ is bounded by $2\log_\gamma d$, i.e. $$\label{eq:ahlfors} \#\left\{z\in{\mathbb{C}}\mid p(z)=z\wedge |p'(z)|<1\right\}\leq 2\log_\gamma d.$$ Zeros of the Poincaré function and Julia sets {#sec:zeros-poinc-funct} ============================================= In this section we relate the distribution of zeros of the Poincaré function in angular regions to geometric properties of the Julia set ${\mathcal{J}}(p)$ of the polynomial $p$. \[thm8\] Let $p$ be a real polynomial with $p(0)=0$ and $p'(0)=\lambda>1$. Then the following are equivalent 1. \[thm8.1\] $\displaystyle{\forall r>0: W_{\alpha,\beta}\cap {\mathcal{J}}(p)\cap B(0,r)\neq\emptyset}$ 2. \[thm8.3\] $W_{\alpha,\beta}$ contains a zero of $f$. 3. \[thm8.2\] $W_{\alpha,\beta}$ contains infinitely many zeros of $f$. We first remark that \[thm8.3\] and \[thm8.2\] are trivially equivalent, since $f(z_0)=0$ implies that $f(\lambda^n z_0)=0$. For the proof of “\[thm8.1\]$\Rightarrow$ \[thm8.3\]” we take $0<{\varepsilon}<\frac{\beta-\alpha}2$ so small that $$\forall r>0: W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap {\mathcal{J}}(p)\cap B(0,r)\neq\emptyset.$$ Then we take $r>0$ so small that $$\label{eq:angle} W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap B(0,r)\subset f\left(W_{\alpha,\beta}\right),$$ which is possible by conformity of $f$ and $f'(0)=1$. Since the preimages of $0$ are dense in ${\mathcal{J}}(p)$, there exists $\eta\in W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap B(0,r)$ and $n\in{\mathbb{N}}$ such that $p^{(n)}(\eta)=0$. By there exists $\xi\in W_{\alpha,\beta}$ such that $f(\xi)=\eta$, from which we obtain $$f(\lambda^n\xi)=p^{(n)}(f(\xi))=p^{(n)}(\eta)=0.$$ For the proof of “\[thm8.2\]$\Rightarrow$ \[thm8.1\]” we take $z_0\in W_{\alpha,\beta}$ with $f(z_0)=0$. Then $$\forall n\in{\mathbb{N}}: f(\lambda^{-n}z_0)\in{\mathcal{J}}(p).$$ For any $r>0$ and $n$ large enough $f(\lambda^{-n}z_0)\in W_{\alpha,\beta}\cap B(0,r)$, which gives \[thm8.1\]. Similar arguments show \[thm:zeros-on-line\] Let $p$ be a real polynomial with $p(0)=0$ and $p'(0)=\lambda>1$. Then $$\label{eq:negative-zeros} {\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}\Leftrightarrow \text{ all zeros of }f \text{ are non-positive real}$$ and $$\label{eq:real-zeros} {\mathcal{J}}(p)\subset{\mathbb{R}}\Leftrightarrow \text{ all zeros of }f\text{ are real.}$$ Real Julia set {#sec:real-julia-set} ============== \[lem1\] Let $p$ be a real polynomial of degree $d>1$. Then the Julia-set ${\mathcal{J}}(p)$ is real, if and only if there exists an interval $[a,b]$ such that $$\label{eq:inv-int} p^{(-1)}\left([a,b]\right)\subseteq[a,b].$$ Assume first that ${\mathcal{J}}(p)\subset{\mathbb{R}}$ and take the interval $[a,b]=[\min{\mathcal{J}}(p),\max{\mathcal{J}}(p)]$. Let ${\varepsilon}>0$. Since ${\mathcal{J}}(p)$ is perfect, there exist $\xi,\eta\in{\mathcal{J}}(p)$ with $a<\xi<a+{\varepsilon}<b-{\varepsilon}<\eta<b$. All preimages of $\xi$ and $\eta$ are in ${\mathcal{J}}(p)$ by the invariance of ${\mathcal{J}}(p)$. Furthermore, all these preimages are distinct. Therefore, every value $x\in[\xi,\eta]$ has exactly $d$ distinct preimages in $[a,b]$ by continuity of $p$. Since ${\varepsilon}$ was arbitrary and the two points $a,b$ also have all their preimages in ${\mathcal{J}}(p)\subset[a,b]$, we have proved . Assume on the other hand that $[a,b]$ satisfies . Since the map $p$ has only finitely many critical values, there exists $x\in[a,b]$ such that the backward iterates of $x$ are dense in the Julia set. By all these backward iterates are real; therefore ${\mathcal{J}}(p)$ is real. \[rem1\] By the above proof we can always assume $[a,b]=[\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)]$. Furthermore, we have $$p\left(\{\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)\}\right)\subseteq\{\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)\},$$ which implies that at least one of the two end points of this interval is either a fixed point, or they form a cycle of length $2$. \[thm1\] Let $p$ be a polynomial of degree $d>1$ with real Julia set ${\mathcal{J}}(p)$. Then for any fixed point $\xi$ of $p$ with $\min {\mathcal{J}}(p)<\xi<\max {\mathcal{J}}(p)$ we have $|p'(\xi)|\geq d$. Furthermore, $|p'(\min{\mathcal{J}}(p))|\geq d^2$ and $|p'(\max {\mathcal{J}}(p))|\geq d^2$. Equality in one of these inequalities implies that $p$ is linearly conjugate to the Chebyshev polynomial $T_d$ of degree $d$. This theorem can be compared to [@Buff2003:bieberbach_conjecture_dynamics Theorem 2] and [@Levin1991:pommerenke's_inequality; @Pommerenke1986:conformal_mapping_iteration], where estimates for the derivative of $p$ for connected Julia sets are derived. Furthermore, in [@Eremenko_Levin1992:estimation_characteristic_exponents] estimates for $\frac1n\log|(p^{(n)})'(z)|$ for periodic points of period $n$ are given. Before we give a proof of the theorem, we present a lemma, which is of some interest on its own. A similar result is given in [@Levin1980:distribution_zeros_entire Chapter V, Section 2, Lemma 3]. \[lem2\] Let $f$ be holomorphic in the angular region $W_{\alpha,\beta}$ If there exists a positive constant $M$ such that $$\forall z\in W_{\alpha,\beta}:|f(z)|\geq M,$$ then $$\forall {\varepsilon}>0\,\,\, \exists A,B>0\,\,\, \forall z\in W_{\alpha+{\varepsilon},\beta-{\varepsilon}}: |f(z)|\leq B\exp(A|z|^\kappa)$$ with $\kappa=\frac\pi{\beta-\alpha}$. Without loss of generality we can assume that $M=1$, $\alpha=-\frac\pi2$, and $\beta=\frac\pi2$. In this case $\kappa=1$. The function $$v(z)=\log|f(z)|$$ is a positive harmonic function in the right half-plane. Thus it can be represented by the Nevanlinna formula (cf. [@Levin1996:lectures_entire_functions p.100]) $$\label{eq:nevanlinna} v(x+iy)=\frac x\pi\int_{-\infty}^\infty\frac{d\nu(t)}{|z-it|^2}+\sigma x,$$ where $\nu$ denotes a measure satisfying $$\int_{-\infty}^\infty\frac{d\nu(t)}{1+t^2}<\infty$$ and $\sigma\geq0$. In the region given by $|\arg z|\leq\frac\pi2-{\varepsilon}$ and $|z|>1$ we have $$|z-it|\geq\max(|t|\sin{\varepsilon},|z|\sin{\varepsilon})\geq\max(1,|t|)\sin{\varepsilon}.$$ From this it follows that $$|z-it|^2\geq\frac12(1+t^2)\sin^2{\varepsilon},$$ which gives $$\int_{-\infty}^\infty\frac{d\nu(t)}{|z-it|^2}\leq \frac2{\sin^2{\varepsilon}}\int_{-\infty}^\infty\frac{d\nu(t)}{1+t^2}\leq B_{\varepsilon}$$ for $|z|\geq1$ and some $B_{\varepsilon}>0$. Setting $A=\frac1\pi B_{\varepsilon}+\sigma$ and observing that $x\leq|z|$ completes the proof. Without loss of generality we may assume that the fixed point $\xi=0$. Then we consider the solution $f$ of the Poincaré equation $$f(\lambda z)=p(f(z))$$ with $\lambda=p'(0)$. We assume first that $\lambda>0$. First we consider the case $\min {\mathcal{J}}(p)<\xi<\max {\mathcal{J}}(p)$. In this case the function $f(z)/z$ tends to infinity uniformly for $z\to\infty$ in the region ${\varepsilon}\leq\arg z\leq\pi-{\varepsilon}$ for any ${\varepsilon}>0$ by Theorem \[thm:poincare-asymp\]. Furthermore, we know that $$|f(z)|\geq C\exp(A|z|^{\log_\lambda d})$$ in this region for some positive constants $A$ and $C$. Since $f(z)/z$ does not vanish at $z=0$, this function satisfies the hypothesis of Lemma \[lem2\], from which we derive that $$\log_\lambda d\leq\frac\pi{\pi-2{\varepsilon}}$$ holds for any ${\varepsilon}>0$, which implies $\lambda=p'(0)\geq d$. The proof in the case $\xi=\max {\mathcal{J}}(p)$ runs along the same lines. The function $f(z)/z$ tends to infinity uniformly in any region $|\arg z|\leq\pi-{\varepsilon}$ in this case, which by Lemma \[lem2\] implies $$\log_\lambda d\leq\frac\pi{2\pi-2{\varepsilon}}$$ for all ${\varepsilon}>0$, and consequently $\lambda=p'(0)\geq d^2$. For negative $\lambda=p'(0)$ we apply the same arguments to $p^{(2)}$. For the proof of the second assertion of the theorem, we first assume that the fixed point $\xi=0$ satisfies $a=\min {\mathcal{J}}(p)<0<\max {\mathcal{J}}(p)=b$ and that $p'(0)=d$. We know that for a suitable linear conjugate $q$ of the Chebyshev polynomial $T_d$ we have $q'(0)=d$ and ${\mathcal{J}}(q)=[a,b]$ with $0\in(a,b)$. Let us assume now that $p'(0)=d$ and ${\mathcal{J}}(p)$ is a Cantor subset of the real line, or after a rotation that ${\mathcal{J}}(p)$ is a Cantor subset of the imaginary axis (this makes notation slightly simpler). By arguments, similar to those in the beginning of Section \[sec:furth-analys-peri\] we can write $$\label{eq:harmonic} H(z)=\Re\log g(f(z))=\int_{{\mathcal{J}}(p)}\log|f(z)-x|\,d\mu(x).$$ Since $\Re\log g(.)$ is the Green function of ${\mathcal{J}}(p)$ with pole at $\infty$ (cf. [@Beardon1991:iteration_rational_functions Lemma 9.5.5] or [@Ransford1995:potential_theory_complex_plane]), we know that $H(z)\geq0$ for all $z\in{\mathbb{C}}$ and $H(z)=0$, if and only if $f(z)\in{\mathcal{J}}(p)$ (since ${\mathcal{K}}(p)={\mathcal{J}}(p)$ in the present case). By Theorem \[thm:poincare-asymp\] we have $$\label{eq:periodic} H(z)=\Re \left(zF(\log_d z)\right)=x\Re(F(\log_d z))-y\Im(F(\log_d z)) \text{ for }z=x+iy,$$ and by Theorem \[thm:constant\] the function $F$ is not constant in the present case. The periodic function $\Im F(t+i{\varphi})$ has zero mean, since the mean of $F$ is real. Thus $\Im F(t+i{\varphi})$ attains positive and negative values for any ${\varphi}$. We now take $z=iy\in i{\mathbb{R}}^+$ to obtain $$H(iy)=-y\Im(F(\log_d y+i\frac\pi{2\log d})).$$ Since $\Im F$ attains positive values by the above argument, we get a contradiction to $H(z)\geq0$ for all $z$. A similar argument shows that for $0=\max {\mathcal{J}}(p)$ and $p'(0)=d^2$ the assumption that the Julia set is not an interval leads to the same contradiction. \[rem10\] Lemma 6.4 in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] proves Theorem \[thm1\] for the special case of quadratic polynomials. The proof given in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] is purely geometrical. \[rem11\] We have a purely real analytic proof for $|p'(\max{\mathcal{J}}(p))|\geq d^2$, which is motivated by the proof of the extremality of the Chebyshev polynomials of the first kind given in [@Rivlin1974:chebyshev_polynomials]. However, we could not find a similar proof for the other assertions of the theorem. The Julia set is a subset of the negative reals {#sec:negative-julia-set} ----------------------------------------------- As a consequence of Lemma \[lem2\] we get that any solution of the Poincaré equation for a polynomial with Julia set contained in the negative real axis has order $\leq\frac12$. The only solutions of a Poincaré equation with order $\frac12$ in this situation are the functions $$f(z)=\frac1a\left(\cosh\sqrt{2az}-1\right)$$ for $$p(z)=(T_d(az+1)-1)/a,$$ where $a\in{\mathbb{R}}^+$ and $T_d$ denotes the Chebyshev polynomial of the first kind of degree $d$. This is also the only case where the periodic function $F$ in is constant in this situation. \[cor15\] Assume that $p$ is a real polynomial such that ${\mathcal{J}}(p)$ is real and all coefficients $p_i$ ($i\geq2$) of $p$ are non-negative. Then ${\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}$ and therefore $$\label{eq:simple-asymp} f(z)\sim\exp\left(z^\rho F\left(\frac{\log z}{\log\lambda}\right)\right)$$ for $z\to\infty$ and $|\operatorname{\mathrm{arg}}z|<\pi$. Here $F$ is a periodic function of period $1$ holomorphic in the strip given by $|\Im w|<\frac\pi{\log\lambda}$. Furthermore, for every ${\varepsilon}>0$ $\Re e^{i\rho\operatorname{\mathrm{arg}}z}F(\frac{\log z}{\log\lambda})$ is bounded between two positive constants for $|\operatorname{\mathrm{arg}}z|\leq\pi-{\varepsilon}$. From [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Lemmas 6.4 and 6.5] it follows that $f(z)$ has only non-positive real zeros. Then by Theorem \[thm:zeros-on-line\] ${\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}$. Finally, the assertion follows by applying [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Theorem 7.5]. In order to illustrate the above results, we shall turn to the equation $$f(5z)=4f(z)^2-3f(z),$$ which arises in the description of Brownian motion on the Sierpiński gasket [@Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Kroen2002:green_functions_self; @Kroen_Teufl2004:asymptotics_transition_probabilities; @Teplyaev2004:spectral_zeta_function]. Here $p(z)=4z^2-3z$, and the fixed point of interest is $f(0)=1$. This fits into the assumptions of Section \[sec:assumptions\] only after substituting $g(z)=4(f(z)-1)$, where $g$ satisfies $$g(5z)=g(z)^2+5g(z).$$ Now Corollary \[cor15\] may be applied to this equation (the preimages of $0$ are real by [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Lemma 6.7]) to give . Note also that $p'(0)=5>4=2^2$ in accordance with Theorem \[thm1\]. The Julia set has positive and negative elements {#sec:julia-set-has} ------------------------------------------------ Again as a consequence of Theorem \[thm1\] the solution of the Poincaré equation for a polynomial with real Julia set with positive and negative elements has order $\leq1$. The only solution of a Poincaré equation of order $1$ in this situation are the functions $$f(z)=\frac1a\left(\cos\left(a\frac{z-\frac{2k\pi}{d-1}} {\sin\frac{k\pi}{d-1}}\right)-\xi_k\right)$$ for $$p(z)=\frac1a\left(T_d(a(z+\xi_k))-\xi_k\right),$$ where $a\in{\mathbb{R}}^+$ and $\xi_k=\cos\frac{k\pi}{d-1}$ for $1\leq k<\frac{d-1}2$. This is again the only case where the periodic function $F$ in is constant in this situation. The Zeta function of the Poincaré function {#sec:zeta-funct-poinc} ========================================== In [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] the zeta function of a fractal Laplace operator was related to the zeta function of certain Poincaré functions. Asymptotic expansions for the Poincaré functions were then used to give a meromorphic continuation of these zeta functions as well as information on the location of their poles and values of residues. In this section we give a generalisation of these results to polynomials whose Fatou set contains an angular region $W_{-\alpha,\alpha}$ around the positive real axis. In this case the solution $f$ of has no zeros in an angular region $W_{-\alpha,\alpha}$. Furthermore, from the Hadamard factorisation theorem we get $$\label{eq:hadamard} f(z)=z\exp\left(\sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell z^\ell}\ell\right) \prod_{\substack{f(-\xi)=0\\\xi\neq0}}\!\!\left(1+\frac z\xi\right) \exp\left(-\frac z\xi+\frac {z^2}{2\xi^2}+\cdots+ (-1)^{k-1}\frac{z^k}{k\xi^k}\right),$$ where $k=\lfloor\log_\lambda d\rfloor$. By the discussion in [@Derfel_Grabner_Vogl2008:zeta_function_laplacian Section 5] the values $e_1,\ldots,e_k$ are given by the first $k$ terms of the Taylor series of $\log\frac{f(z)}z$ $$\log\frac{f(z)}z=\sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell z^\ell}\ell +{\mathcal{O}}(z^{k+1}).$$ The zeta function of $f$ is now defined as $$\label{eq:zeta_f} \zeta_f(s)=\sum_{\substack{f(-\xi)=0\\\xi\neq0}}\xi^{-s},$$ where $\xi^{-s}$ is defined using the principal value of the logarithm, which is sensible, since $\xi$ is never negative real by our assumption on ${\mathcal{F}}_\infty(p)$. The function $\zeta_f(s)$ is holomorphic in the half plane $\Re s>\rho$. In [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] we used the equation $$\label{eq:zeta-weierstrass} \int_0^\infty\left(\log f(x)-\log x- \sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell x^\ell}\ell\right) x^{-s-1}\,dx=\zeta_f(s)\frac\pi{s\sin\pi s},$$ which holds for $\rho<\Re s<k+1$, to derive the existence of a meromorphic continuation of $\zeta_f$ to the whole complex plane. There ([@Derfel_Grabner_Vogl2008:zeta_function_laplacian Theorem 8]) we obtained $$\operatorname*{\mathrm{Res}}_{s=\rho+\frac{2k\pi i}{\log\lambda}}\zeta_f(s)= -\frac{f_k}\pi\left(\rho+\frac{2\pi ik}{\log\lambda}\right) \sin\pi\left(\rho+\frac{2\pi ik}{\log\lambda}\right),$$ where $f_k$ is given by . From this we get $$\label{eq:Res-zeta_f} \operatorname*{\mathrm{Res}}_{s=\rho+\frac{2k\pi i}{\log\lambda}}\zeta_f(s)= -\operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}M_\mu(s).$$ This shows that the function $$\label{eq:zeta_f-M_mu} \zeta_f(s)-M_\mu(-s)$$ is holomorphic in $\rho-1<\Re s<\rho+1$, since the single poles on the line $\Re s=\rho$ cancel. This fact was used in [@Grabner1997:functional_iterations_stopping] to derive an analytic continuation for $\zeta_f(s)$. \[thm9\] Let $f$ be the entire solution of and assume that $p$ is neither linearly conjugate to a Chebyshev polynomial nor to a monomial and that $W_{-\alpha,\alpha}\subset{\mathcal{F}}_\infty(p)$ for some $\alpha>0$. Then the following assertions hold 1. \[enum2\] the limit $\lim_{t\to\infty}t^{-\rho}\log f(t)$ does not exist. 2. \[enum1\] $\zeta_f(s)$ has at least two non-real poles in the set $\rho+2\pi i\sigma{\mathbb{Z}}$ $\sigma=\frac1{\log\lambda}$. 3. \[enum5\] the limit $\lim_{x\to0}x^{-\rho}G(x)$ with $G$ given by does not exist. Equation in Theorem \[thm:poincare-asymp\] (see also [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions]) implies that $$z^{-\rho}\log f(z)=F(\log_\lambda z)+o(1)\text{ for }z\to\infty\text{ and } z\in W_{-\alpha,\alpha}$$ with a periodic function $F$ of period $1$. Theorem \[thm:constant\] implies that $F$ is a non-constant . Thus the limit in \[enum2\] does not exist. Since the periodic function $F$ is non-constant, there exists a $k_0\neq0$ such that the Fourier-coefficients $f_{\pm k_0}$ do not vanish. By we have $$\log f(z)=z^\rho\sum_{k\in{\mathbb{Z}}}f_k z^{\frac{2k\pi i}{\log\lambda}}+ {\mathcal{O}}(z^{-M})$$ for any $M>0$. By properties of the Mellin transform (cf. [@Paris_Kaminski2001:asymptotics_mellin_barnes]), every term $Az^{\rho+i\tau}$ in the asymptotic expansion of $\log f(z)$ corresponds to a first order pole of the Mellin transform of $\log f(z)$ with residue $A$ at $s=\rho+i\tau$. Since $f_{k_0}\neq0$, from we have simple poles of $\zeta_f(s)$ at $s=\rho\pm\frac{2k_0\pi i}{\log\lambda}$. Assertion \[enum5\] follows from \[enum2\] by . In the following we consider the zero counting function of $f$ $$\label{eq:N(x)} N_f(x)=\sum_{\substack{|\xi|<x\\f(\xi)=0}}1.$$ \[thm11\] Let $f$ be the entire solution of . Then the following are equivalent 1. \[enum3\] the limit $\lim_{x\to\infty}x^{-\rho}N_f(x)$ does not exist. 2. \[enum4\] the limit $\lim_{t\to0}t^{-\rho}\mu(B(0,t))$ does not exist. For the proof of the equivalence of \[enum3\] and \[enum4\] we observe that by the fact that $f'(0)=1$, there is an $r_0>0$ such that $f:B(0,r_0)\to{\mathbb{C}}$ is invertible. For the following we choose $n=\lfloor\log_\lambda(x/r_0)\rfloor+k$ and let the integer $k>0$ be fixed for the moment. Then we use the functional equation for $f$ to get $$N_f(x)=\#\left\{\xi\mid f(\lambda^n\xi)=p^{(n)}(f(\xi))=0 \wedge|\xi|<x\lambda^{-n}\right\} \!=\!\#\!\left(p^{(-n)}(0)\cap f(B(0,x\lambda^{-n}))\right).$$ This last expression can now be written in terms of the discrete measure $\mu_n$ given in $$N_f(x)=d^n\mu_n\left(f(B(0,x\lambda^{-n}))\right).$$ By the weak convergence of the measures $\mu_n$ (cf. [@Brolin1965:invariant_sets_under]) we get for $x\to\infty$ (equivalently $n\to\infty$) $$N_f(x)=d^n\mu(f(B(0,x\lambda^{-n})))+o(d^n)= x^\rho(x\lambda^{-n})^{-\rho}\mu(f(B(0,x\lambda^{-n})))+o(x^\rho).$$ By our choice of $n$ we have $r_0\lambda^{-k-1}\leq x\lambda^{-n}\leq r_0\lambda^{-k-1}$, which makes the first term dominant. From this it is clear that the existence of the limit $$\lim_{x\to\infty}x^{-\rho}N_f(x)=C$$ is equivalent to $$\mu(f(B(0,t)))=Ct^\rho\text{ for }r_0\lambda^{-k}\leq t< r_0\lambda^{-(k-1)}.$$ Since $k$ was arbitrary this implies $$\label{eq:mu(B)} \mu(f(B(0,t)))=Ct^\rho\text{ for }0< t< r_0.$$ It follows from $f'(0)=1$ that $$\label{eq:mu(f(B))-mu(B))} \forall{\varepsilon}>0:\exists\delta>0:\forall t<\delta: B(0,(1-{\varepsilon})t)\subset f(B(0,t))\subset B(0,(1+{\varepsilon})t).$$ Thus the existence of the limit in assertion \[enum4\] is equivalent to $$\lim_{t\to0}t^{-\rho}\mu(f(B(0,t)))=C.$$ Thus \[enum3\] and \[enum4\] are equivalent. If ${\mathcal{J}}(p)$ is real and disconnected then the limits in Theorem \[thm11\] do not exist. Furthermore, it is known that the limit $$\lim_{t\to0}t^{-\rho}\mu(f(B(w,t)))=C$$ does not exist for $\mu$-almost all $w\in{\mathcal{J}}(p)$ (cf. [@Mattila1995:geometry_sets_measures Theorem 14.10]), if $\rho$ is not an integer. This motivates the following conjecture. The limits in Theorem \[thm11\] exist, if and only if $p$ is either linearly conjugate to a Chebyshev polynomial or a monomial. This research was initiated during the second author’s visit to the Ben Gurion University of the Negev with support by the Center of Advanced Studies in Mathematics.\ It was completed during the second author’s visit at the Center for Constructive Approximation and the Department of Mathematics at Vanderbilt University, Nashville, Tennessee. He is especially thankful to Edward B. Saff for the invitation and the great hospitality.\ The first author wants to thank Alexandre Erëmenko, Genadi Levin, and Mikhail Sodin for interesting discussions.\ The authors are indebted to an anonymous referee for valuable remarks. [10]{} . Diffusions on fractals. In *Lectures on probability theory and statistics (Saint-Flour, 1995)*, pages 1–121 (Springer Verlag, Berlin, 1998). . Brownian motion on the [S]{}ierpiński gasket. *Probab. Theory Relat. Fields* **79** (1988) 543–623. . *Iteration of [R]{}ational [F]{}unctions* (Springer, Berlin, New York, 1991). . Mellin transforms associated with [J]{}ulia sets and physical applications. *J. Statist. Phys.* **34** (1984) 75–110. . Complex analytic dynamics on the [R]{}iemann sphere. *Bull. Amer. Math. Soc. (N.S.)* **11** (1984) 85–141. . Beiträge zur [T]{}heorie der [I]{}terationsrechnung. *Bull. Kasan Math. Soc.* **14** (1905) 176. . Invariant sets under iteration of rational functions. *Ark. Mat.* **6** (1965) 103–144. . On the [B]{}ieberbach conjecture and holomorphic dynamics. *Proc. Amer. Math. Soc.* **131** (2003) 755–759 (electronic). . Asymptotics of the [P]{}oincaré functions. In [D. Dawson, V. Jaksic B. Vainberg]{}, editors, *Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov*, volume 42 of *CRM Proceedings and Lecture Notes*, pages 113–130 (Centre de Recherches Mathématiques, Montreal, 2007). . The [Z]{}eta function of the [L]{}aplacian on certain fractals. *Trans. Amer. Math. Soc.* **360** (2008) 881–897 (electronic). . *Handbuch der [L]{}aplace-[T]{}ransformation. [B]{}and [I]{}: [T]{}heorie der [L]{}aplace-[T]{}ransformation* (Birkhäuser Verlag, Basel, 1971). Verbesserter Nachdruck der ersten Auflage 1950, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 14. . Periodic points of polynomials ([R]{}ussian). *Ukrain. Mat. Zh.* **41** (1989) 1467–1471, 1581. Translation in Ukrainian Math. J. **41** (1989), 1258–1262. . Estimation of the characteristic exponents of a polynomial ([R]{}ussian). *Teor. Funktsiĭ Funktsional. Anal. i Prilozhen.* pages 30–40 (1993). Translation in J. Math. Sci. (New York) **85** (1997), 2164–2171. . Iterations of rational functions and the distribution of the values of [P]{}oincaré functions ([R]{}ussian. *Teor. Funktsiĭ Funktsional. Anal. i Prilozhen.* **53** (1990) 18–25. Translation in J. Soviet Math. **58** (1992), 504–509. . *Fractal [G]{}eometry* (John Wiley & Sons Inc., Hoboken, NJ, 2003). Mathematical foundations and applications. . *Harmonic [M]{}easure*, volume 2 of *New Mathematical Monographs* (Cambridge University Press, Cambridge, 2005). . *Geometric [T]{}heory of [F]{}unctions of a [C]{}omplex [V]{}ariable*. Translations of Mathematical Monographs, Vol. 26 (American Mathematical Society, Providence, R.I., 1969). . Functional iterations and stopping times for [B]{}rownian motion on the [S]{}ierpiński gasket. *Mathematika* **44** (1997) 374–400. . Length of [J]{}ulia curves. *Pacific J. Math.* **169** (1995) 75–93. . *The [T]{}heory of [B]{}ranching [P]{}rocesses* (Springer, Berlin, New York, 1963). . Borel and [J]{}ulia directions of meromorphic [S]{}chröder functions. *Math. Proc. Cambridge Philos. Soc.* **139** (2005) 139–147. . Recherches sur les intégrales de certaines équations fonctionelles. *Ann. Sci. Ec. Norm. Super. III. Ser.* **1** (1884) 3–41. . Nouvelles recherches sur les Žequations fonctionnelles. *Ann. Sci. Ec. Norm. Super. III. Ser.* **2** (1885) 385–404. . Green functions on self-similar graphs and bounds for the spectrum of the [L]{}aplacian. *Ann. Inst. Fourier (Grenoble)* **52** (2002) 1875–1900. . Asymptotics of the transition probabilities of the simple random walk on self-similar graphs. *Trans. Amer. Math. Soc.* **356** (2004) 393–414 (electronic). . *Iterative [F]{}unctional [E]{}quations*, volume 32 of *Encyclopedia of Mathematics and its Applications* (Cambridge University Press, Cambridge, 1990). . *Distribution of [Z]{}eros of [E]{}ntire [F]{}unctions*, volume 5 of *Translations of Mathematical Monographs* (American Mathematical Society, Providence, R.I., 1980), revised edition. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. . *Lectures on [E]{}ntire [F]{}unctions*, volume 150 of *Translations of Mathematical Monographs* (American Mathematical Society, Providence, RI, 1996). In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by V. Tkachenko. . On [P]{}ommerenke’s inequality for the eigenvalues of fixed points. *Colloq. Math.* **62** (1991) 167–177. . *Brownian [M]{}otion on [N]{}ested [F]{}ractals*, volume 420 of *Mem. Amer. Math. Soc.* (Amer. Math. Soc., 1990). . Self-similarity, operators and dynamics. *Math. Phys. Anal. Geom.* **6** (2003) 201–218. . *Geometry of [S]{}ets and [M]{}easures in [E]{}uclidean [S]{}paces, [F]{}ractals and [R]{}ectifiability*, volume 44 of *Cambridge Studies in Advanced Mathematics* (Cambridge University Press, Cambridge, 1995). . *Dynamics in one complex variable*, volume 160 of *Annals of Mathematics Studies* (Princeton University Press, Princeton, NJ, 2006), third edition. . *Tables of [M]{}ellin [T]{}ransforms* (Springer-Verlag, New York, 1974). . *Asymptotics and [M]{}ellin-[B]{}arnes [I]{}ntegrals*, volume 85 of *Encyclopedia of Mathematics and its Applications* (Cambridge University Press, Cambridge, 2001). . Sur une classe étendue de transcendantes uniformes. *C. R. Acad. Sci. Paris* **103** (1886) 862–864. . Sur une classe nouvelle de transcendantes uniformes. *J. Math. Pures Appl. IV. Ser.* **6** (1890) 316–365. . On conformal mapping and iteration of rational functions. *Complex Variables Theory Appl.* **5** (1986) 117–126. . *Potential [T]{}heory in the [C]{}omplex [P]{}lane*, volume 28 of *London Mathematical Society Student Texts* (Cambridge University Press, Cambridge, 1995). . *The [C]{}hebyshev [P]{}olynomials* (Wiley-Interscience \[John Wiley & Sons\], New York, 1974). Pure and Applied Mathematics. . Long time properties of solutions of simplest $q$-difference equations ([R]{}ussian) (2000). Preprint. . Über iterierte [F]{}unktionen. *Math. Ann.* **3** (1871) 296–322. . Spectral zeta function of symmetric fractals. In [C. Bandt, U. Mosco M. Z[ä]{}hle]{}, editors, *Fractal geometry and stochastics III*, volume 57 of *Progr. Probab.*, pages 245–262 (Birkhäuser, Basel, 2004). . Spectral zeta functions of fractals and the complex dynamics of polynomials. *Trans. Amer. Math. Soc.* **359** (2007) 4339–4358 (electronic). . *Lectures on the [G]{}eneral [T]{}heory of [I]{}ntegral [F]{}unctions* (Private, Toulouse, 1923). . *Fonctions [A]{}nalytiques* (Presses Universitaires de France, Paris, 1954). [^1]: This author is supported by the Austrian Science Foundation FWF, project S9605, part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

ArXiv

--- bibliography: - 'short.bib' title: Mathematical Formulae in Wikimedia Projects 2020 ---

ArXiv

--- abstract: 'Notwithstanding the big efforts devoted to the investigation of the mechanisms responsible for the high-energy ($E>100$ MeV) $\gamma-$ray emission in active galactic nuclei (AGN), the definite answer is still missing. The X-ray energy band ($0.4-10$ keV) is crucial for this type of study, since both synchrotron and inverse Compton emission can contribute to the formation of the continuum. Within an ongoing project aimed at the investigation of the $\gamma-$ray emission mechanism acting in the AGN detected by the EGRET telescope onboard CGRO, we firstly focused on the sources for which X-ray and optical/UV data are available in the *XMM-Newton* public archive. The preliminary results are outlined here.' author: - 'L. Foschini' - 'G. Ghisellini' - 'C.M. Raiteri' - 'F. Tavecchio' - 'M. Villata' - 'M. Dadina' - 'G. Di Cocco' - 'G. Malaguti' - 'L. Maraschi' - 'E. Pian' - 'G. Tagliaferri' title: 'The $XMM-Newton$ view of $\gamma-$ray loud active nuclei' --- Introduction ============ The discovery of $\gamma-$ray loud AGN dates back to the dawn of $\gamma-$ray astronomy, when the European satellite *COS-B* ($1975-1982$) detected photons in the $50-500$ MeV range from 3C273 (Swanenburg et al. 1978). However, 3C273 remained the only AGN detected by *COS-B*. A breakthrough in this research field came later with the Energetic Gamma Ray Experiment Telescope (EGRET) on board the *Compton Gamma-Ray Observatory* (CGRO, 1991-2000). The third catalog of point sources contains $271$ sources detected at energies greater than $100$ MeV and $93$ of them are identified with blazars ($66$ at high confidence and $27$ at low confidence), and $1$ with the nearby radiogalaxy Centaurus A (Hartman et al. 1999). Therefore, EGRET discovered that the blazar type AGN are the primary source of high-energy cosmic $\gamma-$rays (von Montigny et al. 1995). Later on, Ghisellini et al. (1998) and Fossati et al. (1998) proposed a unified scheme for $\gamma-$ray loud blazars, based on their physical properties (see, however, Padovani et al. 2003). Specifically, the blazars are classified according to a sequence going from BL Lac to flat-spectrum radio quasar depending on the increase of the observed luminosity, which in turn leads to a decrease of the synchrotron and inverse Compton peak frequencies, and an increase of the ratio between the emitted radiation at low and high frequencies. In other words, the spectral energy distribution (SED) of blazars is typically composed of two peaks, one due to synchrotron emission and the other to inverse Compton radiation. Low luminosity blazars have the synchrotron peak in the UV-soft X-ray energy band and therefore are “high-energy peaked” (HBL). As the synchrotron peak shifts to low energies (near infrared, “low-energy peaked”, LBL), the luminosity increases and the X-ray emission can be due to synchrotron or inverse Compton or a mixture of both. For the Flat-Spectrum Radio-Quasars (FSRQ), the blazars with the highest luminosity, the synchrotron peak is in the far infrared and the X-ray emission is due to inverse Compton. Moreover, the two-peaks SED is a dynamic picture of the blazar behaviour: indeed, these AGN are characterized by strong flares during which the SED can change dramatically. The X-ray energy band can therefore be crucial to understand the blazars behaviour and to improve the knowledge of high-energy emission. Sample selection and data analysis ================================== To investigate the X-ray and optical/UV characteristics of $\gamma-$ray loud AGN in order to search for specific issues conducive to the $\gamma-$ray loudness, we cross correlated the $3^{\rm rd}$ EGRET Catalog (Hartman et al. 1999), updated with the identifications performed to date, with the public observations available in the *XMM-Newton* Science Archive to search for spatial coincidences within $10'$ of the boresight of the EPIC camera. Fourteen AGN have been found (Table 1) as of April $14^{\rm th}$, 2005, for a total of $43$ observations. For three of them there are several observations available: 15 for 3C $273$, 6 for Mkn $421$, 9 for PKS $2155-304$. The data from $6$ sources of the present sample are analyzed here for the first time and, among them, one has never been observed in X-rays before (PKS $1406-706$). Data from the EPIC camera (MOS, Turner et al. 2001; PN, Strüder et al. 2001) and the Optical Monitor (Mason et al. 2001) have been analyzed with `XMM SAS 6.1` and `HEASoft 6.0`, together with the latest calibration files available at April $14^{\rm th}$, 2005, and by following the standard procedures described in Snowden et al. (2004). In addition, the Optical Monitor makes it possible to have optical/UV data simultaneous to X-ray for most of the selected sources, with the only exception of PKS $0521-365$, Mkn $421$, and Cen A. 3EG Counterpart Type$^{\mathrm{*}}$ Redshift -------------- ---------------- --------------------- ------------------------- J$0222+4253$ $0219+428$ LBL $0.444$ J$0237+1635$ AO $0235+164$ LBL $0.94$ J$0530-3626$ PKS $0521-365$ FSRQ $0.05534$ J$0721+7120$ S5 $0716+714$ LBL $>0.3$ J$0845+7049$ S5 $0836+710$ FSRQ $2.172$ J$1104+3809$ Mkn $421$ HBL $0.03002$ J$1134-1530$ PKS $1127-145$ FSRQ $1.184$ J$1222+2841$ ON $231$ LBL $0.102$ J$1229+0210$ 3C $273$ FSRQ $0.15834$ J$1324-4314$ Cen A RG $0.00182^{\mathrm{**}}$ J$1339-1419$ PKS $1334-127$ FSRQ $0.539$ J$1409-0745$ PKS $1406-076$ FSRQ $1.494$ J$1621+8203$ NGC $6251$ RG $0.0247$ J$2158-3023$ PKS $2155-304$ HBL $0.116$ : Main characteristics of the observed AGN. \[tab:host\] Main Results ============ The main findings of this study can be summarized as follows: \(i) the EGRET blazars studied here have spectral characteristics in agreement with the unified sequence of Ghisellini et al. (1998) and Fossati et al. (1998); \(ii) no evident characteristics conducive to the $\gamma-$ray loudness have been found: the photon indices are generally consistent with what is expected for this type of sources, with FSRQ that are harder than BL Lac; there are hints of some differences in the photon indices when compared with other larger catalogs (e.g. *BeppoSAX* Giommi et al. 2002), particularly for FSRQ: the sources best fit with a simple power law model show a harder photon index ($1.39\pm 0.09$ vs $1.59\pm 0.05$); however, the statistics is too poor to make firm conclusions (3 sources vs 26 in the *BeppoSAX* catalog); \(iii) three sources show Damped Lyman $\alpha$ systems along the line of sight (AO $0235+164$, PKS $1127-145$, S5 $0836+710$), but it is not clear if the intervening galaxies can generate gravitational effects altering the characteristics of the blazars so to enhance the $\gamma-$ray loudness; \(iv) no evidence of peculiar X-ray spectral features has been found, except for the emission lines of the iron complex in Cen A. More details of the analysis will be available in Foschini et al. (2005). Acknowledgments {#acknowledgments .unnumbered} =============== This work is based on public observations obtained with *XMM–Newton*, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This work was partly supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00321 and by the Italian Space Agency (ASI). Foschini L. et al., 2005, submitted Fossati G. et al., 1998, MNRAS 299, 433 Ghisellini G. et al., 1998, MNRAS 301, 451 Giommi P. et al., 2002, astro-ph/0209596 Hartman R.C. et al., 1999, ApJS 123, 79 Mason K.O. et al., 2001, A&A 365, L36 Padovani P. et al., 2003, ApJ 588, 128 Snowden S. et al., 2004. An introduction to XMM-Newton data analysis. Version 2.01, 23 July 2004. Strüder L. et al., 2001, A&A 365, L18 Swanenburg B.N. et al., 1978, Nature 275, 298 Turner M.J. et al., 2001, A&A 365, L27 von Montigny C. et al., 1995, ApJ 440, 525

ArXiv

--- abstract: 'This paper has been addressed to a very old but burning problem of energy in General Relativity. We evaluate energy and momentum densities for the static and axisymmetric solutions. This specializes to two metrics, i.e., Erez-Rosen and the gamma metrics, belonging to the Weyl class. We apply four well-known prescriptions of Einstein, Landau-Lifshitz, Papaterou and M$\ddot{o}$ller to compute energy-momentum density components. We obtain that these prescriptions do not provide similar energy density, however momentum becomes constant in each case. The results can be matched under particular boundary conditions.' author: - | M. Sharif [^1] and Tasnim Fatima\ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan. title: '**Energy Distribution associated with Static Axisymmetric Solutions**' --- [**Keywords:**]{} Energy-momentum, axisymmetric spacetimes. Introduction ============ The problem of energy-momentum of a gravitational field has always been an attractive issue in the theory of General Relativity (GR). The notion of energy-momentum for asymptotically flat spacetime is unanimously accepted. Serious difficulties in connection with its notion arise in GR. However, for gravitational fields, this can be made locally vanish. Thus one is always able to find the frame in which the energy-momentum of gravitational field is zero, while in other frames it is not true. Noether’s theorem and translation invariance lead to the canonical energy-momentum density tensor, $T_a^b$, which is conserved. $$T^b_{a;b}=0,\quad (a,b=0,1,2,3).$$ In order to obtain a meaningful expression for energy-momentum, a large number of definitions for the gravitation energy-momentum in GR have been proposed. The first attempt was made by Einstein who suggested an expression for energy-momentum density \[1\]. After this, many physicists including Landau-Lifshitz \[2\], Papapetrou \[3\], Tolman \[4\], Bergman \[5\] and Weinburg \[6\] had proposed different expressions for energy-momentum distribution. These definitions of energy-momentum complexes give meaningful results when calculations are performed in Cartesian coordinates. However, the expressions given by M$\ddot{o}$ller \[7,8\] and Komar \[9\] allow one to compute the energy-momentum densities in any spatial coordinate system. An alternate concept of energy, called quasi-local energy, does not restrict one to use particular coordinate system. A large number of definitions of quasi-local masses have been proposed by Penrose \[10\] and many others \[11,12\]. Chang et al. \[13\] showed that every energy-momentum complex can be associated with distinct boundary term which gives the quasi-local energy-momentum. There is a controversy with the importance of non-tensorial energy-momentum complexes whose physical interpretation has been a problem for the scientists. There is a uncertainity that different energy-momentum complexes would give different results for a given spacetime. Many researchers considered different energy-momentum complexes and obtained encouraging results. Virbhadra et al. \[14-18\] investigated several examples of the spacetimes and showed that different energy-momentum complexes could provide exactly the same results for a given spacetime. They also evaluated the energy-momentum distribution for asymptotically non-flat spacetimes and found the contradiction to the previous results obtained for asymptotically flat spacetimes. Xulu \[19,20\] evaluated energy-momentum distribution using the M$\ddot{o}$ller definition for the most general non-static spherically symmetric metric. He found that the result is different in general from those obtained using Einstein’s prescription. Aguirregabiria et al. \[21\] proved the consistency of the results obtained by using the different energy-momentum complexes for any Kerr-Schild class metric. On contrary, one of the authors (MS) considered the class of gravitational waves, G$\ddot{o}$del universe and homogeneous G$\ddot{o}$del-type metrics \[22-24\] and used the four definitions of the energy-momentum complexes. He concluded that the four prescriptions differ in general for these spacetimes. Ragab \[25,26\] obtained contradictory results for G$\ddot{o}$del-type metrics and Curzon metric which is a special solution of the Weyl metrics. Patashnick \[27\] showed that different prescriptions give mutually contradictory results for a regular MMaS-class black hole. In recent papers, we extended this procedure to the non-null Einstein-Maxwell solutions, electromagnetic generalization of G$\ddot{o}$del solution, singularity-free cosmological model and Weyl metrics \[28-30\]. We applied four definitions and concluded that none of the definitions provide consistent results for these models. This paper continues the study of investigation of the energy-momentum distribution for the family of Weyl metrics by using the four prescriptions of the energy-momentum complexes. In particular, we would explore energy-momentum for the Erez-Rosen and gamma metrics. The paper has been distributed as follows. In the next section, we shall describe the Weyl metrics and its two family members Erez-Rosen and gamma metrics. Section 3 is devoted to the evaluation of energy-momentum densities for the Erez-Rosen metric by using the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M$\ddot{o}$ller. In section 4, we shall calculate energy-momentum density components for the gamma metric. The last section contains discussion and summary of the results. The Weyl Metrics ================ Static axisymmetric solutions to the Einstein field equations are given by the Weyl metric \[31,32\] $$ds^2=e^{2\psi}dt^2-e^{-2\psi}[e^{2\gamma}(d\rho^2+dz^2) +\rho^2d\phi^2]$$ in the cylindrical coordinates $(\rho,~\phi,~z)$. Here $\psi$ and $\gamma$ are functions of coordinates $\rho$ and $z$. The metric functions satisfy the following differential equations $$\begin{aligned} \psi_{\rho\rho}+\frac{1}{\rho}\psi_{\rho}+\psi_{zz}=0,\\ \gamma_{\rho}=\rho(\psi^2_{\rho}-\psi^2_{z}),\quad \gamma_{z}=2\rho\psi_{\rho}\psi_{z}.\end{aligned}$$ It is obvious that Eq.(3) represents the Laplace equation for $\psi$. Its general solution, yielding an asymptotically flat behaviour, will be $$\psi=\sum^\infty_{n=0}\frac{a_n}{r^{n+1}}P_n(\cos\theta),$$ where $r=\sqrt{\rho^2+z^2},~\cos\theta=z/r$ are Weyl spherical coordinates and $P_n(\cos\theta)$ are Legendre Polynomials. The coefficients $a_n$ are arbitrary real constants which are called [*Weyl moments*]{}. It is mentioned here that if we take $$\begin{aligned} \psi=-\frac{m}{r},\quad\gamma=-\frac{m^2\rho^2}{2r^4},\quad r=\sqrt{\rho^2+z^2}\end{aligned}$$ then the Weyl metric reduces to special solution of Curzon metric \[33\]. There are more interesting members of the Weyl family, namely the Erez-Rosen and the gamma metric whose properties have been extensively studied in the literature \[32,34\]. The Erez-Rosen metric \[32\] is defined by considering the special value of the metric function $$2\psi=ln(\frac{x-1}{x+1})+q_2(3y^2-1)[\frac{1}{4}(3x^2-1) ln(\frac{x-1}{x+1})+\frac{3}{2}x],$$ where $q_2$ is a constant. Energy and Momentum for the Erez-Rosen Metric ============================================= In this section, we shall evaluate the energy and momentum density components for the Erez-Rosen metric by using different prescriptions. To obtain meaningful results in the prescriptions of Einstein, Ladau-Lifshitz’s and Papapetrou, it is required to transform the metric in Cartesian coordinates. This can be done by using the transformation equations $$x=\rho cos\theta,\quad y=\rho sin\theta.$$ The resulting metric in these coordinates will become $$ds^2=e^{2\psi}dt^2-\frac{e^{2(\gamma-\psi)}}{\rho^2}(xdx+ydy)^2\nonumber\\ -\frac{e^{-2\psi}}{\rho^2}(xdy-ydx)^2-e^{2(\gamma-\psi)}dz^2.$$ Energy and Momentum in Einstein’s Prescription ---------------------------------------------- The energy-momentum complex of Einstein \[1\] is given by $$\Theta^b_a= \frac{1}{16 \pi}H^{bc}_{a,c},$$ where $$H^{bc}_a=\frac{g_{ad}}{\sqrt{-g}}[-g(g^{bd}g^{ce} -g^{be}g^{cd})]_{,e},\quad a,b,c,d,e = 0,1,2,3.$$ Here $\Theta^0_{0}$ is the energy density, $\Theta^i_{0}~ (i=1,2,3)$ are the momentum density components and $\Theta^0_{i}$ are the energy current density components. The Einstein energy-momentum satisfies the local conservation laws $$\frac{\partial \Theta^b_a}{\partial x^{b}}=0.$$ The required components of $H_a^{bc}$ are the following $$\begin{aligned} H^{01}_{0}&=&\frac{4y}{\rho^2}e^{2\gamma}(y\psi_{,x}-x\psi_{,y}) +\frac{4x}{\rho^2}(x\psi_{,x}+y\psi_{,y})\nonumber\\ &-&\frac{x}{\rho^2}-2x\psi^2_{,\rho}+\frac{x}{\rho^2}e^{2\gamma},\\ H^{02}_{0}&=&\frac{4x}{\rho^2}e^{2\gamma}(x\psi_{,y}-y\psi_{,x}) +\frac{4y}{\rho^2}(x\psi_{,x}+y\psi_{,y})\nonumber\\ &-&\frac{y}{\rho^2}-2y\psi^2_{,\rho}+\frac{y}{\rho^2}e^{2\gamma}.\end{aligned}$$ Using Eqs.(13)-(14) in Eq.(10), we obtain the energy and momentum densities in Einstein’s prescription $$\begin{aligned} \Theta^0_{0}&=&\frac{1}{8\pi \rho^2}[e^{2\gamma}\{\rho^2\psi^2_{,\rho}+2(x^2\psi_{,yy}+y^2\psi_{xx} -x\psi_{,x}-y\psi_{,y})\}\nonumber\\ &+&2\{x^2\psi_{,xx}+y^2\psi_{,yy}+x\psi_{,x}+y\psi_{,y} -\rho^2\psi_{,\rho}(\psi_{,\rho}+\rho \psi_{,\rho\rho})\}].\end{aligned}$$ All the momentum density components turn out to be zero and hence momentum becomes constant. Energy and Momentum in Landau-Lifshitz’s Prescription ----------------------------------------------------- The Landau-Lifshitz \[2\] energy-momentum complex can be written as $$L^{ab}= \frac{1}{16 \pi}\ell^{acbd}_{,cd},$$ where $$\ell^{acbd}= -g(g^{ab}g^{cd}-g^{ad}g^{cb}).$$ $L^{ab}$ is symmetric with respect to its indices. $L^{00}$ is the energy density and $L^{0i}$ are the momentum (energy current) density components. $\ell^{abcd}$ has symmetries of the Riemann curvature tensor. The local conservation laws for Landau-Lifshitz energy-momentum complex turn out to be $$\frac{\partial L^{ab}}{\partial x^{b}}=0.$$ The required non-vanishing components of $\ell^{acbd}$ are $$\begin{aligned} \ell^{0101}&=&-\frac{y^2}{\rho^2}e^{4\gamma-4\psi} -\frac{x^2}{\rho^2}e^{2\gamma-4\psi},\\ \ell^{0202}&=&-\frac{x^2}{\rho^2}e^{4\gamma-4\psi} -\frac{y^2}{\rho^2}e^{2\gamma-4\psi},\\ \ell^{0102}&=&\frac{xy}{\rho^2}e^{4\gamma-4\psi} -\frac{xy}{\rho^2}e^{2\psi-4\gamma}.\end{aligned}$$ Using Eqs.(19)-(21) in Eq.(16), we get $$\begin{aligned} L^{00}&=&\frac{e^{2\gamma-4\psi}}{8\pi \rho^2}[e^{2\gamma}\{2\rho^2\psi^2_{,\rho}-8(y^2\psi^2_{,x} +x^2\psi^2_{,y})+2(x^2\psi_{,xx}+y^2\psi_{,yy}\nonumber\\ &-&x\psi_{,x}-y\psi_{,y})+16xy\psi_{,x}\psi_{,y}-4xy\psi_{,xy}\}\nonumber\\ &-&\rho^2\psi_{,\rho}(3\psi_{,\rho} +2\rho^2\psi^3_{,\rho}+2\rho \psi_{,\rho\rho})-8\rho^2\psi^2_{,\rho}(x\psi_{,x}+y\psi_{,y})\nonumber\\ &-&8(x^2\psi^2_{,x}+y^2\psi^2_{,y})+2(x^2\psi_{,xx}+y^2\psi_{,yy}\nonumber\\ &+&x\psi_{,x}+y\psi_{,y})-16xy\psi_{,x}\psi_{,y}+4xy\psi_{,xy}].\end{aligned}$$ The momentum density vanishes and hence momentum becomes constant. Energy and Momentum in Papapetrou’s Prescription ------------------------------------------------ We can write the prescription of Papapetrou \[3\] energy-momentum distribution in the following way $$\Omega^{ab}=\frac{1}{16\pi}N^{abcd}_{,cd},$$ where $$N^{abcd}=\sqrt{-g}(g^{ab}\eta^{cd}-g^{ac}\eta^{bd} +g^{cd}\eta^{ab}-g^{bd}\eta^{ac}),$$ and $\eta^{ab}$ is the Minkowski spacetime. It follows that the energy-momentum complex satisfies the following local conservation laws $$\frac{\partial \Omega^{ab}}{\partial x^b}=0.$$ $\Omega^{00}$ and $\Omega^{0i}$ represent the energy and momentum (energy current) density components respectively. The required components of $N^{abcd}$ are $$\begin{aligned} N^{0011}&=&-\frac{y^2}{\rho^2}e^{2\gamma}-\frac{x^2}{\rho^2} -e^{2\gamma-4\psi},\\ N^{0022}&=&-\frac{x^2}{\rho^2}e^{2\gamma}-\frac{y^2}{\rho^2} -e^{2\gamma-4\psi},\\ N^{0012}&=&-\frac{xy}{\rho^2}e^{2\gamma}-\frac{xy}{\rho^2}.\end{aligned}$$ Substituting Eqs.(26)-(28) in Eq.(23), we obtain the following energy density $$\begin{aligned} \Omega^{00}&=&\frac{e^{2\gamma}}{8\pi}[\psi^2_{,\rho} -e^{-4\psi}\{\psi^2_{,\rho}+2\rho^2\psi^4_{,\rho} +2\rho \psi_{,\rho}\psi_{,\rho\rho}\nonumber\\ &-&8\psi^2_{,\rho}(x\psi_{,x}+y\psi_{,y})+ 8(\psi^2_{,x}+\psi^2_{,y})-2(\psi_{,xx}+\psi_{,yy})\}].\end{aligned}$$ The momentum density vanishes. Energy and Momentum in Möller’s Prescription -------------------------------------------- The energy-momentum density components in Möller’s prescription \[7,8\] are given as $$M^b_a= \frac{1}{8\pi}K^{bc}_{a,c},$$ where $$K_a^{bc}= \sqrt{-g}(g_{ad,e}-g_{ae,d})g^{be}g^{cd}.$$ Here $K^{bc}_{ a}$ is symmetric with respect to the indices. $M^0_{0}$ is the energy density, $M^i_{0}$ are momentum density components, and $M^0_{i}$ are the components of energy current density. The Möller energy-momentum satisfies the following local conservation laws $$\frac{\partial M^b_a}{\partial x^b}=0.$$ Notice that Möller’s energy-momentum complex is independent of coordinates. The components of $K^{bc}_a$ for Erez-Rosen metric is the following $$\begin{aligned} K^{01}_0&=&2\rho \psi_{,\rho}.\end{aligned}$$ Substitute Eq.(33) in Eq.(30), we obtain $$\begin{aligned} M^0_0&=&\frac{1}{4\pi}[\psi_{,\rho}+\rho\psi_{,\rho\rho}].\end{aligned}$$ Again, we get momentum constant. The partial derivatives of the function $\psi$ are given by $$\begin{aligned} \psi_{,x}&=&\frac{1}{x^2-1}+\frac{q_2}{4}(3y^2-1)[3x ln(\frac{x-1}{x+1})+\frac{3x^2-1}{x^2-1}+3],\\ \psi_{,y}&=&\frac{3yq_2}{4}[(3x^2-1) ln(\frac{x-1}{x+1})+6x],\\ \psi_{,xx}&=&\frac{-2x}{(x^2-1)^2}+\frac{q_2}{4}(3y^2-1)[3 ln(\frac{x-1}{x+1})+2x\frac{3x^2-5}{(x^2-1)^2}],\\ \psi_{,yy}&=&\frac{3q_2}{4}[(3x^2-1) ln(\frac{x-1}{x+1})+6x],\\ \psi_{,xy}&=& U_{,yx}=\frac{3yq_2}{4}[3x ln(\frac{x-1}{x+1})+2\frac{3x^2-2}{x^2-1}],\\ \psi_{,\rho}&=& \frac{\rho}{x(x^2-1)}+\frac{\rho q_2}{4x}[3x(3\rho^2-2) ln(\frac{x-1}{x+1})\nonumber\\ &+&2\frac{(3x^2-1)(3y^2-1)}{x^2-1}+18x^2],\end{aligned}$$ $$\begin{aligned} \psi_{,\rho\rho}&=&\frac{1}{x(x^2-1)}-\frac{2\rho^2}{x(x^2-1)^2} +\frac{q_2}{4x^2}(3y^2-1)[3(\rho^2+x^2) ln(\frac{x-1}{x+1})\nonumber\\ &+&\frac{2x}{x^2-1}(3x^2-2+\frac{\rho^2(3x^2-5)}{x^2-1})] +\frac{3\rho q_2}{4}(1+\frac{\rho}{y^2})\nonumber\\ &\times&[(3x^2-1)ln(\frac{x-1}{x+1})+6x]+\frac{3\rho^2q_2}{x}[3 ln(\frac{x-1}{x+1})+2\frac{3x^2-1}{x^2-1}].\end{aligned}$$ Energy and Momentum for the Gamma Metric ======================================== A static and asymptotically flat exact solution to the Einstein vacuum equations is known as the gamma metric. This is given by the metric \[34\] $$ds^2=(1-\frac{2m}{r})^{\gamma}dt^2-(1-\frac{2m}{r})^{-\gamma} [(\frac{\Delta}{\Sigma})^{\gamma^2-1}dr^2+\frac{\Delta^{\gamma^2}} {\Sigma^{\gamma^2-1}}d\theta^2+\Delta\sin^2\theta d\phi^2],$$ where $$\begin{aligned} \Delta &=& r^2-2mr,\\ \Sigma &=& r^2-2mr+m^2sin^2\theta,\end{aligned}$$ $m$ and $\gamma$ are constant parameters. $m=0$ or $\gamma=0$ gives the flat spacetime. For $|\gamma|=1$ the metric is spherically symmetric and for $|\gamma|\neq1$, it is axially symmetric. $\gamma=1$ gives the Schwarzschild spacetime in the Schwarzschild coordinates. $\gamma=-1$ gives the Schwarzschild spacetime with negative mass, as putting $m=-M(m>0)$ and carrying out a non-singular coordinate transformation $(r\rightarrow R=r+2M)$ one gets the Schwarzschild spacetime (with positive mass) in the Schwarzschild coordinates $(t,R,\theta,\Phi)$. In order to have meaningful results in the prescriptions of Einstein, Landau-Lifshitz and Papapetrou, it is necessary to transform the metric in Cartesian coordinates. We transform this metric in Cartesian coordinates by using $$x=rsin\theta\cos\phi,\quad y=rsin\theta\sin\phi,\quad z=rcos\theta.$$ The resulting metric in these coordinates will become $$\begin{aligned} ds^2&=&(1-\frac{2m}{r})^{\gamma}dt^2-(1-\frac{2m}{r})^{-\gamma} [(\frac{\Delta}{\Sigma})^{\gamma^2-1}\frac{1}{r^2} \{xdx+ydy+zdz\}^2\nonumber\\ &+&\frac{\Delta^{\gamma^2}}{\Sigma^{\gamma^2-1}} \{\frac{xzdx+yzdy-(x^2+y^2)dz}{r^2\sqrt{x^2+y^2}}\}^2 +\frac{\Delta(xdy-ydx)^2}{r^2(x^2+y^2)}.\end{aligned}$$ Now we calculate energy-momentum densities using the different prescriptions given below. Energy and Momentum in Einstein’s Prescription ---------------------------------------------- The required non-vanishing components of $H^{bc}_{a}$ are $$\begin{aligned} H^{01}_{0}&=&4\gamma m \frac{x}{r^3}+(\frac{\Delta} {\Sigma})^{\gamma^2-1}\frac{x}{x^2+y^2}-(\gamma^2+1) (1-\frac{m}{r})\frac{2x}{r^2}\nonumber\\ &+& (\gamma^2-1)(1-\frac{m}{r})\frac{2\Delta x}{\Sigma r^2}+\frac{2\Delta x}{r^4} +(\gamma^2-1)\frac{2m^2xz^2}{\Sigma r^4}\nonumber\\ &-&\frac{xz^2}{r^2(x^2+y^2)}+\frac{x}{r^2},\\ H^{02}_{0}&=&4\gamma m \frac{y}{r^3}+(\frac{\Delta} {\Sigma})^{\gamma^2-1}\frac{y}{x^2+y^2}-(\gamma^2+1) (1-\frac{m}{r})\frac{2y}{r^2}\nonumber\\ &+& (\gamma^2-1)(1-\frac{m}{r})\frac{2\Delta y}{\Sigma r^2}+\frac{2\Delta y}{r^4} +(\gamma^2-1)\frac{2m^2yz^2}{\Sigma r^4}\nonumber\\ &-&\frac{xyz^2}{r^2(x^2+y^2)}+\frac{y}{r^2},\\ H^{03}_{0}&=&4\gamma m \frac{z}{r^3}-(\gamma^2+1)(1-\frac{m}{r})\frac{2z}{r^2}+ (\gamma^2-1)(1-\frac{m}{r})\frac{2\Delta z}{\Sigma r^2}\nonumber\\&+&\frac{2\Delta z}{r^4} -(\gamma^2-1)(x^2+y^2)\frac{2m^2z}{\Sigma r^4}+\frac{2z}{r^2}.\end{aligned}$$ Using Eqs.(47)-(49) in Eq.(10), we obtain non-vanishing energy density in Einstein’s prescription given as $$\begin{aligned} \Theta^0_{0}&=&\frac{1}{8\pi \Sigma^{\gamma^2}r^6}[(\gamma^2-1)\Sigma \Delta^{\gamma^2-2}r^5(r-m)-(\gamma^2-1)\Delta^{\gamma^2-1}r^2\nonumber\\ &\times&\{r^4-mr^3+m^2r^2-m^2(x^2+y^2)\} -(\gamma^2+1)\Sigma^{\gamma^2}r^4\nonumber\\ &+&2(\gamma^2-1)\Sigma^{\gamma^2-1} r^4(r-m)^2 +(\gamma^2-1)m\Delta\Sigma^{\gamma^2-1}r^2\nonumber\\ &-&(\gamma^2-1)\Delta^{\gamma^2-2}\Delta r^4(r-m)^2+(\gamma^2-1)\Delta^{\gamma^2-1}\nonumber\\&\times& \Delta r^5(r-m)+2\Sigma^{\gamma^2}r^3(r-m)-\Sigma^{\gamma^2}\Delta r^2+\Sigma^{\gamma^2}r^4\nonumber\\ &+&3(\gamma^2-1)\Sigma^{\gamma^2}r^2m^2z^2 -(\gamma^2-1)\Sigma^{\gamma^2-1}r^4m^2\nonumber\\ &-&2(\gamma^2-1) \Sigma^{\gamma^2-2}m^4z^2(x^2+y^2)].\end{aligned}$$ The momentum density components become zero and consequently momentum is constant. Energy and Momentum in Landau-Lifshitz’s Prescription ----------------------------------------------------- The required non-vanishing components of $\ell^{acbd}$ are $$\begin{aligned} \ell^{0101}&=&-(1-\frac{2m}{r})^{-2\gamma}[\frac{y^2\Delta^{2\gamma^2-1}} {r^2(x^2+y^2)\Sigma^{2(\gamma^2-1)}} +\frac{x^2\Delta^{\gamma^2+1}}{r^6\Sigma^{\gamma^2-1}}\nonumber\\& +&\frac{\Delta^{\gamma^2}x^2z^2} {r^4(x^2+y^2)\Sigma^{\gamma^2-1}}],\\ \ell^{0202}&=&-(1-\frac{2m}{r})^{-2\gamma}[\frac{x^2\Delta^{2\gamma^2-1}} {r^2(x^2+y^2)\Sigma^{2(\gamma^2-1)}} +\frac{y^2\Delta^{\gamma^2+1}}{r^6\Sigma^{\gamma^2-1}}\nonumber\\& +&\frac{\Delta^{\gamma^2}y^2z^2} {r^4(x^2+y^2)\Sigma^{\gamma^2-1}}],\\ \ell^{0303}&=&-(1-\frac{2m}{r})^{-2\gamma}[\frac{z^2\Delta^{\gamma^2+1}} {r^6\Sigma^{\gamma^2-1}}+\frac{(x^2+y^2)\Delta^{\gamma^2}} {r^4\Sigma^{\gamma^2-1}}],\\ \ell^{0102}&=&(1-\frac{2m}{r})^{-2\gamma}[\frac{xy\Delta^{2\gamma^2-1}} {r^2(x^2+y^2)\Sigma^{2(\gamma^2-1)}} -\frac{xy\Delta^{\gamma^2+1}}{r^6\Sigma^{\gamma^2-1}}\nonumber\\& -&\frac{\Delta^{\gamma^2}xyz^2} {r^4(x^2+y^2)\Sigma^{\gamma^2-1}}],\\ \ell^{0103}&=&-(1-\frac{2m}{r})^{-2\gamma}[\frac{xz\Delta^{\gamma^2+1}} {r^6\Sigma^{\gamma^2-1}}-\frac{xz\Delta^{\gamma^2}} {r^4\Sigma^{\gamma^2-1}}],\\ \ell^{0203}&=&-(1-\frac{2m}{r})^{-2\gamma}[\frac{yz\Delta^{\gamma^2+1}} {r^6\Sigma^{\gamma^2-1}}-\frac{yz\Delta^{\gamma^2}} {r^4\Sigma^{\gamma^2-1}}].\end{aligned}$$ When we substitute these values in Eq.(16), it follows that the energy density remains non-zero while momentum density components vanish. This is given as follows $$\begin{aligned} L^{00}&=&\frac{(1-\frac{2m}{r})^{-2\gamma}}{8\pi }[-\frac{4\gamma m^2}{r^4}(\frac{\Delta}{\Sigma})^{\gamma^2-1}(2\gamma+1)+\frac{2\gamma m}{r^7}\{-(\frac{\Delta}{\Sigma})^{2(\gamma^2-1)}r^4\nonumber\\ &+&4(\gamma^2+1)(1-\frac{m}{r})(\frac{\Delta}{\Sigma})^{\gamma^2-1}r^4 -4(\gamma^2-1)(1-\frac{m}{r})(\frac{\Delta}{\Sigma})^{\gamma^2}r^4\nonumber\\ &-&\frac{\Delta^{\gamma^2}}{\Sigma^{\gamma^2-1}}r^2- (\frac{\Delta}{\Sigma})^{\gamma^2-1}r^4+4({\gamma^2-1}) \frac{\Delta^{\gamma^2-1}}{\Sigma^{\gamma^2}}m^2z^2(x^2+y^2)\}\nonumber\\ &+&\frac{1}{r^2}(2\gamma^2-1)(1-\frac{m}{r})(\frac{\Delta} {\Sigma})^{2(\gamma^2-1)}-\frac{2}{r^2}(\gamma^2-1)(1-\frac{m}{r}\nonumber\\ &+&\frac{m^2}{r^2}-\frac{m^2(x^2+y^2)}{r^4})(\frac{\Delta} {\Sigma})^{2\gamma^2-1}-\frac{\Delta^{2\gamma^2-1}} {r^4\Sigma^{2(\gamma^2-1)}}\nonumber\\ &-&\frac{2\gamma^2}{r^2}(\gamma^2+1)(1-\frac{m}{r})^2(\frac{\Delta} {\Sigma})^{\gamma^2-1}+\frac{4}{r^2}({\gamma^4-1}) (1-\frac{m}{r})^2(\frac{\Delta}{\Sigma})^{\gamma^2}\nonumber\\ &-&\frac{2}{r^2}({\gamma^2-1})(1-\frac{m}{r})^2(\frac{\Delta} {\Sigma})^{\gamma^2+1}-2({\gamma^2-1})(1-\frac{m}{r}+\frac{m^2}{r^2}\nonumber\\ &-&\frac{m^2(x^2+y^2)}{r^4})\frac{\Delta^{\gamma^2+1}}{\Sigma^{\gamma^2}}- (\gamma^2+1)\frac{m\Delta^{\gamma^2}}{r^5\Sigma^{\gamma^2}-1}\nonumber\\ &+&3(\gamma^2+1)\frac{\Delta^{\gamma^2}}{r^4\Sigma^{\gamma^2-1}}(1-\frac{m}{r}) +(\gamma^2-1)\frac{m\Delta^{\gamma^2+1}}{r^5\Sigma^{\gamma^2}}(1-\frac{2m}{r}\nonumber\\ &-& \frac{m^2(x^2+y^2)}{r^3}) -2(\gamma^2-1)(x^2+y^2) \frac{m^2z^2\Delta^{\gamma^2+1}}{r^{10}\Sigma^{\gamma^2}}\nonumber\\&-&\frac{3\Delta^ {\gamma^2+1}}{r^6\Sigma^{\gamma^2-1}}+\gamma^2(\frac{\Delta} {\Sigma})^{\gamma^2-1}(1-\frac{m}{r})(\frac{1}{r^2}+\frac{2z^2}{r^4}) -2\gamma^2(\gamma^2-1)\nonumber\\&\times&(x^2+y^2)\frac{m^4z^2\Delta^{\gamma^2}} {r^8\Sigma^{\gamma^2+1}} +2(\gamma^2-1)(x^2+y^2)(\frac{\Delta} {\Sigma})^{\gamma^2}\frac{m^2z^2}{r^8}\nonumber\\&-&(\gamma^2-1)(\frac{\Delta} {\Sigma})^{\gamma^2}(1-\frac{m}{r}+\frac{m^2}{r^2})\frac{1}{r^2}.\end{aligned}$$ As momentum density vanishes hence it is constant. Energy and Momentum in Papapetrou’s Prescription ------------------------------------------------ The required non-vanishing components of $N^{abcd}$ are given by $$\begin{aligned} N^{0011}&=&-(\frac{\Delta}{\Sigma})^{\gamma^2-1}\frac{y^2}{x^2+y^2} -\frac{\Delta x^2}{r^4}-\frac{x^2z^2}{r^2(x^2+y^2)}\nonumber\\ &-&(1-\frac{2m}{r})^{-2\gamma} \frac{\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^{2}-1}},\\ N^{0022}&=&-(\frac{\Delta}{\Sigma})^{\gamma^2-1}\frac{x^2}{x^2+y^2} -\frac{\Delta y^2}{r^4}-\frac{y^2z^2}{r^2(x^2+y^2)}\nonumber\\ &-&(1-\frac{2m}{r})^{-2\gamma} \frac{\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^{2}-1}},\\ N^{0033}&=&-\frac{\Delta z^2}{r^4}-\frac{x^2+y^2}{r^2} -(1-\frac{2m}{r})^{-2\gamma} \frac{\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^{2}-1}},\\ N^{0012}&=&(\frac{\Delta}{\Sigma})^{\gamma^2-1}\frac{xy}{x^2+y^2} -\frac{\Delta xy}{r^4}-\frac{xy}{r^2(x^2+y^2)}\nonumber\\ &-&(1-\frac{2m}{r})^{-2\gamma} \frac{\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^{2}-1}},\\ N^{0013}&=&-\frac{\Delta xz}{r^4}+\frac{xz}{r^2},\\ N^{0023}&=&-\frac{\Delta yz}{r^4}+\frac{yz}{r^2}.\end{aligned}$$ Substituting Eqs.(58)-(63) in Eq.(23), we obtain the following energy density and momentum density components $$\begin{aligned} \Omega^{00}&=&\frac{(1-\frac{2m}{r})^{-2\gamma}}{8\pi}[-4\gamma m^2(2\gamma+1)\frac{\Delta^{\gamma^2-2}} {r\Sigma^{\gamma^2-1}}+8\gamma m\{\frac{\Delta^{\gamma^2-2}} {r\Sigma^{\gamma^2-1}}(1-\frac{m}{r})\nonumber\\ &-&(\gamma^2-1)(1-\frac{m}{r})\frac{\Delta^{\gamma^2-1}} {r\Sigma^{\gamma^2}}-(\frac{\Delta}{\Sigma})^{\gamma^2}\frac{1}{r^3}\}\nonumber\\ &-&2\gamma^2(\gamma^2-1)(1-\frac{m}{r})^2\frac{\Delta^{\gamma^2-2}} {\Sigma^{\gamma^2-1}}+4\gamma^2(\gamma^2-1)(1-\frac{m}{r})^2 \frac{\Delta^{\gamma^2-1}}{\Sigma^{\gamma^2}}\nonumber\\ &+&(1-\frac{m}{r})\frac{\gamma^2}{r^2} (\frac{\Delta}{\Sigma})^{\gamma^2-1}-2\gamma^2(\gamma^2-1) \frac{(x^2+y^2)\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^2+1}}(1-\frac{m}{r}\nonumber\\ &+&\frac{m^2}{r^2}-\frac{m^2(x^2+y^2)}{r^4})^2-2\gamma^2(\gamma^2-1) \frac{z^2\Delta^{\gamma^2}}{r^2\Sigma^{\gamma^2+1}} (1-\frac{m}{r}\nonumber\\&-&\frac{m^2(x^2+y^2)}{r^4})^2- \frac{\gamma^2-1}{r^2}(\frac{\Delta}{\Sigma})^{\gamma^2} (1-\frac{m}{r}-\frac{2m^2}{r^2}-\frac{3m^2(x^2+y^2)}{r^4})\nonumber\\&-& \frac{\Delta^{\gamma^2}}{r^4\Sigma^{\gamma^2-1}}]+\frac{1}{8\pi}[ (\gamma^2-1)(1-\frac{m}{r})\{\frac{2x^2}{x^2+y^2}-1\} \frac{\Delta^{\gamma^2-2}}{\Sigma^{\gamma^2-1}}\nonumber\\ &+&(\gamma^2-1)\frac{\Delta^{\gamma^2-1}}{\Sigma^{\gamma^2}} (1-\frac{m}{r}+\frac{m^2}{r^2}-\frac{m^2(x^2+y^2)}{r^4}) \{1-\frac{2x^2}{x^2+y^2}\}\nonumber\\ &+&(\frac{\Delta}{\Sigma})^{\gamma^2-1}\{\frac{1}{x^2+y^2} -\frac{2x^2}{(x^2+y^2)^2}\}-\frac{4}{r^2}(1-\frac{m}{r})+\frac{6\Delta}{r^4}].\end{aligned}$$ Energy and Momentum in Möller’s Prescription -------------------------------------------- For the gamma metric, we obtain the following non-vanishing components of $K^{bc}_a$ $$K^{01}_0=-2m\gamma sin\theta.$$ When we make use of Eq.(65) in Eq.(30), the energy and momentum density components turn out to be $$M^0_0=0.$$ and $$M^i_0=0=M^0_i.$$ This shows that energy and momentum turn out to be constant. Conclusion ========== Energy-momentum complexes provide the same acceptable energy-momentum distribution for some systems. However, for some systems \[22-30\], these prescriptions disagree. The debate on the localization of energy-momentum is an interesting and a controversial problem. According to Misner et al. \[35\], energy can only be localized for spherical systems. In a series of papers \[36\] Cooperstock et al. has presented a hypothesis which says that, in a curved spacetime, energy and momentum are confined to the regions of non-vanishing energy-momentum tensor $T_a^b$ of the matter and all non-gravitational fields. The results of Xulu \[19,20\] and the recent results of Bringley \[37\] support this hypothesis. Also, in the recent work, Virbhadra and his collaborators \[14-18\] have shown that different energy-momentum complexes can provide meaningful results. Keeping these points in mind, we have explored some of the interesting members of the Weyl class for the energy-momentum distribution. In this paper, we evaluate energy-momentum densities for the two solutions of the Weyl metric, i.e., Erez-Rosen and the gamma metrics. We obtain this target by using four well-known prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M$\ddot{o}$ller. From Eqs.(15), (22), (29), (34), (50), (57), (64) and (67), it can be seen that the energy-momentum densities are finite and well defined. We also note that the energy density is different for the four different prescriptions. However, momentum density components turn out to be zero in all the prescriptions and consequently we obtain constant momentum for these solutions. The results of this paper also support the Cooperstock’s hypothesis \[36\] that energy is localized to the region where the energy-momentum tensor is non-vanishing. We would like to mention here that the results of energy-momentum distribution for different spacetimes are not surprising rather they justify that different energy-momentum complexes, which are pseudo-tensors, are not covariant objects. This is in accordance with the equivalence principle \[35\] which implies that the gravitational field cannot be detected at a point. These examples indicate that the idea of localization does not follow the lines of pseudo-tensorial construction but instead it follows from the energy-momentum tensor itself. This supports the well-defined proposal developed by Cooperstock \[36\] and verified by many authors \[22-30\]. In GR, many energy-momentum expressions (reference frame dependent pseudo-tensors) have been proposed. There is no consensus as to which is the best. Hamiltonian’s principle helps to solve this enigma. Each expression has a geometrically and physically clear significance associated with the boundary conditions. [**Acknowledgment**]{} We would like to thank for the anonymous referee for his useful comments. [**References**]{} [\[1\]]{} Trautman, A.: [*Gravitation: An Introduction to Current Research*]{} ed. Witten, L. (Wiley, New York, 1962)169. [\[2\]]{} Landau, L.D. and Lifshitz, E.M.: [*The Classical Theory of Fields*]{} (Addison-Wesley Press, 1962). [\[3\]]{} Papapetrou, A.: [*Proc. R. Irish Acad*]{} [**A52**]{}(1948)11. [\[4\]]{} Tolman R. C: Relativity, Thermodynamics and Cosmology, (Oxford University Press, Oxford, 1934)227. [\[5\]]{} Bergman P.G: and Thompson R. Phys. Rev. [**89**]{}(1958)400. [\[6\]]{} Weinberg, S.: [*Gravitation and Cosmology*]{} (Wiley, New York, 1972). [\[7\]]{} Möller, C.: Ann. Phys. (NY) [**4**]{}(1958)347. [\[8\]]{} Möller, C.: Ann. Phys. (NY) [**12**]{}(1961)118. [\[9\]]{} Komar, A. Phys. Rev. [**113**]{}(1959)934. [\[10\]]{} Penrose, R.: [*Proc. Roy. Soc.*]{} London [**A388**]{}(1982)457;\ [*GR10 Conference*]{}, eds. Bertotti, B., de Felice, F. and Pascolini, A. Padova [**1**]{} (1983)607. [\[11\]]{} Brown, J.D. and York, J,W.: Phys. Rev. [**D47**]{}(1993)1407. [\[12\]]{} Hayward, S.A.: Phys. Rev. [**D49**]{}(1994)831. [\[13\]]{} Chang, C.C., Nester, J.M. and Chen, C.: Phys. Rev. Lett. [**83**]{}(1999)1897. [\[14\]]{} Virbhadra, K.S.: Phys. Rev. [**D42**]{}(1990)2919. [\[15\]]{} Virbhadra, K.S.: Phys. Rev. [**D60**]{}(1999)104041. [\[16\]]{} Rosen, N. and Virbhadra, K.S.: Gen. Relati. Gravi. [**25**]{}(1993)429. [\[17\]]{} Virbhadra, K.S. and Parikh, J.C.: Phys. Lett. [**B317**]{}(1993)312. [\[18\]]{} Virbhadra, K.S. and Parikh, J.C.: Phys. Lett. [**B331**]{}(1994)302. [\[19\]]{} Xulu, S.S.: Int. J. of Mod. Phys. [**A15**]{}(2000)2979; Mod. Phys. Lett. [**A15**]{}(2000)1151 and reference therein. [\[20\]]{} Xulu, S.S.: Astrophys. Space Sci. [**283**]{}(2003)23. [\[21\]]{} Aguirregabiria, J.M., Chamorro, A. and Virbhadra, K.S,: [*Gen. Relativ. and Grav.*]{} [**17**]{}, 927 [**28**]{}(1996)1393. [\[22\]]{} Sharif, M.: Int. J. of Mod. Phys. [**A17**]{}(2002)1175. [\[23\]]{} Sharif, M.: Int. J. of Mod. Phys. [**A18**]{}(2003)4361; Errata [**A19**]{}(2004)1495. [\[24\]]{} Sharif, M.: Int. J. of Mod. Phys. [**D13**]{}(2004)1019. [\[25\]]{} Gad, R.M: Astrophys. Space Sci. [**293**]{}(2004)453. [\[26\]]{} Gad, R.M: Mod. Phys. Lett. [**A19**]{}(2004)1847. [\[27\]]{} Patashnick, O.: Int. J. of Mod. Phys. [**D**]{}(2005) (gr-qc/0408086). [\[28\]]{} Sharif, M. and Fatima, Tasnim: Int. J. of Mod. Phys. [**A20**]{}(2005)4309. [\[29\]]{} Sharif, M. and Fatima, Tasnim: Nuovo Cimento [**B**]{}(2005). [\[30\]]{} Fatima, Tasnim: M.Phil. Thesis (University of the Punjab, Lahore, 2004). [\[31\]]{} Weyl, H.: Ann. Phys. (Leipzig) [**54**]{}(1917)117; [**59**]{}(1919)185;\ Civita, Levi, L.: Atti. Acad. Naz. Lince Rend. Classe Sci. Fis. Mat. e Nat., [**28**]{}(1919)101;\ Synge, J.L.: [*Relativity, the General Theory*]{} (North-Holland Pub. Co. Amsterdam, 1960). [\[32\]]{} Kramer, D., Stephani, H., MacCallum, M.A.H. and Hearlt, E.: [*Exact Solutions of Einstein’s Field Equations*]{} (Cambridge University Press, 2003). [\[33\]]{} Curzon, H.E.J.: [*Proc. Math. Soc.*]{} London [**23**]{}(1924)477. [\[34\]]{} Esposito, F. and Witten, L.: Phys. Lett. [**B58**]{}(1975)357;\ Virbhadra, K.S.: gr-qc/9606004;\ Herrera, L., A Di Prisco, A.Di. and Fuenmayor, E.: Class. Quant. Grav. [**20**]{}(2003)1125. [\[35\]]{} Misner,C.W., Thorne, K.S. and Wheeler, J.A. [*Gravitation*]{} (W.H. Freeman, New York, 1973)603. [\[36\]]{} Cooperstock, F.I. and Sarracino, R.S. [*J. Phys. A.: Math. Gen.*]{} [**11**]{}(1978)877.\ Cooperstock, F.I.: in [*Topics on Quantum Gravity and Beyond*]{}, Essays in honour of Witten, L. on his retirement, ed. Mansouri, F. and Scanio, J.J. (World Scientific, Singapore, 1993); Mod. Phys. Lett. [**A14**]{}(1999)1531; Annals of Phys. [**282**]{}(2000)115;\ Cooperstock, F.I. and Tieu, S.: Found. Phys. [**33**]{}(2003)1033. [\[37\]]{} Bringley, T.: Mod. Phys. Lett. [**A17**]{}(2002)157. [^1]: e-mail: [email protected]

ArXiv

--- abstract: 'In this paper, we present a method of applying integral action to enhance the robustness of energy shaping controllers for underactuated mechanical systems with matched disturbances. Previous works on this problem have required a number of technical assumptions to be satisfied, restricting the class of systems for which the proposed solution applies. The design proposed in this paper relaxes some of these technical assumptions.' author: - 'Joel Ferguson$^{1}$, Alejandro Donaire$^{2}$, Romeo Ortega$^{3}$ and Richard H. Middleton$^{1}$[^1][^2][^3]' bibliography: - 'libraryURLRemoved.bib' title: '**Matched disturbance rejection for energy-shaping controlled underactuated mechanical systems** ' --- at (current page.south) ; Introduction ============ Interconnection and damping assignment passivity-based control (IDA-PBC) is a nonlinear control method whereby the closed-loop system is a passive port-Hamiltonian (pH) system with desired characteristics to comply with the control objectives [@Ortega2004]. Many systematic solutions have been proposed for the stabilization of nonlinear systems using IDA-PBC, but the general procedure is still limited by the designers ability to solve the so called *matching equations*. Although the matching equation are difficult to solve in some cases, IDA-PBC has been successful applied to a variety of nonlinear systems such as electrical machines [@Petrovic2001; @Gonzalez2008], power converters [@Rodriguez2000; @Rodriguez2001] and underactuated mechanical systems [@Acosta2005]-[@Donaire2016a]. In general, the equilibrium of a mechanical system stabilised with IDA-PBC will be shifted when an external disturbance acts on the system. In this paper we are interested in robustifying IDA-PBC [*vis-á-vis*]{} constant external disturbances. A general design for the addition of integral action to pH systems with the objective of rejecting disturbances was first presented in [@Donaire2009] and further discussed in [@Ortega2012]. The approach relies on a (possibly implicit) change of coordinates to satisfy the matching equations. The integral action scheme was tailored to fully actuated mechanical systems in [@Romero2013a] and underactuated mechanical systems in [@Donaire2016]. While in both cases the required change of coordinates to satisfy the matching equations were given explicitly, a number of technical assumption were imposed to do so. In both cases, the proposed integral action controllers were shown to preserve the desired equilibrium of the system, rejecting the effects of an unknown matched disturbance. More recently, an alternative method for the addition of integral action to pH systems was presented in [@Ferguson2015], [@Ferguson]. In these works, the controller is constructed from the open-loop dynamics of the plant. The energy function of the controller is chosen such that it couples the plant and controller states, which allows the matching equations to be satisfied by construction. In addition, the control system studied in [@Ferguson] has a physical interpretation and is shown to be equivalent to a control by interconnection (CbI) scheme, another PBC technique [@Ortega2007]. The method in [@Ferguson2015] was shown to be applicable to mechanical systems with constant mass matrix. In this paper, we extend the integral action design proposed in [@Ferguson2015] to underactuated mechanical systems subject to matched disturbances. The assumption of a constant mass matrix is relaxed, and general mechanical systems are considered. The method proposed in this paper is constructed to directly satisfy the matching equations without the need of the technical assumptions previously used in [@Donaire2016]. Specifically, the presented scheme allows the open-loop mass matrix, shaped mass matrix and input mapping matrix to be state dependant. [**Notation.**]{} In this paper we use the following notation: Let $x \in\mathbb{R}^n$, $x_1\in\mathbb{R}^m$, $x_2\in\mathbb{R}^s$. For real valued function $\mathcal{H}(x)$, $\nabla\mathcal{H}\triangleq \left(\frac{\partial \mathcal{H}}{\partial x}\right)^\top$. For functions $\mathcal{G}(x_1,x_2)\in\mathbb{R}$, $\nabla_{x_i}\mathcal{G}\triangleq \left(\frac{\partial \mathcal{G}}{\partial x_i}\right)^\top$ where $i \in\{1,2\}$. For fixed elements $x^\star\in \mathbb{R}^n$, we denote $\nabla \mathcal{H}^\star\triangleq \nabla \mathcal{H}(x)|_{x=x^\star}$. For vector valued functions $\mathcal{C}(x)\in\mathbb{R}^m$, $\nabla_x \mathcal{C}$ denotes the transposed Jacobian matrix $\left(\frac{\partial \mathcal{C}}{\partial x}\right)^\top$. Problem Formulation {#ProbForm} =================== In this paper, we consider mechanical systems that have been stabilised using IDA-PBC. This class of systems can be expressed as[^4]: $$\label{mecdist} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{\bp} \end{bmatrix} &= \underbrace{ \begin{bmatrix} 0_{n\times n} & M^{-1}(q)\mathbf{M}_d(q) \\ -\mathbf{M}_d(q)M^{-1}(q) & \mathbf{J}_2(q,\bp)-R_d(q) \end{bmatrix}}_{F_m(q,\mathbf{p})} \begin{bmatrix} \nabla_q \mathbf{H}_d \\ \nabla_\mathbf{p} \mathbf{H}_d \end{bmatrix} \\ &\phantom{---}+ \underbrace{ \begin{bmatrix} 0_{m\times n} & G^\top(q) \end{bmatrix}^\top}_{G_m(q)} (u-d) \\ \mathbf{y} &= G^\top(q)\nabla_\mathbf{p} \mathbf{H}_d, \end{split}$$ with Hamiltonian $$\label{mechOLHam} \mathbf{H}_d(q,\mathbf{p})= \frac 12 \bp^\top \mathbf{M}_d^{-1}(q) \bp + V_d(q),$$ where $q,\mathbf{p} \in \mathbb{R}^n$ are the generalised configuration and momentum vectors respectively, $n$ is the number of degrees of freedom of the system, $u\in\mathbb{R}^m$ is the input, $y\in\mathbb{R}^m$ is the output, $d\in\mathbb{R}^m$ is a constant disturbance, $M(q) > 0$ and $\mathbf{M}_d(q) >0$ are the open-loop and shaped mass matrices of the system respectively, $V_d(q)$ is the shaped potential energy, $G(q)$ is the full-rank input matrix, $R_d (q)= G(q)K_p(q)G^\top(q)$ for some $K_p(q) \geq 0$ is the damping matrix and $\mathbf{J}_2(q,\mathbf{p}) = -\mathbf{J}_2^\top(q,\mathbf{p})$ is a skew-symmetric matrix. We assume that has a strict minimum at the desired operating point $(q,\mathbf{p}) = (q^\star,0_{n\times 1})$. For the remainder of the paper, the explicit state dependency of terms and various mapping are assumed and omitted. The control objective is to develop a dynamic controller $u=\beta(q,\bp,\zeta)$, where $\zeta \in \mathbb{R}^m$ is the state of the controller, that ensures asymptotic stability of the desired equilibrium $(q,\bp,\zeta) = (q^{\star},0,\zeta^\star)$, for some $\zeta^\star \in \mathbb{R}^m$, even under the action of constant disturbances $d$. Previous Work {#PrevSol} ============= A nonlinear PID controller was proposed in [@Donaire2016] as a solution to the matched disturbance rejection problem. Under the assumptions: - $G$ and $\mathbf{M}_d$ are constant - $G^\perp\nabla_q(\bp^\top M^{-1}\bp) = 0_{(n-m)\times 1}$, the control law was proposed to be $$\begin{split} u = &-\left[ K_pG^\top \mathbf{M}_d^{-1}GK_1G^\top M^{-1} + K_1G^\top \dot{M}^{-1} + K_2K_I \right. \\ &\left. \times(K_2^\top + K_3^\top G^\top \mathbf{M}_d^{-1}GK_1)G^\top M^{-1} \right]\nabla V_d \\ & -\left[K_1G^\top M^{-1}\nabla^2V_dM^{-1} + (G^\top G)^{-1}G^\top J_2\mathbf{M}_d^{-1} \right. \\ &\left. + K_2K_IK_3^\top G^\top \mathbf{M}_d^{-1} \right]\bp \\ & -(K_P G^\top \mathbf{M}_d^{-1} GK_2 + K_3)K_I\zeta \\ \dot{\zeta} = &(K_2^\top G^\top M^{-1} + K_3^\top G^\top \mathbf{M}_d^{-1}GK_1G^\top M^{-1})\nabla V_d \\ &+ K_3^\top G^\top \mathbf{M}_d^{-1}\bp, \end{split}$$ where $K_1 > 0$, $K_P > 0$, $K_I > 0$, $K_3 > 0$ and $$K_2 = (G^\top \mathbf{M}_d^{-1}G)^{-1}.$$ The resulting closed-loop can be expressed as $$\label{DonaireCL} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{z}_2 \\ \dot{\zeta} \end{bmatrix} &= \begin{bmatrix} -\Gamma_1 & M^{-1}M_d & -\Gamma_2 \\ -M_d M^{-1} & -GK_pG^\top & -GK_3 \\ \Gamma_2^\top & K_3^\top G^\top & -K_3^\top \end{bmatrix} \nabla H_z \\ H_z &= \frac12 z_2^\top\mathbf{M}_d^{-1}z_2 + V_d(q) + \frac{1}{2}(\zeta-\alpha)^\top K_I(\zeta-\alpha) \end{split}$$ where, $$\label{Donairez2} \begin{split} z_2 &= \bp+GK_1G^\top M^{-1}\nabla V_d+GK_2K_I(\zeta-\alpha) \\ \Gamma_1 &= M^{-1}GK_1G^\top M^{-1} \\ \Gamma_2 &= M^{-1}GK_2 \\ \alpha &= K_I^{-1}(K_p + K_3)^{-1}d. \end{split}$$ The closed-loop system was shown to have a stable equilibrium at $(q,\bp,\zeta) = (q^\star,0_{n\times 1},\alpha)$. Furthermore, if the output signal $$y_{d3} = \begin{bmatrix} G^\top M^{-1}\nabla V_d \\ G^\top \mathbf{M}_d^{-1}z_2 \\ K_I(z_3-\alpha) \end{bmatrix}$$ is detectable, then the equilibrium is asymptotically stable. The assumptions P.1 and P.2 are necessary to ensure that the dynamics of $z_2$ in match the dynamics of $\mathbf{p}$ in , using the transformation . Integral Action for Underactuated Mechanical Systems ==================================================== In this section we propose an alternative method to add integral action to mechanical systems. This is achieved by first performing a momentum transformation such that the disturbance is pre-multiplied by the identity, rather than $G$. The integral action control law is then defined in the transformed coordinates. The resulting closed-loop is shown to be unique and preserves the desired operating point $q^\star$ of the original system. Momentum transformation {#momentumTransform} ----------------------- To solve the integral action problem, we transform the dynamics such that the disturbance is pre-multiplied by the identity, rather than $G$. Such a transformation is always possible utilising the following matrix: $$\label{Ttransform} T(q) = \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix},$$ where $G^\perp \in \mathbb{R}^{m\times n}$ is a full-rank, left annihilator of $G$. \[momLemma\] Consider the system under the change of momentum coordinates $p= T\bp$. The dynamics can be equivalently expressed as $$\label{mecp} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{p}_1 \\ \dot{p}_2 \end{bmatrix} &= \begin{bmatrix} 0_{n \times n} & S_1 & S_2 \\ -S_1^\top & S_{31}-K_p & S_{32} \\ -S_2^\top & -S_{32}^\top & S_{34} \end{bmatrix} \begin{bmatrix} \nabla_{q}\calH_d \\ \nabla_{p_1}\calH_d \\ \nabla_{p_2}\calH_d \end{bmatrix} \\ &\phantom{---} + \begin{bmatrix} 0_{m\times n} & I_{m\times m} & 0_{m\times s} \end{bmatrix}^\top (u-d) \\ y &= \nabla_{p_1}\calH_d \\ \calH_d&=\frac12 p^\top M_d^{-1}(q) p + V_d(q), \end{split}$$ where $p = \operatorname{col}(p_1,p_2)$, $p_1 \in \mathbb R^m$, $p_2 \in \mathbb R^s$, $s=n-m$, $$\label{s3} \begin{split} M_d &= T \mathbf{M}_d T^{\top} \\ S_1 &= M^{-1}\mathbf{M}_dG\{G^\top G\}^{-1} \\ S_2 &= M^{-1}\mathbf{M}_dG^{\perp\top} \\ S_{31} &= \{G^\top G\}^{-1}G^\top J_p G\{G^\top G\}^{-1} \\ S_{32} &= \{G^\top G\}^{-1}G^\top J_p G^{\perp\top} \\ S_{34} &= G^{\perp}J_p G^{\perp\top} \\ \end{split}$$ and $J_p$ is defined by $$\label{Jp} \begin{split} J_p &= \mathbf{M}_d M^{-1}\nabla_q^\top(T^{-1}p) -\nabla_q(T^{-1}p) M^{-1}\mathbf{M}_d \\ &\phantom{---}+\mathbf J_2(q,T^{-1}p). \end{split}$$ As $J_p = -J_p^\top$, both $S_{31}$ and $S_{34}$ are skew-symmetric. The proof of this lemma follows along the lines of the proof of [@Fujimoto2001a Lemma 2], [@Venkatraman2010a Proposition 1] and [@Duindam208 Theorem 1], therefore the full proof is omitted. An outline of the proof, however, can be found in the Appendix. Importantly, the output of the system under the change of momentum, $y$, remains unchanged. Indeed, $$\label{outputEquv} \begin{split} \mathbf{y} &= G^\top\nabla_\mathbf{p} \mathbf{H}_d \\ &= G^\top T^\top\nabla_{p}\calH_d \\ &= G^\top \begin{bmatrix} G\{G^\top G\}^{-1} & (G^\perp)^\top \end{bmatrix} \nabla_{p}\calH_d \\ &= \begin{bmatrix} I_{m} & 0_{m\times s} \end{bmatrix} \nabla_{p}\calH_d \\ &= y. \end{split}$$ Integral action control law --------------------------- The integral action control law is now proposed for the underactuated mechanical system described in $(q,p)$ coordinates by . \[propiacl\] Consider the system in closed-loop with the controller \[controlLaw\] $$\begin{aligned} u &= (-S_{31}+K_p+J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1} \calH_d \nonumber \\ &\phantom{---}+ (J_{c_1}-R_{c_1}) \nabla_{p_1}\calH_c \label{iacontroller} \\ \dot{\zeta}&=-R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d, \label{iazeta} \end{aligned}$$ where $$\mathcal{H}_c = \frac12(p_1-\zeta)^\top K_I(p_1-\zeta),$$ $\zeta \in \mathbb{R}^m$, $K_I > 0$ and $J_{c_1}=-J_{c_1}^\top$, $R_{c_1} > 0$ $R_{c_2} > 0$ are constant matrices free to be chosen. Then, the closed-loop dynamics can be written in the pH form, $$\label{iacl} \begin{bmatrix} \dot{q} \\ \dot{p}_1 \\ \dot{p}_2 \\ \dot{\zeta} \end{bmatrix} = F(x) \begin{bmatrix} \nabla_{q}\mathcal{H}_{cl} \\ \nabla_{p_1}\mathcal{H}_{cl} \\ \nabla_{p_2}\mathcal{H}_{cl} \\ \nabla_{ \zeta}\mathcal{H}_{cl} \end{bmatrix} - \begin{bmatrix} 0_{n\times 1} \\ d \\ 0_{s\times 1} \\ 0_{m\times 1} \end{bmatrix},$$ where $$\label{Fx} F(x)= \begin{bmatrix} 0_{n \times n} & S_1 & S_2 & S_1 \\ -S_1^\top & J_{c_1}-R_{c_1}-R_{c_2} & S_{32} & -R_{c_2} \\ -S_2^\top & -S_{32}^\top & S_{34} & -S_{32}^\top \\ -S_1^\top & -R_{c_2} & S_{32} & -R_{c_2} \\ \end{bmatrix}$$ and $\mathcal{H}_{cl}:\mathbb{R}^{2n+m}\to\mathbb{R}$ is the closed-loop Hamiltonian defined as $$\mathcal{H}_{cl}(q,p_1,p_2,\zeta)= \mathcal{H}_d(q,p_1,p_2) + \mathcal{H}_c(p_1,\zeta).$$ First notice that $\nabla_{p_1}\mathcal{H}_c = -\nabla_{\zeta}\mathcal{H}_c$. Due to this relationship, the dynamics of $q$ and $p_2$ in are equivalent to the dynamics of $q$ and $p_2$ in . Considering the dynamics of $\zeta$ in and using $\nabla_{p_1}\mathcal{H}_c = -\nabla_{\zeta}\mathcal{H}_c$ yields $$\begin{split} \dot{\zeta} &= -R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d \\ &= -R_{c_2}(\nabla_{p_1}\mathcal{H}_d + \nabla_{p_1}\mathcal{H}_c - \nabla_{p_1}\mathcal{H}_c) -S_1^\top\nabla_{q}\mathcal{H}_d \\ &\phantom{---} + S_{32}\nabla_{p_2}\mathcal{H}_d \\ &= -R_{c_2}\nabla_{p_1}\mathcal{H}_{cl} -R_{c_2}\nabla_{\zeta}\mathcal{H}_{cl} -S_1^\top\nabla_{q}\mathcal{H}_{cl} + S_{32}\nabla_{p_2}\mathcal{H}_{cl} \\ \end{split}$$ which matches the dynamics of $\zeta$ in . Finally, considering the dynamics of $p_1$ in , $$\begin{split} \dot{p}_1 &= -S_1^\top\nabla_{q}\calH_d + (S_{31}-K_p)\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---}+ u - d \\ &= -S_1^\top\nabla_{q}\calH_d + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---} + (J_{c_1}-R_{c_1}) \nabla_{p_1}\calH_c - d \\ &= -S_1^\top\nabla_{q}\calH_d + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---} + (J_{c_1}-R_{c_1}-R_{c_2}) \nabla_{p_1}\calH_c + R_{c_2} \nabla_{p_1}\calH_c - d \\ &= -S_1^\top\nabla_{q}\calH_{cl} + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_{cl} \\ &\phantom{---} + S_{32}\nabla_{p_2}\calH_{cl}- R_{c_2} \nabla_{\zeta}\calH_{cl} - d, \end{split}$$ which is equivalent to the dynamics of $p_1$ in . In the case that $S_{31}$ and $K_v$ are constant, The choice $J_{c_1} = S_{31}$, $R_{c_1} = K_v$ can be made to simplify the control law . Stability --------- For the remainder of this section, the stability properties of the closed-loop system are considered. It is shown that the integral action control preserves the desired operating point $q^\star$ of the open-loop system. Further, if the original system is detectable, then the closed-loop system is asymptotically stable. The closed-loop system has an isolated equilibrium point $$\label{equilibrium} (q,p,\zeta) = (q^\star,0_{n\times 1},-K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d).$$ The dynamics of $q$ in can be simplified to $$\begin{split} \dot{q} &= M^{-1}\mathbf{M}_d T^\top\nabla_p\mathcal{H}_d \\ &=M^{-1}\mathbf{M}_d T^\top M_d^{-1}p. \end{split}$$ As $M, \mathbf{M}_d, M_d, T$ are full-rank, $p=0_{n\times 1}$ and $\nabla_p \mathcal{H}_{d} = 0_{n\times 1}$ at any equilibrium. As $\nabla_{p_2} \mathcal{H}_{cl} = \nabla_{p_2} \mathcal{H}_{d}$, $$\label{equli3} \nabla_{p_2} \mathcal{H}_{cl} = 0_{s\times 1}.$$ The difference between the dynamics of $p_1$ and $\zeta$ are given by $\dot{p}_1 - \dot{\zeta} = (J_{c_1} - R_{c_1})\nabla_{p_1}\mathcal{H}_{cl} - d$. As $\nabla_p\mathcal{H}_d = 0_{n\times 1}$, $$\label{equli1} \nabla_{p_1}\mathcal{H}_{cl} = -\nabla_{\zeta}\mathcal{H}_{cl} = (J_{c_1} - R_{c_1})^{-1}d.$$ Recalling that $-\nabla_{\zeta}\mathcal{H}_{cl} = -\nabla_{\zeta}\mathcal{H}_{c} = K_I(p_1-\zeta)$ and $p_1 = 0$, can be rearranged to find $\zeta = -K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d$. Substituting the equilibrium gradients and into and considering the dynamics of $p$, it results in $$\dot{p} = -\begin{bmatrix} S_1 & S_2 \end{bmatrix}^\top\nabla_q\mathcal{H}_{cl},$$ which implies that $\nabla_q\mathcal{H}_{cl} = \nabla_q\mathcal{H}_{d} = 0_{n\times 1}$ at any equilibrium as $\begin{bmatrix} S_1 & S_2 \end{bmatrix}$ is full-rank. The equilibrium gradient $\nabla_q\mathcal{H}_{d} = 0_{n\times 1}$ is satisfied by $q = q^\star$. \[propmatched\] Consider system subject to unknown matched disturbance in closed-loop with the controller . The following properties hold: (i) The equilibrium of the closed-loop system is stable. \[stability\]\ (ii) If the output $$\label{detectOutput} y_{p_1}=\begin{bmatrix} \nabla_{p_1}\mathcal{H}_d \\ \nabla_{p_1} \calH_{c} - (J_{c_1} - R_{c_1})^{-1}d \end{bmatrix}$$ is detectable, the equilibrium is asymptotically stable. \[asympStability\]\ (iii) If the shaped potential energy $V_d$ is radially unbounded, then the stability properties are global. \[globStable\] To verify , consider the function $$\label{iamatchlyap} \mathcal{W} = \mathcal{H}_d(q,p_1,p_2) + \frac12(z - z^\star)^\top K_I(z - z^\star),$$ where $z = p_1 - \zeta$ and $z^\star = p_1^\star - \zeta^\star = K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d$, as a Lyapunov candidate for the system. $\mathcal{W}$ has a strict minimum at as $\mathcal{H}_d$ is strictly minimised by $(q,p)=(q^\star,0_{n\times 1})$ and $K_I > 0$. Defining $w = \operatorname{col}(q,p_1,p_2,\zeta)$, the closed-loop dynamics can be equivalently expressed as $$\label{iaclMod} \dot w = F(x) \underbrace{ \begin{bmatrix} \nabla_{q}\mathcal{H}_{cl} \\ \nabla_{p_1}\mathcal{H}_{cl} - (J_{c_1}-R_{c_1})^{-1}d \\ \nabla_{p_2}\mathcal{H}_{cl} \\ \nabla_{ \zeta}\mathcal{H}_{cl} + (J_{c_1}-R_{c_1})^{-1}d \end{bmatrix}}_{\nabla_w\mathcal{W}}.$$ The equilibrium is stable since $F+F^\top \leq 0$, which implies that $\dot{\mathcal{W}}\leq 0$ along the trajectories of the closed-loop system. The claim follows by considering the structure of $F$ and invoking LaSalle’s invariance principle. Finally, to verify , first note that the component of $\mathcal{W}$ associated with the controller and disturbance, $\frac12(z - z^\star)^\top K_I(z - z^\star)$, is radially unbounded in $z$. Then, recalling that $\mathcal{H}_d$ is of the form and $M_d^{-1} > 0$, it is clear that $\mathcal{H}_d$ is radially unbounded in $p$. Finally, if $V_d$ is radially unbounded in $q$, then $\mathcal{W}$ is radially unbounded. This implies that the closed-loop system is globally stable. \[CorrDetect\] If the output of the system is detectable when $d=0_{m\times 1}$ and $u=0_{m\times 1}$, then the closed-loop system is asymptotically stable. By Proposition \[propmatched\], the equilibrium of the closed-loop is asymptotically stable if $y_{p_1}$ is detectable. The control action , evaluated at $y_{p_1} = 0_{2m\times 1}$ is $u = d$. Further, using , the output of resolves to be $\mathbf{y} = y = \nabla_{p_1}\mathcal{H}_d = 0_{m\times 1}$. Substituting $u = d$ and $\mathbf{y} = 0_{m\times 1}$ into recovers the zero dynamics of the original, undisturbed system. Thus, if is detectable when $d=0_{m\times 1}$ and $u=0_{m\times 1}$, then the closed-loop system is asymptotically stable. Cart Pendulum Example ===================== In this section, we apply the presented integral action scheme to the cart pendulum system. For the existing IDA-PBC laws, the shaped mass matrix $\mathbf{M}_d$ is not constant so the integral action scheme of [@Donaire2016] cannot be used. Stabilisation control of the cart pendulum using IDA-PBC was solved in [@Acosta2005]. After partial feedback linearisation, the cart pendulum can be modelled as a pH system of the form $$\label{PendubotOL} \begin{split} \begin{bmatrix} \dot q \\ \dot{\mathbf{p}} \end{bmatrix} &= \begin{bmatrix} 0_{2\times 2} & I_{2\times 2} \\ -I_{2\times 2} & 0_{2\times 2} \end{bmatrix} \nabla \mathcal{H} \\ &\phantom{---} + \begin{bmatrix} 0_{2\times 1} \\ \mathbf{G} \end{bmatrix} \left(u-d\frac{1}{m_c+m_p\sin^2\theta}\right) \\ \mathcal{H} &= \frac12 \mathbf{p}^\top M^{-1}\mathbf{p} + \mathcal{V}, \end{split}$$ where $q = \begin{bmatrix} q_1 & q_2 \end{bmatrix}^\top$ is the configuration vector containing the angle of the pendulum from vertical and the horizontal position of the car respectively, $\mathbf{p} = \begin{bmatrix} \mathbf{p}_1 & \mathbf{p}_2 \end{bmatrix}^\top$ is the generalised momenta, $$\begin{split} M &= I_{2\times 2} \\ \mathbf{G} &= \begin{bmatrix} -b\cos(q_1) \\ 1 \end{bmatrix} \\ \mathcal{V} &= a\cos(q_1), \end{split}$$ $m_c$ and $m_p$ are the masses of the cart and pendulum respectively, $a = \frac{g}{l}$, $b = \frac{1}{l}$, $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. The disturbance $d$ is an unknown constant force collocated with the input $u$. Note that the system is not in the form as the disturbance is not constant. In the remainder of this section, the undisturbed system will be stabilised using IDA-PBC and the resulting closed-loop will be converted into the form by defining a new input mapping matrix and input. Energy shaping -------------- In the case that $d = 0_{m\times 1}$, the cart pendulum can be stabilised around a desired equilibrium $(q_1,q_2,\mathbf{p}) = (0,q_{2}^\star,0_{2\times 1})$ using the IDA-PBC law $$\label{PendubotESCtrl} \begin{split} u &= \{\mathbf{G}^\top \mathbf{G}\}^{-1}\mathbf{G}^\top\{ \nabla_q\mathcal{H} - \mathbf{M}_dM^{-1}\nabla_q\mathbf{H}_d + \mathbf{J}_2\mathbf{M}_d^{-1}\mathbf{p} \} \\ &\phantom{---}- \frac{1}{(m_c+m_p\sin^2\theta)^2}K_p\mathbf{G}^\top \mathbf{M}_d^{-1}\mathbf{p} + u', \end{split}$$ where $$\begin{split} \mathbf{M}_d &= \begin{bmatrix} \frac{kb^2}{3}\cos^3q_1 & -\frac{kb}{2}\cos^2q_1 \\ -\frac{kb}{2}\cos^2q_1 & k\cos q_1 + m_{22}^0 \end{bmatrix} \\ V_d &= \frac{3a}{kb^2cos^2 q_1} + \frac{P}{2}\left[q_2 - q_{2}^\star + \frac{3}{b}\log\left(\sec q_1 + \tan q_1\right) \right. \\ &\left.\phantom{---------} + \frac{6m_{22}^0}{kb}\tan^2 q_1\right] \\ \mathbf{J}_2 &= \mathbf{p}^\top \mathbf{M}_d^{-1}\alpha \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \\ \alpha &= \frac{k\gamma_1}{2}\sin q_1 \begin{bmatrix} -b\cos q_1 \\ 1 \end{bmatrix} \\ \gamma_1 &= -\frac{kb^2}{6}\cos^3q_1, \end{split}$$ $P > 0$, $k>0$, $m_{22}^0>0$ are tuning parameters, $K_p > 0$ is a constant used for damping injection and $u'$ is an additional input for further control design. The cart pendulum , together with the control law , results in the closed-loop $$\label{ESCarkDist} \begin{split} \begin{bmatrix} \dot q \\ \dot{\mathbf{p}} \end{bmatrix} &= \begin{bmatrix} 0_{2\times 2} & M^{-1}\mathbf{M}_d \\ -\mathbf{M}_dM^{-1} & \mathbf{J}_2 - GK_p G^\top \end{bmatrix} \nabla \mathbf{H}_d \\ &\phantom{---}+ \begin{bmatrix} 0_{1\times 2} & G^\top \end{bmatrix}^\top (\tilde u - d) \\ \mathbf{H}_d &= \frac12 \mathbf{p}^\top M_d^{-1}(q)\mathbf{p} + V_d(q), \end{split}$$ where $\tilde u = (M+m\sin^2\theta)u'$, and $G = \frac{1}{m_c+m_p\sin^2\theta}\mathbf{G}$ Clearly, the closed-loop system is of the form . Integral action --------------- Before the integral action control law can be applied, the momentum must be transformed as per Section \[momentumTransform\]. Taking $G^\perp = (m_c+m_p\sin^2\theta)\begin{bmatrix} 1 & b\cos q_1 \end{bmatrix}$, the necessary momentum transformation is $p = T\mathbf{p}$ with $$T(q) = (m_c+m_p\sin^2\theta) \begin{bmatrix} \frac{-b\cos q_1}{b^2\cos^2 q_1 + 1} & \frac{1}{b^2\cos^2 q_1 + 1} \\ 1 & b\cos q_1 \end{bmatrix},$$ and results in the transformed Hamiltonian $$\calH_d=\frac12 p^\top \underbrace{T^{-\top}\mathbf M_d^{-1}(q) T^{-1}}_{M_d^{-1}(q)}p + V_d(q).$$ In the new momentum coordinates, the $S$ matrices can be resolved as per : $$\begin{split} S_1 &= \frac{m_c+m_p\sin^2\theta}{b^2\cos^2 q_1 + 1} \begin{bmatrix} -\frac{kb^3}{3}\cos^4q_1 -\frac{kb}{2}\cos^2q_1 \\ \frac{kb^2}{2}\cos^3q_1 + k\cos q_1 + m_{22}^0 \end{bmatrix} \\ S_{31} &= 0 \\ S_{32} &= \frac{1}{b^2\cos^2 q_1 + 1} \begin{bmatrix} -b\cos(q_1) \\ 1 \end{bmatrix}^\top\\ &\phantom{--}\left[\mathbf{M}_d M^{-1}\nabla_q^\top(T^{-1}(q)p)-\nabla_q(T^{-1}(q)p) M^{-1}\mathbf{M}_d\right.\\ &\left.\phantom{---} + J_2(q,p)\right] \begin{bmatrix} 1 \\ b\cos(q_1) \end{bmatrix}. \end{split}$$ where $$\begin{split} J_2 &= p^\top T^{-\top} \mathbf{M}_d^{-1}\alpha \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}. \\ \end{split}$$ As $S_{31} = 0$ and $K_p$ is constant, the integral control law is simplified by making the selection $R_{c_1} = K_p, J_{c_1} = 0$ which results in \[CartPendIntLaw\] $$\begin{aligned} \tilde u &= -R_{c_2}\nabla_{p_1} \calH_d -K_p \nabla_{p_1}\calH_c \\ \dot{\zeta}&=-R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d. \end{aligned}$$ As discussed in [@Acosta2005], $V_d$ is radially unbounded on the domain $Q = \left\lbrace(-\frac{\pi}{2},\frac{\pi}{2})\times\mathbb{R}\right\rbrace$ and the system with $\tilde u = d = 0$ is detectable. Thus, by Proposition \[propmatched\] and Corollary \[CorrDetect\], the closed-loop system is asymptotically stable with region of attraction given by the set $\{Q\times \mathbb{R}^2\times \mathbb{R}\}$. Numerical simulation -------------------- The cart pendulum was simulated using the following plant parameters: $g = 9.8, M = I_{2\times 2}, l = 1, m_c = 1, m_p = 1$. The desired cart position was selected to be $q_2^\star = 0$ and the energy shaping control law was implemented with the controller parameters $k=1, m_{22}^0 = 1, P = 1, K_p = 10$. To reject the effects the disturbance $d$, the control law was applied with the controller storage function $\mathcal{H}_c(p_1,\zeta) = \frac12 K_I(p_1-\zeta)^2$ and $K_I = 0.05$. The system was simulated for 60 seconds with state of the plant initialised at $(q_1(0),q_2(0),p(0)) = (0,1,0_{2\times 1})$ and the controller initialised at $\zeta(0) = 0$. For the time interval $t\in[0,30)$ the disturbance was set to $d=0$. At $t=30s$, a disturbance of $d=2$ was applied for the remainder of the simulation. Figure \[Cartfig2\] shows that the cart pendulum, together with the integral action control law, tends towards the desired equilibrium on the time interval $t\in[0,30)$. At $t=30$, the disturbance $d=2$ is applied and the states move away from the desired equilibrium. On the time interval $t\in[30,60]$, the integral control compensates for the disturbance and the system again approaches the desired equilibrium. ![The cart pendulum in closed-loop with an energy shaping controller and integral action subject to a constant disturbance. The system tends toward the desired final position $(q_1,q_2) = (0,0)$ on the interval $t\in[0,30)$. At $t=30$, a disturbance is applied to the system. The integral control compensates for the disturbance and the system tends toward the equilibrium.[]{data-label="Cartfig2"}](DistRejc){width="49.00000%"} Conclusions =========== In this paper, a method to robustify IDA-PBC via the addition of integral action to underactuated mechanical systems was presented. The method relaxes technical assumptions required by previous solutions. The control scheme preserves the desired equilibrium of the open-loop system, rejecting the effects of an unknown matched disturbance. Further, the closed-loop system was shown to be asymptotically stable provided that the passive output of the open-loop system is detectable. \[app1\] *Proof of Lemma \[momLemma\]:* Let $x_m = \operatorname{col}(q,p)$, $\mathbf{x}_m = \operatorname{col}(q,\mathbf{p})$ and $x_m = g_t(\mathbf{x}_m) = (q,T\mathbf{p})$. The transformed Hamiltonian is defined as $$\begin{split} \mathcal{H}_d(q,p) &= \mathbf{H}_d(q,T^{-1}(q)p) \\ &= \frac 12 p^\top \underbrace{T^{-\top}(q) \mathbf{M}_d^{-1}(q) T^{-1}(q)}_{M_d^{-1}(q)}p + V_d(q). \end{split}$$ Utilising the differential of $g_t$ (see [@lee2012introduction]) can be equivalently expressed in $x_m$ as $$\label{dynXm} \begin{split} \dot{x}_m &= \left\lbrace \nabla_{\mathbf{x}_m}^\top g_t F_m \nabla_{\mathbf{x}_m}g_t \right\rbrace \big|_{\mathbf{x}_m = g_t^{-1}(q,p)} \nabla_{x_m}\mathcal{H}_d \\ &\phantom{---}+ \left\lbrace \nabla_{\mathbf{x}_m}^\top g_tG_m \right\rbrace \big|_{\mathbf{x}_m = g_t^{-1}(q,p)} (u-d_m) \\ &= \left\lbrace \begin{bmatrix} I_{l\times l} & 0_{l\times l} \\ \nabla_q^\top\left(T\mathbf{p}\right) & T \end{bmatrix} \begin{bmatrix} 0_{l\times l} & M^{-1}\mathbf{M}_d \\ -\mathbf{M}_dM^{-1} & \mathbf{J}_2-R_d \end{bmatrix} \right. \\ &\phantom{} \left. \times \begin{bmatrix} I_{l\times l} & \nabla_q\left(T\mathbf{p}\right) \\ 0_{l\times l} & T^\top \end{bmatrix} \right\rbrace \bigg|_{\mathbf{x}_m = g_t^{-1}(q,p)} \begin{bmatrix} \nabla_q \mathcal{H}_d \\ \nabla_p \mathcal{H}_d \end{bmatrix} \\ & + \left\lbrace \begin{bmatrix} I_{l\times l} & 0_{l\times l} \\ \nabla_q^\top\left(T\mathbf{p}\right) & T \end{bmatrix} \begin{bmatrix} 0_{l\times m} \\ G \end{bmatrix} \right\rbrace \bigg|_{\mathbf{x}_m = g_t^{-1}(q,p)} (u-d_m) \\ &= \begin{bmatrix} 0_{n\times n} & M^{-1}\mathbf{M}_dT^\top \\ -T\mathbf{M}_dM^{-1} & T(J_p-R_d)T^\top \end{bmatrix} \begin{bmatrix} \nabla_q \mathcal{H}_d \\ \nabla_p \mathcal{H}_d \end{bmatrix} \\ &\phantom{---}+ \begin{bmatrix} 0_{n\times m} \\ TG \end{bmatrix} (u-d), \end{split}$$ where $J_p$ is defined in . Recalling that $R_d (q)= G(q)K_p(q)G^\top(q)$, the term $TR_dT^\top$ can be simplified to $$\begin{split} TR_dT^\top &= \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix} GK_pG^\top \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix}^\top \\ &= \begin{bmatrix} K_p & 0_{m\times s} \\ 0_{s\times m} & 0_{s\times s} \end{bmatrix}. \end{split}$$ Finally, subdividing the momentum variable of into $p = \operatorname{col}(p_1,p_2)$ and substituting $T$ by its definition recovers the dynamics .   [^1]: $^{1}$Joel Ferguson and Richard H. Middleton are with School of Electrical Engineering and Computing and PRC CDSC, The University of Newcastle, Callaghan, NSW 2308, Australia. [Email: [email protected], [email protected]]{} [^2]: $^{2}$Alejandro Donaire is with the Department of Electrical Engineering and Information Theory and PRISMA Lab, University of Naples Federico II, Napoli 80125, Italy, and with the School of Electrical Eng. and Comp. Sc. of the Queensland University of Technology, Brisbane, QLD, Australia. [Email: [email protected]]{} [^3]: $^{3}$Romeo Ortega is with Laboratoire des Signaux et Systèmes, CNRS-SUPELEC, 91192, Gif-sur-Yvette, France [Email: [email protected]]{} [^4]: See [@Donaire2016] for the detailed explanation and motivation of the problem formulation.

ArXiv

--- abstract: 'It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^n$ tiles $\mathbb{Z}^d$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots,k\} \subset \mathbb{Z}$ with its middle point removed: they showed that this tiles $\mathbb{Z}^d$ for $d = 2k^2$, and they asked if the dimension needed tends to infinity with $k$. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles $\mathbb{Z}^4$.' author: - Harry Metrebian title: Tiling with punctured intervals --- Introduction ============ A *tile* is a finite non-empty subset of $\mathbb{Z}^n$ for some $n$. We say that a tile $T$ *tiles* $\mathbb{Z}^d$ if $\mathbb{Z}^d$ can be partitioned into copies of $T$, that is, subsets that are translations, rotations or reflections, or any combination of these, of $T$. For example, the tile $\texttt{X.X} = \{-1,1\} \subset \mathbb{Z}$ tiles $\mathbb{Z}$. The tile $\texttt{XX.XX} = \{-2,-1,1,2\} \subset \mathbb{Z}$ does not tile $\mathbb{Z}$, but we can also regard it as a tile in $\mathbb{Z}^2$, and indeed it tiles $\mathbb{Z}^2$, as shown, for example, in [@gltan16]. Chalcraft [@chalcraft1; @chalcraft2] conjectured that, for any tile $T \subset \mathbb{Z}^n$, there is some dimension $d$ for which $T$ tiles $\mathbb{Z}^d$. This was proved by Gruslys, Leader and Tan [@gltan16]. The first non-trivial case is the *punctured interval* $T = \underbrace{\texttt{XXXXX}}_{k}\!\texttt{.}\!\underbrace{\texttt{XXXXX}}_{k}$. The authors of [@gltan16] showed that $T$ tiles $\mathbb{Z}^d$ for $d = 2k^2$, but they were unable to prove that the smallest required dimension $d$ was quadratic in $k$, or even that $d \to \infty$ as $k \to \infty$. They therefore asked the following question: Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, and let $d$ be the least number such that $T$ tiles $\mathbb{Z}^d$. Does $d \to \infty$ as $k \to \infty$? In this paper we will show that, rather unexpectedly, $d$ does not tend to $\infty$: \[mainthm\] Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$. Then $T$ tiles $\mathbb{Z}^4$. Furthermore, if $k$ is odd or congruent to $4 \pmod 8$, then $T$ tiles $\mathbb{Z}^3$. We have already noted that `X.X` tiles $\mathbb{Z}$, and `XX.XX` tiles $\mathbb{Z}^2$ but not $\mathbb{Z}$. It can be shown via case analysis that, for $k \geq 3$, the tile $T$ does not tile $\mathbb{Z}^2$. However, this proof is tedious and provides little insight, and since it is not the focus of this paper, we omit it. For odd $k \geq 3$ and for $k \equiv 4 \pmod 8$, 3 is therefore the least $d$ such that $T$ tiles $\mathbb{Z}^d$. For the remaining cases, namely $k \equiv 0, 2, 6 \pmod 8$, $k \geq 6$, it is unknown whether the least possible $d$ is 3 or 4. In this paper, we will first prove the result for odd $k$. This will introduce some key ideas, which we will develop to prove the result for general $k$, and then to improve the dimension from 4 to 3 for $k \equiv 4 \pmod 8$. Finally, we give some background. Tilings of $\mathbb{Z}^2$ by polyominoes (edge-connected tiles in $\mathbb{Z}^2$) have been thoroughly investigated. For example, Golomb [@golomb70] showed that results of Berger [@berger66] implied that there is no algorithm which decides whether copies of a given finite set of polyominoes tile $\mathbb{Z}^2$. It is unknown whether the same is true for tilings by a single polyomino. For tilings of $\mathbb{Z}$ by sets of general one-dimensional tiles, such an algorithm does exist, as demonstrated by Adler and Holroyd [@ah81]. Kisisel [@kisisel01] introduced an ingenious technique for proving that certain tiles do not tile $\mathbb{Z}^2$ without having to resort to case analysis. A similar problem is to consider whether a tile $T$ tiles certain finite regions, such as cuboids. There is a significant body of research, sometimes involving computer searches, on tilings of rectangles in $\mathbb{Z}^2$ by polyominoes (see, for example, Conway and Lagarias [@cl90] and Dahlke [@dahlke]). Friedman [@friedman] has collected some results on tilings of rectangles by small one-dimensional tiles. More recently, Gruslys, Leader and Tomon [@gltomon16] and Tomon [@tomon16] considered the related problem of partitioning the Boolean lattice into copies of a poset, and similarly Gruslys [@gruslys16] and Gruslys and Letzter [@gl16] have worked on the problem of partitioning the hypercube into copies of a graph. Preliminaries and the odd case ============================== We begin with the case of $k$ odd. This is technically much simpler than the general case, and allows us to demonstrate some of the main ideas in the proof of Theorem \[mainthm\] in a less complicated setting. \[kodd\] Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, with $k$ odd. Then $T$ tiles $\mathbb{Z}^3$. Throughout this section, $T$ is fixed, and $k \geq 3$. We will not yet assume that $k$ is odd, because the tools that we are about to develop will be relevant to the general case too. We start with an important definition from [@gltan16]: a *string* is a one-dimensional infinite line in $\mathbb{Z}^d$ with every $(k+1)$th point removed. Crucially, a string is a disjoint union of copies of $T$. We cannot tile $\mathbb{Z}^d$ with strings, as each string intersects $[k+1]^d$ in either 0 or $k$ points, and $(k+1)^d$ is not divisible by $k$. However, we could try to tile $\mathbb{Z}^d$ by using strings in $d-1$ of the $d$ possible directions, leaving holes that can be filled with copies of $T$ in the final direction. We therefore consider $\mathbb{Z}^d$ as consisting of slices equivalent to $\mathbb{Z}^{d-1}$, each of which will be partially tiled by strings. Any partial tiling of the discrete torus $\mathbb{Z}_{k+1}^{d-1} = (\mathbb{Z}/(k+1)\mathbb{Z})^{d-1}$ by lines with one point removed corresponds to a partial tiling of $\mathbb{Z}^{d-1}$ by strings. We will restrict our attention to these tilings at first, as they are easy to work with. We will call a set $X \subset \mathbb{Z}_{k+1}^{d-1}$ a *hole* in $\mathbb{Z}_{k+1}^{d-1}$ if $\mathbb{Z}_{k+1}^{d-1} \setminus X$ can be tiled with strings. One particularly useful case of this is when $d = 3$ and $X$ either has exactly one point in each row of $\mathbb{Z}_{k+1}^2$ or exactly one point in each column of $\mathbb{Z}_{k+1}^2$. Then $X$ is clearly a hole, since a string in $\mathbb{Z}_{k+1}^2$ is just a row or column minus a point. The following result will allow us to fill the gaps in the final direction, assuming we have chosen the partial tilings of the $\mathbb{Z}^{d-1}$ slices carefully: \[biglemma\] Let $S \subset \mathbb{Z}^d$, $|S| = 3$. Then there exists $Y \subset S \times \mathbb{Z}$ such that $T$ tiles $Y$, and for every $n \in \mathbb{Z}$, $|Y \cap (S \times \{n\})| = 2$. Let $S = \{x_1, x_2, x_3\}$. For $i = 1,2,3$, place a copy of $T$ beginning at $\{x_i\} \times \{n\}$ for every $n \equiv ik \pmod {3k}$. The union $Y$ of these tiles has the required property:\ For $n \equiv 0, k+1, \ldots, 2k-1 \pmod{3k}$, $Y \cap (S \times \{n\}) = \{x_1, x_3\} \times \{n\}$.\ For $n \equiv k, 2k+1, \ldots, 3k-1 \pmod{3k}$, $Y \cap (S \times \{n\}) = \{x_1, x_2\} \times \{n\}$.\ For $n \equiv 2k, 1, \ldots, k-1 \pmod{3k}$, $Y \cap (S \times \{n\}) = \{x_2, x_3\} \times \{n\}$.\ We will now prove Theorem \[kodd\]. We know that if $X \subset \mathbb{Z}_{k+1}^2$ has one point in each row or column then $X$ is a hole of size $k+1$. Since $k+1$ is even, we can try to choose $X_n$ in each slice $\mathbb{Z}_{k+1}^2 \times \{n\}$ so that $\bigcup_{n\in\mathbb{Z}}X_n$ is the disjoint union of $\frac{k+1}{2}$ sets $Y_i$ of the form in Lemma \[biglemma\]. We can do this as follows:\ For $n \equiv 0, k+1, \ldots, 2k-1 \pmod{3k}$, let $X_n = \{(0,0),(1,1),\ldots,(k-1,k-1),(k,k)\}$.\ For $n \equiv k, 2k+1, \ldots, 3k-1 \pmod{3k}$, let $X_n = \{(0,0),(0,1),(2,2),(2,3),\ldots,(k-1,k-1),\newline(k-1,k)\}$.\ For $n \equiv 2k, 1, \ldots, k-1 \pmod{3k}$, let $X_n = \{(0,1),(1,1),(2,3),(3,3),\ldots,(k-1,k),(k,k)\}$.\ Then let $X = \bigcup\limits_{n\in\mathbb{Z}} (X_n \times \{n\}) \subset \mathbb{Z}_{k+1}^2 \times \mathbb{Z}$. Each $X_n$ is a hole, so we can tile $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z})\setminus X$ with strings. Also, $X$ is the disjoint union of sets of the form $Y$ from Lemma \[biglemma\]: for $0 \leq i \leq \frac{k-1}{2}$, let $S_i = \{(2i,2i),(2i,2i+1),(2i+1,2i+1)\}$. Then $X \cap (S_i \times \mathbb{Z})$ is precisely the set $Y$ generated from $S_i$ in the proof of Lemma \[biglemma\]. Hence $T$ tiles $X$. Since $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z})\setminus X$ can be tiled with strings, we can partially tile $\mathbb{Z}^3$ with strings, leaving a copy of $X$ empty in each copy of $\mathbb{Z}_{k+1}^2 \times \mathbb{Z}$. We can tile all of these copies of $X$ with $T$, so $T$ tiles $\mathbb{Z}^3$, completing the proof of Theorem \[kodd\]. The general case ================ We now move on to general $k$: \[generalk\] Let $T$ be the tile $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$. Then $T$ tiles $\mathbb{Z}^4$. We will assume throughout that $T$ is fixed and $k \geq 3$. For even $k$, the construction used to prove Theorem \[kodd\] does not work, as all holes in $\mathbb{Z}_{k+1}^2$ have size $(k+1)^2-mk$ for some $m$, and this is always odd, so we cannot use Lemma \[biglemma\]. The same is true if we replace 2 with a larger dimension, or if, as in [@gltan16], we use strings in which every $(2k+1)$th point, rather than every $(k+1)$th point, is removed. We will therefore need a new idea. Instead of using strings in $d-1$ out of $d$ directions, we could only use them in $d-2$ directions and fill the gaps with copies of $T$ in the 2 remaining directions. We will show that this approach works in the case $d = 2$, giving a tiling of $\mathbb{Z}^4$. The strategy will be to produce a partial tiling of each $\mathbb{Z}^3$ slice and use the construction from Lemma \[biglemma\] to fill the gaps with tiles in the fourth direction. We will again build partial tilings of $\mathbb{Z}^{2}$, and therefore of higher dimensions, from partial tilings of the discrete torus $\mathbb{Z}_{k+1}^{2}$. The following result is a special case of one proved in [@gltan16]: \[onepoint\] If $x \in \mathbb{Z}_{k+1}^{2}$, then $\mathbb{Z}_{k+1}^{2}\setminus\{x\}$ can be tiled with strings. Let $x = (x_1,x_2)$, where the first coordinate is horizontal and the second vertical. Since a string is a row or column minus one point, we can place a string $(\{n\} \times \mathbb{Z}_{k+1})\setminus\{(n,x_2)\}$ in each column, leaving only the row $\mathbb{Z}_{k+1} \times \{x_2\}$ empty. Placing the string $(\mathbb{Z}_{k+1} \times \{x_2\})\setminus \{x\}$ in this row completes the tiling of $\mathbb{Z}_{k+1}^{2}\setminus\{x\}$. The sets $S$ of size 3 that we will use in Lemma \[biglemma\] will have 2 points, say $x_1$ and $x_2$, in one $\mathbb{Z}_{k+1}^{2}$ layer and one point, say $x_3$, in another layer. Every layer will contain points from exactly one such set $S$. Let $Y$ be the set constructed from $S$ in the proof of Lemma \[biglemma\]. In a given slice $\mathbb{Z}^3 \times \{n\}$, there are therefore two cases: 1. $Y \cap (S \times \{n\}) = \{x_1, x_3\} \times \{n\}$ or $\{x_2, x_3\} \times \{n\}$. 2. $Y \cap (S \times \{n\}) = \{x_1, x_2\} \times \{n\}$. In Case 1, each $\mathbb{Z}_{k+1}^{2}$ layer contains exactly one point of $Y$. $T$ then tiles the rest of the layer by Proposition \[onepoint\]. In Case 2, some of the layers contain two points of $Y$, and some of the layers contain no points. Holes of size 0 and 2 do not exist, so we will need copies of $T$ in the third direction to fill some gaps (where $Y$ consists of copies of $T$ in the fourth direction). The following lemma provides us with a way to do this: \[otherlemma\] Let $A \subset \mathbb{Z}^d$, $|S| = 3k$. Then there exists $B \subset S \times \mathbb{Z}$ such that $T$ tiles $B$, and $$|B \cap (S \times \{n\})| = \begin{cases} k+1 & \text{\emph{if} } n \equiv 1, \ldots, k \pmod{2k}\\ k-1 & \text{\emph{if} } n \equiv k+1, \ldots, 2k \pmod{2k} \end{cases}$$ Let $A = \{a_1, \ldots, a_{3k}\}$. Then:\ For $i = 1, \ldots, k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i \pmod{6k}$.\ For $i = k+1, \ldots, 2k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i+k \pmod{6k}$.\ For $i = 2k+1, \ldots, 3k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i+2k \pmod{6k}$.\ We now observe that the union $B$ of these tiles has the required property.\ For $n \equiv 1, \ldots, k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{2k+n}, \ldots, a_{3k}, a_1, \ldots, a_n\}$ (size $k+1$).\ For $n \equiv k+1, \ldots, 2k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_1, \ldots, a_k\}\setminus\{a_{n-k}\}$ (size $k-1$).\ For $n \equiv 2k+1, \ldots, 3k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{n-2k}, \ldots, a_{n-k}\}$ (size $k+1$).\ For $n \equiv 3k+1, \ldots, 4k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{k+1}, \ldots, a_{2k}\}\setminus\{a_{n-2k}\}$ (size $k-1$).\ For $n \equiv 4k+1, \ldots, 5k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{n-3k}, \ldots, a_{n-2k}\}$ (size $k+1$).\ For $n \equiv 5k+1, \ldots, 6k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{2k+1}, \ldots, a_{3k}\}\setminus\{a_{n-3k}\}$ (size $k-1$). The reasoning behind this lemma is that there exist sets $X \subset \mathbb{Z}_{k+1}^{2} \times \mathbb{Z}$ that are missing exactly $k+1$ points in every $\mathbb{Z}_{k+1}^{2}$ layer and can be tiled with strings. If we take $d = 2$ in Lemma \[otherlemma\], we would like to choose such a set $X$ and a set $A \subset \mathbb{Z}_{k+1}^{2}$ (abusing notation slightly, as $\mathbb{Z}_{k+1}^{2}$ is not actually a subset of $\mathbb{Z}^2$) such that the resulting $B$ in Lemma \[otherlemma\] is disjoint from $X$. Then $(\mathbb{Z}_{k+1}^{2} \times \mathbb{Z})\setminus(B \cup X)$ contains either 2 or 0 points in each $\mathbb{Z}_{k+1}^{2}$ layer, which is what we wanted. In order for this construction to work, we need the set $B \cap (A \times \{n\})$ to be a hole whenever it has size $k+1$, and to be a subset of a hole of size $k+1$ whenever it has size $k-1$, so that we actually can tile the required points with strings. By observing the forms of the sets $B \cap (A \times \{n\})$ in the proof of Lemma \[otherlemma\], we see that it is sufficient to choose the $a_n$ such that for all $n$, $\{a_n, \ldots, a_{n+k}\}$ is a hole. Here we regard the indices $n$ of the points $a_n$ of $A$ as integers mod $3k$, so $a_{3k+1} = a_1$ and so on. The following proposition says that we can do this. \[anprop\] There exists a set $A = \{a_1, \ldots, a_{3k}\} \subset \mathbb{Z}_{k+1}^{2}$ such that for all $n$, $\{a_n, \ldots, a_{n+k}\}$ contains either one point in every row or one point in every column. Here the indices are regarded as integers *mod* $3k$. For $n = 1, \ldots, k+1$, let $a_n = (n-1,n-1)$.\ For $n = k+2, \ldots, 2k-1$, let $a_n = (n-k-2,n-k-1)$.\ For $n = 2k, 2k+1, 2k+2$, let $a_n = (n-k-2,n-2k)$.\ For $n = 2k+3, \ldots, 3k$, let $a_n = (n-2k-3,n-2k)$.\ Note that all the $a_n$ are distinct. Let us regard the first coordinate as horizontal and the second as vertical.\ Then, for $n = 1, \ldots, 2k$, $\{a_n, \ldots, a_{n+k}\}$ contains one point in every column.\ For $n = 2k+1, \ldots, 3k$, $\{a_n, \ldots, a_{n+k}\}$ contains one point in every row. From now on, $a_n$ refers to the points defined in the above proof. This proposition is the motivation for choosing the value $6k$ in the proof of Lemma \[otherlemma\]. We can now prove Theorem \[generalk\]. We will need 3 distinct partial tilings of $\mathbb{Z}^3$ slices, corresponding to the 3 cases in the proof of Lemma \[biglemma\] with $d = 3$. The repeating unit in each of these partial tilings will have size $(k+1) \times (k+1) \times 6k$, so we will work in $\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k}$. We start by choosing the sets $S$ as in Lemma \[biglemma\]. These will be as follows:\ For $n = 1, \ldots, k$, $S_n = \{(0,0,n),(a_n,n+k),(a_{k+1},n+k)\}$.\ For $n = k+1, \ldots, 2k$, $S_n = \{(0,0,n+k),(a_n,n+2k),(a_{2k+1},n+2k)\}$.\ For $n = 2k+1, \ldots, 3k$, $S_n = \{(0,0,n+2k),(a_n,n+3k),(a_1,n+3k)\}$.\ We will refer to the points in $S_n$ as $x_{n,1},x_{n,2},x_{n,3}$ in the order given. We can construct a set $Y_n \subset \mathbb{Z}^4$ from each $S_n$ using the construction in the proof of Lemma \[biglemma\]. Let $Y = \bigcup_{1 \leq n \leq 3k} Y_n$. For a given $m \in \mathbb{Z}$, there are two possibilities for the structure of $Y \cap (\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\})$: 1. $Y \cap (\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\})$ consists of pairs of the form $\{x_{n,1},x_{n,2}\}$ or $\{x_{n,1},x_{n,3}\}$. Then it contains exactly one point in each $\mathbb{Z}_{k+1}^2$ layer. We can therefore tile $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\}) \setminus Y$ entirely with strings, by Proposition \[onepoint\]. 2. $Y \cap (\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\})$ consists of pairs of the form $\{x_{n,2},x_{n,3}\}$. Then it contains either 2 or 0 points in each $\mathbb{Z}_{k+1}^2$ layer.\ If $A = \{a_1, \ldots, a_{3k}\}$, and $B$ is the set constructed from $A$ in the proof of Lemma \[otherlemma\], then, by the choice of the $S_n$, the sets $B$ and $Y \cap (\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\})$ are disjoint. Furthermore, if $C$ is the union of these two sets, then, for every $n$, $C \cap (\mathbb{Z}_{k+1}^2 \times \{n\} \times \{m\}) = \{a_r, \ldots, a_{r+k}\}$ for some $r$, and by Proposition \[anprop\], this contains either one point in every row or one point in every column and is therefore a hole.\ Since $T$ tiles $B$, it also tiles $(\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \{m\}) \setminus Y$. $T$ tiles $Y$ by Lemma \[biglemma\]. Hence $T$ tiles $\mathbb{Z}_{k+1}^2 \times \mathbb{Z}_{6k} \times \mathbb{Z}$, and therefore also $\mathbb{Z}^4$, completing the proof of Theorem \[generalk\]. The 4 mod 8 case ================ To finish the proof of Theorem \[mainthm\], all that remains is to prove the following: \[4mod8\] Let $T$ be the tile $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, with $k \equiv 4 \pmod 8$. Then $T$ tiles $\mathbb{Z}^3$. We will prove this by constructing partial tilings of each $\mathbb{Z}^2$ slice and filling in the gaps using the construction from the proof of Lemma \[biglemma\]. We will define 3 subsets $X_1$, $X_2$, $X_3$ of $\mathbb{Z}^2$ and show that $T$ tiles each of them. However, two of these tilings will not make use of strings. Let $S_1 = \{(x,x+n(k+1)) \; | \; n \in \mathbb{Z}, x \equiv 2n,2n+1,2n+2,2n+3 \pmod 8\}$. Let $S_2 = \{(x,x+n(k+1)) \; | \; n\in \mathbb{Z}, x \equiv 2n+4,2n+5,2n+6,2n+7 \pmod 8\}$. Let $S_3 = \{(x,x+n(k+1)+1) \; | \; n \in \mathbb{Z}, x \equiv 2n+2,2n+3,2n+4,2n+5 \pmod 8\}$. Let $X_1 = \mathbb{Z}^2 \setminus (S_2 \cup S_3)$, $X_2 = \mathbb{Z}^2 \setminus (S_1 \cup S_3)$, $X_3 = \mathbb{Z}^2 \setminus (S_1 \cup S_2)$. Let the first coordinate be horizontal and the second vertical. $X_3$ is $\mathbb{Z}^2$ with every $(k+1)$th diagonal removed, so each row (or column) is $Z$ with every $(k+1)$th point removed, that is, a string. Hence $T$ tiles $X_3$. We will show that $X_1$ can be tiled with vertical copies of $T$ and $X_2$ can be tiled with horizontal copies of $T$. Note that $(x,x+n(k+1))+(2,k+3) = (x+2,(x+2)+(n+1)(k+1))$. Also, if $x \equiv 2n+r \pmod 8$, then $x+2 \equiv 2(n+1)+r \pmod 8$. Hence, by the definitions of $S_2$ and $S_3$, we see that $X_1$ is invariant under translation by $(2,k+3)$. To show that vertical copies of $T$ tile $X_1$, it therefore suffices to show that $T$ tiles the columns $X_1 \cap (\{0\} \times \mathbb{Z})$ and $X_1 \cap (\{1\} \times \mathbb{Z})$. But in fact, if $(0,y) \in S_2$, then $0 \equiv 2n+4$ or $2n+6 \pmod 8$, so $1 \equiv 2n+5$ or $2n+7 \pmod 8$, so also $(1,y+1) \in S_2$. The converse also holds, and the same is true for $S_3$. Thus we only need to check the case $x = 0$. $(0,n(k+1)) \in S_2$ for $n \equiv 1,2,5,6 \pmod 8$, that is, $n \equiv 1,2 \pmod 4$. $(0,n(k+1)+1) \in S_3$ for $n \equiv 2,3,6,7 \pmod 8$, that is, $n \equiv 2,3 \pmod 4$. Therefore $(0,y) \notin X_1$ for $y \equiv k+1, 2(k+1), 2(k+1)+1, 3(k+1)+1 \pmod{4(k+1)}$, so copies of $T$ beginning at positions $1$ and $2(k+1)+2 \pmod{4(k+1)}$ tile $X_1 \cap (\{0\} \times \mathbb{Z})$. Hence $T$ tiles $X_1$. Note that $(x,x+n(k+1))+(k+2,1) = (x+k+2,(x+k+2)+(n-1)(k+1))$.\ Since $k \equiv 4 \pmod 8$, if $x \equiv 2n+r \pmod 8$ then $x+k+2 \equiv 2(n-1)+r \pmod 8$. Hence $X_2$ is invariant under translation by $(k+2,1)$, by the definitions of $S_1$ and $S_3$. To show that horizontal copies of $T$ tile $X_2$, it is therefore enough to show that $T$ tiles the row $X_2 \cap (\mathbb{Z} \times \{0\})$. We can express $S_1$ as $\{(y-n(k+1),y) \; | \; y \equiv -n,1-n,2-n,3-n \pmod 8\}$. Similarly $S_3 = \{(y-n(k+1)-1,y) \; | \; y \equiv 3-n,4-n,5-n,6-n \pmod 8\}$. Therefore $(-n(k+1),0) \in S_1$ for $n \equiv 0,1,2,3 \pmod 8$, and $(-n(k+1)-1,0) \in S_3$ for $n \equiv 3,4,5,6 \pmod 8$. Hence $(x,0) \notin X_2$ for $x \equiv 0, 2(k+1)-1, 3(k+1)-1, 4(k+1)-1, 5(k+1)-1, 5(k+1), 6(k+1), \newline 7(k+1) \pmod{8(k+1)}$, so copies of $T$ beginning at positions $k+1, 3(k+1), 5(k+1)+1, 7(k+1)+1 \pmod{8(k+1)}$ tile $X_2 \cap (\mathbb{Z} \times \{0\})$. Hence $T$ tiles $X_2$. $S_1 \cup S_2 \cup S_3$ can be partitioned into sets of the form $S = \{x_1, x_2, x_3\}$, where $x_1 = (x,y) \in S_1$, $x_2 = (x+4,y+4) \in S_2$, $x_3 = (x+2,y+3) \in S_3$. Then $|S| = 3$, so we can construct the corresponding set $Y \subset \mathbb{Z}^3$ as in Lemma \[biglemma\]. Now, given $n \in \mathbb{Z}$, $(S \times \{n\}) \setminus Y = \{x_i\}$ for some $i \in \{1,2,3\}$. Then $Y \cap (X_i \times \{n\}) = \emptyset$. If we do this for all such sets $S$, and let $U$ be the (disjoint) union of the resulting sets $Y$, then $U \cap (X_i \times \{n\}) = \emptyset$, and $\mathbb{Z}^2 \times \{n\} \subset U \cup (X_i \times \{n\})$. Recall that $T$ tiles each $Y$ and therefore $U$. We can do this for every $n$, choosing a partial tiling $X_i$ for the corresponding $\mathbb{Z}^2$ layer. Together with $U$, these form a tiling of $\mathbb{Z}^3$ by $T$. This completes the proof of Theorem \[4mod8\], and therefore also the proof of Theorem \[mainthm\]. Open problems ============= Theorem \[mainthm\], together with the result that a punctured interval $T = \underbrace{\texttt{XXXXX}}_{k}\!\texttt{.}\!\underbrace{\texttt{XXXXX}}_{k}$ does not tile $\mathbb{Z}^2$ for $k \geq 3$, determines the smallest dimension $d$ such that $T$ tiles $\mathbb{Z}^d$ in the cases $k$ odd and $k \equiv 4 \pmod 8$. However, for other values of $k$, it is still unknown whether the smallest such dimension $d$ is 3 or 4: Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, where $k \equiv 0, 2, 6 \pmod 8$, $k \geq 6$. Does $T$ tile $\mathbb{Z}^3$? It is also natural to consider more general tiles. The next non-trivial case is that of an interval with a non-central point removed. One might wonder if there is an analogue of Theorem \[mainthm\] for these tiles: Does there exist a number $d$ such that, for any tile $T$ consisting of an interval in $\mathbb{Z}$ with one point removed, $T$ tiles $\mathbb{Z}^d$? For general one-dimensional tiles, Gruslys, Leader and Tan [@gltan16] conjectured that there is a bound on the dimension in terms of the size of the tile: For any positive integer $t$, there exists a number $d$ such that any tile $T \subset \mathbb{Z}$ with $|T| \leq t$ tiles $\mathbb{Z}^d$. This conjecture remains unresolved. The authors of [@gltan16] showed that if $d$ always exists then $d \to \infty$ as $t \to \infty$, by exhibiting a tile of size $3d-1$ that does not tile $\mathbb{Z}^d$. This gives a simple lower bound on $d$; better bounds would be of great interest. Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank Vytautas Gruslys for suggesting this problem and for many helpful discussions, and Imre Leader for his encouragement and useful comments. [99]{} A. Adler and F. C. Holroyd, ‘Some results on one-dimensional tilings’, *Geom. Dedicata* 10 (1981) 49–58. R. Berger, ‘The undecidability of the domino problem’, *Mem. Amer. Math. Soc.* 66 (1966) 1–72. J. H. Conway and J. C. Lagarias, ‘Tiling with polyominoes and combinatorial group theory’, *J. Combin. Theory Ser. A* 53 (1990) 183–208. K. Dahkle, ‘Tiling rectangles with polyominoes’, http://eklhad.net/polyomino/index.html (retrieved 7 May 2018) E. Friedman, ‘Problem of the Month (February 1999)’,\ https://www2.stetson.edu/\~efriedma/mathmagic/0299.html (retrieved 7 May 2018) S. W. Golomb, ‘Tiling with sets of polyominoes’, *J. Combin. Theory* 9 (1970) 60–71. V. Gruslys, ‘Decomposing the vertex set of a hypercube into isomorphic subgraphs’, arXiv:1611.02021. V. Gruslys, I. Leader, T. S. Tan, ‘Tiling with arbitrary tiles’, *Proc. London Math. Soc.* (3) 112 (2016) 1019–1039. V. Gruslys, I. Leader, I. Tomon, ‘Partitioning the Boolean lattice into copies of a poset’, arXiv:1609.02520. V. Gruslys, S. Letzter, ‘Almost partitioning the hypercube into copies of a graph’, arXiv:1612.04603. A. U. O. Kisisel, ‘Polyomino convolutions and tiling problems’, *J. Combin. Theory Ser. A* 95 (2001) 373–380. The Math Forum, ‘Two tiling problems’,\ http://mathforum.org/kb/message.jspa?messageID=6223965 (retrieved 7 May 2018) MathOverflow, ‘Does every polyomino tile $\mathbb{R}^n$ for some $n$?’\ https://mathoverflow.net/questions/49915/does-every-polyomino-tile-rn-for-some-n (retrieved 7 May 2018) I. Tomon, ‘Almost tiling of the Boolean lattice with copies of a poset’, arXiv:1611.06842. Harry Metrebian\ Trinity College\ Cambridge\ CB2 1TQ\ United Kingdom [email protected]

ArXiv

--- abstract: 'In this universe, governed fundamentally by quantum mechanical laws, characterized by indeterminism and distributed probabilities, classical deterministic laws are applicable over a wide range of time, place, and scale. We review the origin of these deterministic laws in the context of the quantum mechanics of closed systems, most generally, the universe as a whole. In this formulation of quantum mechanics, probabilities are predicted for the individual members of sets of alternative histories of the universe that decohere, for which there is negligible interference between pairs of histories in the set as measured by a decoherence functional. An expansion of the decoherence functional in the separation between histories allows the form of the phenomenological, deterministic equations of motion to be derived for suitable coarse grainings of a class of non-relativistic systems, including ones with general non-linear interactions. More coarse graining is needed to achieve classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed to effect decoherence, and coarse graining beyond that to achieve the inertia necessary to resist the noise that mechanisms of decoherence produce. Sets of histories governed largely by deterministic laws constitute the quasiclassical realm of everyday experience which is an emergent feature of the closed system’s initial condition and Hamiltonian. We analyse the question of the sensitivity of the existence of a quasiclassical realm to the particular form of the initial condition. We find that almost any initial condition will exhibit a quasiclassical realm of some sort, but only a small fraction of the total number of possible initial states could reproduce the everyday quasiclassical realm of our universe.' author: - 'James B. Hartle' title: 'Quasiclassical Realms In A Quantum Universe[^1]' --- \#1\#2[[\#1 \#2]{}]{} Introduction {#sec:I} ============ In cosmology we confront a problem which is fundamentally different from that encountered elsewhere in physics. This is the problem of providing a theory of the initial condition of the universe. The familiar laws of physics describe evolution in time. The evolution of a plasma is described by the classical laws of electrodynamics and mechanics and the evolution of an atomic state by Schrödinger’s equation. These dynamical laws require boundary conditions and the laws which govern the evolution of the universe — the classical Einstein equation, for instance — are no exception. There are no particular laws governing these boundary conditions; they summarize our observations of the universe outside the subsystem whose evolution we are studying. If we don’t see any radiation coming into a room, then we solve Maxwell’s equations inside with no-incoming-radiation boundary conditions. If we prepare an atom in a certain way, then we solve Schrödinger’s equation with the corresponding initial condition. In cosmology, however, by definition, there is no rest of the universe to pass the specification of the boundary conditions off to. The boundary conditions must be part of the laws of physics themselves. Constructing a theory of the initial condition of the universe, effectively its initial quantum state, and examining its observational consequences is the province of that area of astrophysics that has come to be called quantum cosmology.[^2] This talk will consider one manifest feature of the quantum universe and its connection to the theory of the initial condition. This is the applicability of the deterministic laws of classical physics to a wide range of phenomena in the universe ranging from the cosmological expansion itself to the turbulent and viscous flow of water through a pipe. This quasiclassical realm[^3] is one of the most immediate facts of our experience. Yet what we know of the basic laws of physics suggests that we live in a quantum mechanical universe, characterized by indeterminacy and distributed probabilities, where classical laws can be but approximations to the unitary evolution of the Schrödinger equation and the reduction of the wave packet. What is the origin of this wide range of time, place, and scale on which classical determinism applies? How can we derive the form of the phenomenological classical laws, say the Navier-Stokes equations, from a distantly related fundamental quantum mechanical theory which might, after all, be heterotic, superstring theory? What features of these laws can be traced to their quantum mechanical origins? It is such old questions that will be examined anew in this lecture from the perspective of quantum cosmology, reporting largely on joint work with Murray Gell-Mann [@GH93a]. Standard derivations of classical behavior from the laws of quantum mechanics are available in many quantum mechanics texts. One popular approach is based on Ehrenfest’s theorem relating the acceleration of the expected value of position to the expected value of the force: $$m\ \frac{d^2\langle x\rangle}{dt^2} = - \left\langle\frac{\partial V}{\partial x}\right\rangle\ , \label{oneone}$$ (written here for one-dimensional motion). Ehrenfest’s theorem is true in general, but for certain states, typically narrow wave packets, we may approximately replace the expected value of the force with the force evaluated at the expected value of position, thereby obtaining a classical equation of motion for that expected value: $$m\ \frac{d^2\langle x\rangle}{dt^2} = - \frac{\partial V(\langle x \rangle)}{\partial x}\ . \label{onetwo}$$ This equation shows that the center of a narrow wave packet moves on an orbit obeying Newton’s laws. More precisely, if we make a succession of position and momentum measurements that are crude enough not to disturb the approximation that allows (1.2) to replace (1.1), the expected values of the results will be correlated by Newton’s deterministic law. This kind of elementary derivation is inadequate for the type of classical behavior that we hope to discuss in quantum cosmology for the following reasons: - The behavior of expected or average values is not enough to define classical behavior. In quantum mechanics, the statement that the moon moves on a classical orbit is properly the statement that, among a set of alternative histories of its position as a function of time, the probability is high for those histories exhibiting the correlations in time implied by Newton’s law of motion and near zero for all others. To discuss classical behavior, therefore, we should be dealing with the probabilities of individual time histories, not with expected or average values. - The Ehrenfest theorem derivation deals with the results of “measurements” on an isolated system with a few degrees of freedom. However, in quantum cosmology we are interested in classical behavior in much more general situations, over cosmological stretches of space and time, and over a wide range of subsystems, [*independent*]{} of whether these subsystems are receiving attention from observers. Certainly we imagine that our observations of the moon’s orbit, or a bit of the universe’s expansion, have little to do with the classical behavior of those systems. Further, we are interested not just in classical behavior as exhibited in a few variables and at a few times of our choosing, but in as refined a description as possible, so that classical behavior becomes a feature of the systems themselves and not a choice of observers. - The Ehrenfest theorem derivation relies on a close connection between the equations of motion of the fundamental action and the phenomenological deterministic laws that govern classical behavior. But when we speak of the classical behavior of the moon, or of the cosmological expansion, or even of water in a pipe, we are dealing with systems with many degrees of freedom whose phenomenological classical equations of motion may be only distantly related to the underlying fundamental theory, say superstring theory. We need a derivation which derives the [*form*]{} of the equations as well as the probabilities that they are satisfied. - The Ehrenfest theorem derivation posits the variables — the position $x$ — in which classical behavior is exhibited. But, as mentioned above, classical behavior is most properly defined in terms of the probabilities and properties of histories. In a closed system we should be able to [*derive*]{} the variables that enter into the deterministic laws, especially because, for systems with many degrees of freedom, these may be only distantly related to the coördinates entering the fundamental action. Despite these shortcomings, the elementary Ehrenfest analysis already exhibits two necessary requirements for classical behavior: Some coarseness is needed in the description of the system as well as some restriction on its initial condition. Not every initial wave function permits the replacement of (\[oneone\]) by (\[onetwo\]) and therefore leads to classical behavior; only for a certain class of wave functions will this be true. Even given such a suitable initial condition, if we follow the system too closely, say by measuring position exactly, thereby producing a completely delocalized state, we will invalidate the approximation that allows (1.2) to replace (1.1) and classical behavior will not be expected. Some coarseness in the description of histories is therefore needed. For realistic systems we therefore have the important questions of [*how restricted*]{} is the class of initial conditions which lead to classical behavior and [*what*]{} and [*how large*]{} are the coarse grainings necessary to exhibit it. Before pursuing these questions in the context of quantum cosmology I would like to review a derivation of classical equations of motion and the probabilities they are satisfied in a simple class of model systems, but before doing [*that*]{} I must review, even more briefly, the essential elements of the quantum mechanics of closed systems [@Gri84; @Omnsum; @GH90a]. [=2.00in ]{} The Quantum Mechanics of Closed Systems {#sec:II} ======================================= Most generally we aim at predicting the probabilities of alternative time histories of a closed system such as the universe as a whole. Alternatives at a moment of time are represented by an exhaustive set of orthogonal projection operators $\{P^k_{\alpha_k} (t_k)\}$. For example, these might be projections on a set of alternative intervals for the center of mass position of a collection of particles, or projections onto alternative ranges of their total momentum. The superscript denotes the set of alternatives a certain set of position ranges or a certain set of momentum ranges, the discrete index $\alpha_k = 1,2,3 \cdots$ labels the particular alternative, a particular range of position, and $t_k$ is the time. A set of alternative histories is defined by giving a series of such alternatives at a sequence of times, say $t_1, \cdots, t_n$. An individual history is a sequence of alternatives $(\alpha_1, \cdots, \alpha_n)\equiv \alpha$ and is represented by the corresponding chain of projections. $$C_\alpha \equiv P^n_{\alpha_n} (t_n) \cdots P^1_{\alpha_1} (t_1)\ . \label{twoone}$$ Such a set is said to be “coarse-grained” because the $P$’s do not restrict all possible variables and because they do not occur at all possible times. The decoherence functional $$D\left(\alpha^\prime, \alpha\right) = Tr\,\bigl[C_{\alpha^\prime} \rho C^\dagger_\alpha\bigr] \label{twotwo}$$ measures the amount of quantum mechanical interference between pairs of histories in a universe whose initial condition is represented by a density matrix $\rho$. When, for a given set, the interference between all pairs of distinct histories is sufficiently low, $$D\left(\alpha^\prime, \alpha\right) \approx 0\quad , \quad {\rm all} \ \alpha^\prime \not= \alpha \label{twothree}$$ the set of alternative histories is said to [*decohere*]{}, and probabilities can be consistently assigned to its individual members. The probability of an individual history $\alpha$ is just the corresponding diagonal element of $D$, [*viz.*]{} $$p(\alpha) = D(\alpha, \alpha)\ . \label{twofour}$$ Describe in terms of operators, check decoherence and evaluate probabilities — that is how predictions are made for a closed system, whether the alternatives are participants in a measurement situation or not. When the projections at each time are onto the ranges $\{\Delta_\alpha\}$ of some generalized coördinates $q^i$ the decoherence functional can be written in a convenient path integral from $$D\left(\alpha^\prime, \alpha\right) = \int_{\alpha^\prime} \delta q^\prime \int_\alpha \delta q\, \delta \bigl(q^\prime_f - q_f\bigr) e^{i(S[q^\prime(\tau)] - S[q(\tau)])/\hbar} \rho \left(q^\prime_0, q_0\right) \label{twofive}$$ where the integral is over the paths that pass through the intervals defining the histories (Fig. 1). This form will be useful in what follows. Classical Behavior in a Class of Model Quantum Systems {#sec:III} ====================================================== The class of models we shall discuss are defined by the following features: - We restrict attention to coarse grainings that follow a fixed subset of the fundamental coördinates $q^i$, say the center of mass position of a massive body, and ignore the rest. We denote the followed variables by $x^a$ and the ignored ones by $Q^A$ so that $q^i=(x^a, Q^A)$. We thus posit, rather than derive, the variables exhibiting classical behavior, but we shall derive, rather than posit, the form of their phenomenological equations of motion. - We suppose the action is the sum of an action for the $x$’s, an action for the $Q$’s, and an interaction between them that is the integral of a local Lagrangian free from time derivatives. That is, $$S[q(\tau)] = S_{\rm free} [x(\tau)] + S_0 [Q(\tau)] + S_{\rm int} [x(\tau), Q(\tau)] \label{threeone}$$ suppressing indices where clarity is not diminished. - We suppose the initial density matrix factors into a product of one depending on the $x$’s and another depending on the ignored $Q$’s which are often called the “bath” or the “environment”. $$\rho\left(q^\prime_0, q_0\right) = \bar\rho \left(x^\prime_0, x_0\right)\, \rho_B \left(Q^\prime_0, Q_0\right)\ . \label{threetwo}$$ Under these conditions the integral over the $Q$’s in (2.5) can be carried out to give a decoherence functional just for coarse-grained histories of the $x$’s of the form: $$D\left(\alpha^\prime, \alpha\right) = \int_{\alpha^\prime} \delta x^\prime \int_{\alpha} \delta x\, \delta\bigl(x^\prime_f - x_f\bigr) \exp \biggl\{i\Bigl(S_{\rm free} [x^\prime (\tau)] - S_{\rm free} [x(\tau)] + W \left[x^\prime (\tau), x(\tau)\right]\Bigr)/\hbar\biggr\}\, \bar\rho\left(x^\prime_0, x_0\right) \label{threethree}$$ where $W [x^\prime(\tau), x(\tau)]$, called the Feynman-Vernon influence phase, summarizes the results of integrations over the $Q$’s. [=2.00in ]{} The influence phase $W$ generally possesses a positive imaginary part [@Bru93]. If that grows as $|x^\prime-x|$ increases, it will effect decoherence because there will then be negligible contribution to the integral (3.3) for $x^\prime \not= x$ or $\alpha^\prime \not= \alpha$. That, recall, is the definition of decoherence (2.3). Let us suppose this to be the case, as is true in many realistic examples. Then we can make an important approximation, which is a [*decoherence*]{} [*expansion*]{}. Specifically, introduce coördinates which measure the average and difference between $x^\prime$ and $x$ (Fig. 2) $$X = \half \left(x^\prime + x\right)\ , \quad \xi = x^\prime - x\ . \label{threefour}$$ The integral defining the diagonal elements of $D$, which are the probabilities of the histories, receives a significant contribution only for small $\xi(t)$. We can thus expand the exponent of the integrand of (\[threethree\]) in powers of $\xi(t)$ and legitimately retain only the lowest, say up to quadratic, terms. The result for the exponent is $$\begin{aligned} S[x(\tau) & + & \xi(\tau)/2] - S[x(\tau) -\xi(\tau)/2] + W[x(\tau), \xi(\tau)] \nonumber\\ & = & -\xi_0 P_0 + \int^T_0 dt\, \xi(t)\ \left[\frac{\delta S}{\delta X(t)} + \left(\frac{\delta W}{\delta\xi(t)}\right)_{\xi(t)=0}\right] + \half \int^T_0 dt' \int^T_0 dt\ \xi(t^\prime) \left(\frac{\delta^2 W}{\delta \xi(t') \delta \xi(t)}\right)_{\xi(t)=0} \ \xi(t) + \cdots\ . \label{threefive}\end{aligned}$$ The essentially unrestricted integrals over the $\xi(t)$ can then be carried out to give the following expression for the probabilities $$p(\alpha) = \int_\alpha \delta X\, ({\rm det}\ K_I/4\pi)^{-\half} \exp\Bigl[-\frac{1}{\hbar} \int^T_0 dt' \int^T_0 dt\ {\cal E} (t', X(\tau)]\, K^{\rm inv}_I \left(t', t; X(\tau)\right]\ {\cal E} (t, X(\tau)]\Bigr]\ \bar w \left(X_0, P_0\right)\ . \label{threesix}$$ Here, $${\cal E}(t, X(\tau)] \equiv \frac{\delta S}{\delta X(t)} + \left\langle F(t, X(\tau)]\right\rangle \label{threeseven}$$ where $\langle F(t, X (\tau)]\rangle$ has been written for $(\delta W/\delta\xi(t))_{\xi=0}$ because it can be shown to be the expected value of the force arising from the ignored variables in the state of the bath. $K^{\rm inv}(t', t, X(\tau)]$ is the inverse of $(2\hbar/i)(\delta^2 W/\delta\xi(t')\delta\xi(t))$ which turns out to be real and positive. Finally $\bar w(X,P)$ is the Wigner distribution for the density matrix $\bar\rho$: $$w(X, P) = \frac{1}{2\pi} \int d\xi\, e^{i P\xi /\hbar}\rho (X+\xi/2, X- \xi/2)\ . \label{threeeight}$$ This expression shows that, when $K^{\rm inv}_I$ is sufficiently large, the probabilities for histories of $X(t)$ are peaked about those which satisfy the equation of motion $${\cal E} (t, X(\tau)] = \frac{\delta S}{\delta X(t)} + \langle F(t, X(\tau)]\rangle = 0\ . \label{threenine}$$ and the initial conditions of these histories are distributed according to the Wigner distribution. The Wigner distribution is not generally positive, but, up to the accuracy of the approximations, this integral of it must be [@Hal92]. Thus we derive the form of the phenomenological equations of motion for this class of models. It is the equation of motion of the fundamental action $S[X(t)]$ corrected by phenomenological forces arising from the interaction with the bath. These depend not only on the form of the interaction Hamiltonian but also on the initial state of the bath, $\rho_B$. These forces are generally non-local in time, depending at a given instant on the whole trajectory $X(\tau)$. It can be shown that quantum mechanical causality implies that they depend only on part of path $X(\tau)$ to the past of $t$. Thus quantum mechanical causality implies classical causality. It is important to stress that the expansion of the decoherence functional has enabled us to consider the equations of motion for fully non-linear systems, not just the linear oscillator models that have been widely studied. The equation of motion (\[threenine\]) is not predicted to be satisfied [*exactly*]{}. The probabilities are [*peaked*]{} about ${\cal E}=0 $ but distributed about that value with a width that depends on the size of $K^{\rm inv}$. That is quantum noise whose spectrum and properties can be derived from (\[threethree\]). The fact that both the spectrum of fluctuations and the phenomenological forces can be derived from the same influence phase is the origin of the fluctuation dissipation theorem for linear systems. Simple examples of this analysis are the linear oscillator models that have been studied using path integrals by Feynman and Vernon [@FV63], Caldeira and Leggett [@CL83], Unruh and Zurek [@UZ89], and many others. For these, the $x$’s describe a distinguished harmonic oscillator linearly coupled to a bath of many others. If the initial state of the bath is a thermal density matrix, then the decoherence expansion is exact. In the especially simple case of a cut-off continuum of bath oscillators and high bath temperature, there are the following results: The imaginary part of the influence phase is given by $$ImW[x'(\tau),x(\tau)]= \frac{2M\gamma kT_B}{\hbar} \int^T_0 dt \left(x^\prime(t) - x(t)\right)^2 \label{threeten}$$ where $M$ is the mass of the $x$-oscillator, $\gamma$ is a measure of the strength of its coupling to the bath, and $T_B$ is the temperature of the bath. The exponent of the expression (\[threeseven\]) giving the probabilities for histories is $$-\frac{M}{8\gamma kT_B} \int^T_0 dt\, \left[\ddot X + \omega^2 X + 2\gamma \dot X\right]^2 \label{threeeleven}$$ where $\omega$ is the frequency of the $x$-oscillator renormalized by its interaction with the bath. The phenomenological force is friction, and the occurrence of $\gamma$, both in that force and the constant in front of (\[threeeleven\]), whose size governs the deviation from classical predictability, is a simple example of the fluctuation-dissipation theorem. In this simple case, an analysis of the the requirements for classical behavior is straightforward. To achieve decoherence we need high values of $\gamma kT_B$. That is, strong coupling is needed if interference phases are to be dissipated efficiently into the bath. However, the larger the value of $\gamma kT_B$ the smaller the coefficient of front of (\[threeeleven\]), decreasing the size of the exponential and [*increasing*]{} deviations from classical predictability. This is reasonable: the stronger the coupling to the bath the more noise is produced by the interactions that are carrying away the phases. To counteract that, and achieve a sharp peaking about the classical equation of motion, $M$ must be large so that $M/\gamma kT_B$ is large. That is, high inertia is needed to resist the noise that arises from the interactions with the bath. Thus, much more coarse graining is needed to ensure classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed to effect decoherence, and coarse graining beyond that to achieve the inertia necessary to resist the noise that the mechanisms of decoherence produce. Quasiclassical Realms in Quantum Cosmology {#sec:IV} ========================================== As observers of the universe, we deal every day with coarse-grained histories that exhibit classical correlations. Indeed, only by extending our direct perceptions with expensive and delicate instruments can we exhibit [*non*]{}-classical behavior. The coarse grainings that we use individually and collectively are, of course, characterized by a large amount of ignorance, for our observations determine only a very few of the variables that describe the universe and those only very imprecisely. Yet, we have the impression that the universe exhibits a much finer-grained set of histories, [*independent of our choice*]{}, defining an always decohering “quasiclassical realm”, to which our senses are adapted but deal with only a small part of. If we are preparing for a journey to a yet unseen part of the universe, we do not believe that we need to equip our spacesuits with detectors, say sensitive to coherent superpositions of position or other unfamiliar quantum operators. We expect that histories of familiar quasiclassical operators will decohere and exhibit patterns of classical correlation there as well as here. Roughly speaking, a quasiclassical realm is a set of decohering histories, that is maximally refined with respect to decoherence, and whose individual histories exhibit as much as possible patterns of deterministic correlation. At present we lack satisfactory measures of maximality and classicality with which to make the existence of one or more quasiclassical realms into quantitative questions in quantum cosmology [@GH90a; @PZ93]. We therefore do not know whether the universe exhibits a [*unique*]{} class of roughly equivalent sets of histories with high levels of classicality constituting the quasiclassical realm of familiar experience, or whether there might be other essentially inequivalent quasiclassical realms [@GH94]. However, even in the absence of such measures and such analyses, we can make an argument for the form of at least some of the operators we expect to occur over and over again in histories defining one kind of quasiclassical realm — operators we might call “quasiclassical”. In the earliest instants of the history of the universe, the coarse grainings defining spacetime geometry on scales above the Planck scale must emerge as quasiclassical. Otherwise, our theory of the initial condition is simply inconsistent with observation in a manifest way. Then, when there is classical spacetime geometry we can consider the conservation of energy, and momentum, and of other quantities which are conserved by virtue of the equations of quantum fields. Integrals of densities of conserved or nearly conserved quantities over suitable volumes are natural candidates for quasiclassical operators. Their approximate conservation allow them to resist deviations from predictability caused by “noise” arising from their interactions with the rest of the universe that accomplish decoherence. Such “hydrodynamic” variables [*are*]{} among the principal variables of classical theories. This argument is not unrelated to a standard one in classical statistical mechanics that seeks to identify the variables in which a hydrodynamic description of non-equilibrium systems may be expected. All isolated systems approach equilibrium — that is statistics. With certain coarse grainings this approach to equilibrium may be approximately described by hydrodynamic equations, such as the Navier-Stokes equation, incorporating phenomenological descriptions of dissipation, viscosity, heat conduction, diffusion, etc. The variables that characterize such hydrodynamic descriptions are the local quantities which very most [*slowly*]{} in time — that is, averages of densities of approximately conserved quantities over suitable volumes. The volumes must be large enough that statistical fluctuations in the values of the averages are small, but small enough that equilibrium is established within each volume in a time short compared to the dynamical times on which the variables vary. The constitutive relations defining coefficients of viscosity, diffusion, etc. are then defined and independent of the initial condition, permitting the closure of the set of hydrodynamic equations. Local equilibrium being established, the further equilibration of the volumes among themselves is described by the hydrodynamic equations. In the context of quantum cosmology, coarse grainings by averages of densities of approximately conserved quantities not only permit local equilibrium and resist gross statistical fluctuations leading to high probabilities for deterministic histories as in this argument, they also, as described above, resist the fluctuations arising from the mechanicsms of decoherence necessary for predicting probabilities of any kind in quantum mechanics. In this way we can sketch how a quasiclassical realm consisting of histories of ranges of values of quasiclassical operators, extended over cosmological dimensions both in space and in time, but highly refined with respect to those scales, is a feature of our universe and thus must be a prediction of its quantum initial condition. It may seem strange to attribute the classical behavior of everyday objects to the initial condition of the universe some 12 billion years ago, but, in this connection, two things should be noted: First, we are not just speaking of the classical behavior of a few objects described in a very coarse graining of our choosing, but of a much more refined feature of the universe extending over cosmological dimensions and indeed including the classical behavior of the cosmological geometry itself all the way back to the briefest of moments after the big bang. Second, at the most fundamental level the [*only*]{} ingredients entering into quantum mechanics are the theory of the initial condition and the theory of dynamics, so that [*any*]{} feature of the universe must be traceable to these two starting points and the accidents of our particular history. Put differently (neglecting quantum gravity) the possible classical behavior of a set of histories represented by strings of projection operators as in (\[twoone\]) does not depend on the operators alone except in trivial cases. Rather, like decoherence itself, classicality depends on the relation of those operators to the initial state $|\Psi\rangle$ through which we calculate the decoherence and probabilities of sets of histories by which classical behavior is defined. Yet it is reasonable to ask — how sensitive is the existence of a quasiclassical realm to the particular form of the initial condition? In seeking to answer this question it is important to recognize that there are two things it might mean. First, we might ask whether [*given*]{} an initial state $|\Psi\rangle$, there is always a set of histories which decoheres and exhibits deterministic correlations. There is, trivially. Consider the set of histories which just consists of projections down on ranges $\{\Delta E_\alpha\}$ of the [*total*]{} energy (or any other conserved quantity) at a sequence of times $$C_\alpha = P^H_{\alpha_n} (t_n) \cdots P^H_{\alpha_1} (t_1) \ .\label{fourone}$$ Since the energy is conserved these operators are independent of time, commute, and $C_\alpha$ is merely the projection onto the intersection of the intervals $\Delta E_{\alpha_`}, \cdots, \Delta E_{\alpha_n}$. The set of histories represented by (\[fourone\]) thus [*exactly*]{} decoheres $$D\left(\alpha^\prime, \alpha\right) = Tr\bigl[C_{\alpha^\prime} |\Psi\rangle\langle\Psi|C^\dagger_\alpha\bigr] = \langle\Psi|C^\dagger_\alpha C_{\alpha^\prime} |\Psi \rangle \propto \delta_{\alpha\alpha^\prime}\ , \label{fourtwo}$$ and exhibits deterministic correlations — the total energy today is the same as it was yesterday. Of course, such a set is far from maximal, but imagine subdividing the total volume again and again and considering the set of histories which results from following the values of the energy in each subvolume over the sequence of times. If the process of subdividing is followed until we begin to lose decoherence we might hope to retain some level of determinism while moving towards maximality. Thus, it seems likely that, for most initial $|\Psi\rangle$, we may find [*some*]{} sets of histories which constitute a quasiclassical realm. However, we might ask about the sensitivity of a quasiclassical realm to initial condition in a different way. We might fix the chains of projections that describe [*our*]{} highly refined quasiclassical realm and ask for how many [*other*]{} initial states does this set of histories decohere and exhibit the same classical correlations. This amounts to asking, for a given set of alternative histories $\{C_\alpha\}$, how many initial states $|\Psi\rangle$ will have the same decoherence functional? Expand $|\Psi\rangle$ in some generic basis in Hilbert space, $|i\rangle$: $$|\Psi\rangle = \sum\nolimits_i c_i | i\rangle\ . \label{fourthree}$$ The condition that $|\Psi\rangle$ result in a given decoherence functional $D(\alpha^\prime, \alpha)$ is $$\sum\nolimits_{ij} c^*_i c_j\ \bigl\langle i|C^\dagger_{\alpha^\prime} C_\alpha | j\bigr\rangle = D\left(\alpha^\prime, \alpha\right)\ . \label{fourfour}$$ Unless the $C_\alpha$ are such that decoherence and correlations are trivially implied by the operators (as is the above example of chains of projections onto a total conserved energy), the matrix elements $\langle i|C^\dagger_{\alpha^\prime}\, C_\alpha | j\rangle$ will not vanish indentically. Equation (\[fourfour\]) is therefore (number of histories $\alpha)^2$ equations for (dimension of Hilbert space) coefficients. When that dimension is made finite, say by limiting the total volume and energy, we expect a solution only when $$\left({\rm number\ of\ histories}\atop{\rm in\ the\ quasiclassical \ realm}\right)^2 \ltwid \left({\rm dim} \ {\cal H}\right)\ . \label{fourfive}$$ As the set of histories becomes increasingly refined, so that there are more and more alternative cases, the two sides may come closer to equality. The number of states $|\Psi\rangle$ which reproduce the [*particular*]{} maximal quasiclassical realm of our universe may thus be large but still small compared to the total number of states in Hilbert space. The Main Points Again {#sec:V} ===================== - Classical behavior of quantum systems is defined through the probabilities of deterministic correlations of individual time histories of a closed system. - Classical predictability requires coarse graining to accomplish decoherence, and coarse graining beyond that to achieve the necessary inertia to resist the noise which mechanisms of decoherence produce. - The maximally refined quasiclassical realm of familiar experience is an emergent feature, not of quantum evolution alone, but of that evolution, coupled to a specific theory of the universe’s initial condition. Whether the whole closed system exhibits a quasiclassical realm like ours, and indeed whether it exhibits more than one essentially inequivalent realm, are calculable questions in quantum cosmology if suitable measures of maximality and classicality can be supplied. - A generic initial state will exhibit some sort of quasiclassical realm, but the maximally refined quasiclassical realm of familiar experience will be an emergent feature of only a small fraction of the total possible initial states of the universe. Most of this paper reports joint work with M. Gell-Mann. The author’s research was supported in part by NSF grant PHY90-08502. [99]{} J. Halliwell, in [*Quantum Cosmology and Baby Universes: Proceedings of the 1989 Jerusalem Winter School for Theoretical Physics*]{}, S. Coleman, J.B. Hartle, T. Piran, and S. Weinberg, World Scientific, Singapore (1991) pp. 65-157. M. Gell-Mann and J.B. Hartle, [*Phys. Rev. D*]{} [**47**]{}, 3345 (1993). R. Griffiths, [*J. Stat. Phys.*]{} [**36**]{} 219 (1984). R. Omnès, [*J. Stat. Phys.*]{} [**53**]{}, 893 (1988); [*ibid*]{} [**53**]{}, 933 (1988); [*ibid*]{} [**53**]{}, 957 (1988); [*ibid*]{} [**57**]{}, 357 (1989); [*Rev. Mod. Phys.*]{} [**64**]{}, 339 (1992). M. Gell-Mann and J.B. Hartle in [*Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity*]{}, Vol. VIII, W. Zurek, Addison Wesley, Reading (1990) or in [*Proceedings of the 3rd International Symposium on the Foundations of Quantum Mechanics in the Light of New Technology*]{} S. Kobayashi, H. Ezawa, Y. Murayama, and S. Nomura, Physical Society of Japan, Tokyo (1990). T. Brun, [*Phys. Rev. D*]{} [**47**]{} 3383 (1993). J. Halliwell, [*Phys. Rev. D*]{} [**46**]{}, 1610 (1992). R.P. Feynman and J.R. Vernon, [*Ann. Phys. (N.Y.)*]{} [**24**]{}, 118 (1963). A. Caldeira and A. Leggett, [*Physica*]{} [**121A**]{}, 587 (1983). W. Unruh and W. Zurek, [*Phys. Rev. D*]{} [**40**]{}, 1071 (1989). J.P. Paz and W.H. Zurek, [*Phys. Rev. D*]{} [**48**]{}, 2728, (1993). M. Gell-Mann and J.B. Hartle, [*Equivalent Sets of Histories and Multiple Quasiclassical Domains*]{}; gr-qc/9404013. [^1]: Talk given at the Lanczos Centenary Meeting, North Carolina State University, December 15, 1993 [^2]: For a recent review see [@Hal91] [^3]: Earlier work, e.g. [@GH90a] called this the ‘quasiclassical domain’, but this risks confusion the usage in condensed matter physics.

ArXiv

--- abstract: 'We evaluate the elastic scattering cross section of vector dark matter with nucleon based on the method of effective field theory. The dark matter is assumed to behave as a vector particle under the Lorentz transformation and to interact with colored particles including quarks in the Standard Model. After formulating general formulae for the scattering cross sections, we apply them to the case of the first Kaluza-Klein photon dark matter in the minimal universal extra dimension model. The resultant cross sections are found to be larger than those calculated in previous literature.' address: - '$^1$ Department of Physics, Nagoya University, Nagoya 464-8602, Japan' - '$^2$ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan' author: - 'Natsumi Nagata$^{1, 2}$' title: 'A calculation for vector dark matter direct detection[^1]' --- Introduction ============ The existence of dark matter (DM) has been established by cosmological observations [@Komatsu:2010fb]. One of the most attractive candidates is what we call Weakly Interacting Massive Particles (WIMPs), which are stable particles with masses of the electroweak scale and weakly interact with ordinary matters. This interactions enable us to search for WIMP DM by using the scattering signal of DM with nuclei on the earth. Such kind of experiments are called the direct detection experiments of WIMP DM. For the past years, a lot of efforts have been dedicated to the direct detection of WIMP DM, and their sensitivities have been extremely improving. The XENON100 Collaboration, for example, gives a severe constraint on the spin-independent (SI) elastic scattering cross section of WIMP DM with nucleon $\sigma^{\rm SI}_N$ ($\sigma^{\rm SI}_N < 2.0\times 10^{-45}~{\rm cm}^2$ for WIMPs with a mass of 55 GeV$/c^2$) [@Aprile:2012nq]. Moreover, ton-scale detectors for the direct detection experiments are now planned and expected to have significantly improved sensitivities. In order to study the nature of DM based on these experiments, we need to evaluate the WIMP-nucleon elastic scattering cross section precisely. In this work, we assume the WIMP DM to be a vector particle, and evaluate its cross section scattering off a nucleon. Several candidates for vector DM have been proposed in various models, and there have been a lot of previous work computing the scattering cross sections [@Cheng:2002ej; @Servant:2002hb; @Birkedal:2006fz]. However, we found that in the calculations some of the leading contributions to the scattering cross section are not evaluated correctly, or in some cases completely neglected. Taking such situation into account, we study the way of evaluating the cross section systematically by using the method of effective field theory. Direct detection of vector dark matter ====================================== In this section we discuss the way of evaluating the elastic scattering cross section of vector DM with nucleon. First, we write down the effective interactions of vector DM ($B_\mu$) with light quarks and gluon [@Hisano:2010yh]: $$\mathcal{L}^{\mathrm{eff}}=\sum_{q=u,d,s}\mathcal{L}^{\mathrm{eff}}_q +\mathcal{L}^{\mathrm{eff}}_G,$$ with $$\begin{aligned} \mathcal{L}^{\mathrm{eff}}_q &=& f_q m_q B^{\mu}B_{\mu}\bar{q}q+ \frac{d_q}{M} \epsilon_{\mu\nu\rho\sigma}B^{\mu}i\partial^{\nu}B^{\rho} \bar{q}\gamma^{\sigma}\gamma^{5}q+\frac{g_q}{M^2} B^{\rho}i\partial^{\mu}i\partial^{\nu}B_{\rho}\mathcal{O}^q_{\mu\nu}, \label{eff_lagq} \\ \mathcal{L}^{\mathrm{eff}}_G&=&f_G B^{\rho}B_{\rho}G^{a\mu\nu}G^a_{\mu\nu}, \label{eff_lagG}\end{aligned}$$ where $m_q$ are the masses of light quarks, $M$ is the DM mass, and $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor defined as $\epsilon^{0123}=+1$. The covariant derivative is defined as $D_\mu\equiv\partial_\mu+i g_sA^a_\mu T_a$, with $g_s$, $T_a$ and $A^a_\mu$ being the strong coupling constant, the SU(3)$_C$ generators, and the gluon fields, respectively. The gluon field strength tensor is denoted by $G^a_{\mu\nu}$, and $\mathcal{O}^q_{\mu\nu}\equiv\frac12 \bar{q} i \left(D_{\mu}\gamma_{\nu} + D_{\nu}\gamma_{\mu} -\frac{1}{2}g_{\mu\nu}{{\ooalign{\hfil/\hfil\crcr$D$}}} \right) q $ are the twist-2 operators of light quarks. When we write down the effective Lagrangian, we consider the fact that the scattering process is non-relativistic. The coefficients of the operators are to be determined by integrating out the heavy particles in high energy theory. The second term in Eq. (\[eff\_lagq\]) gives rise to the spin-dependent (SD) interaction, while the other terms yield the spin-independent (SI) interactions. We focus on the SI interactions hereafter, because the experimental constraint is much severe for the SI interactions, rather than for the SD interactions. In order to obtain the effective coupling of the vector DM with nucleon induced by the effective Lagrangian, we need to evaluate the nucleon matrix elements of the quark and gluon operators in Eqs.(\[eff\_lagq\]) and (\[eff\_lagG\]). First, the nucleon matrix elements of the scalar-type quark operators are parametrized as $$f_{Tq}\equiv \langle N \vert m_q \bar{q} q \vert N\rangle/m_N~,$$ with $\vert N\rangle$ and $m_N$ the one-particle state and the mass of nucleon, respectively. The parameters are called the mass fractions and their values are obtained from the lattice simulations [@Young:2009zb; @:2012sa]. Second, for the quark twist-2 operators, we can use the parton distribution functions (PDFs): $$\begin{aligned} \langle N(p)\vert {\cal O}_{\mu\nu}^q \vert N(p) \rangle &=&\frac{1}{m_N} (p_{\mu}p_{\nu}-\frac{1}{4}m^2_N g_{\mu\nu})\ (q(2)+\bar{q}(2)) \ ,\end{aligned}$$ where $q(2)$ and $\bar{q}(2)$ are the second moments of PDFs of quark $q(x)$ and anti-quark $\bar{q}(x)$, respectively, which are defined as $q(2)+ \bar{q}(2) =\int^{1}_{0} dx ~x~ [q(x)+\bar{q}(x)]$. These values are obtained from Ref. [@Pumplin:2002vw]. Finally, the matrix element of gluon field strength tensor can be evaluated by using the trace anomaly of the energy-momentum tensor in QCD [@Shifman:1978zn]. The resultant expression is given as $$\langle N\vert G^a_{\mu\nu}G^{a\mu\nu}\vert N\rangle =-\frac{8\pi}{9\alpha_s} m_N f_{TG}$$ with $f_{TG}\equiv 1-\sum_{q=u,d,s}f_{Tq}$. Note that the right hand side of the expression is divided by the strong coupling constant, $\alpha_s$. For this reason, although the gluon contribution is induced by higher loop diagrams, it can be comparable to the quark contributions [@Hisano:2010ct]. Briefly speaking, the enhancement comes from the large gluon contribution to the mass of nucleon. As a result, the SI effective coupling of vector DM with nucleon, $f_N$, is given as $$\begin{aligned} f_N/m_N&=&\sum_{q=u,d,s} f_q f_{Tq} +\sum_{q=u,d,s,c,b} \frac{3}{4} \left(q(2)+\bar{q}(2)\right)g_q -\frac{8\pi}{9\alpha_s}f_{TG} f_G ~. \label{f}\end{aligned}$$ Using the effective coupling, we eventually obtain the SI scattering cross section of DM with nucleon: $$\sigma^{\rm (SI)}_{N}= \frac{1}{\pi}\biggl(\frac{m_N}{M+m_N}\biggr)^2~\vert f_N\vert ^2~.$$ Now, all we have to do reduces to evaluate the coefficients of the effective operators by integrating out the heavy fields in the high-energy theories. For example, we take the case where the interaction Lagrangian of the vector DM has a generic form as $$\begin{aligned} \mathcal{L}= \bar{\psi}_2 ~(a_{\psi_2\psi_1} \gamma^{\mu}+b_{\psi_2\psi_1} \gamma^{\mu}\gamma_5)\psi_1 B_{\mu} + \mathrm{h.c.}~, \label{eq:simpleL}\end{aligned}$$ where $\psi_1$ and $\psi_2$ are colored fermions with masses $m_1$ and $m_2$ ($m_1<m_2$), respectively. ![Tree-level diagrams of exchanging colored fermion $\psi_2$ to generate interaction of vector dark matter with light quarks.[]{data-label="fig:tree_general"}](tree_general.eps){height="4cm"} In this case, the vector DM is scattered by light quarks at tree-level. The relevant interaction Lagrangian is given by taking $\psi_1=q$ in Eq. (\[eq:simpleL\]), and the corresponding diagrams are shown in Fig. \[fig:tree\_general\]. After integrating out the heavy particle $\psi_2$, we obtain $$\begin{aligned} f_q&=&\frac{a^2_{\psi_2 q}-b^2_{\psi_2 q}}{m_q}\frac{m_{2}}{m^2_{2}-M^2} -(a^2_{\psi_2 q}+b^2_{\psi_2 q}) \frac{m^2_{2}}{2(m^2_{2}-M^2)^2}, \label{tree_general_1} \\ g_q&=&-\frac{2M^2(a^2_{\psi_2 q}+b^2_{\psi_2 q})}{(m^2_{2}-M^2)^2}. \label{tree_general_3}\end{aligned}$$ One can easily find that the effective couplings obtained here are enhanced when the vector DM and the heavy colored fermion are degenerate in mass [@Hisano:2011um]. ![One-loop contributions to scalar-type effective coupling with gluon. []{data-label="fig:loop_general"}](loop_general.eps){height="4cm"} The effective coupling of the vector DM with gluon is induced by 1-loop diagrams illustrated in Fig. \[fig:loop\_general\]. In those processes, all the particles $\psi_1$ and $\psi_2$ which couple with $B_{\mu}$ run in the loop. The resultant expressions are somewhat complicated, and thus we just quote Ref. [@Hisano:2010yh] for their complete formulae as well as their derivation. Application and Results ======================= Next, we deal with a particular model for vector DM as an application. We carry out the calculation for the first Kaluza-Klein (KK) photon DM [@Cheng:2002ej; @Servant:2002hb] in the minimal universal extra dimension (MUED) model [@Appelquist:2000nn; @Cheng:2002iz]. In this model, an extra dimension is compactified on an $S^1/\mathbb{Z}_2$ orbifold with the compactification radius $R$, and all of the Standard Model (SM) particles propagate in the dimension. The lightest KK-odd particle (LKP) is prevented from decaying to the SM particles, so it becomes DM. This model has just three undetermined parameters: the radius of the extra dimension $R$, the mass of Higgs boson $m_h$, and the cutoff scale $\Lambda$. In general, extra dimensional models give rise to the degenerate mass spectrum at tree-level, which is broken by radiative corrections. By evaluating the radiative corrections [@Cheng:2002iz], one finds that the first KK photon $B^{(1)}$ is the lightest KK-odd particle, thus, becomes the DM in the Universe. Moreover, since the mass difference is induced by radiative corrections, the mass degeneracy is tight for a small cut-off scale. In such a case, although it is difficult to probe the MUED model at the LHC because of the soft QCD jets, the direct detection rate of dark matter is expected to be enhanced [@Hisano:2011um; @Arrenberg:2008wy], as we shall see soon later[^2]. ![Tree-level diagrams for the elastic scattering of the KK photon DM $B^{(1)}$ with light quarks: (a) Higgs boson exchange contribution, and (b) KK quark exchange contributions.[]{data-label="fig:tree"}](tree.eps){height="3.5cm"} Now we evaluate the SI scattering cross section of the KK photon DM with nucleon. The effective interaction of the KK photon DM $B^{(1)}$ with light quarks is induced by the tree-level diagrams shown in Fig. \[fig:tree\]. Here, $h^0$ and $q^{(1)}$ are the Higgs boson and the first KK quark, respectively. ![One-loop diagrams for the effective interaction of $B^{(1)}$ with gluon.[]{data-label="fig:qloop"}](1loop.eps){height="4cm"} Also, there are one-loop diagrams we should evaluate for the gluon contribution. They are illustrated in Fig. \[fig:qloop\]. In these diagrams, all of the KK quarks run in the loop. Taking all the contributions into account, we obtain the effective interactions. Their expressions are given in Ref. [@Hisano:2010yh]. ![Each contribution in effective coupling $f_N/m_N$ given in Eq. (\[f\]). Here we set $m_h=125~{\rm GeV}$ and $(m_{\rm 1st}-M)/M=0.1$.[]{data-label="fig:fN_10"}](fN.eps){height="7cm"} ![Spin-independent cross section with proton for $m_h = 125$ GeV. Each line corresponds to $\Lambda = 5/R$, $20/R$, and $50/R$, respectively. []{data-label="mhfixed"}](sigmasi.eps){height="7.5cm"} In Fig. \[fig:fN\_10\], we plot each contribution to the SI effective coupling of DM with a proton. The solid line shows the contribution of the Higgs boson exchanging diagrams, the dashed line indicates the twist-2 type contribution, the dash-dotted line corresponds to the scalar-type contribution (except for the Higgs-exchanging contribution), and the dotted line represents the gluon contribution (except for the Higgs-exchanging contribution). Here we set the Higgs boson mass[^3] equal to 125 GeV, and the mass difference between the DM and the first KK quark to be 10 % by hand. We find that all of the contributions have the same sign thus they are constructive. The twist-2 contribution is dominant when $M\lesssim 800~{\rm GeV}$, while that of Higgs boson exchanging process becomes dominant above it. Moreover, although the tree-level quark contributions are dominant, it is found that the gluon contribution is not negligible at all. By using the effective couplings obtained above, we evaluate the SI scattering cross sections. In Fig. \[mhfixed\], we plot the SI cross section of KK photon DM with a proton as a function of the DM mass. Here again, the Higgs boson mass is set to be 125 GeV. We take the cut-off scale $\Lambda=5/R,~20/R, ~50/R$ from top to bottom. We find that the scattering cross sections reduce as the cut-off scale is taken to be large, as expected. As a result, we obtain the SI scattering cross section which is larger than those obtained by the previous calculations. Note that the DM mass with which the thermal relic abundance is preferred by the WMAP observation [@Komatsu:2010fb] is around 1300 GeV [@Belanger:2010yx]. It corresponds to $\sigma^{\rm SI}_N=$(2.5-5)$\times 10^{-47}$ cm$^{2}$, so the direct detection experiments with ton-scale detectors might be able to probe the DM in the future. Conclusion ========== We calculate the spin independent elastic scattering cross sections of vector dark matter with nucleon based on the effective field theory. It is found that the interaction of dark matter with gluon as well as quarks yields sizable contribution to the scattering cross section, though the gluon contribution is induced at loop level. The scattering cross section of the first Kaluza-Klein photon dark matter in the MUED model turns out to be larger than those obtained by the previous calculations. The author would like to thank Junji Hisano, Koji Ishiwata, and Masato Yamanaka for collaboration. This work is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. References {#references .unnumbered} ========== [99]{} J. Hisano, K. Ishiwata, N. Nagata and M. Yamanaka, Prog. Theor. Phys.  [**126**]{}, 435 (2011) . E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl.  [**192**]{}, 18 (2011) . E. Aprile [*et al.*]{} \[XENON100 Collaboration\], arXiv:1207.5988 \[astro-ph.CO\]. H. C. P. Cheng, J. L. Feng and K. T. Matchev, Phys. Rev. Lett.  [**89**]{}, 211301 (2002). G. Servant and T. M. P. Tait, New J. Phys.  [**4**]{}, 99 (2002). A. Birkedal, A. Noble, M. Perelstein and A. Spray, Phys. Rev.  D [**74**]{}, 035002 (2006). R. D. Young and A. W. Thomas, Phys. Rev. D [**81**]{}, 014503 (2010) . H. Ohki , [*et al.*]{} \[JLQCD Collaboration\], arXiv:1208.4185 \[hep-lat\]. J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung, JHEP [**0207**]{}, 012 (2002). M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. B [**78**]{}, 443 (1978). J. Hisano, K. Ishiwata and N. Nagata, Phys. Rev. D [**82**]{}, 115007 (2010) . J. Hisano, K. Ishiwata and N. Nagata, Phys. Lett. B [**706**]{}, 208 (2011) . T. Appelquist, H. -C. Cheng and B. A. Dobrescu, Phys. Rev. D [**64**]{}, 035002 (2001) . H. -C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D [**66**]{}, 036005 (2002) . S. Arrenberg, L. Baudis, K. Kong, K. T. Matchev and J. Yoo, Phys. Rev. D [**78**]{}, 056002 (2008) . M. Asano, T. Bringmann and C. Weniger, Phys. Lett. B [**709**]{}, 128 (2012). M. Garny, A. Ibarra, M. Pato and S. Vogl, arXiv:1207.1431 \[hep-ph\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Lett. B [**716**]{}, 1 (2012) . S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**716**]{}, 30 (2012) . G. Belanger, M. Kakizaki and A. Pukhov, JCAP [**1102**]{}, 009 (2011) . [^1]: This talk is based on the work with Junji Hisano, Koji Ishiwata, and Masato Yamanaka [@Hisano:2010yh]. [^2]: Indirect DM searches also might be a powerful alternative in such a case [@Asano:2011ik; @Garny:2012eb]. [^3]: Recent searches for the Standard Model Higgs boson at the LHC indicate its mass to be around 125 GeV [@:2012gk; @:2012gu].

ArXiv

--- abstract: 'We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal ${O\left({T^{1/2}}\right)}$ *worst case* regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is *predictable*, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle.' author: - | Arun Sai Suggala\ Carnegie Mellon University\ `[email protected]`\ Praneeth Netrapalli\ Microsoft Research, India\ `[email protected]`\ bibliography: - 'local.bib' title: 'Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games' --- Introduction {#sec:intro} ============ In this work, we consider the problem of online learning, where in each iteration, the learner chooses an action and observes a loss function. The goal of the learner is to choose a sequence of actions which minimizes the cumulative loss suffered over the course of learning. The paradigm of online learning has many theoretical and practical applications and has been widely studied in a number of fields, including game theory and machine learning. One of the popular applications of online learning is in solving minimax games arising in various contexts such as boosting [@freund1996game], robust optimization [@chen2017robust], Generative Adversarial Networks [@goodfellow2014generative]. In recent years, a number of efficient algorithms have been developed for regret minimization. These algorithms fall into two broad categories, namely, Follow the Regularized Leader (FTRL) [@mcmahan2017survey] and FTPL [@kalai2005efficient] style algorithms. When the sequence of loss functions encountered by the learner are convex, both these algorithms are known to achieve the optimal ${O\left({T^{1/2}}\right)}$ worst case regret [@cesa2006prediction; @hazan2016introduction]. While these algorithms have similar regret guarantees, they differ in computational aspects. Each iteration of FTRL involves implementation of an expensive projection step. In contrast, each step of FTPL involves solving a linear optimization problem, which can be implemented efficiently for many problems of interest [@garber2013playing; @gidel2016frank; @hazan2020projection]. This crucial difference between FTRL and FTPL makes the latter algorithm more attractive in practice. Even in the more general nonconvex setting, where the loss functions encountered by the learner can potentially be nonconvex, FTPL algorithms are attractive. In this setting, FTPL requires access to an offline optimization oracle which computes the perturbed best response, and achieves ${O\left({T^{1/2}}\right)}$ worst case regret [@suggala2019online]. Furthermore, these optimization oracles can be efficiently implemented for many problems by leveraging the rich body of work on global optimization [@horst2013handbook]. Despite its importance and popularity, FTPL has been mostly studied for the worst case setting, where the loss functions are assumed to be adversarially chosen. In a number of applications of online learning, the loss functions are actually benign and predictable [@rakhlin2012online]. In such scenarios, FTPL can not utilize the predictability of losses to achieve tighter regret bounds. While [@rakhlin2012online; @suggala2019online] study variants of FTPL which can make use of predictability, these works either consider restricted settings or provide sub-optimal regret guarantees (see Section \[sec:bg\] for more details). This is unlike FTRL, where optimistic variants that can utilize the predictability of loss functions have been well understood [@rakhlin2012online; @rakhlin2013optimization] and have been shown to provide faster convergence rates in applications such as minimax games. In this work, we aim to bridge this gap and study a variant of FTPL called Optimistic FTPL (OFTPL), which can achieve better regret bounds, while retaining the optimal worst case regret guarantee for unpredictable sequences. The main challenge in obtaining these tighter regret bounds is handling the stochasticity and optimism in the algorithm, which requires different analysis techniques to those commonly used in the analysis of FTPL. In this work, we rely on the dual view of perturbation as regularization to derive regret bounds of OFTPL. To demonstrate the usefulness of OFTPL, we consider the problem of solving minimax games. A widely used approach for solving such games relies on online learning algorithms [@cesa2006prediction]. In this approach, both the minimization and the maximization players play a repeated game against each other and rely on online learning algorithms to choose their actions in each round of the game. In our algorithm for solving games, we let both the players use OFTPL to choose their actions. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the optimization oracle. While there are prior algorithms that achieve these convergence rates [@he2015semi; @suggala2019online], an important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle. We note that such parallelizable algorithms are especially useful in large-scale machine learning applications such as training of GANs, adversarial training, which often involve huge datasets such as ImageNet [@russakovsky2015imagenet]. Preliminaries and Background Material {#sec:bg} ===================================== #### Online Learning. The online learning framework can be seen as a repeated game between a learner and an adversary. In this framework, in each round $t$, the learner makes a prediction for some compact set ${\mathcal{X}}$, and the adversary simultaneously chooses a loss function and observe each others actions. The goal of the learner is to choose a sequence of actions $\{{{\ensuremath{\mathbf{x}}}}_t\}_{t=1}^T$ so that the following notion of regret is minimized: When the domain ${\mathcal{X}}$ and loss functions $f_t$ are convex, a number of efficient algorithms for regret minimization have been studied. Some of these include deterministic algorithms such as Online Mirror Descent, Follow the Regularized Leader (FTRL) [@hazan2016introduction; @mcmahan2017survey], and stochastic algorithms such as Follow the Perturbed Leader (FTPL) [@kalai2005efficient]. In FTRL, one predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}{\left\langle {\nabla}_i, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, for some strongly convex regularizer $R$, where ${\nabla}_i = {\nabla}f_i({{\ensuremath{\mathbf{x}}}}_i)$. FTRL is known to achieve the optimal $O(T^{1/2})$ worst case regret in the convex setting [@mcmahan2017survey]. In FTPL, one predicts ${{\ensuremath{\mathbf{x}}}}_t$ as $m^{-1}\sum_{j=1}^m{{\ensuremath{\mathbf{x}}}}_{t,j}$, where ${{\ensuremath{\mathbf{x}}}}_{t,j}$ is a minimizer of the following linear optimization problem: ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle \sum_{i=1}^{t-1}{\nabla}_i - \sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$ Here, $\{\sigma_{t,j}\}_{j=1}^m$ are independent random perturbations drawn from some appropriate probability distribution such as exponential distribution or uniform distribution in a hyper-cube. Various choices of perturbation distribution gives rise to various FTPL algorithms. When the loss functions are linear, @kalai2005efficient show that FTPL achieves ${O\left({T^{1/2}}\right)}$ expected regret, irrespective of the choice of $m$. When the loss functions are convex, @hazan2016introduction showed that the deterministic version of FTPL (*i.e.,* as $m \to \infty$) achieves ${O\left({T^{1/2}}\right)}$ regret. While projection free methods for online convex learning have been studied since the early work of [@hazan2012projection], surprisingly, regret bounds of FTPL for finite $m$ have only been recently studied [@hazan2020projection]. @hazan2020projection show that for Lipschitz and convex functions, FTPL achieves ${O\left({T^{1/2} + m^{-1/2}T }\right)}$ expected regret, and for smooth convex functions, the algorithm achieves ${O\left({T^{1/2} + m^{-1}T}\right)}$ expected regret. When either the domain ${\mathcal{X}}$ or the loss functions $f_t$ are non-convex, no deterministic algorithm can achieve $o(T)$ regret [@cesa2006prediction; @suggala2019online]. In such cases, one has to rely on randomized algorithms to achieve sub-linear regret. In randomized algorithms, in each round $t$, the learner samples the prediction ${{\ensuremath{\mathbf{x}}}}_t$ from a distribution $P_t \in {\mathcal{P}}$, where ${\mathcal{P}}$ is the set of all probability distributions supported on ${\mathcal{X}}$. The goal of the learner is to choose a sequence of distributions $\{P_t\}_{t=1}^T$ to minimize the expected regret $ \sum_{t = 1}^T {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_t}\left[f_t({{\ensuremath{\mathbf{x}}}})\right]} - \inf_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}). $ A popular technique to minimize the expected regret is to consider a linearized problem in the space of probability distributions with losses $\Tilde{f}_t(P) = {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f_t({{\ensuremath{\mathbf{x}}}})\right]}$ and perform FTRL in this space. In such a technique, $P_t$ is computed as: $ {\mathop{\rm argmin}}_{P \in {\mathcal{P}}} \sum_{i=1}^{t-1} \Tilde{f}_i(P) + R(P), $ for some strongly convex regularizer $R(P).$ When $R(P)$ is the negative entropy of $P$, the algorithm is called entropic mirror descent or continuous exponential weights. This algorithm achieves ${O\left({T^{1/2}}\right)}$ expected regret for bounded loss functions $f_t$. Another technique to minimize expected regret is to rely on FTPL [@gonen2018learning; @suggala2019online]. Here, the learner generates the random prediction ${{\ensuremath{\mathbf{x}}}}_t$ by first sampling a random perturbation $\sigma$ and then computing the perturbed best response, which is defined as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}}) - {\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}$. In a recent work, @suggala2019online show that this algorithm achieves ${O\left({T^{1/2}}\right)}$ expected regret, whenever the sequence of loss functions are Lipschitz. We now briefly discuss the computational aspects of FTRL and FTPL. Each iteration of FTRL (with entropic regularizer) requires sampling from a non-logconcave distribution. In contrast, FTPL requires solving a nonconvex optimization problem to compute the perturbed best response. Of these, computing the perturbed best response seems significantly easier since standard algorithms such as gradient descent seem to be able to find approximate global optima reasonably fast, even for complicated tasks such as training deep neural networks. #### Online Learning with Optimism. When the sequence of loss functions are convex and predictable, @rakhlin2012online [@rakhlin2013optimization] study optimistic variants of FTRL which can exploit the predictability to obtain better regret bounds. Let $g_t$ be our guess of ${\nabla}_t$ at the beginning of round $t$. Given $g_t$, we predict ${{\ensuremath{\mathbf{x}}}}_t$ in Optimistic FTRL (OFTRL) as $ {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle \sum_{i=1}^{t-1}{\nabla}_i + g_t, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}). $ Note that when $g_t=0$, OFTRL is equivalent to FTRL. [@rakhlin2012online; @rakhlin2013optimization] show that the regret bounds of OFTRL only depend on $(g_t-{\nabla}_t)$. Moreover, these works show that OFTRL provides faster convergence rates for solving smooth convex-concave games. In contrast to FTRL, the optimistic variants of FTPL have been less well understood. [@rakhlin2012online] studies OFTPL for linear loss functions. But they consider restrictive settings and their algorithms require the knowledge of sizes of deviations $(g_t-\nabla_t)$. [@suggala2019online] studies OFTPL for the more general nonconvex setting. The algorithm predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}}) +g_t({{\ensuremath{\mathbf{x}}}}) - {\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $g_t$ is our guess of $f_t$. However, the regret bounds of [@suggala2019online] are sub-optimal and weaker than the bounds we obtain in our work (see Theorem \[thm:oftpl\_noncvx\_regret\]). Moreover, [@suggala2019online] does not provide any consequences of their results to minimax games. We note that their sub-optimal regret bounds translate to sub-optimal rates of convergence for solving smooth minimax games. #### Minimax Games. Consider the following problem, which we refer to as minimax game: $\min_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\max_{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}} f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})$. In these games, we are often interested in finding a Nash Equilibrium (NE). A pair $(P,Q)$, where $P$ is a probability distribution over ${\mathcal{X}}$ and $Q$ is a probability distribution over ${\mathcal{Y}}$, is called a NE if: $\sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}\leq{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P, {{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]} \leq \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}.$ A standard technique for finding a NE of the game is to rely on no-regret algorithms [@cesa2006prediction; @hazan2016introduction]. Here, both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players play a repeated game against each other and use online learning algorithms to choose their actions. The average of the iterates generated via this repeated game can be shown to converge to a NE. #### Projection Free Learning. Projection free learning algorithms are attractive as they only involve solving linear optimization problems. Two broad classes of projection free techniques have been considered for online convex learning and minimax games, namely, Frank-Wolfe (FW) methods and FTPL based methods. @garber2013playing consider the problem of online learning when the action space ${\mathcal{X}}$ is a polytope. They provide a FW method which achieves ${O\left({T^{1/2}}\right)}$ regret using $T$ calls to the linear optimization oracle. @hazan2012projection provide a FW technique which achieves ${O\left({T^{3/4}}\right)}$ regret for general online convex learning with Lipschitz losses and uses $T$ calls to the linear optimization oracle. In a recent work, @hazan2020projection show that FTPL achieves ${O\left({T^{2/3}}\right)}$ regret for online convex learning with smooth losses, using $T$ calls to the linear optimization oracle. This translates to ${O\left({T^{-1/3}}\right)}$ rate of convergence for solving smooth convex-concave games. Note that, in contrast, our algorithm achieves ${O\left({T^{-1/2}}\right)}$ convergence rate in the same setting. @gidel2016frank study FW methods for solving convex-concave games. When the constraint sets ${\mathcal{X}},{\mathcal{Y}}$ are *strongly convex*, the authors show geometric convergence of their algorithms. In a recent work, @he2015semi propose a FW technique for solving smooth convex-concave games which converges at a rate of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the linear optimization oracle. We note that our simple OFTPL based algorithm achieves these rates, with the added advantage of parallelizability. That being said, @he2015semi achieve dimension free convergence rates in the Euclidean setting, where the smoothness is measured w.r.t $\|\cdot\|_2$ norm. In contrast, the rates of convergence of our algorithm depend on the dimension. #### Notation. $\|\cdot\|$ is a norm on some vector space, which is typically $\mathbb{R}^d$ in our work. $\|\cdot\|_{*}$ is the dual norm of $\|\cdot\|$, which is defined as $\|{{\ensuremath{\mathbf{x}}}}\|_{*} = \sup\{{\left\langle {{\ensuremath{\mathbf{u}}}}, {{\ensuremath{\mathbf{x}}}}\right\rangle}: {{\ensuremath{\mathbf{u}}}}\in\mathbb{R}^d, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1\}$. We use ${\Psi_1}, {\Psi_2}$ to denote norm compatibility constants of $\|\cdot\|$, which are defined as ${\Psi_1}= \sup_{{{\ensuremath{\mathbf{x}}}}\neq 0} \|{{\ensuremath{\mathbf{x}}}}\|/\|{{\ensuremath{\mathbf{x}}}}\|_2,\ {\Psi_2}= \sup_{{{\ensuremath{\mathbf{x}}}}\neq 0} \|{{\ensuremath{\mathbf{x}}}}\|_2/\|{{\ensuremath{\mathbf{x}}}}\|.$ We use the notation $f_{1:t}$ to denote $\sum_{i=1}^tf_i$. In some cases, when clear from context, we overload the notation $f_{1:t}$ and use it to denote the set $\{f_1,f_2\dots f_t\}$. For any convex function $f$, $\partial f({{\ensuremath{\mathbf{x}}}})$ is the set of all subgradients of $f$ at ${{\ensuremath{\mathbf{x}}}}$. For any function $f:{\mathcal{X}}\times {\mathcal{Y}}\to \mathbb{R}$, $f(\cdot,{{\ensuremath{\mathbf{y}}}}), f({{\ensuremath{\mathbf{x}}}},\cdot)$ denote the functions ${{\ensuremath{\mathbf{x}}}}\rightarrow f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}), {{\ensuremath{\mathbf{y}}}}\rightarrow f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}).$ For any function $f:{\mathcal{X}}\to\mathbb{R}$ and any probability distribution $P$, we let $f(P)$ denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]}.$ Similarly, for any function $f:{\mathcal{X}}\times{\mathcal{Y}}\to\mathbb{R}$ and any two distributions $P,Q$, we let $f(P,Q)$ denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P,{{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}.$ For any set of distributions $\{P_j\}_{j=1}^m$, $\frac{1}{m}\sum_{j=1}^mP_j$ is the mixture distribution which gives equal weights to its components. We use $\text{Exp}(\eta)$ to denote the exponential distribution, whose CDF is given by $P(Z\leq s) =1-\exp(-s/\eta).$ Dual view of Perturbation as Regularization {#sec:duality} =========================================== In this section, we present a key result which shows that when the sequence of loss functions are convex, every FTPL algorithm is an FTRL algorithm. Our analysis of OFTPL relies on this dual view to obtain tight regret bounds. This duality between FTPL and FTRL was originally studied by @hofbauer2002global, where the authors show that any FTPL algorithm, with perturbation distribution admitting a strictly positive density on $\mathbb{R}^d$, is an FTRL algorithm w.r.t some convex regularizer. However, many popular perturbation distributions such as exponential and uniform distributions don’t have a strictly positive density. In a recent work, @abernethy2016perturbation point out that the duality between FTPL and FTRL holds for very general perturbation distributions. However, the authors do not provide a formal theorem showing this result. Here, we provide a proposition formalizing the claim of [@abernethy2016perturbation]. \[prop:ftpl\_ftrl\_connection\] Consider the problem of online convex learning, where the sequence of loss functions $\{f_t\}_{t=1}^T$ encountered by the learner are convex. Consider the deterministic version of FTPL algorithm, where the learner predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{1:t-1}-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}$. Suppose the perturbation distribution is absolutely continuous w.r.t the Lebesgue measure. Then there exists a convex regularizer $R:\mathbb{R}^d\to \mathbb{R}\cup\{\infty\}$, with domain ${\text{dom}(R)}\subseteq {\mathcal{X}}$, such that $ {{\ensuremath{\mathbf{x}}}}_t = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle {\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle}+R({{\ensuremath{\mathbf{x}}}}). $ Moreover, and ${{\ensuremath{\mathbf{x}}}}_t = \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$ where $\partial R^{-1}$ is the inverse of $\partial R$ in the sense of multivalued mappings. Online Learning with OFTPL {#sec:onlinelearning} ========================== Online Convex Learning ---------------------- **Input:** Perturbation Distribution ${P_{\text{PRTB}}},$ number of samples $m,$ number of iterations $T$ Denote ${\nabla}_0 = 0$ Let $g_t$ be the guess for ${\nabla}_t$ Sample $\sigma_{t,j}\sim {P_{\text{PRTB}}}$ ${{\ensuremath{\mathbf{x}}}}_{t,j}\in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{0:t-1}+ g_t-\sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ Play ${{\ensuremath{\mathbf{x}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{x}}}}_{t,j}$ Observe loss function $f_t$ In this section, we present the OFTPL algorithm for online convex learning and derive an upper bound on its regret. The algorithm we consider is similar to the OFTRL algorithm (see Algorithm \[alg:oftpl\_cvx\]). Let $g_t[f_1\dots f_{t-1}]$ be our guess for ${\nabla}_t$ at the beginning of round $t$, with $g_1 = 0$. To simplify the notation, in the sequel, we suppress the dependence of $g_t$ on $\{f_{i}\}_{i=1}^{t-1}$. Given $g_t$, we predict ${{\ensuremath{\mathbf{x}}}}_t$ in OFTPL as follows. We sample independent perturbations $\{\sigma_{t,j}\}_{j=1}^m$ from the perturbation distribution ${P_{\text{PRTB}}}$ and compute ${{\ensuremath{\mathbf{x}}}}_t$ as $m^{-1}\sum_{j=1}^m{{\ensuremath{\mathbf{x}}}}_{t,j}$, where ${{\ensuremath{\mathbf{x}}}}_{t,j}$ is a minimizer of the following linear optimization problem $${{\ensuremath{\mathbf{x}}}}_{t,j} \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle {\nabla}_{1:t-1} + g_t - \sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ We now present our main theorem which bounds the regret of OFTPL. A key quantity the regret depends on is the *stability* of predictions of the deterministic version of OFTPL. Intuitively, an algorithm is stable if its predictions in two consecutive iterations differ by a small quantity. To capture this notion, we first define function ${{\nabla}\Phi}:\mathbb{R}^d \to \mathbb{R}^d$ as: $ {{\nabla}\Phi\left(g\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\left\langle g-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}. $ Observe that ${{\nabla}\Phi\left(\nabla_{1:t-1} + g_t\right)}$ is the prediction of the deterministic version of OFTPL. We say the predictions of OFTPL are stable, if ${{\nabla}\Phi}$ is a Lipschitz function. The predictions of OFTPL are said to be $\beta$-stable w.r.t some norm $\|\cdot\|$, if $$\forall g_1,g_2\in\mathbb{R}^d \quad \|{{\nabla}\Phi\left(g_1\right)} - {{\nabla}\Phi\left(g_2\right)}\|_{*} \leq \beta \|g_1-g_2\|.$$ \[thm:oftpl\_regret\] Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is absolutely continuous w.r.t Lebesgue measure. Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|$, which is defined as $D= \sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|.$ Let and suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$, where ${C}$ is a constant that depends on the set $\mathcal{X}.$ Finally, suppose the sequence of loss functions $\{f_t\}_{t=1}^T$ are Holder smooth and satisfy $$\forall {{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in {\mathcal{X}}\quad \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_1)-{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_2)\|_* \leq L\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|^{\alpha},$$ for some constant $\alpha \in [0,1]$. Then the expected regret of Algorithm \[alg:oftpl\_cvx\] satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|{\nabla}_t-g_{t}\|_{*}^2\right]} - \sum_{t=1}^T\frac{\eta}{2{C}} {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]} \\ &\quad + LT\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}.\end{aligned}$$ where ${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty} = {\mathbb{E}\left[\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}|f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}$ denotes the prediction in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_cvx\], if guess $g_{t}=0$ was used. Here, ${\Psi_1}, {\Psi_2}$ denote the norm compatibility constants of $\|\cdot\|.$ Regret bounds that hold with high probability can be found in Appendix \[sec:hp\_bounds\]. The above Theorem shows that the regret of OFTPL only depends on $\|{\nabla}_t-g_{t}\|_{*}$, which quantifies the accuracy of our guess $g_t$. In contrast, the regret of FTPL depends on $\|{\nabla}_t\|_{*}$ [@hazan2016introduction]. This shows that for predictable sequences, with an appropriate choice of $g_t$, OFTPL can achieve better regret guarantees than FTPL. As we demonstrate in Section \[sec:games\], this helps us design faster algorithms for solving minimax games. Note that the above result is very general and holds for any absolutely continuous perturbation distribution. The key challenge in instantiating this result for any particular perturbation distribution is in showing the stability of predictions. Several past works have studied the stability of FTPL for various perturbation distributions such as uniform, exponential, Gumbel distributions [@kalai2005efficient; @hazan2016introduction; @hazan2020projection]. Consequently, the above result can be used to derive tight regret bounds for all these perturbation distributions. As one particular instantiation of Theorem \[thm:oftpl\_regret\], we consider the special case of $g_t = 0$ and derive regret bounds for FTPL, when the perturbation distribution is the uniform distribution over a ball centered at the origin. \[cor:ftpl\_cvx\_gaussian\] Suppose the perturbation distribution is equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|_2$. Then ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} = \eta$, and the predictions of OFTPL are $dD\eta^{-1}$-stable w.r.t $\|\cdot\|_2$. Suppose, the sequence of loss functions $\{f_t\}_{t=1}^T$ are $G$-Lipschitz and satisfy $\sup_{{{\ensuremath{\mathbf{x}}}}\in \mathcal{X}} \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}})\|_2 \leq G$. Moreover, suppose $f_t$ satisfies the Holder smooth condition in Theorem \[thm:oftpl\_regret\] w.r.t $\|\cdot\|_2$ norm. Then the expected regret of Algorithm \[alg:oftpl\_cvx\], with guess $g_t = 0$, satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \frac{dDG^2 T}{2\eta} + LT\left(\frac{D}{\sqrt{m}}\right)^{1+\alpha}.\end{aligned}$$ This recovers the regret bounds of FTPL for general convex loss functions, derived by [@hazan2020projection]. Online Nonconvex Learning {#sec:online_noncvx} ------------------------- **Input:** Perturbation Distribution ${P_{\text{PRTB}}},$ number of samples $m$, number of iterations $T$ Denote $f_0=0$ Let $g_t$ be the guess for $f_t$ Sample $\sigma_{t,j}\sim {P_{\text{PRTB}}}$ ${{\ensuremath{\mathbf{x}}}}_{t,j} \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}f_{0:t-1}({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}})$ Let $P_t $ be the empirical distribution over $\{{{\ensuremath{\mathbf{x}}}}_{t,1}, {{\ensuremath{\mathbf{x}}}}_{t,2} \dots {{\ensuremath{\mathbf{x}}}}_{t,m}\}$ Play ${{\ensuremath{\mathbf{x}}}}_t$, a random sample generated from $P_t$ Observe loss function $f_t$ We now study OFTPL in the nonconvex setting. In this setting, we assume the sequence of loss functions belong to some function class ${\mathcal{F}}$ containing real-valued measurable functions on ${\mathcal{X}}$. Some popular choices for ${\mathcal{F}}$ include the set of Lipschitz functions, the set of bounded functions. The OFTPL algorithm in this setting is described in Algorithm \[alg:oftpl\_noncvx\]. Similar to the convex case, we first sample random perturbation functions $\{\sigma_{t,j}\}_{j=1}^m$ from some distribution ${P_{\text{PRTB}}}$. Some examples of perturbation functions that have been considered in the past include $\sigma_{t,j}({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle},$ for some random vector $\bar{\sigma}_{t,j}$ sampled from exponential or uniform distributions [@gonen2018learning; @suggala2019online]. Another popular choice for $\sigma_{t,j}$ is the Gumbel process, which results in the continuous exponential weights algorithm [@maddison2014sampling]. Letting, $g_t$ be our guess of loss function $f_t$ at the beginning of round $t$, the learner first computes ${{\ensuremath{\mathbf{x}}}}_{t,j}$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\sum_{i = 1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}}).$ We assume access to an optimization oracle which computes a minimizer of this problem. We often refer to this oracle as the *perturbed best response* oracle. Let $P_t$ denote the empirical distribution of $\{{{\ensuremath{\mathbf{x}}}}_{t,j}\}_{j=1}^m$. The learner then plays an ${{\ensuremath{\mathbf{x}}}}_t$ which is sampled from $P_t$. Algorithm \[alg:oftpl\_noncvx\] describes this procedure. We note that for the online learning problem, $m=1$ suffices, as the expected loss suffered by the learner in each round is independent of $m$; that is ${\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)\right]} = {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_{t,1})\right]}$. However, the choice of $m$ affects the rate of convergence when Algorithm \[alg:oftpl\_noncvx\] is used for solving nonconvex nonconcave minimax games. Before we present the regret bounds, we introduce the *dual space* associated with ${\mathcal{F}}$. Let $\|\cdot\|_{{\mathcal{F}}}$ be a seminorm associated with ${\mathcal{F}}$. For example, when ${\mathcal{F}}$ is the set of Lipschitz functions, $\|\cdot\|_{{\mathcal{F}}}$ is the Lipschitz seminorm. Various choices of $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$ induce various distance metrics on ${\mathcal{P}}$, the set of all probability distributions on ${\mathcal{X}}$. We let ${\gamma_{{\mathcal{F}}}}$ denote the Integral Probability Metric (IPM) induced by $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$, which is defined as $${\gamma_{{\mathcal{F}}}}(P,Q) = \sup_{f\in{\mathcal{F}},\|f\|_{{\mathcal{F}}} \leq 1}\Big|{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]} - {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}})\right]}\Big|.$$ We often refer to $({\mathcal{P}}, {\gamma_{{\mathcal{F}}}})$ as the dual space of $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$. When ${\mathcal{F}}$ is the set of Lipschitz functions and when $\|\cdot\|_{{\mathcal{F}}}$ is the Lipschitz seminorm, ${\gamma_{{\mathcal{F}}}}$ is the Wasserstein distance. Table \[tab:ipm\] in Appendix \[sec:primal\_dual\_spaces\] presents examples of ${\gamma_{{\mathcal{F}}}}$ induced by some popular function spaces. Similar to the convex case, the regret bounds in the nonconvex setting depend on the stability of predictions of OFTPL. Suppose the perturbation function $\sigma({{\ensuremath{\mathbf{x}}}})$ is sampled from ${P_{\text{PRTB}}}$. For any $f\in{\mathcal{F}}$, define random variable ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}}).$ Let ${{\nabla}\Phi\left(f\right)}$ denote the distribution of ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$. The predictions of OFTPL are said to be $\beta$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$ if $$\forall f,g\in{\mathcal{F}}\quad {\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(f\right)}, {{\nabla}\Phi\left(g\right)}) \leq \beta \|f-g\|_{{\mathcal{F}}}.$$ \[thm:oftpl\_noncvx\_regret\] Suppose the sequence of loss functions $\{f_t\}_{t=1}^T$ belong to $({\mathcal{F}}, \|\cdot\|_{{\mathcal{F}}})$. Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is such that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one, for any $f\in{\mathcal{F}}$. Let ${\mathcal{P}}$ be the set of probability distributions over ${\mathcal{X}}$. Define the diameter of ${\mathcal{P}}$ as $D= \sup_{P_1,P_2\in{\mathcal{P}}} {\gamma_{{\mathcal{F}}}}(P_1,P_2).$ Let $\eta={\mathbb{E}\left[\|\sigma\|_{{\mathcal{F}}}\right]}$. Suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$, for some constant ${C}$ that depends on ${\mathcal{X}}$. Then the expected regret of Algorithm \[alg:oftpl\_noncvx\] satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t)-f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]} -\sum_{t=1}^T \frac{\eta}{2{C}}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]},\end{aligned}$$ where $P_t^{\infty} = {\mathbb{E}\left[P_t|g_t, f_{1:t-1}, P_{1:t-1}\right]}, $ $\Tilde{P}_{t}^{\infty} = {\mathbb{E}\left[\Tilde{P}_{t-1}|f_{1:t-1}, P_{1:t-1}\right]}$ and $\Tilde{P}_{t-1}$ is the empirical distribution computed in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_noncvx\], if guess $g_t = 0$ was used. We note that, unlike the convex case, there are no known analogs of Fenchel duality for infinite dimensional function spaces. As a result, more careful analysis is needed to obtain the above regret bounds. Our analysis mimics the arguments made in the convex case, albeit without explicitly relying on duality theory. As in the convex case, the key challenge in instantiating the above result for any particular perturbation distribution is in showing the stability of predictions. In a recent work, [@suggala2019online] consider linear perturbation functions $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle},$ for $\bar{\sigma}$ sampled from exponential distribution, and show stability of FTPL. We now instantiate the above Theorem for this setting. \[cor:ftpl\_noncvx\_exp\] Consider the setting of Theorem \[thm:oftpl\_noncvx\_regret\]. Let ${\mathcal{F}}$ be the set of Lipschitz functions and $\|\cdot\|_{{\mathcal{F}}}$ be the Lipschitz seminorm, which is defined as $\|f\|_{{\mathcal{F}}}=\sup_{{{\ensuremath{\mathbf{x}}}}\neq {{\ensuremath{\mathbf{y}}}}\text{ in }{\mathcal{X}}} |f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|_1$. Suppose the perturbation function is such that $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is a random vector whose entries are sampled independently from $\text{Exp}(\eta)$. Then ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta\log{d}$, and the predictions of OFTPL are ${O\left({d^2D\eta^{-1}}\right)}$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$. Moreover, the expected regret of Algorithm \[alg:oftpl\_noncvx\] is upper bounded by We note that the above regret bounds are tighter than the regret bounds of [@suggala2019online], where the authors show that the regret of OFTPL is bounded by ${O\left({\eta D\log{d} + \sum_{t=1}^T \frac{d^2D }{\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]}}\right)}$. These tigher bounds help us design faster algorithms for solving minimax games in the nonconvex setting. Minimax Games {#sec:games} ============= We now consider the problem of solving minimax games of the following form $$\label{eqn:minimax_game} \min_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \max_{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}} f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}).$$ Nash equilibria of such games can be computed by playing two online learning algorithms against each other [@cesa2006prediction; @hazan2016introduction]. In this work, we study the algorithm where both the players employ OFTPL to decide their actions in each round. For convex-concave games, both the players use the OFTPL algorithm described in Algorithm \[alg:oftpl\_cvx\] (see Algorithm \[alg:oftpl\_cvx\_games\] in Appendix \[sec:cvx-games\]). The following theorem derives the rate of convergence of this algorithm to a Nash equilibirum (NE). \[thm:oftpl\_cvx\_smooth\_games\_uniform\] Consider the minimax game in Equation . Suppose both the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$, with diameter Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is smooth w.r.t $\|\cdot\|_2$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Suppose Algorithm \[alg:oftpl\_cvx\_games\] is used to solve the minimax game. Suppose the perturbation distributions used by both the players are the same and equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_cvx\_games\] is run with $\eta = 6dD(L+1), m = T$, then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}= {O\left({\frac{dD^2(L+1)}{T}}\right)}.\end{aligned}$$ Rates of convergence which hold with high probability can be found in Appendix \[sec:hp\_bounds\]. We note that Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] can be extended to more general noise distributions and settings where gradients of $f$ are Holder smooth w.r.t non-Euclidean norms, and ${\mathcal{X}},{\mathcal{Y}}$ lie in spaces of different dimensions (see Theorem \[thm:oftpl\_cvx\_smooth\_games\] in Appendix). The above result shows that for smooth convex-concave games, Algorithm \[alg:oftpl\_cvx\_games\] converges to a NE at ${O\left({T^{-1}}\right)}$ rate using $T^2$ calls to the linear optimization oracle. Moreover, the algorithm runs in ${O\left({T}\right)}$ iterations, with each iteration making ${O\left({T}\right)}$ parallel calls to the optimization oracle. We believe the dimension dependence in the rates can be removed by appropriately choosing the perturbation distributions based on domains ${\mathcal{X}}, {\mathcal{Y}}$ (see Appendix \[sec:pert\_dist\_choice\]). We now consider the more general nonconvex-nonconcave games. In this case, both the players use the nonconvex OFTPL algorithm described in Algorithm \[alg:oftpl\_noncvx\] to choose their actions. Instead of generating a single sample from the empirical distribution $P_t$ computed in $t^{th}$ iteration of Algorithm \[alg:oftpl\_noncvx\], the players now play the entire distribution $P_t$ (see Algorithm \[alg:oftpl\_noncvx\_games\] in Appendix \[sec:ncvx-games\]). Letting $\{P_t\}_{t=1}^T, \{Q_t\}_{t=1}^T$, be the sequence of iterates generated by the ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players, the following theorem shows that $\left(\frac{1}{T}\sum_{t=1}^TP_t,\frac{1}{T}\sum_{t=1}^TQ_t\right)$ converges to a NE. \[thm:oftpl\_noncvx\_smooth\_games\_exp\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$ with diameter $D = \max\{\sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1, \sup_{{{\ensuremath{\mathbf{y}}}}_1,{{\ensuremath{\mathbf{y}}}}_2\in{\mathcal{Y}}} \|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|_1\}$. Suppose $f$ is Lipschitz w.r.t $\|\cdot\|_1$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} + \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_1 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_1.\end{aligned}$$ Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\_games\] to solve the game with linear perturbation functions $\sigma({{\ensuremath{\mathbf{z}}}})={\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{z}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is such that each of its entries is sampled independently from $\text{Exp}(\eta)$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_noncvx\_games\] is run with $\eta = 10d^2D(L+1), m = T$, then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]}& = {O\left({\frac{d^2D^2(L+1)\log{d}}{T}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2L, \frac{d^2G^2\log{T}}{LT}\right\rbrace}\right)}.\end{aligned}$$ More general versions of the Theorem, which consider other function classes and general perturbation distributions, can be found in Appendix \[sec:ncvx-games\]. The above result shows that Algorithm \[alg:oftpl\_noncvx\_games\] converges to a NE at ${\Tilde{O}\left({T^{-1}}\right)}$ rate using $T^2$ calls to the perturbed best response oracle. This matches the rates of convergence of FTPL [@suggala2019online]. However, the key advantage of our algorithm is that it is highly parallelizable and runs in ${O\left({T}\right)}$ iterations, in contrast to FTPL, which runs in ${O\left({T^2}\right)}$ iterations. Conclusion {#sec:conclusion} ========== We studied an optimistic variant of FTPL which achieves better regret guarantees when the sequence of loss functions is predictable. As one specific application of our algorithm, we considered the problem of solving minimax games. For solving convex-concave games, our algorithm requires access to a linear optimization oracle and for nonconvex-nonconcave games our algorithm requires access to a more powerful perturbed best response oracle. In both these settings, our algorithm achieves ${O\left({T^{-1/2}}\right)}$ convergence rates using $T$ calls to the oracles. Moreover, our algorithm runs in ${O\left({T^{1/2}}\right)}$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle. We believe our improved algorithms for solving minimax games are useful in a number of modern machine learning applications such as training of GANs, adversarial training, which involve solving nonconvex-nonconcave minimax games and often deal with huge datasets. Dual view of Perturbations as Regularization ============================================ Proof of Theorem \[prop:ftpl\_ftrl\_connection\] ------------------------------------------------ We first define a convex function $\Psi:\mathbb{R}^d \to \mathbb{R}$ as $$\Psi(f)={\mathbb{E}_{\sigma}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]} = {\mathbb{E}_{\sigma}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]},$$ where perturbation $\sigma$ follows probability distribution ${P_{\text{PRTB}}}$ which is absolutely continuous w.r.t the Lebesgue measure. For our choice of ${P_{\text{PRTB}}}$, we now show that $\Psi$ is differentiable. Consider the function $\psi(g) = \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle}$. Since $\psi(g)$ is a proper convex function, we know that it is differentiable almost everywhere, except on a set of Lebesgue measure $0$ [see Theorem 25.5 of @rockafellar1970convex]. Moreover, it is easy to verify that ${\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle} \in \partial \psi(g).$ These two observations, together with the fact that ${P_{\text{PRTB}}}$ is absolutely continuous, show that the $\sup$ expression inside the expectation of $\Psi$ has a unique maximizer with probability one. Since the sup expression inside the expectation has a unique maximizer with probability $1$, we can swap the expectation and gradient to obtain [see Proposition 2.2 of @bertsekas1973stochastic] $$\label{eqn:phi_gradient} {\nabla}\Psi(f) = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}.$$ Note that ${\nabla}\Psi$ is related to the prediction of deterministic version of FTPL. Specifically, ${\nabla}\Psi(-{\nabla}_{1:t-1})$ is the prediction of deterministic FTPL in the $t^{th}$ iteration. We now show that ${\nabla}\Psi(f) = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle -f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, for some convex function $R$. Since all differentiable functions are closed, $\Psi(f)$ is a proper, closed and differentiable convex function over $\mathbb{R}^d$. Let $R({{\ensuremath{\mathbf{x}}}})$ denote the Fenchel conjugate of $\Psi(f)$ $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f\in {\text{dom}(\Phi)}} {\left\langle {{\ensuremath{\mathbf{x}}}}, f \right\rangle} - \Psi(f),$$ where ${\text{dom}(\Psi)}$ denotes the domain of $\Psi$. Following Theorem \[thm:fenchel\_prop1\] (see Appendix \[sec:fenchel\_conjugate\]), $\Psi(f)$ is the Fenchel conjugate of $R({{\ensuremath{\mathbf{x}}}})$ $$\begin{aligned} \Psi(f) = \sup_{{{\ensuremath{\mathbf{x}}}}\in {\text{dom}(R)}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).\end{aligned}$$ Furthermore, from Theorem \[thm:fenchel\_prop3\] we have $${\nabla}\Psi(f) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\text{dom}(R)}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).$$ We now show that the domain of $R$ is a subset of ${\mathcal{X}}$. This, together with the previous two equations, would then immediately imply $$\begin{aligned} \Psi(f) = \sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}),\\ \label{eqn:ftpl_ftrl_connection_proof} {\nabla}\Psi(f) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).\end{aligned}$$ From Theorem \[thm:fenchel\_prop2\], we know that the domain of $R$ satisfies $$\text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\text{dom}(R)},$$ where $\text{ri}(A)$ denotes the relative interior of a set $A$. Moreover, from the definition of ${\nabla}\Psi(f)$ in Equation , we have $\text{range} {\nabla}\Psi \subseteq {\mathcal{X}}$. Combining these two properties, we can show that one of the following statements is true $$\begin{aligned} \text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\mathcal{X}}\subseteq {\text{dom}(R)},\\ \text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\text{dom}(R)} \subseteq {\mathcal{X}}.\end{aligned}$$ Suppose the first statement is true. Since ${\mathcal{X}}$ is a compact set, it is easy to see that If the second statement is true, then ${\text{dom}(R)} \subseteq {\mathcal{X}}$. Together, these two statements imply ${\text{dom}(R)} \subseteq {\mathcal{X}}$. #### Connecting back to FTPL. We now connect the above results to FTPL. From Equation , we know that the prediction at iteration $t$ of deterministic FTPL is equal to ${\nabla}\Psi(-{\nabla}_{1:t-1}).$ From Equation , ${\nabla}\Psi(-{\nabla}_{1:t-1})$ is defined as $${{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1}) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle -{\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).$$ This shows that $${{\ensuremath{\mathbf{x}}}}_t = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ So the prediction of FTPL can also be obtained using FTRL for some convex regularizer $R({{\ensuremath{\mathbf{x}}}})$. Finally, to show that $-{\nabla}_{1:t-1} \in \partial R({{\ensuremath{\mathbf{x}}}}_t), {{\ensuremath{\mathbf{x}}}}_t = \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$ we rely on Theorem \[thm:fenchel\_prop4\]. Since ${{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1})$, from Theorem \[thm:fenchel\_prop4\], we have $$-{\nabla}_{1:t-1} \in \partial R({{\ensuremath{\mathbf{x}}}}_t),\quad {{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1})= \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$$ where $\partial R^{-1}$ is the inverse of $\partial R$ in the sense of multivalued mappings. Note that, even though $\partial R$ can be a multivalued mapping, its inverse $\partial R^{-1} = {\nabla}\Psi$ is a singlevalued mapping (this follows form differentiability of $\Psi$). This finishes the proof of the Theorem. Online Convex Learning ====================== Proof of Theorem \[thm:oftpl\_regret\] -------------------------------------- Before presenting the proof of the Theorem, we introduce some notation. ### Notation We define functions $\Phi:\mathbb{R}^d\to \mathbb{R}$, $R:\mathbb{R}^d\to\mathbb{R}$ as follows $$\begin{aligned} &\Phi(f) = {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle f-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]},\quad R({{\ensuremath{\mathbf{x}}}}) = \sup_{f\in \mathbb{R}^d} {\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + \Phi(-f).\end{aligned}$$ Note that $\Phi$ is related to the function $\Psi$ defined in the proof of Proposition \[prop:ftpl\_ftrl\_connection\]. To be precise, $\Psi(f) = -\Phi(-f)$. Moreover, $R({{\ensuremath{\mathbf{x}}}})$ is the Fenchel conjugate of $\Psi$. For our choice of perturbation distribution, $\Psi$ is differentiable (see proof of Proposition \[prop:ftpl\_ftrl\_connection\]). This implies $\Phi$ is also differentiable with gradient ${{\nabla}\Phi}$ defined as $$\begin{aligned} {{\nabla}\Phi\left(f\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle f-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}.\end{aligned}$$ Note that ${\nabla}\Phi$ is the prediction of deterministic version of FTPL. In Proposition \[prop:ftpl\_ftrl\_connection\] we showed that $${{\nabla}\Phi\left(f\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ ### Main Argument Since ${{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the prediction of deterministic version of FTPL, following FTPL-FTRL duality proved in Proposition \[prop:ftpl\_ftrl\_connection\], ${{\ensuremath{\mathbf{x}}}}_t^{\infty}$ can equivalently be written as $${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {{\nabla}\Phi\left({\nabla}_{1:t-1} + g_t\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle {\nabla}_{1:t-1} + g_t, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ Similarly, $\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}$ can be written as $$\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}={{\nabla}\Phi\left({\nabla}_{1:t}\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle {\nabla}_{1:t}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ We use the notation ${\nabla}_{1:0}=0$. So $\Tilde{{{\ensuremath{\mathbf{x}}}}}_0^{\infty},{{\ensuremath{\mathbf{x}}}}_1^{\infty}$ are equal to ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} R({{\ensuremath{\mathbf{x}}}}).$ From the first order optimality conditions, we have $$-{\nabla}_{1:t-1} - g_t \in \partial R\left({{\ensuremath{\mathbf{x}}}}_t^{\infty}\right),\quad -{\nabla}_{1:t} \in \partial R\left(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\right).$$ Define functions $B(\cdot,{{\ensuremath{\mathbf{x}}}}_t^{\infty}), B(\cdot,\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})$ for any $t\in[T]$ as $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) &= R({{\ensuremath{\mathbf{x}}}}) - R({{\ensuremath{\mathbf{x}}}}_t^{\infty}) + {\left\langle {\nabla}_{1:t-1} + g_t, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_t^{\infty} \right\rangle},\\ B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) &= R({{\ensuremath{\mathbf{x}}}}) - R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) + {\left\langle {\nabla}_{1:t}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty} \right\rangle}.\end{aligned}$$ From the stability of predictions of OFTPL we know that: $ \|{{\nabla}\Phi\left(g_1\right)} - {{\nabla}\Phi\left(g_2\right)}\| \leq C\eta^{-1}\|g_1-g_2\|_{*}. $ Following our connection between $\Psi,\Phi$, this implies $ \|{\nabla}\Psi(g_1) - {\nabla}\Psi(g_2)\| \leq C\eta^{-1}\|g_1-g_2\|_{*}. $ This implies the following smoothness condition on $\Psi$ [see Lemma 15 of @shalev2007thesis] $$\Psi(g_2) \leq \Psi(g_1) + {\left\langle {\nabla}\Psi(g_1), g_2-g_1 \right\rangle} + \frac{C\eta^{-1}}{2}\|g_1-g_2\|_{*}^2.$$ Since $\Psi$ is $C\eta^{-1}$-smooth w.r.t $\|\cdot\|_{*}$, following duality between strong convexity and strong smoothness properties (see Theorem \[thm:fenchel\_strong\_convex\_weak\]), we can infer that $R$ is ${C}^{-1}\eta$- strongly convex w.r.t $\|\cdot\|$ norm and satisfies $$B({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) \geq \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2, \quad B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) \geq \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|^2.$$ We now go ahead and bound the regret of the learner. For any ${{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}$, we have $$\begin{aligned} f_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) \stackrel{(a)}{\leq} {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle} & = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty} - {{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle}\\ &= {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, {\nabla}_t-g_{t} \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} \\ &\quad +{\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle}\\ & \leq {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} \\ &\quad +{\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle},\end{aligned}$$ where $(a)$ follows from convexity of $f$. Next, a simple calculation shows that $$\begin{aligned} {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} &= B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\\ {\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle} &= B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})-B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}).\end{aligned}$$ Substituting this in the previous inequality gives us $$\begin{aligned} f_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) & \leq {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ &\quad +B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})-B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ & = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ & \stackrel{(a)}{\leq}{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) - \frac{\eta\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2}{2{C}}-\frac{ \eta\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2}{2{C}},\end{aligned}$$ where $(a)$ follows from strongly convexity of $R$. Summing over $t=1,\dots T$, gives us $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \underbrace{B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})}_{S_1} \\ &\quad + \sum_{t=1}^T \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left(\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2 + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right).\end{aligned}$$ #### Bounding $S_1$. We now bound $B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})$. From the definition of $B$, we have $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} -R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) + {\left\langle {\nabla}_{1:0}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle}.\end{aligned}$$ Note that ${\nabla}_{1:0} = 0.$ This gives us $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} -R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}).\end{aligned}$$ We now use duality to convert the RHS of the above equation, which is currently in terms of $R$, into a quantity which depends on $\Phi$. From Proposition \[prop:ftpl\_ftrl\_connection\] we have $$\Phi(g) = -\Psi(-g)=\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ Since $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}$ is the minimizer of ${\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, we have $\Phi({\nabla}_{1:T}) = {\left\langle {\nabla}_{1:T}, \Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} + R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})$. Similarly, $\Phi(0) = R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_0^{\infty}).$ Substituting these in the previous equation gives us $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= \Phi({\nabla}_{1:T}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} -\Phi(0)\\ &={\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle {\nabla}_{1:T}-\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]} - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle -\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[\left\langle {\nabla}_{1:T}-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}- {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle -\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]}\\ &= {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]} - {\mathbb{E}_{\sigma}\left[\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}\\ &\leq D{\mathbb{E}_{\sigma}\left[\|\sigma\|_{*}\right]} = \eta D\end{aligned}$$ #### Bounding Regret. Substituting this in our regret bound and taking expectation on both sides gives us $$\begin{aligned} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \sum_{t=1}^T{\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}\right]} + \eta D + \sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left({\mathbb{E}\left[\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2\right]} + {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right)\\ &\leq \sum_{t=1}^T{\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}\right]} + \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|{\nabla}_t-g_{t}\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\\\end{aligned}$$ To finish the proof, we make use of the Holder’s smoothness assumption on $f_t$ to bound the first term in the RHS above. From Holder’s smoothness assumption, we have $${\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t - {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} \leq L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}.$$ Using this, we get $$\begin{aligned} {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}|g_t, {{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]} &\leq {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} + L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(a)}{=} L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(b)}{\leq} {\Psi_1}^{1+\alpha}L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(c)}{\leq} {\Psi_1}^{1+\alpha}L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^2|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}^{(1+\alpha)/2}\\ & \stackrel{(d)}{\leq} L\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha},\end{aligned}$$ where $(a)$ follows from the fact that ${\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]} = 0$, $(b)$ follows from the definition of norm compatibility constant ${\Psi_1}$, $(c)$ follows from Holders inequality and $(d)$ uses the fact that conditioned on $\{g_t, {{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\}$, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ i.i.d bounded mean $0$ random variables, the variance of which scales as $O(D^2/m)$. Substituting this in the above regret bound gives us the required result. Proof of Corollary \[cor:ftpl\_cvx\_gaussian\] ---------------------------------------------- We first bound ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]}$. Relying on spherical symmetry of the perturbation distribution and the fact that the density of ${P_{\text{PRTB}}}$ on the spherical shell of radius $r$ is proportional to $r^{d-1}$, we get $$\begin{aligned} {\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} = \frac{\int_{r=0}^{(1+d^{-1})\eta} r\times r^{d-1} dr}{\int_{r=0}^{(1+d^{-1})\eta} r^{d-1} dr} = \eta.\end{aligned}$$ We now bound the stability of predictions of OFTPL. Our technique for bounding the stability uses similar arguments as  @hazan2020projection (see Lemma 4.2 of [@hazan2020projection]). Recall, to bound stability, we need to show that $\Phi(g) = {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\left\langle g-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}$ is smooth. Let $\phi_0(g) = \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}$, where ${{\ensuremath{\mathbf{x}}}}_{00}$ is an arbitrary point in ${\mathcal{X}}$. We can rewrite $\Phi(g)$ as $$\Phi(g) = {\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]} + {\left\langle g, {{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}.$$ Since the second term in the RHS above is linear in $g$, any upper bound on the smoothness of ${\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$ is also a bound on the smoothness of $\Phi(g)$. So we focus on bounding the smoothness of ${\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$. First note that $\phi_0(g)$ is $D$ Lipschitz and satisfies the following for any $g_1,g_2\in\mathbb{R}^d$ $$\begin{aligned} \phi_0(g_1) - \phi_0(g_2) & = \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle -g_2, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}-\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle -g_1, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}\\ & \leq \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g_1-g_2, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}\\ &\leq D\|g_1-g_2\|_2.\end{aligned}$$ Letting $\Phi_0(g) = {\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$, Lemma 4.2 of @hazan2020projection shows that $\Phi_0(g)$ is smooth and satisfies $$\|{\nabla}\Phi_0(g_1) - {\nabla}\Phi_0(g_2)\|_2 \leq dD\eta^{-1}\|g_1-g_2\|_2.$$ This shows that the predictions of OFTPL are $dD\eta^{-1}$ stable. The rest of the proof involves substituting $C=dD$ in the regret bound of Theorem \[thm:oftpl\_regret\] and setting $g_t = 0$ and using the fact that $\|{\nabla}_t\|_2\leq G$. Online Nonconvex Learning {#online-nonconvex-learning} ========================= Proof of Theorem \[thm:oftpl\_noncvx\_regret\] ---------------------------------------------- Before we present the proof of the Theorem, we introduce some notation and present some useful intermediate results. We note that unlike the convex case, there are no know Fenchel duality theorems for infinite dimensional setting. So more careful arguments are need to obtain tight regret bounds. Our proof mimics the proof of Theorem \[thm:oftpl\_regret\]. ### Notation Let ${\mathcal{P}}$ be the set of all probability measures on ${\mathcal{X}}$. We define functions $\Phi:{\mathcal{F}}\to \mathbb{R}$, $R:{\mathcal{P}}\to\mathbb{R}$ as follows $$\begin{aligned} &\Phi(f) = {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}\right]},\\ &R(P) = \sup_{f\in {\mathcal{F}}} -{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]} + \Phi(f).\end{aligned}$$ Also, note that the function ${{\nabla}\Phi}:{\mathcal{F}}\to{\mathcal{P}}$ defined in Section \[sec:online\_noncvx\] can be written as $$\begin{aligned} {{\nabla}\Phi\left(f\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}\right]}.\end{aligned}$$ Note that, ${{\nabla}\Phi\left(f\right)}$ is well defined because from our assumption on the perturbation distribution, the minimization problem inside the expectation has a unique minimizer with probability one. To simplify the notation, in the sequel, we use the shorthand notation ${\left\langle P, f \right\rangle}$ to denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]}$, for any $P\in{\mathcal{P}}$ and $f\in {\mathcal{F}}$. Similarly, for any $P_1,P_2\in {\mathcal{P}}$ and $f\in {\mathcal{F}}$, we use the notation ${\left\langle P_1-P_2, f \right\rangle}$ to denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_1}\left[f({{\ensuremath{\mathbf{x}}}})\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_2}\left[f({{\ensuremath{\mathbf{x}}}})\right]}$. ### Intermediate Results \[lem:noncvx\_gradient\] For any $g\in {\mathcal{F}}$, $R({{\nabla}\Phi\left(g\right)}) = -{\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle} + \Phi(g)$. Define $P_{g,\sigma}$ as $$P_{g,\sigma}={\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[g({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}.$$ Note that ${{\nabla}\Phi\left(g\right)} = {\mathbb{E}_{\sigma}\left[P_{g,\sigma}\right]}$. For any $g,h \in {\mathcal{F}}$, we have $$\begin{aligned} \Phi(h) &= {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, h-\sigma \right\rangle}\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, h-\sigma \right\rangle}\right]}\\ &= {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, g-\sigma \right\rangle}\right]} + {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, h-g \right\rangle}\right]}\\ & = \Phi(g) + {\left\langle {{\nabla}\Phi\left(g\right)}, h-g \right\rangle}.\end{aligned}$$ This shows that for any $g,h \in {\mathcal{F}}$ $$\label{eqn:noncvx_phi_convex} \Phi(h) - {\left\langle {{\nabla}\Phi\left(g\right)}, h \right\rangle} \leq \Phi(g) - {\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle}.$$ Taking supremum over $h$ of the LHS quantity gives us $$R({{\nabla}\Phi\left(g\right)})=\sup_{h\in {\mathcal{F}}}\Phi(h) - {\left\langle {{\nabla}\Phi\left(g\right)}, h \right\rangle} = \Phi(g) - {\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle}.$$ \[lem:noncvx\_phi\_smooth\] The function $-\Phi$ is convex and strongly smooth and satisfies the following inequality for any $g_1,g_2\in {\mathcal{F}}$ $$-\Phi(g_2)\leq -\Phi(g_1) - {\left\langle {{\nabla}\Phi\left(g_1\right)}, g_2-g_1 \right\rangle} + \frac{C}{2\eta}\|g_2-g_1\|_{{\mathcal{F}}}^2.$$ Let $g_1,g_2\in{\mathcal{F}}$ and $\alpha \in [0,1]$. Then $$\begin{aligned} \Phi(\alpha g_1 + (1-\alpha)g_2) &= {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, \alpha g_1 + (1-\alpha)g_2-\sigma \right\rangle}\right]}\\ & \geq \alpha{\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, g_1-\sigma \right\rangle}\right]} + (1-\alpha){\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, g_2-\sigma \right\rangle}\right]}\\ & = \alpha\Phi(g_1) + (1-\alpha)\Phi(g_2).\end{aligned}$$ This shows that $-\Phi$ is convex. To show smoothness, we rely on the following stability property $$\forall g_1,g_2\in{\mathcal{F}}\quad {\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(g_1\right)}, {{\nabla}\Phi\left(g_2\right)}) \leq \frac{C}{\eta} \|g_1-g_2\|_{{\mathcal{F}}}.$$ Let $T$ be an arbitrary positive integer and for $t\in \{0,1,\dots T\}$, define $\alpha_t = t/T$. Let $h = g_2-g_1$. We have $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &= \Phi(g_1 + \alpha_0h)-\Phi(g_1 + \alpha_Th)\\ &=\sum_{t=0}^{T-1}\left(\Phi(g_1 + \alpha_{t}h)-\Phi(g_1 + \alpha_{t+1}h)\right)\end{aligned}$$ Since $-\Phi$ is convex and satisfies Equation , we have $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &=\sum_{t=0}^{T-1}\left(\Phi(g_1 + \alpha_{t}h)-\Phi(g_1 + \alpha_{t+1}h)\right)\\ &\leq -\sum_{t=0}^{T-1} \frac{1}{T}{\left\langle {{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle}\end{aligned}$$ Using stability, we get $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &\leq -\sum_{t=0}^{T-1} \frac{1}{T}{\left\langle {{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle}\\ & = \sum_{t=0}^{T-1} \frac{1}{T}\left({\left\langle {{\nabla}\Phi\left(g_1\right)}-{{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle} - {\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle}\right)\\ & \stackrel{(a)}{\leq} -{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{1}{T}{\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(g_1\right)},{{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)})\|h\|_{{\mathcal{F}}} \\ &\stackrel{(b)}{\leq} -{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{C}{T\eta}\|\alpha_{t+1}h\|_{{\mathcal{F}}}\|h\|_{{\mathcal{F}}}\\ &=-{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{C\alpha_{t+1}}{T\eta}\|h\|^2_{{\mathcal{F}}}\\ &=-{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} +\frac{C}{\eta}\frac{T+1}{2T}\|h\|^2_{{\mathcal{F}}},\end{aligned}$$ where $(a)$ follows from the definition of ${\gamma_{{\mathcal{F}}}}$ and $(b)$ follows from the stability assumption. Taking $T\to\infty$, we get $$-\Phi(g_2)\leq -\Phi(g_1) - {\left\langle {{\nabla}\Phi\left(g_1\right)}, g_2-g_1 \right\rangle} + \frac{C}{2\eta}\|g_2-g_1\|_{{\mathcal{F}}}^2.$$ \[lem:noncvx\_reg\_strong\_cvx\] For any $P\in {\mathcal{P}}$ and $g \in {\mathcal{F}}$, $R$ satisfies the following inequality $$R(P)\geq R({{\nabla}\Phi\left(g\right)}) +{\left\langle {{\nabla}\Phi\left(g\right)}-P, g \right\rangle} + \frac{\eta}{2{C}}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(g\right)})^2.$$ From Lemma \[lem:noncvx\_phi\_smooth\] we know that the following holds for any $g,h\in{\mathcal{F}}$ $$\Phi(g)\geq \underbrace{\Phi(h) + {\left\langle {{\nabla}\Phi\left(h\right)}, g-h \right\rangle} - \frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2}_{\Phi_{\text{lb}, h}(g)}.$$ Define $R_{\text{lb},h}(P)$ as $$R_{\text{lb}}(P) =\sup_{g\in{\mathcal{F}}} -{\left\langle P, g \right\rangle} + \Phi_{\text{lb},h}(g).$$ Since $\Phi(g) \geq \Phi_{\text{lb},h}(g)$ for all $g\in{\mathcal{F}}$, $R(P) \geq R_{\text{lb},h}(P)$ for all $P$. We now derive an expression for $R_{\text{lb},h}(P)$. Note that from Lemma \[lem:noncvx\_gradient\] we have $R({{\nabla}\Phi\left(h\right)}) = -{\left\langle {{\nabla}\Phi\left(h\right)}, h \right\rangle} + \Phi(h)$. Using this, we get $$\begin{aligned} R_{\text{lb},h}(P) &= \sup_{g\in{\mathcal{F}}} -{\left\langle P, g \right\rangle} + \Phi_{\text{lb},h}(g)\\ & \stackrel{(a)}{=} \sup_{g\in{\mathcal{F}}} \left(-{\left\langle P, g \right\rangle} + \Phi(h) + {\left\langle {{\nabla}\Phi\left(h\right)}, g-h \right\rangle} - \frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2\right)\\ &\stackrel{(b)}{=}R({{\nabla}\Phi\left(h\right)}) + \sup_{g\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g \right\rangle}-\frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2\right),\end{aligned}$$ where $(a)$ follows from the definition of $\Phi_{\text{lb},h}(g)$ and $(b)$ follows from Lemma \[lem:noncvx\_gradient\]. We now do a change of variables in the supremum of the above expression. Substituting $g' = g - h$, we get $$\begin{aligned} R_{\text{lb},h}(P) & = R({{\nabla}\Phi\left(h\right)}) + {\left\langle {{\nabla}\Phi\left(h\right)}-P, h \right\rangle} + \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right).\end{aligned}$$ We now show that $$\sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ To this end, we choose a $g'' \in {\mathcal{F}}$ such that $$\label{eqn:noncvx_sc_g} \|g''\|_{{\mathcal{F}}} = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}),\quad {\left\langle {{\nabla}\Phi\left(h\right)}-P, g'' \right\rangle} = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ If such a $g''$ can be found, we have $$\begin{aligned} \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) &\geq {\left\langle {{\nabla}\Phi\left(h\right)}-P, g'' \right\rangle}-\frac{C}{2\eta}\|g''\|_{{\mathcal{F}}}^2\\ & = \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ This would then imply the main claim of the Lemma. $$\begin{aligned} R(P) \geq R_{\text{lb},h}(P) \geq R({{\nabla}\Phi\left(h\right)}) + {\left\langle {{\nabla}\Phi\left(h\right)}-P, h \right\rangle} + \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ #### Finding $g''$. We now construct a $g''$ which satisfies Equation . From the definition of ${\gamma_{{\mathcal{F}}}}$ we know that $${\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}) = \sup_{\|g'\|_{{\mathcal{F}}}\leq 1} |{\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}|$$ Suppose the supremum is achieved at $g^*$. Define $g''$ as $\frac{\eta s}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})g^*$, where $s = \text{sign}({\left\langle {{\nabla}\Phi\left(h\right)}-P, g^* \right\rangle})$. It can be easily verified that $g''$ satifies Equation . If the supremum is never achieved, the same argument as above can still be made using a sequence of functions $\{g_{n}\}_{n=1}^{\infty}$ such that $$\|g_n\|_{{\mathcal{F}}}\leq 1,\quad \lim_{n\to\infty} |{\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle}| = {\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}).$$ Define $g''_n$ as $\frac{\eta s_n}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})g_n$, where $s_n = \text{sign}({\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle})$. Since $\lim_{n\to\infty}\|g_n\|_{{\mathcal{F}}} = 1$, we have $\lim_{n \to \infty}\|g''_{n}\|_{{\mathcal{F}}} = \frac{\eta}{C} {\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})$. Moreover, $$\lim_{n\to\infty} {\left\langle {{\nabla}\Phi\left(h\right)}-P, g''_n \right\rangle} = \lim_{n\to\infty} \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}) \Big|{\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle}\Big| = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ This shows that $$\begin{aligned} \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) &\geq \lim_{n\to\infty}{\left\langle {{\nabla}\Phi\left(h\right)}-P, g''_n \right\rangle}-\frac{C}{2\eta}\|g''_n\|_{{\mathcal{F}}}^2\\ & = \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ This finishes the proof of the Lemma. ### Main Argument We are now ready to prove Theorem \[thm:oftpl\_noncvx\_regret\]. Our proof relies on Lemma \[lem:noncvx\_reg\_strong\_cvx\] and uses similar arguments as used in the proof of Theorem \[thm:oftpl\_regret\]. We first rewrite $P_t, \Tilde{P}_t$ as $$\begin{aligned} P_t &= \frac{1}{m}\sum_{j=1}^m{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[\sum_{i = 1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}})\right]},\\ \Tilde{P}_t &= \frac{1}{m}\sum_{j=1}^m{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[\sum_{i = 1}^{t}f_i({{\ensuremath{\mathbf{x}}}})-\sigma'_{t,j}({{\ensuremath{\mathbf{x}}}})\right]}.\end{aligned}$$ Note that $$\begin{aligned} P_t^{\infty} &= {\mathbb{E}\left[P_t|g_t,f_{1:t-1},P_{1:t-1}\right]} = {{\nabla}\Phi\left(f_{1:t-1} + g_t\right)},\\ \Tilde{P}_t^{\infty} &= {\mathbb{E}\left[\Tilde{P}_t|f_{1:t-1},P_{1:t-1}\right]} = {{\nabla}\Phi\left(f_{1:t}\right)},\end{aligned}$$ with $P_1^{\infty} = \Tilde{P}_0^{\infty} ={{\nabla}\Phi\left(0\right)}$. Define functions $B(\cdot,P_t^{\infty}), B(\cdot, \Tilde{P}_t^{\infty})$ as $$\begin{aligned} B(P,P_t^{\infty}) &= R(P) - R(P_t^{\infty}) + {\left\langle P-P_t^{\infty}, f_{1:t-1}+g_t \right\rangle},\\ B(P,\Tilde{P}_t^{\infty}) &= R(P) - R(\Tilde{P}_t^{\infty}) + {\left\langle P-\Tilde{P}_t^{\infty}, f_{1:t} \right\rangle}.\end{aligned}$$ From Lemma \[lem:noncvx\_reg\_strong\_cvx\], we have $$B(P,P_t^{\infty}) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P, P_t^{\infty})^2,\quad B(P,\Tilde{P}_t^{\infty}) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P, \Tilde{P}_t^{\infty})^2.$$ For any $P\in {\mathcal{P}}$, we have $$\begin{aligned} {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &= {\mathbb{E}\left[f_t(P_t)-f_t(P)\right]} \\ & = {\mathbb{E}\left[{\left\langle P_t-P, f_t \right\rangle}\right]}\\ &={\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]} + {\mathbb{E}\left[{\left\langle P_t^{\infty}-P, f_t \right\rangle}\right]}\\ &={\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]} + {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, f_t-g_t \right\rangle}\right]} \\ &\quad+ {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_t \right\rangle}\right]}+{\mathbb{E}\left[{\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle}\right]}\\ &\stackrel{(a)}{\leq}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]}+ {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_t \right\rangle}\right]} \\ &\quad+{\mathbb{E}\left[{\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle}\right]},\end{aligned}$$ where $(a)$ follows from the fact that ${\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}|g_t, f_{1:t-1}, P_{1:t-1}\right]} = 0$ and as a result ${\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]}=0$. Next, a simple calculation shows that $$\begin{aligned} {\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_{t} \right\rangle} &= B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\\ {\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle} &= B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})-B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}).\end{aligned}$$ Substituting this in the previous regret bound gives us $$\begin{aligned} {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} & \leq {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} + {\mathbb{E}\left[B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ &\quad +{\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})-B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ & = {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} \\ &\quad + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ & \stackrel{(a)}{\leq}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} \\ &\quad + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})\right]} - {\mathbb{E}\left[\frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(\Tilde{P}_t^{\infty}, P_t^{\infty})^2 + \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\\ & \stackrel{(b)}{\leq} \frac{C}{2\eta}{\mathbb{E}\left[\|f_t-g_t\|_{{\mathcal{F}}}^2\right]} + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})\right]} - {\mathbb{E}\left[ \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\end{aligned}$$ where $(a)$ follows from Lemma \[lem:noncvx\_reg\_strong\_cvx\], and $(b)$ uses the fact that $|xy|\leq \frac{1}{2c}|x|^2 + \frac{c}{2}|y|^2$, for any $x,y$, $c> 0$. Summing over $t=1,\dots T$ gives us $$\begin{aligned} \sum_{t=1}^T {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &\leq \underbrace{{\mathbb{E}\left[B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty})\right]}}_{S_1}+ \sum_{t=1}^T\frac{C}{2\eta}{\mathbb{E}\left[\|f_t-g_t\|_{{\mathcal{F}}}^2\right]}\\ &\quad- \sum_{t=1}^T\frac{\eta}{2C}{\mathbb{E}\left[ {\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\end{aligned}$$ To finish the proof of the Theorem, we need to bound $S_1$. #### Bounding $S_1$. From the definition of $B$, we have $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= R(\Tilde{P}_{T}^{\infty}) - {\left\langle P-\Tilde{P}_T^{\infty}, f_{1:T} \right\rangle}-R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}),\end{aligned}$$ where we used the fact that $f_{1:0} = 0$. We now rely on Lemma \[lem:noncvx\_gradient\] to convert the above equation, which is currently in terms of $R$, into a quantity which depends on $\Phi$. Using Lemma \[lem:noncvx\_gradient\], we get $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= \Phi(f_{1:T}) - {\left\langle P, f_{1:T} \right\rangle}-\Phi(0).\end{aligned}$$ From the definition of $\Phi$ we have $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= \Phi(f_{1:T}) - {\left\langle P, f_{1:T} \right\rangle}-\Phi(0)\\ &={\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', f_{1:T}-\sigma \right\rangle}\right]} - {\left\langle P, f_{1:T} \right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', -\sigma \right\rangle}\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[{\left\langle P, f_{1:T}-\sigma \right\rangle}\right]} - {\left\langle P, f_{1:T} \right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', -\sigma \right\rangle}\right]}\\ &= {\mathbb{E}_{\sigma}\left[\sup_{P' \in {\mathcal{P}}}{\left\langle P', \sigma \right\rangle}\right]} - {\mathbb{E}_{\sigma}\left[\left\langle P,\sigma\right\rangle\right]}\\ &\leq D{\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta D,\end{aligned}$$ where the last inequality follows from our bound on the diameter of ${\mathcal{P}}$. Substituting this in the above regret bound gives us the required result. Proof of Corollary \[cor:ftpl\_noncvx\_exp\] -------------------------------------------- To prove the corollary we first show that for our choice of perturbation distribution, ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one, for any $f\in{\mathcal{F}}$. Next, we show that the predictions of OFTPL are stable. ### Intermediate Results Suppose the perturbation function is such that $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is a random vector whose entries are sampled independently from $\text{Exp}(\eta)$. Then, for any $f\in{\mathcal{F}}$, ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one. Define ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$ as $${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}) \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ For any $\bar{\sigma}_1,\bar{\sigma}_2$ we now show that ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$ satisfies the following monotonicity property $${\left\langle {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)-{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2), \bar{\sigma}_1-\bar{\sigma}_2 \right\rangle} \geq 0.$$ From the optimality of ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1),{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)$ we have $$\begin{aligned} f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)) - {\left\langle \bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle} &\leq f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)) - {\left\langle \bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}\\ & = f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)) - {\left\langle \bar{\sigma}_2, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle} + {\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}\\ &\leq f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)) - {\left\langle \bar{\sigma}_2, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle}+ {\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}.\end{aligned}$$ This shows that ${\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)-{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle} \geq 0$. To finish the proof of Lemma, we rely on Theorem 1 of @zarantonello1973dense, which shows that the set of points for which a monotone operator is not single-valued has Lebesgue measure zero. Since the distribution of $\bar{\sigma}$ is absolutely continuous w.r.t Lebesgue measure, this shows that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one. ### Main Argument For our choice of perturbation distribution, ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = {\mathbb{E}_{\bar{\sigma}}\left[\|\bar{\sigma}\|_{\infty}\right]} = \eta\log{d}$. We now bound the stability of predictions of OFTPL. First note that for our choice of primal space $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$, ${\gamma_{{\mathcal{F}}}}$ is the Wasserstein-1 metric, which is defined as $${\gamma_{{\mathcal{F}}}}(P_1,P_2) = \sup_{f\in{\mathcal{F}}, \|f\|_{{\mathcal{F}}}\leq 1} \Big|{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_1}\left[f({{\ensuremath{\mathbf{x}}}})\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_2}\left[f({{\ensuremath{\mathbf{x}}}})\right]}\Big| = \inf_{Q\in\Gamma(P_1,P_2)}{\mathbb{E}_{({{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2)\sim Q}\left[\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1\right]},$$ where $\Gamma(P_1,P_2)$ is the set of all probability measures on ${\mathcal{X}}\times{\mathcal{X}}$ with marginals $P_1,P_2$ on the first and second factors respectively. Define ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$ as $${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}) \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ Note that ${{\nabla}\Phi\left(f\right)}$ is the distribution of random variable ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$. @suggala2019online show that for any $f,g\in{\mathcal{F}}$ $${\mathbb{E}_{\bar{\sigma}}\left[\|{{\ensuremath{\mathbf{x}}}}_{f}(\bar{\sigma})-{{\ensuremath{\mathbf{x}}}}_{g}(\bar{\sigma})\|_1\right]} \leq \frac{125d^2D}{\eta}\|f-g\|_{{\mathcal{F}}}.$$ Since ${\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(f\right)},{{\nabla}\Phi\left(g\right)}) \leq {\mathbb{E}_{\bar{\sigma}}\left[\|{{\ensuremath{\mathbf{x}}}}_{f}(\bar{\sigma})-{{\ensuremath{\mathbf{x}}}}_{g}(\bar{\sigma})\|_1\right]}$, this shows that OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$. Substituting the stability bound in the regret bound of Theorem \[thm:oftpl\_noncvx\_regret\] shows that $$\begin{aligned} \sup_{P\in{\mathcal{P}}}{\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &= \eta D\log{d} \\ &\quad +{O\left({ \sum_{t=1}^T \frac{d^2D }{\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]} -\sum_{t=1}^T \frac{\eta}{d^2D }{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]}}\right)}.\end{aligned}$$ Convex-Concave Games {#sec:cvx-games} ==================== Our algorithm for convex-concave games is presented in Algorithm \[alg:oftpl\_cvx\_games\]. Before presenting the proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\], we first present a more general result in Section \[sec:cvx\_games\_general\]. Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] immediately follows from our general result by instantiating it for the uniform noise distribution. **Input:** Perturbation Distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players, number of samples $m,$ iterations $T$ Sample $\{\sigma_{1,j}^1\}_{j=1}^m,$ $\{\sigma_{1,j}^2\}_{j=1}^m$ from ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_1 = \frac{1}{m}\sum_{j=1}^m\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle -\sigma_{1,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right], {{\ensuremath{\mathbf{y}}}}_1 = \frac{1}{m}\left[\sum_{j=1}^m{\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\left\langle \sigma_{1,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}\right]$ **continue** `//Compute guesses` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}= \underset{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{{\mathop{\rm argmin}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i) -\sigma_{t,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}= \underset{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}}{{\mathop{\rm argmax}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i) +\sigma_{t,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1} = \frac{1}{m}\sum_{j=1}^m\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}$, $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1} = \frac{1}{m}\sum_{j=1}^m\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}$ `//Use the guesses to compute the next action` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{t,j}= \underset{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{{\mathop{\rm argmin}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i)+ {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})-\sigma_{t,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ ${{\ensuremath{\mathbf{y}}}}_{t,j}= \underset{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}}{{\mathop{\rm argmax}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i)+ {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})+\sigma_{t,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}$ ${{\ensuremath{\mathbf{x}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{x}}}}_{t,j}, {{\ensuremath{\mathbf{y}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{y}}}}_{t,j}$ $\{({{\ensuremath{\mathbf{x}}}}_t, {{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ General Result {#sec:cvx_games_general} -------------- \[thm:oftpl\_cvx\_smooth\_games\] Consider the minimax game in Equation . Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is Holder smooth w.r.t some norm $\|\cdot\|$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L_1\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|^{\alpha} + L_2\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|^{\alpha},\\ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L_2\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|^{\alpha}+L_1\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|^{\alpha}.\end{aligned}$$ Define diameter of sets ${\mathcal{X}},{\mathcal{Y}}$ as Let $L=\{L_1,L_2\}$. Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_cvx\] to solve the minimax game. Suppose the perturbation distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2,$ used by ${{\ensuremath{\mathbf{x}}}}$, ${{\ensuremath{\mathbf{y}}}}$ players are absolutely continuous and satisfy ${\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^1}\left[\|\sigma\|_{*}\right]} = {\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^2}\left[\|\sigma\|_{*}\right]}=\eta$. Suppose the predictions of both the players are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used in that iteration. Then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ generated by the OFTPL based algorithm satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & 2L_1\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+\frac{2\eta D}{T} \\ &+ \frac{20{C}L^2}{\eta} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}+10L\left(\frac{5{C}L}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}\end{aligned}$$ Since both the players are responding to each others actions using OFTPL, using Theorem \[thm:oftpl\_regret\], we get the following regret bounds for the players $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D \\ &\quad+ \frac{{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_*^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D \\ &\quad+ \frac{{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. Since $\|a_1+\dots +a_5\|^2 \leq 5(\|a_1\|^2\dots + \|a_5\|^2)$, we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2 \leq &5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t)\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\\ &\stackrel{(a)}{\leq} 5L_1^2\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2\alpha}+5L_1^2\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\\ &\quad + 5L_2^2\|{{\ensuremath{\mathbf{y}}}}_t-{{\ensuremath{\mathbf{y}}}}_t^{\infty}\|^{2\alpha}+5L_2^2\|\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\\ &\quad + 5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2.\end{aligned}$$ where $(a)$ follows from the Holder’s smoothness of $f$. Using a similar technique as in the proof of Theorem \[thm:oftpl\_regret\], relying on Holders inequality, we get $$\begin{aligned} {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2\alpha}|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]} &\leq {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2}|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]}^{\alpha}\\ &\leq {\Psi_1}^{2\alpha}{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2}_2|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]}^{\alpha}\\ &\stackrel{(a)}{\leq} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha},\end{aligned}$$ where $(a)$ follows from the fact that conditioned on past randomness, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ i.i.d bounded mean $0$ random variables, the variance of which scales as $O(D^2/m)$. A similar bound holds for the expectation of other quantities appearing in the RHS of the above equation. Using this, the regret of ${{\ensuremath{\mathbf{x}}}}$ player can be upper bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\mathbb{E}\left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+\eta D + \frac{10{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ Similarly, the regret of ${{\ensuremath{\mathbf{y}}}}$ player can be bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D + \frac{10{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ Summing the above two inequalities, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}+{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right).\end{aligned}$$ From Holder’s smoothness assumption on $f$, we have $$\begin{aligned} {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} & \leq 2{\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]}\\ &\quad + 2{\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]}\\ &\stackrel{(a)}{\leq} 2L^2 {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + 2L^2{\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]},\end{aligned}$$ Using a similar argument, we get $$\begin{aligned} {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \leq 2L^2 {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + 2L^2{\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]}.\end{aligned}$$ Plugging this in the previous bound, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}\\ &\quad+\frac{10CL^2}{\eta}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]}\right)\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}+{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right).\end{aligned}$$ #### Case $\alpha=1$. We first consider the case of $\alpha = 1$. In this case, choosing $\eta> \sqrt{20}CL$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}.\end{aligned}$$ #### General $\alpha$. The more general case relies on AM-GM inequality. Consider the following $$\begin{aligned} \frac{10CL^2}{\eta}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha} &= \left((2\alpha C)^{\frac{\alpha}{1-\alpha}}\eta^{-\frac{1+\alpha}{1-\alpha}}(10CL^2)^{\frac{1}{1-\alpha}}\right)^{1-\alpha}\left(\frac{\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2}{2\alpha C\eta^{-1}}\right)^{\alpha}\\ &\stackrel{(a)}{\leq} (1-\alpha) \left((2\alpha C)^{\frac{\alpha}{1-\alpha}}\eta^{-\frac{1+\alpha}{1-\alpha}}(10CL^2)^{\frac{1}{1-\alpha}}\right)+ \frac{\eta}{2C}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\\ &= \sqrt{20}L \left( \frac{\sqrt{20}CL}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}+ \frac{\eta}{2C}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\end{aligned}$$ where $(a)$ follows from AM-GM inequality. Plugging this in the previous bound, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]}\leq & 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+2\eta D \\ &+ \frac{20{C}L^2T}{\eta} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}+4\sqrt{5}LT\left(\frac{\sqrt{20}{C}L}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}.\end{aligned}$$ The claim of the theorem then follows from the observation that $$\begin{aligned} {\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]} \leq \frac{1}{T}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]}.\end{aligned}$$ Proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] ----------------------------------------------------------- To prove the Theorem, we instantiate Theorem \[thm:oftpl\_cvx\_smooth\_games\] for the uniform noise distribution. As shown in Corollary \[cor:ftpl\_cvx\_gaussian\], the predictions of OFTPL are $dD\eta^{-1}$-stable in this case. Plugging this in the bound of Theorem \[thm:oftpl\_cvx\_smooth\_games\] and using the fact that ${\Psi_1}={\Psi_2}=1$ and $\alpha =1$ gives us $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & 2L\left(\frac{D}{\sqrt{m}}\right)^{2}+\frac{2\eta D}{T} \\ &+ \frac{20dD L^2}{\eta} \left(\frac{D}{\sqrt{m}}\right)^{2}+10L\left(\frac{5dD L}{\eta}\right)^{\infty}.\end{aligned}$$ Plugging in $\eta = 6dD(L+1)$, $m=T$ in the above bound gives us $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & {O\left({\frac{dD^2(L+1)}{T}}\right)}.\end{aligned}$$ Nonconvex-Nonconcave Games {#sec:ncvx-games} ========================== Our algorithm for nonconvex-nonconcave games is presented in Algorithm \[alg:oftpl\_noncvx\_games\]. Note that in each iteration of this game, both the players play empirical distributions $(P_t,Q_t)$. Before presenting the proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\], we first present a more general result in Section \[sec:noncvx\_games\_general\]. Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\] immediately follows from our general result by instantiating it for exponential noise distribution. **Input:** Perturbation Distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players, number of samples $m,$ iterations $T$ Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{1,j} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} -\sigma_{1,j}^1({{\ensuremath{\mathbf{x}}}})$ ${{\ensuremath{\mathbf{y}}}}_{1,j} = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \sigma_{1,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $P_1,Q_1 $ be the empirical distributions over $\{{{\ensuremath{\mathbf{x}}}}_{1,j}\}_{j=1}^m, \{{{\ensuremath{\mathbf{y}}}}_{1,j}\}_{j=1}^m$ **continue** `//Compute guesses` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}= {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{i = 1}^{t-1}f({{\ensuremath{\mathbf{x}}}},Q_i) -\sigma_{t,j}^1({{\ensuremath{\mathbf{x}}}})$ $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}= {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{i = 1}^{t-1}f(P_i,{{\ensuremath{\mathbf{y}}}}) +\sigma_{t,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $\Tilde{P}_{t-1} $, $\Tilde{Q}_{t-1} $ be the empirical distributions over $\{\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}\}_{j=1}^m, \{\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}\}_{j=1}^m$ `//Use the guesses to compute the next action` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{t,j}= {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{i = 1}^{t-1}f({{\ensuremath{\mathbf{x}}}},Q_i) + f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}) -\sigma_{t,j}^1({{\ensuremath{\mathbf{x}}}})$ ${{\ensuremath{\mathbf{y}}}}_{t,j}= {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{i = 1}^{t-1}f(P_i,{{\ensuremath{\mathbf{y}}}}) + f(\Tilde{P}_{t-1},{{\ensuremath{\mathbf{y}}}}) +\sigma_{t,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $P_t, Q_t$ be the empirical distributions over $\{{{\ensuremath{\mathbf{x}}}}_{t,j}\}_{j=1}^m, \{{{\ensuremath{\mathbf{y}}}}_{t,j}\}_{j=1}^m$ $\{(P_t, Q_t)\}_{t=1}^T$ Primal Dual Spaces {#sec:primal_dual_spaces} ------------------ In this section, we present some integral probability metrics induced by popular choices of functions spaces $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$. [||c |c c||]{} ${\gamma_{{\mathcal{F}}}}(P, Q)$ & $\|f\|_{{\mathcal{F}}}$&${\mathcal{F}}$\ \[0.5ex\] Dudley Metric & $\text{Lip}(f)+\|f\|_{\infty}$& $\{f:\text{Lip}(f) + \|f\|_{\infty} < \infty\}$\ ------------------------- Kantorovich Metric (or) Wasserstein-1 Metric ------------------------- : Table showing some popular Integral Probability Metrics. Here $\text{Lip}(f)$ is the Lipschitz constant of $f$ which is defined as $\sup_{{{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{X}}}|f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|$ and $\|f\|_{\infty}$ is the supremum norm of $f$.[]{data-label="tab:ipm"} & $\text{Lip}(f)$ & $\{f:\text{Lip}(f) < \infty\}$\ Total Variation (TV) Distance & $\|f\|_{\infty}$ & $\{f:\|f\|_{\infty} < \infty\}$\ -------------------------------- Maximum Mean Discrepancy (MMD) for RKHS $\mathcal{H}$ -------------------------------- : Table showing some popular Integral Probability Metrics. Here $\text{Lip}(f)$ is the Lipschitz constant of $f$ which is defined as $\sup_{{{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{X}}}|f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|$ and $\|f\|_{\infty}$ is the supremum norm of $f$.[]{data-label="tab:ipm"} & $\|f\|_{\mathcal{H}}$ & $\{f:\|f\|_{\mathcal{H}} < \infty\}$\ \[1ex\] General Result {#sec:noncvx_games_general} -------------- \[thm:oftpl\_noncvx\_smooth\_games\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$. Let ${\mathcal{F}},{\mathcal{F}}'$ be the set of Lipschitz functions over ${\mathcal{X}},{\mathcal{Y}}$, and $\|g_1\|_{{\mathcal{F}}},\|g_2\|_{{\mathcal{F}}'}$ be the Lipschitz constants of functions $g_2:{\mathcal{Y}}\to\mathbb{R}$ w.r.t some norm $\|\cdot\|$. Suppose $f$ is such that $\max\{\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|f(\cdot,{{\ensuremath{\mathbf{y}}}})\|_{{\mathcal{F}}}, \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|f({{\ensuremath{\mathbf{x}}}},\cdot)\|_{{\mathcal{F}}'}\}\leq G$ and satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\| + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|,\\ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|+L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|.\end{aligned}$$ Let ${\mathcal{P}},{\mathcal{Q}}$ be the set of probability distributions over ${\mathcal{X}},{\mathcal{Y}}$. Define diameter of ${\mathcal{P}},{\mathcal{Q}}$ as $D = \max\{\sup_{P_1,P_2\in{\mathcal{P}}} {\gamma_{{\mathcal{F}}}}(P_1,P_2), \sup_{Q_1,Q_2\in{\mathcal{Q}}} {\gamma_{{\mathcal{F}}'}}(Q_1,Q_2)\}$. Suppose both ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\] to solve the game. Suppose the perturbation distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2,$ used by ${{\ensuremath{\mathbf{x}}}}$, ${{\ensuremath{\mathbf{y}}}}$ players are such that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}}), {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} f({{\ensuremath{\mathbf{y}}}})+\sigma({{\ensuremath{\mathbf{y}}}})$ have unique optimizers with probability one, for any $f$ in ${\mathcal{F}},{\mathcal{F}}'$ respectively. Moreover, suppose ${\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^1}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = {\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^2}\left[\|\sigma\|_{{\mathcal{F}}'}\right]}=\eta$ and predictions of both the players are ${C}\eta^{-1}$-stable w.r.t norms $\|\cdot\|_{{\mathcal{F}}}, \|\cdot\|_{{\mathcal{F}}'}$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. Then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ generated by the Algorithm \[alg:oftpl\_cvx\_games\] satisfy the following, for $\eta > \sqrt{3}{C}L$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{\eta D}{T} + \frac{{C}D^2L^2}{\eta m}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d{C}{\Psi_1}^2 {\Psi_2}^2G^2 \log(2m)}{\eta m}, \frac{CD^2L^2}{\eta}\right\rbrace}\right)}.\end{aligned}$$ The proof of this Theorem uses similar arguments as Theorem \[thm:oftpl\_cvx\_smooth\_games\]. Since both the players are responding to each others actions using OFTPL, using Theorem \[thm:oftpl\_noncvx\_regret\], we get the following regret bounds for the players $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t)\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|_{{\mathcal{F}}}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]},\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t)\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f(P_t,\cdot)-f(\Tilde{P}_{t-1},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]},\end{aligned}$$ where $P_t^{\infty},\Tilde{P}_{t-1}^{\infty}, Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}$ are as defined in Theorem \[thm:oftpl\_noncvx\_regret\]. First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. We upper bound $\|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}$ as $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \\ &\quad + 3\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}\\ &\quad + 3\|f(\cdot, \Tilde{Q}_{t-1}^{\infty})-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}.\end{aligned}$$ We now show that ${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\Tilde{P}_{t-1}, \Tilde{Q}_{t-1},P_{1:t-1},Q_{1:t-1}\right]}$ is $O(1/m)$. To simplify the notation, we let Let ${\mathcal{N}}_{\epsilon}$ be the $\epsilon$-net of ${\mathcal{X}}$ w.r.t $\|\cdot\|$. Then $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &\stackrel{(a)}{=} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*\\ & \stackrel{(b)}{\leq} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_* + 2L\epsilon,\end{aligned}$$ where $(a)$ follows from the definition of Lipschitz constant and $(b)$ follows from our smoothness assumption on $f$. Using this, we get $$\begin{aligned} &{\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq 2{\mathbb{E}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*^2\Big|\zeta_t\right]} + 8L^2\epsilon^2,\end{aligned}$$ Since $f$ is Lipschitz, $\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_*$ is bounded by $G$. So is bounded by $2G$ and is bounded by $2{\Psi_1}G$. Moreover, conditioned on past randomness ($\zeta_t$), $\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})$ is a sub-Gaussian random vector and satisfies the following bound $$\begin{aligned} {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty}) \right\rangle}|\zeta_t\right]} \leq \exp\left(2{\Psi_1}^2 G^2\|{{\ensuremath{\mathbf{u}}}}\|_2^2/m\right).\end{aligned}$$ From tail bounds of sub-Gaussian random vectors [@hsu2012tail], we have $$\begin{aligned} {\mathbb{P}}\left(\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4{\Psi_1}^2G^2}{m}(d+2\sqrt{ds} + 2s)\Big|\zeta_t\right) \leq e^{-s},\end{aligned}$$ for any $s>0$. Using union bound, and the fact that $\log|{\mathcal{N}}_{\epsilon}|$ is upper bounded by $d\log\left(1+2D/\epsilon\right)$, we get $$\begin{aligned} {\mathbb{P}}\left(\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4{\Psi_1}^2G^2}{m}(d+2\sqrt{ds} + 2s)\Big|\zeta_t\right) \leq e^{-s+d\log(1+2D/\epsilon)}.\end{aligned}$$ Let $Z = \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2$. The expectation of $Z$ can be bounded as follows $$\begin{aligned} {\mathbb{E}\left[Z|\zeta_t\right]} &= {\mathbb{P}}(Z \leq a|\zeta_t) {\mathbb{E}\left[Z|\zeta_t, Z \leq a\right]} + {\mathbb{P}}(Z > a|\zeta_t) {\mathbb{E}\left[Z|\zeta_t, Z > a\right]}\\ &\leq a + 4{\Psi_1}^2G^2{\mathbb{P}}(Z > a|\zeta_t).\end{aligned}$$ Choosing $\epsilon=Dm^{-1/2}, s = 3d\log(1+2m^{1/2})$, and $a= \frac{44d{\Psi_1}^2G^2 \log(1+2m^{1/2})}{m}$, we get $${\mathbb{E}\left[Z|\zeta_t\right]} \leq \frac{48d{\Psi_1}^2G^2 \log(1+2m^{1/2})}{m}.$$ This shows that ${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq \frac{96d{\Psi_1}^2{\Psi_2}^2G^2 \log(1+2m^{1/2})}{m} + \frac{8D^2L^2}{m}$. Note that another trivial upper bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}}$ is $DL$, which can obtained as follows $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*\\ & = \|{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}_1\sim Q_t,{{\ensuremath{\mathbf{y}}}}_2\sim Q_t^{\infty}}\left[{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\right]}\|_*\\ &\stackrel{(a)}{\leq} LD,\end{aligned}$$ where $(a)$ follows from the smoothness assumption on $f$ and the fact that the diameter of ${\mathcal{X}}$ is $D$. When $L$ is close to $0$, this bound can be much better than the above bound. So we have $${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq \min\left(\frac{96d{\Psi_1}^2{\Psi_2}^2G^2 \log(1+2m^{1/2})}{m} + \frac{8D^2L^2}{m}, L^2D^2\right).$$ Using this, the regret of the ${{\ensuremath{\mathbf{x}}}}$ player can be bounded as follows $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t)\right]} &\leq \eta D + \frac{24{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{288d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{3CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]}.\end{aligned}$$ A similar analysis shows that the regret of ${{\ensuremath{\mathbf{y}}}}$ player can be bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t)\right]} &\leq \eta D + \frac{24{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{288d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{3CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]},\end{aligned}$$ Summing the above two inequalities, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P,Q_t)\right]}&\leq 2\eta D + \frac{48{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{576d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{6CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}^2\right]}\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]} + {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]}\right).\end{aligned}$$ From our assumption on smoothness of $f$, we have $$\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}} \leq L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}),\quad \|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'} \leq L {\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty}).$$ To see this, consider the following $$\begin{aligned} \|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}&=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}^{\infty})\|_{*}\\ &= \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1} {\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}^{\infty}) \right\rangle}\\ &= \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1} {\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim Q_t^{\infty}}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}) \right\rangle}\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim \Tilde{Q}_{t-1}^{\infty}}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}) \right\rangle}\right]}\\ &\leq {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1}\|{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\cdot) \right\rangle}\|_{{\mathcal{F}}'}\\ &= {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1}\left(\sup_{{{\ensuremath{\mathbf{y}}}}_1\neq {{\ensuremath{\mathbf{y}}}}_2 \in {\mathcal{Y}}}\frac{|{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) \right\rangle}-{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2) \right\rangle}|}{\|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|}\right)\\ &\leq {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\left(\sup_{{{\ensuremath{\mathbf{y}}}}_1\neq {{\ensuremath{\mathbf{y}}}}_2 \in {\mathcal{Y}}}\frac{\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\|_{*}}{\|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|}\right)\\ &\stackrel{(a)}{\leq} L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty}),\end{aligned}$$ where $(a)$ follows from smoothness of $f$. Substituting this in the previous equation, and choosing $\eta > \sqrt{3}{C}L$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P,Q_t)\right]}&\leq 2\eta D + \frac{48{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{576d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{6CD^2L^2T}{\eta}\right)\\\end{aligned}$$ This finishes the proof of the Theorem. We note that a similar result can be obtained for other choice of function classes such as the set of all bounded and Lipschitz functions. The only difference between proving such a result vs. proving Theorem \[thm:oftpl\_noncvx\_smooth\_games\] is in bounding $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}}$. Proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\] ---------------------------------------------------------- To prove the Theorem, we instantiate Theorem \[thm:oftpl\_noncvx\_smooth\_games\] for exponential noise distribution. Recall, in Corollary \[cor:ftpl\_noncvx\_exp\], we showed that ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta\log{d}$ and OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$, for this choice of perturbation distribution (similar results hold for $({\mathcal{F}}',\|\cdot\|_{{\mathcal{F}}'})$). Substituting this in the bounds of Theorem \[thm:oftpl\_noncvx\_smooth\_games\] and using the fact that ${\Psi_1}=\sqrt{d},{\Psi_2}= 1$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{\eta D\log{d}}{T} + \frac{d^2 D^3L^2}{\eta m}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d^4DG^2 \log(2m)}{\eta m}, \frac{d^2D^3L^2}{\eta}\right\rbrace}\right)}.\end{aligned}$$ Choosing $\eta = 10d^2D(L+1), m=T$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{d^2 D^2(L+1)\log{d}}{T}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d^2G^2 \log(T)}{ LT}, D^2L\right\rbrace}\right)}.\end{aligned}$$ Choice of Perturbation Distributions {#sec:pert_dist_choice} ==================================== #### Regularization of some Perturbation Distributions. We first study the regularization effect of various perturbation distributions. Table \[tab:reg\_linf\] presents the regularizer $R$ corresponding to some commonly used perturbation distributions, when the action space ${\mathcal{X}}$ is $\ell_{\infty}$ ball of radius $1$ centered at origin. ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- Perturbation Distribution ${P_{\text{PRTB}}}$ Regularizer \[0.5ex\] Uniform over $[0,\eta]^d$ $\eta \|{{\ensuremath{\mathbf{x}}}}-1\|_2^2$ Exponential $P(\sigma > t)=\exp(-t/\eta)$ $\displaystyle\sum_i\eta({{\ensuremath{\mathbf{x}}}}_i+1)\left[\log({{\ensuremath{\mathbf{x}}}}_i+1) - (1+\log 2)\right]$ Gaussian $P(\sigma =t)\propto e^{-t^2/2\eta^2}$ $\displaystyle\sum_i \sup_{u\in\mathbb{R}} u\left[{{\ensuremath{\mathbf{x}}}}_i-1+2F(-u/\eta)\right]$ ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- : Regularizers corresponding to various perturbation distributions used in FTPL when the action space ${\mathcal{X}}$ is $\ell_{\infty}$ ball of radius $1$ centered at origin. Here, $F$ is the CDF of a standard normal random variable.[]{data-label="tab:reg_linf"} #### Dimension independent rates. Recall, the OFTPL algorithm described in Algorithm \[alg:oftpl\_cvx\_games\] converges at ${O\left({d/T}\right)}$ rate to a Nash equilibrium of smooth convex-concave games (see Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\]). We now show that for certain constraint sets ${\mathcal{X}}, {\mathcal{Y}}$, by choosing the perturbation distributions appropriately, the dimension dependence in the rates can *potentially* be removed. Suppose the action set is ${\mathcal{X}}=\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq 1\}$. Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is the multivariate Gaussian distribution with mean $0$ and covariance $\eta^{2}I_{d\times d}$, where $I_{d\times d}$ is the identity matrix. We now try to explicitly compute the reguralizer corresponding to this perturbation distribution and action set. Define function $\Psi$ as $$\Psi(f) = {\mathbb{E}_{\sigma}\left[\max_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}= {\mathbb{E}_{\sigma}\left[\|f+\sigma\|_2\right]}.$$ As shown in Proposition \[prop:ftpl\_ftrl\_connection\], the regularizer $R$ corresponding to any perturbation distribution is given by the Fenchel conjugate of $\Psi$ $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \Psi(f).$$ Since getting an exact expression for $R$ is a non-trivial task, we only compute an *approximate expression* for $R$. Consider the high dimensional setting (*i.e.,* very large $d$). In this setting, $\|f+\sigma\|_2$, for $\sigma$ drawn from ${\mathcal{N}}(0,\eta^2I_{d\times d})$, can be approximated as follows $$\begin{aligned} \|f+\sigma\|_2 &= \sqrt{\|f\|_2^2 + \|\sigma\|_2^2 + 2{\left\langle f, \sigma \right\rangle}}\\ &\stackrel{(a)}{\approx} \sqrt{\|f\|_2^2 + \eta^2d + 2{\left\langle f, \sigma \right\rangle}}\\ &\stackrel{(b)}{\approx} \sqrt{\|f\|_2^2 + \eta^2d}\end{aligned}$$ where $(a)$ follows from the fact that $\|\sigma\|_2^2$ is highly concentrated around $\eta^2d$ [@hsu2012tail]. To be precise $$\mathbb{P}(\|\sigma\|_2^2 \geq \eta^2(d + 2\sqrt{dt} + 2t)) \leq e^{-t}.$$ A similar bound holds for the lower tail. Approximation $(b)$ follows from the fact that ${\left\langle f, \sigma \right\rangle}$ is a Gaussian random variable with mean $0$ and variance $\eta^2\|f\|_2^2$, and with high probability its magnitude is upper bounded by $\Tilde{O}(\eta\|f\|_2)$. Since $\eta\|f\|_2 \ll \sqrt{d}\eta\|f\|_2 \leq \|f\|_2^2 + \eta^2d$, approximation $(b)$ holds. This shows that $\Psi(f)$ can be approximated as $$\Psi(f) \approx \sqrt{\|f\|_2^2 + \eta^2d}.$$ Using this approximation, we now compute the reguralizer corresponding to the perturbation distribution $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \Psi(f) \approx \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \sqrt{\|f\|_2^2 + \eta^2d} = -\eta\sqrt{d}\sqrt{1-\|{{\ensuremath{\mathbf{x}}}}\|_2^2}.$$ This shows that $R$ is $\eta\sqrt{d}$-strongly convex w.r.t $\|\cdot\|_2$ norm. Following duality between strong convexity and strong smoothness, $\Psi(f)$ is $(\eta^2d)^{-1/2}$ strongly smooth w.r.t $\|\cdot\|_2$ norm and satisfies $$\|{\nabla}\Psi(f_1)-{\nabla}\Psi(f_2)\|_2 \leq (\eta^2d)^{-1/2}\|f_1-f_2\|_2.$$ This shows that the predictions of OFTPL are $(\eta^2d)^{-1/2}$ stable w.r.t $\|\cdot\|_2$ norm. We now instantiate Theorem \[thm:oftpl\_cvx\_smooth\_games\] for this perturbation distribution and for constraint sets which are unit balls centered at origin, and use the above stability bound, together with the fact that ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} \approx \eta\sqrt{d}$. Suppose $f$ is smooth w.r.t $\|\cdot\|_2$ norm and satisfies $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Then Theorem \[thm:oftpl\_cvx\_smooth\_games\] gives us the following rates of convergence to a NE $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & \frac{ 2L_1}{m}+\frac{2\eta\sqrt{d}}{T} \\ &+ \frac{20 L^2}{\eta\sqrt{d}} \left(\frac{ 1}{m}\right)+10L\left(\frac{5 L}{\eta\sqrt{d}}\right)^{\infty}\end{aligned}$$ Choosing $\eta = 6L/\sqrt{d}, m=T$, we get ${O\left({\frac{L}{T}}\right)}$ rate of convergence. Although, these rates are dimension independent, we note that our stability bound is only approximate. More accurate analysis is needed to actually claim that Algorithm \[alg:oftpl\_cvx\_games\] achieves dimension independent rates in this setting. That being said, for general constraints sets, we believe one can get dimension independent rates by choosing the perturbation distribution appropriately. High Probability Bounds {#sec:hp_bounds} ======================= In this section, we provide high probability bounds for Theorems \[thm:oftpl\_regret\], \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. Our results rely on the following concentration inequalities. \[prop:azuma\] Let $X_1,\dots X_K$ be $K$ independent mean $0$ vector-valued random variables such that $\|X_i\|_2\leq B_i$. Then $$\mathbb{P}\left({\|{\sum_{i=1}^KX_i} \|}_2 \geq t\right) \leq 2\exp\left(-c\frac{t^2}{\sum_{i=1}^KB_i^2}\right),$$ where $c>0$ is a universal constant. We also need the following concentration inequality for martingales. \[prop:martingale\_diff\] Let $X_1,\dots X_K \in \mathbb{R}$ be a martingale difference sequence, where ${\mathbb{E}\left[X_i|{\mathcal{F}}_{i-1}\right]} = 0$. Assume that $X_i$ satisfy the following tail condition, for some scalar $B_i>0$ $$\mathbb{P}\left(\Big|\frac{X_i}{B_i}\Big| \geq z\Big| {\mathcal{F}}_{i-1}\right) \leq 2\exp(-z^2).$$ Then $$\mathbb{P}\left(\Big|\sum_{i=1}^K X_i\Big| \geq z\right)\leq 2\exp\left(-c\frac{z^2}{\sum_{i=1}^KB_i^2}\right),$$ where $c>0$ is a universal constant. Online Convex Learning ---------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_regret\]. \[thm:oftpl\_regret\_hp\] Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is absolutely continuous w.r.t Lebesgue measure. Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|$, which is defined as $D= \sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|.$ Let and suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$, where ${C}$ is a constant that depends on the set $\mathcal{X}.$ Suppose, the sequence of loss functions $\{f_t\}_{t=1}^T$ are $G$-Lipschitz w.r.t $\|\cdot\|$ and satisfy $\sup_{{{\ensuremath{\mathbf{x}}}}\in \mathcal{X}} \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}})\|_* \leq G$. Moreover, suppose $\{f_t\}_{t=1}^T$ are Holder smooth and satisfy $$\begin{aligned} \forall {{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}\quad \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_1)-{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_2)\|_* \leq L\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|^{\alpha},\end{aligned}$$ for some constant $\alpha \in [0,1]$. Then the regret of Algorithm \[alg:oftpl\_cvx\] satisfies the following with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \eta D + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\\ &\quad+cGD\sqrt{\frac{T\log{2/\delta}}{m}} + cLT\left(\frac{{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}},\end{aligned}$$ where $c$ is a universal constant, ${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty} = {\mathbb{E}\left[\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}|f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}$ denotes the prediction in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_cvx\], if guess $g_{t}=0$ was used. Here, ${\Psi_1}, {\Psi_2}$ denote the norm compatibility constants of $\|\cdot\|.$ Our proof uses the same notation and similar arguments as in the proof Theorem \[thm:oftpl\_regret\]. Recall, in Theorem \[thm:oftpl\_regret\] we showed that the regret of OFTPL is upper bounded by $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \eta D + \sum_{t=1}^T \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left(\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2 + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right)\\ &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \eta D + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2.\end{aligned}$$ From Holder’s smoothness assumption, we have $${\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t - {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} \leq L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}.$$ Substituting this in the previous bound gives us $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \underbrace{\sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}}_{S_1}+\sum_{t=1}^TL\underbrace{\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}}_{S_2} + \eta D \\ &\quad + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2.\end{aligned}$$ We now provide high probability bounds for $S_1$ and $S_2$. #### Bounding $S_1$. Let $\xi_i = \{g_{i+1}, f_{i+1}, {{\ensuremath{\mathbf{x}}}}_{i}\}$ and let $\xi_{0:t}$ denote the union of sets $\xi_0,\xi_1,\dots, \xi_t$. Let $\zeta_t = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}$ with $\zeta_0=0$. Note that $\{\zeta_t\}_{t=0}^T$ is a martingale difference sequence w.r.t $\xi_{0:T}$. This is because ${\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|\xi_{0:t-1}\right]} = {{\ensuremath{\mathbf{x}}}}_t^{\infty}$ and ${\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty})$ is a deterministic quantity conditioned on $\xi_{0:t-1}$. As a result ${\mathbb{E}\left[\zeta_t|\xi_{0:t-1}\right]}=0$. Moreover, conditioned on $\xi_{0:t-1}$, $\zeta_t$ is the average of $m$ independent mean $0$ random variables, each of which is bounded by $GD$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(|\zeta_t| \geq s\Big| \xi_{0:t-1}\right) \leq 2\exp\left(-\frac{ms^2}{G^2D^2}\right).$$ Using Proposition \[prop:martingale\_diff\] on the martingale difference sequence $\{\zeta_t\}_{t=0}^T$, we get $$\mathbb{P}\left(\Big|\sum_{t=1}^T\zeta_t\Big| \geq s\right)\leq 2\exp\left(-c\frac{ms^2}{G^2D^2T}\right),$$ where $c>0$ is a universal constant. This shows that with probability at least $1-\delta/2$, $S_1$ is upper bounded by $ {O\left({\sqrt{\frac{G^2D^2T\log{\frac{2}{\delta}}}{m}}}\right)}.$ #### Bounding $S_2$. Conditioned on $\{g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\}$, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ independent mean $0$ random variables which are bounded by $D$ in $\|\cdot\|$ norm. From our definition of norm compatibility constant ${\Psi_2}$, this implies the random variables are bounded by ${\Psi_2}D$ in $\|\cdot\|_2$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2 \geq {\Psi_2}D\sqrt{\frac{c\log{4T/\delta}}{m}}\Bigg| g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right) \leq \frac{\delta}{2T}.$$ Since the above bound holds for any set of $\{g_t,f_{1:t}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\}$, the same tail bound also holds without the conditioning. This shows that $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha} \geq \left(\frac{c{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}}\right) \leq \frac{\delta}{2T},$$ where we converted back to $\|\cdot\|$ by introducing the norm compatibility constant ${\Psi_1}$. #### Bounding the regret. Plugging the above high probability bounds for $S_1,S_2$ in the previous regret bound and using union bound, we get the following regret bound which holds with probability at least $1-\delta$ $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq cGD\sqrt{\frac{T\log{2/\delta}}{m}} + cLT\left(\frac{{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}} + \eta D \\ &\quad + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2,\end{aligned}$$ where $c>0$ is a universal constant. Convex-Concave Games {#convex-concave-games} -------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. \[thm:oftpl\_cvx\_smooth\_games\_uniform\_hp\] Consider the minimax game in Equation . Suppose both the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$, with diameter Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is Lipschitz w.r.t $\|\cdot\|_2$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{2}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{2}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ is smooth w.r.t $\|\cdot\|_2$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Suppose Algorithm \[alg:oftpl\_cvx\_games\] is used to solve the minimax game. Suppose the perturbation distributions used by both the players are the same and equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_cvx\_games\] is run with $\eta = 6dD(L+1), m = T$, then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ satisfy the following bound with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]= {O\left({\frac{GD\sqrt{\log{\frac{8}{\delta}}}}{T}+ \frac{D^2(L+1)\left(d + \log{\frac{16T}{\delta}}\right)}{T}}\right)}.\end{aligned}$$ We use the same notation and proof technique as Theorems \[thm:oftpl\_cvx\_smooth\_games\], \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. From Theorem \[cor:ftpl\_cvx\_gaussian\] we know that the predictions of OFTPL are $dD\eta^{-1}$ stable w.r.t $\|\cdot\|_2$, for the particular perturbation distribution we consider here. We use this stability bound in our proof. From Theorem \[thm:oftpl\_regret\_hp\], we have the following regret bound for both the players, which holds with probability at least $1-\delta/2$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + cLT\left(\frac{D^2\log{16T/\delta}}{m}\right) + \eta D \\ &\quad+ \frac{dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_2^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + cLT\left(\frac{D^2\log{16T/\delta}}{m}\right) + \eta D \\ &\quad+ \frac{dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. From the proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\], we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{2}^2 &\leq 5L^2\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2}+5L^2\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2}\\ &\quad + 5L^2\|{{\ensuremath{\mathbf{y}}}}_t-{{\ensuremath{\mathbf{y}}}}_t^{\infty}\|_2^{2}+5L^2\|\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2}\\ &\quad + 5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2.\end{aligned}$$ Moreover, from the proof of Theorem \[thm:oftpl\_regret\_hp\], we know that $\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2}$ satisfies the following tail bound $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2} \geq \frac{c D^2\log{16T/\delta}}{m}\right) \leq \frac{\delta}{8T}.$$ Similar bounds hold for the quantities appearing in the regret bound of ${{\ensuremath{\mathbf{y}}}}$ player. Plugging this in the previous regret bounds, we get the following which hold with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{cD^2\log{16T/\delta}}{m}\right)T \\ &\quad + \eta D+ \frac{5dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_2^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{cD^2\log{16T/\delta}}{m}\right)T \\ &\quad+ \eta D + \frac{5dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ Summing these two regret bounds, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq 2cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{2cD^2\log{16T/\delta}}{m}\right)T + 2\eta D \\ &\quad+ \frac{10dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_2^2\right] \\ &\quad+ \frac{10dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2+\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ From Holder’s smoothness assumption on $f$, we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2 & \leq 2\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\\ &\quad + 2\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\\ &\leq 2L^2 \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2} + 2L^2\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2},\end{aligned}$$ Using a similar argument, we get $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2 \leq 2L^2 \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2} + 2L^2\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2}.\end{aligned}$$ Plugging this in the previous bound, and setting $\eta = 6d D(L+1), m=T$, we get the following bound which holds with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq {O\left({GD\sqrt{\log{\frac{8}{\delta}}}+ D^2(L+1)\left(d + \log{\frac{16T}{\delta}}\right)}\right)}.\end{aligned}$$ Nonconvex-Nonconcave Games {#nonconvex-nonconcave-games} -------------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\]. \[thm:oftpl\_noncvx\_smooth\_games\_exp\_hp\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$ with diameter $D = \max\{\sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1, \sup_{{{\ensuremath{\mathbf{y}}}}_1,{{\ensuremath{\mathbf{y}}}}_2\in{\mathcal{Y}}} \|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|_1\}$. Suppose $f$ is Lipschitz w.r.t $\|\cdot\|_1$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} + \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_1 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_1.\end{aligned}$$ Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\_games\] to solve the game with linear perturbation functions $\sigma({{\ensuremath{\mathbf{z}}}})={\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{z}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is such that each of its entries is sampled independently from $\text{Exp}(\eta)$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_noncvx\_games\] is run with $\eta = 10d^2D(L+1), m = T$, then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ satisfy the following with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{t=1}^Tf(P_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},Q_t)& = {O\left({\frac{d^2D^2(L+1)\log{d}}{T} + \frac{GD}{T}\sqrt{\log{\frac{8}{\delta}}}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2L, \frac{d^2G^2\log{T}+dG^2\log{\frac{8}{\delta}}}{LT}\right\rbrace}\right)}.\end{aligned}$$ We use the same notation used in the proofs of Theorems \[thm:oftpl\_noncvx\_regret\], \[thm:oftpl\_noncvx\_smooth\_games\]. Let ${\mathcal{F}},{\mathcal{F}}'$ be the set of Lipschitz functions over ${\mathcal{X}},{\mathcal{Y}}$, and $\|g_1\|_{{\mathcal{F}}},\|g_2\|_{{\mathcal{F}}'}$ be the Lipschitz constants of functions w.r.t $\|\cdot\|_1$. Recall, in Corollary \[cor:ftpl\_noncvx\_exp\] we showed that for our choice of perturbation distribution, ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta \log{d}$ and OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable. We use this in our proof. From Theorem \[thm:oftpl\_noncvx\_regret\], we know that the regret of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players satisfy $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + \underbrace{\sum_{t=1}^T{\left\langle P_t-P_t^{\infty}, f(\cdot, Q_t) \right\rangle}}_{S_1} \\ &\quad + \sum_{t=1}^T\frac{cd^2D}{2\eta}\underbrace{\|f(\cdot,Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|_{{\mathcal{F}}}^2}_{S_2}\\ &\quad- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\end{aligned}$$ $$\begin{aligned} \sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t) &\leq \eta D\log{d}+ \sum_{t=1}^T{\left\langle Q_t-Q_t^{\infty}, f(P_t,\cdot) \right\rangle}\\ &\quad +\sum_{t=1}^T\frac{cd^2D}{2\eta}\|f(P_t,\cdot)-f(\Tilde{P}_{t-1},\cdot)\|_{{\mathcal{F}}'}^2\\ &\quad- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2,\end{aligned}$$ where $c>0$ is a positive constant. We now provide high probability bounds for $S_1,S_2$. #### Bounding $S_1$. Let $\xi_i = \{\Tilde{P}_{i}, \Tilde{Q}_i, P_{i}, Q_{i+1}\}$ with $\xi_0=\{Q_1\}$ and let $\xi_{0:t}$ denote the union of sets $\xi_0,\dots, \xi_t$. Let $\zeta_t = {\left\langle P_t-P_t^{\infty}, f(\cdot, Q_t) \right\rangle}$ with $\zeta_0 = 0$. Note that $\{\zeta_t\}_{t=0}^T$ is a martingale difference sequence w.r.t $\xi_{0:T}$. This is because ${\mathbb{E}\left[P_t|\xi_{0:t-1}\right]} = P_t^{\infty}$ and $f(\cdot,Q_t)$ is a deterministic quantity conditioned on $\xi_{0:t-1}$. As a result ${\mathbb{E}\left[\zeta_t|\xi_{0:t-1}\right]}=0$. Moreover, conditioned on $\xi_{0:t-1}$, $\zeta_t$ is the average of $m$ independent mean $0$ random variables, each of which is bounded by $2GD$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(|\zeta_t| \geq s\Big| \xi_{0:t-1}\right) \leq 2\exp\left(-\frac{ms^2}{4G^2D^2}\right).$$ Using Proposition \[prop:martingale\_diff\] on the martingale difference sequence $\{\zeta_t\}_{t=0}^T$, we get $$\mathbb{P}\left(\Big|\sum_{t=1}^T\zeta_t\Big| \geq s\right)\leq 2\exp\left(-c\frac{ms^2}{G^2D^2T}\right),$$ where $c>0$ is a universal constant. This shows that with probability at least $1-\delta/8$, $S_1$ is upper bounded by $ {O\left({\sqrt{\frac{G^2D^2T\log{\frac{8}{\delta}}}{m}}}\right)}.$ #### Bounding $S_2$. We upper bound $S_2$ as $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \\ &\quad + 3\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}\\ &\quad + 3\|f(\cdot, \Tilde{Q}_{t-1}^{\infty})-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}.\end{aligned}$$ We first provide a high probability bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}$. A trivial bound for this quantity is $L^2D^2$, which can be obtained as follows $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_{\infty}\\ & = \|{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}_1\sim Q_t,{{\ensuremath{\mathbf{y}}}}_2\sim Q_t^{\infty}}\left[{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\right]}\|_{\infty}\\ &\stackrel{(a)}{\leq} LD,\end{aligned}$$ where $(a)$ follows from the smoothness assumption on $f$ and the fact that the diameter of ${\mathcal{X}}$ is $D$. A better bound for this quantity can be obtained as follows. From proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\], we have $$\begin{aligned} &\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \leq 2\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_{\infty}^2 + 8L^2\epsilon^2.\end{aligned}$$ where ${\mathcal{N}}_{\epsilon}$ be the $\epsilon$-net of ${\mathcal{X}}$ w.r.t $\|\cdot\|$. Recall, in the proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\], we showed the following high probability bound for the RHS quantity $$\begin{aligned} {\mathbb{P}}\left(\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4dG^2}{m}(d+2\sqrt{ds} + 2s)\right) \leq e^{-s+d\log(1+2D/\epsilon)}.\end{aligned}$$ Choosing $\epsilon=Dm^{-1/2}, s = \log{\frac{8}{\delta}}+d\log(1+2m^{1/2})$, we get the following bound for $\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2$ which holds with probability at least $1-\delta/8$ $$\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 \leq \frac{20dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right).$$ Together with our trivial bound of $D^2L^2$, this gives us the following bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} $, which holds with probability at least $1-\delta/8$ $$\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \leq \min\left(\frac{20dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right), D^2L^2\right) + \frac{8D^2L^2}{m}.$$ Next, we bound $\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}$. From our smoothness assumption on $f$, we have $$\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}} \leq L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}).$$ Combining the previous two results, we get the following upper bound for $S_2$ which holds with probability at least $1-\delta/8$ $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3L^2{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2 + \frac{48D^2L^2}{m} \\ &\quad + \min\left(\frac{120dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right), 6D^2L^2\right).\end{aligned}$$ #### Regret bound. Substituting the above bounds for $S_1,S_2$ in the regret bound for ${{\ensuremath{\mathbf{x}}}}$ player gives us the following bound, which holds with probability at least $1-\delta/2$ $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + {O\left({GD\sqrt{\frac{T\log{\frac{8}{\delta}}}{m}}+\frac{d^2D^3L^2T}{\eta m}}\right)} \\ &\quad +{O\left({\min\left(\frac{d^3DG^2T}{\eta m}\left(\log{\frac{8}{\delta}}+d\log(2m)\right), \frac{d^2D^3L^2T}{\eta}\right)}\right)}\\ &\quad+\sum_{t=1}^T\frac{3cd^2DL^2}{2\eta}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\end{aligned}$$ Using a similar analysis, we get the following regret bound for the ${{\ensuremath{\mathbf{y}}}}$ player $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + {O\left({GD\sqrt{\frac{T\log{\frac{8}{\delta}}}{m}}+\frac{d^2D^3L^2T}{\eta m}}\right)} \\ &\quad +{O\left({\min\left(\frac{d^3DG^2T}{\eta m}\left(\log{\frac{8}{\delta}}+d\log(2m)\right), \frac{d^2D^3L^2T}{\eta}\right)}\right)}\\ &\quad+\sum_{t=1}^T\frac{3cd^2DL^2}{2\eta}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\end{aligned}$$ Choosing, $\eta = 10d^2D(L+1), m= T$, and adding the above two regret bounds, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{t=1}^Tf(P_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},Q_t)& = {O\left({d^2D^2(L+1)\log{d} + GD\sqrt{\log{\frac{8}{\delta}}}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2LT, \frac{d^2G^2\log{T}}{L} + \frac{dG^2\log{\frac{8}{\delta}}}{L}\right\rbrace}\right)}.\end{aligned}$$ Background on Convex Analysis ============================= #### Fenchel Conjugate. {#sec:fenchel_conjugate} The Fenchel conjugate of a function $f$ is defined as $$f^*(x^*) = \sup_{x}{\left\langle x, x^* \right\rangle} - f(x).$$ We now state some useful properties of Fenchel conjugates. These properties can be found in @rockafellar1970convex. \[thm:fenchel\_prop1\] Let $f$ be a proper convex function. The conjugate function $f^*$ is then a closed and proper convex function. Moreover, if $f$ is lower semi-continuous then $f^{**} = f$. \[thm:fenchel\_prop3\] For any proper convex function $f$ and any vector $x$, the following conditions on a vector $x^*$ are equivalent to each other - $x^* \in \partial f(x)$ - ${\left\langle z, x^* \right\rangle} - f(z)$ achieves its supremum in $z$ at $z=x$ - $f(x) + f^*(x^*) = {\left\langle x, x^* \right\rangle}$ If $(\text{cl}f)(x) = f(x)$, the following condition can be added to the list - $x\in\partial f^*(x^*)$ \[thm:fenchel\_prop4\] If $f$ is a closed proper convex function, $\partial f^*$ is the inverse of $\partial f$ in the sense of multivalued mappings, *i.e.,* $x \in \partial f^*(x^*)$ iff $x^* \in \partial f(x).$ \[thm:fenchel\_prop2\] Let $f$ be a closed proper convex function. Let $\partial f$ be the subdifferential mapping. The effective domain of $\partial f$, which is the set ${\text{dom}(\partial f)} = \{x|\partial f \neq 0\},$ satisfies $$\text{ri}({\text{dom}( f)}) \subseteq {\text{dom}(\partial f)} \subseteq {\text{dom}( f)}.$$ The range of $\partial f$ is defined as $\text{range} \partial f=\cup\{\partial f(x)|x\in \mathbb{R}^d\}$. The range of $\partial f$ is the effective domain of $\partial f^*$, so $$\text{ri}({\text{dom}( f^*)}) \subseteq \text{range} \partial f \subseteq {\text{dom}( f^*)}.$$ #### Strong Convexity and Smoothness. We now define strong convexity and strong smoothness and show that these two properties are duals of each other. \[def:strong\_convexity\] A function $f:{\mathcal{X}}\to \mathbb{R}\cup\{\infty\}$ is $\beta$-strongly convex w.r.t a norm $\|\cdot\|$ if for all $x,y \in \text{ri}({\text{dom}( f)})$ and $\alpha\in (0,1)$ we have $$f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y) - \frac{1}{2}\beta \alpha (1-\alpha) \|x-y\|^2.$$ This definition of strong convexity is equivalent to the following condition on $f$ [see Lemma 13 of @shalev2007thesis] $$f(y) \geq f(x) + {\left\langle g, y-x \right\rangle} + \frac{1}{2}\beta\|y-x\|^2, \quad \text{for any } x,y\in \text{ri}({\text{dom}( f)}), g\in\partial f(x)$$ \[def:strong\_smoothness\] A function $f:{\mathcal{X}}\to \mathbb{R}\cup\{\infty\}$ is $\beta$-strongly smooth w.r.t a norm $\|\cdot\|$ if $f$ is everywhere differentiable and if for all $x,y$ we have $$f(y) \leq f(x) + {\left\langle {\nabla}f(x), y-x \right\rangle} + \frac{1}{2}\beta\|y-x\|^2.$$ \[thm:fenchel\_strong\_convex\_weak\] Assume that $f$ is a proper closed and convex function. Suppose $f$ is $\beta$-strongly smooth w.r.t a norm $\|\cdot\|$. Then its conjugate $f^*$ satisfies the following for all $a,x$ with $u = {\nabla}f(x)$ $$f^*(a+u) \geq f^*(u) + {\left\langle x, a \right\rangle}+\frac{1}{2\beta}\|a\|_*^2.$$ \[thm:fenchel\_strong\_convex\] Assume that $f$ is a closed and convex function. Then $f$ is $\beta$-strongly convex w.r.t a norm $\|\cdot\|$ iff $f^*$ is $\frac{1}{\beta}$-strongly smooth w.r.t the dual norm $\|\cdot\|_{*}$.

ArXiv

Introduction ============ One of the goals of the recent nuclear physics is to find the equation of state of nuclear matter. Indeed, the dependence of the pressure on the density of nucleons is a crucial input for a hydrodynamical modeling of heavy ion collisions or of astrophysical events like the big bang, supernova explosions and neutron stars [@SG86]. In the absence of any direct measurement, it is hoped that the equation of state can be deduced from heavy ion collisions via the following scheme. Heavy ion collision data are fitted with the Boltzmann equation (BE) $$\begin{aligned} &&{\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k} \nonumber\\ &&=\sum_b\int{dpdq\over(2\pi)^5} \delta\left(\varepsilon_1+\varepsilon_2- \varepsilon_3-\varepsilon_4\right) \nonumber\\ &&\times |T|^2\left(\varepsilon_1+\varepsilon_2,k,p,q,t,r\right) \nonumber\\ &&\times \Bigl[f_3f_4\bigl(1-f_1\bigr)\bigl(1-f_2\bigr)- \bigl(1-f_3\bigr)\bigl(1-f_4\bigr)f_1f_2\Bigr]. \label{1}\end{aligned}$$ Arguments of distributions $f$ and energies $\varepsilon$ are shortened as $f_1\equiv f_a(k,r,t)$, $f_2\equiv f_b(p,r,t)$, $f_3\equiv f_a(k-q,r,t)$, and $f_4\equiv f_b(p+q,r,t)$, with momenta $k,p,q$, coordinate $r$, time $t$, and spin and isospin $a,b$. Once the differential cross sections $|T|^2$ and the functional dependence of energy $\varepsilon$ on the distribution $f$ are fitted, the equation of state is evaluated from the kinetic equation. This scheme has two drawbacks. First, accessible fits of the quasiparticle energy $\varepsilon$ are not sufficiently reliable since two possible fits, momentum-dependent and momentum-independent, result in very contradictory predictions giving hard and soft equations of state, respectively[@BG88]. When this more or less technical problem is resolved in future, one has to face the second drawback: the BE is not thermodynamically consistent with virial corrections to the equation of state. This problem is principal for “how can one infer the equation of state from the BE if the two equations are not consistent?”. A consistency between the kinetic and the thermodynamic theories is a general question for the quantum statistics exceeding the merits of the nuclear matter. Here we approach this question from nonequilibrium Green’s functions. It is shown that the consistency is achieved by a consistent treatment of the quasiclassical limit which results in nonlocal and noninstant corrections to the scattering integral of the BE. The need of nonlocal corrections can be seen on the classical gas of hard spheres. In the scattering integral of (\[1\]), all space arguments of the distributions are identical, i.e., colliding particles $a$ and $b$ are at the same space point $r$. In reality, these particles are displaced by the sum of their radii. This inconsistency has been noticed by Enskog [@CC90] and cured by nonlocal corrections to the scattering integral. The equation of state evaluated from the kinetic equation with the nonlocal scattering integral is of the van der Waals type covering the excluded volume [@CC90; @HCB64]. For nuclear matter, Enskog’s corrections has been first discussed by Malfliet [@M84] and recently implemented by Kortemayer, Daffin and Bauer [@KDB96]. The noninstant corrections are closer to the chemical picture of reacting gases. In the scattering integral of (\[1\]), all time arguments of the distributions are identical what implies that the collision is instant. In reality, the collision has a finite duration which might be quite long when two particles form a resonant state. The resonant two-particle state behaves as an effective short-living molecule. Like in reacting gases [@HCB64], the presence of these molecules reduces the pressure since it reduces the number of freely flying particles. The finite duration of nucleon-nucleon collisions and its thermodynamic consequences has been for the first time discussed only recently by Danielewicz and Pratt [@DP96]. The noninstant scattering integral and its consequencies for the linear response has been also discussed for electrons in semiconductors scattered by resonant levels [@SLM97]. Except for dense Fermi systems, the above intuitively formulated nonlocal and noninstant corrections has been confirmed by systematic approaches. For classical gases, this theory was developed already by Bogoliubov and Green [@B46; @G52]. Obtained gradient contributions to the scattering integral are the lowest order terms of the virial expansion in the kinetic equation [@comdiv]. The first quantum kinetic equation with nonlocal corrections has been derived by Snider [@S60]. Recently, it has been recognized that Snider’s equation is not consistent with the second order virial corrections to equations of state. A consistent quantum mechanical theory of the virial corrections to the BE has been developed from the multiple scattering expansion in terms of Møller operators [@NTL91] and confirmed by Balescu’s formalism [@H90]. Presented treatment extends the nonlocal and noninstant corrections to dense Fermi systems. We follow Baerwinkel [@B69] in starting from nonequilibrium Green’s functions and keeping all gradient contributions to the scattering integral. Baerwinkel’s results are limited to low densities (to avoid medium effects on binary collisions) and not consistent (since he uses the quasiparticle approximation). Here we describe the binary collisions by the Bethe-Goldstone T-matrix which includes the medium effects. Instead of the quasiparticle approximation, the [*extended*]{} quasiparticle approximation is used. This extension is sufficient to gain consistency of the kinetic theory with the virial corrections to thermodynamic quantities. Extended quasiparticle picture ============================== We start our derivation of the kinetic equation from the quasiparticle transport equation first obtained by Kadanoff and Baym [@D84; @SL95] $${\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k}= z_1(1-f_1)\Sigma^<_{1,\varepsilon_1}-z_1f_1\Sigma^>_{1,\varepsilon_1}. \label{2}$$ Like in (\[1\]), quasiparticle distribution $f$, quasiparticle energy $\varepsilon$ and wave-function renormalization $z$ are functions of time $t$, coordinate $r$, momentum $k$ and spin and isospin $a$. Self-energy $\Sigma$, taken from nonequilibrium Green’s function in the notation of Kadanoff and Baym [@D84], is moreover a function of energy $\omega$, however, it enters the transport equation only by its value at pole $\omega=\varepsilon_1$. Particular forms of the quasiparticle energy and the scattering integral we derive for a model and an approximation used in nuclear matter for heavy ion collisions in the non-relativistic energy domain. The system is composed of protons and neutrons of equal mass $m$. They interact via an instant potential $V$. We assume no spin-flipping mechanism. As common, the self-energy is constructed from the two-particle T-matrix $T^R$ in the Bethe-Goldstone approximation [@D84; @MR94] as \[$T^R_{\rm sc}\!(1,2,3,4)\!=\!(1\!-\!\delta_{a_1a_2})T^R\!(1,2,3,4)\!+\! {1\over\sqrt{2}}\delta_{a_1a_2}(T^R\!(1,2,3,4)\!-\!T^R\!(1,2,4,3))$\] $$\begin{aligned} \Sigma^<(1,2)&=& T^R_{\rm sc}(1,\bar 3;\bar 5,\bar 6)T^A_{\rm sc}(\bar 7,\bar 8;2,\bar 4) \nonumber\\ &\times &G^>(\bar 4,\bar 3)G^<(\bar 5,\bar 7)G^<(\bar 6,\bar 8), \label{3}\end{aligned}$$ and $\Sigma^>$ is obtained from (\[3\]) by an interchange $>\leftrightarrow <$. Here, $G$’s are single-particle Green’s functions, numbers are cumulative variables, $1\equiv (t_1,r_1,a_1)$, and bars denote internal variables that are integrated over. Before (\[3\]) is plugged in (\[2\]), it has to be transformed into the mixed representation, \[off-diagonal elements in spin and isospin are excluded, $a_1=a_2=a$\] $$\begin{aligned} \Sigma^<(1,2)&=&\int{d\omega\over 2\pi}{dk\over(2\pi)^3} {\rm e}^{ik(r_1-r_2)-i\omega(t_1-t_2)} \nonumber\\ &\times &\Sigma^<_a\left(\omega,k,r,t\right)_ {r={r_1+r_2\over 2},t={t_1+t_2\over 2}}, \label{4}\end{aligned}$$ and all Green’s functions in (\[3\]), too. The self-energy $\Sigma$ is a functional of Green’s functions $G$. This functional $\Sigma[G]$ is converted to the functional of the quasiparticle distribution $\Sigma_\varepsilon[f]$ via the extended quasiparticle approximation [@SL95; @BKKS96] \[$z_1=1+\left. {\partial\over\partial\omega}{\rm Re}\Sigma_{1\omega}\right|_ {\varepsilon_1}$\] $$G^{\begin{array}{c}>\\[-2mm] <\end{array}}_{1,\omega}= \left(\!\begin{array}{c}1\!-\!f_1\\ f_1\end{array}\!\right) 2\pi z_1\delta(\omega-\varepsilon_1)+{\rm Re}{\Sigma^{\begin{array}{c} >\\[-2mm] <\end{array}}_{1,\omega}\over(\omega-\varepsilon_1)^2}, \label{5}$$ where $G_{1,\omega}\equiv G_a(\omega,k,r,t)$ and similarly $\Sigma$. Unlike the plain quasiparticle approximation (without the second term) used by Baerwinkel [@B69], approximation (\[5\]) leads to the consistent theory. The first term brings the on-shell quasiparticle part, the second term is the off-shell contribution. The off-shell part plays four-fold role. First, it justifies the kinetic equation (\[2\]). Equation (\[2\]) has been originally derived from the plain quasiparticle approximation neglecting the off-shell drift.[^1] The off-shell part of $G^<$ in (\[5\]) compensates the off-shell drift so that (\[2\]) is recovered without uncontrollable neglects [@SL95]. Second, in the quasiparticle energy $\varepsilon_1={k^2\over 2m_a}+{\rm Re}\Sigma^R_ {1,\varepsilon_1}$, the off-shell part brings contributions that are essential for the correct binding energy [@KM93]. Third, (\[5\]) provides Wigner’s distribution $\rho=\int{d\omega\over 2\pi}G^<$ as a functional of the quasiparticle distribution $f$ [@SL95; @KM93]. Fourth, in the scattering integral of (\[2\]), the off-shell part results in sequential three-particle processes with the off-shell propagation between the two composing binary processes. Since the three-particle processes are beyond the scope of the present paper, they are excluded from scattering integral. Non-local scattering integral ============================= Now the approximation is specified and we can start to simplify the scattering integral. In contrast to previous treatments of degenerated systems, we keep all terms linear in gradients. The gradient expansion of the self-energy (\[3\]) is a straightforward but tedious task. It results in a one nongradient and nineteen gradient terms that are analogous to those found within the chemical physics [@NTL91; @H90]. All these terms can be recollected into a nonlocal and noninstant scattering integral that has an intuitively appealing structure of the scattering integral in the BE (\[1\]) with Enskog-type shifts of arguments.[^2] In agreement with [@NTL91; @H90], all gradient corrections result proportional to derivatives of the scattering phase shift , $$\begin{array}{lclrcl}\Delta_t&=&{\displaystyle \left.{\partial\phi\over\partial\Omega} \right|_{\varepsilon_1+\varepsilon_2}}&\ \ \Delta_2&=& {\displaystyle\left({\partial\phi\over\partial p}- {\partial\phi\over\partial q}-{\partial\phi\over\partial k} \right)_{\varepsilon_1+\varepsilon_2}}\\ &&&&&\\ \Delta_E&=& {\displaystyle\left.-{1\over 2}{\partial\phi\over\partial t} \right|_{\varepsilon_1+\varepsilon_2}}&\Delta_3&=& {\displaystyle\left.-{\partial\phi\over\partial k} \right|_{\varepsilon_1+\varepsilon_2}}\\ &&&&&\\ \Delta_K&=& {\displaystyle\left.{1\over 2}{\partial\phi\over\partial r} \right|_{\varepsilon_1+\varepsilon_2}}&\Delta_4&=& {\displaystyle-\left({\partial\phi\over\partial k}+ {\partial\phi\over\partial q}\right)_{\varepsilon_1+\varepsilon_2}}. \end{array} \label{8}$$ After derivatives, $\Delta$’s are evaluated at the energy shell $\Omega\to\varepsilon_1+\varepsilon_2$. The corrected BE with the collected gradient terms then reads \[$\Delta_r={1\over 4}(\Delta_2+\Delta_3+\Delta_4)$\] $$\begin{aligned} &&{\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k} \nonumber\\ &&=\sum_b\int{dpdq\over(2\pi)^5}\delta\left(\varepsilon_1+\varepsilon_2- \varepsilon_3-\varepsilon_4+2\Delta_E\right)\nonumber\\ &&\times z_1z_2z_3z_4 \Biggl(1-{1\over 2}{\partial\Delta_2\over\partial r} -{\partial\bar\varepsilon_2\over\partial r} {\partial\Delta_2\over\partial\omega}\Biggr) \nonumber\\ &&\times |T_{\rm sc}^R|^2\!\left(\varepsilon_1\!+\!\varepsilon_2\!-\! \Delta_E,k\!-\!{\Delta_K\over 2},p\!-\!{\Delta_K\over 2}, q,r\!-\!\Delta_r,t\!-\!{\Delta_t\over 2}\!\right) \nonumber\\ &&\times\Bigl[f_3f_4\bigl(1-f_1\bigr)\bigl(1-f_2\bigr)- \bigl(1-f_3\bigr)\bigl(1-f_4\bigr)f_1f_2\Bigr]. \label{9}\end{aligned}$$ Unlike in (\[1\]), the subscripts denote shifted arguments: $f_1\equiv f_a(k,r,t)$, $f_2\equiv f_b(p,r\!-\!\Delta_2,t)$, $f_3\equiv f_a(k\!-\!q\!-\!\Delta_K,r\!-\!\Delta_3,t\!-\!\Delta_t)$, and $f_4\equiv f_b(p\!+\!q\!-\!\Delta_K,r\!-\!\Delta_4,t\!-\!\Delta_t)$. The $\Delta$’s are effective shifts and they represent mean values of various nonlocalities of the scattering integral. These shifts enter the scattering integral in the form known from the theory of gases [@CC90; @NTL91; @H90], however, the set of shifts is larger due to the medium effects on the binary collision that are dominated by the Pauli blocking of the internal states of the collision. The physical meaning of the $\Delta$’s is best seen on gradually more complex limiting cases:\ (o) Sending all $\Delta$’s to zero, (\[9\]) reduces to the BE (\[1\]).\ (i) In the classical limit for hard spheres of the diameter $d$, the scattering phase shift $\phi\to\pi-|q|d$ gives $\Delta_4=\Delta_2= {q\over|q|}d$ and all other $\Delta$’s are zero. The Enskog’s nonlocal corrections are thus recovered.\ (ii) For a collision of two isolated particles interacting via the single-channel separable potential, the scattering phase shift does not depend on $r$, $t$, $q$ and $k-p$, while it depends on $k+p$ exclusively via the energy dependency, $\phi\to\phi(\Omega- {1\over 4m}(k+p)^2)$. Then $\Delta_{E,K,2}=0$ and $\Delta_{3,4}= {k+p\over 2m}\Delta_t$. Since ${k+p\over 2m}$ is the center-of-mass velocity, the displacements $\Delta_4=\Delta_3$ represent a distance over which particles fly together as a molecule.\ (iii) For two isolated particles interacting via a general spherical potential, the scattering phase shift reflects the translational and the spherical symmetries. From translation of the center of mass during $\Delta_t$ follows the relation for the molecular flight ${1\over 2}(\Delta_4+\Delta_3-\Delta_2)={k+p\over 2m}\Delta_t$. From the spherical symmetry follows that the sum of relative coordinates of the particles $a$ and $b$ at the end and at the beginning of collision has the direction of the transferred momentum ${1\over 2}(\Delta_4-\Delta_3+\Delta_2)={q\over|q|}d$ (Enskog-type shift), and that the difference has the perpendicular in-plane direction ${1\over 2}(\Delta_4-\Delta_3-\Delta_2)={k-p-q\over|k-p-q|}\alpha$ (rotation of the molecule). The nonlocality of the collision is thus given by three scalars $\Delta_t$, $d$ and $\alpha$.\ (iv) For effectively isolated two-particle collisions (no Pauli blocking but mean-field contributions $U_a(r,t)$ to the energy, i.e., $\phi\to \phi(\Omega\!-\!{1\over 4m}(k\!+\!p)^2\!-\!U_a\!-\!U_b,\ldots)$) all $\Delta$’s become nonzero. The space displacements are the same as in (iii). The energy and the momentum corrections $2\Delta_{E,K}=-\Delta_t{\partial\over\partial t,r}(U_a+U_b)$, represent the energy and the momentum which the effective molecule gains during its short life time $\Delta_t$. These corrections are in fact of three-particle nature, however, only on the mean-field level, thus formally within the binary process. The three scalars, $\Delta_t$, $d$ and $\alpha$, are still sufficient to parameterize all $\Delta$’s. We note that in the energy-conserving $\delta$ function all $\Delta$’s and mean-fields compensate $\delta\left(\varepsilon_1\!+\!\varepsilon_2\!-\!\varepsilon_3\!-\! \varepsilon_4\!-\!2\Delta_E\right)\!=\!\delta\left({k^2\over 2m}\!+\! {p^2\over 2m}\!-\!{(k-q)^2\over 2m}\!-\!{(p+q)^2\over 2m}\right)$. This limit describes the dilute quantum gases and (\[9\]) reduces to the kinetic equation found in [@NTL91; @H90].\ (v) With the medium effects on collisions, all $\Delta$’s become independent. A generally nonparabolic momentum-dependency of the quasiparticle energy does not allow to separate the center-of-mass motion, and an anisotropic, inhomogeneous and time-dependent distribution in the Pauli blocking ruins all symmetries of the collision. The energy conservation does not reduce to the simple conservation of the kinetic energy. The uncompensated residuum of the energy gain contributes to a conversion between the kinetic and the configuration energies of the system. The kinetic equation (\[9\]) is numerically tractable by recent Monte Carlo codes. Kortemayer, Daffin and Bauer [@KDB96] have already studied the kinetic equation with an intuitive extension by Enskog-type displacement, claiming only a little increase in numerical demands compared to the BE. The larger set of $\Delta$’s in (\[9\]) does not require principal changes of the numerical method used in [@KDB96]. The functions $\varepsilon$, $z$, $|T_{\rm sc}^R|^2$, and $\Delta$’s should form a consistent set, i.e., it is preferable to obtain them from Green’s function studies, e.g., like [@ARS94]. Eventual fitting parameters should enter directly the effective nucleon-nucleon interaction. Observables =========== Let us presume that a reasonable set of functions $\varepsilon$, $z$, $|T_{\rm sc}^R|^2$, and $\Delta$’s is known. One can then proceed to evaluate observables. Here we present the density $n_a$ of particles $a$, the density of energy $\cal E$, and the stress tensor ${\cal J}_{ij}$. The observables in question are directly obtained from balance equations which also establish conservation laws. Integrating the kinetic equation (\[2\]) over momentum $k$ with factors $\varepsilon_1,k,1$, one finds that each observable has the standard quasiparticle part following from the drift $$\begin{aligned} {\cal E}^{\rm qp} &=&\sum_a\int{dk\over(2\pi)^3}{k^2\over 2m}f_1 \nonumber\\ &+&{1\over 2}\sum_{a,b}\int{dkdp\over(2\pi)^6} T_{\rm ex}(\varepsilon_1+\varepsilon_2,k,p,0)f_1f_2, \nonumber\\ {\cal J}_{ij}^{\rm qp}&=&\sum_a\int{dk\over(2\pi)^3}\left(k_j {\partial\varepsilon_1\over\partial k_i}+ \delta_{ij}\varepsilon_1\right)f_1- \delta_{ij}{\cal E}^{\rm qp}, \nonumber\\ n_a^{\rm qp}&=&\int{dk\over(2\pi)^3}f_1, \label{10a}\end{aligned}$$ and the $\Delta$-contribution following from the nonlocality of the scattering integral $$\begin{aligned} \Delta {\cal E}&=&{1\over 2}\sum_{a,b}\int{dkdpdq\over(2\pi)^9} P (\varepsilon_1+\varepsilon_2)\Delta_t, \nonumber\\ \Delta {\cal J}_{ij}&=&{1\over 2} \sum_{a,b}\int{dkdpdq\over(2\pi)^9} P \left[(p\!+\!q)\Delta_4+(k\!-\!q)\Delta_3-p\Delta_2\right], \nonumber\\ \Delta n_a&=&\sum_b\int{dkdpdq\over(2\pi)^9} P \Delta_t, \label{10}\end{aligned}$$ where $P=|T_{\rm sc}^R|^22\pi\delta(\varepsilon_1\!+\!\varepsilon_2\!- \!\varepsilon_3\!-\!\varepsilon_4)f_1f_2(1\!-\!f_3\!-\!f_4)$. The arguments denoted by numerical subscripts are identical to those used in (\[1\]), for all $\Delta$’s are explicit. The T-matrix $T_{\rm ex}$ used in the quasiparticle part of energy is the real part of the antisymmetrized Bethe-Goldstone T-matrix, $T^R_{\rm ex} =(1-\delta_{ab})T^R_{\rm sc}+\delta_{ab}\sqrt{2}T^R_{\rm sc}$. The actual observables are sum of the quasiparticle part and the $\Delta$-correction. In the low density limit, the $\Delta$-contributions (\[10\]) become proportional to the square of density. Therefore they turn into the second order virial corrections. In the degenerated system, the density dependence cannot be expressed in the power-law expansion since the T-matrix, and consequently all $\Delta$’s, depend on the density. Nevertheless, we find it instructive to call the $\Delta$-contribution the virial corrections because of their similar structure. The total energy ${\cal E}={\cal E}^{\rm qp}+\Delta{\cal E}$ conserves within kinetic equation (\[2\]). This energy conservation law generalizes the result of Bornath, Kremp, Kraeft and Schlanges [@BKKS96] restricted to non-degenerated systems. At degenerated systems, a new mechanism of energy conversion appears due to the medium effect on binary collisions. This mechanism can be seen writing the energy conservation ${\partial\over\partial t}{\cal E}= {\partial\over\partial t}{\cal E}^{\rm qp}+ {\partial\over\partial t}\Delta{\cal E}=0$ in a form $${\partial{\cal E}^{\rm qp}\over\partial t}= \sum_a\int{dk\over(2\pi)^3}\varepsilon{\partial f_1\over\partial t}- \sum_{a,b}\int{dkdpdq\over(2\pi)^9}P\Delta_E. \label{12}$$ The first term on the right hand side is the drift contribution to the energy balance. It is the only mechanism which appears in the absence of the virial corrections. The second term is the mean energy gain. It provides the conversion of the interaction energy controlled by two-particle correlations into energies of single-particle excitations. By this mechanism, the latent heat hidden in the interaction energy is converted into “thermal” excitations. The non-zero energy conversion, $\Delta_E\not=0$, results from the time-dependency of the scattering phase shift on the quasiparticle distribution via the Pauli blocking of internal states. Accordingly, the energy conversion is a consequence of the in-medium effects. The energy gain has its space counterpart in the momentum gain to the stress forces ${\partial\over\partial r_j}{\cal J}_{ij}$. The energy density contributing to the stress tensor (\[10a\]) has the gradient $${\partial{\cal E}^{\rm qp}\over\partial r_i}= \sum_a\int{dk\over(2\pi)^3}\varepsilon{\partial f_1\over\partial r_i}- \sum_{a,b}\int{dkdpdq\over(2\pi)^9}P\Delta_K. \label{13}$$ In the absence of the virial corrections (\[10\]), the first term of (\[13\]) combines together with the derivative of the first term in (\[10a\]) into the standard quasiparticle contribution. In the presence of the virial corrections, the energy gain is necessary to obtain the correct momentum conservation law. The density of energy ${\cal E}={\cal E}^{\rm qp}+\Delta{\cal E}$ given by (\[10a\]) and (\[10\]) alternatively results from Kadanoff and Baym formula, $${\cal E}=\sum_a\int{dk\over(2\pi)^3}\int{d\omega\over 2\pi} {1\over 2}\left(\omega+{k^2\over 2m}\right)G^<(\omega,k,r,t), \label{11}$$ with $G^<$ in the extended quasiparticle approximation (\[5\]). The particle density $n_a=n_a^{\rm qp}+\Delta n_a$ obtained from (\[5\]) via the definition, $n_a=\int{d\omega\over 2\pi} {dk\over(2\pi)^3}G^<$, also confirms (\[10a\]) and (\[10\]). The equivalence of these two alternative approaches confirms that the extended quasiparticle approximation is thermodynamically consistent with the nonlocal corrections to the scattering integral. For equilibrium distributions, formulas (\[10a\]) and (\[10\]) provide equations of state. Two known cases are worthy of comparison. First, the particle density $n_a=n_a^{\rm qp}+\Delta n_a$ is identical to the quantum Beth-Uhlenbeck equation of state [@BKKS96; @MR94], where $n_a^{\rm qp}$ is called the free density and $\Delta n_a$ the correlated density. Second, the virial correction to the stress tensor has a form of the collision flux contribution known in the theory of moderately dense gases [@CC90; @HCB64]. Summary ======= In this Letter we have derived the kinetic equation (\[9\]) which is consistent with thermodynamic observables (\[10a\]) and (\[10\]) up to the second order virial coefficient. The presented theory extends the theory of quantum gases [@NTL91; @H90] and non-ideal plasma [@BKKS96] to degenerated system. The most important new mechanism is the energy conversion which follows from the medium effect on binary collisions. The proposed corrections can be evaluated from known in-medium T-matrices and incorporated into existing Monte Carlo simulation codes, e.g., with the routine used in [@KDB96]. The nonlocal corrections to the scattering integral and corresponding second order virial corrections to thermodynamic observables enlighten the link between the kinetic equation approach and the hydrodynamical modeling. With expected progress in fits of the single-particle energy and other ingredients of the kinetic equation, the virial corrections can improve our ability to infer the equation of state from the heavy ion collision data. The authors are grateful to P. Danielewicz, D. Kremp and G. Röpke for stimulating discussions. This work was supported from Grant Agency of Czech Republic under contracts Nos. 202960098 and 202960021, the BMBF (Germany) under contract Nr. 06R0884, the Max-Planck-Society and the EC Human Capital and Mobility Programme. H. Stöcker, W. Greiner, Physics Reports [**137**]{}, 277 (1986). G. F. Bertsch, S. Das Gupta, Physics Reports [**160**]{}, 189 (1988). S. Chapman, T. G. Cowling, [*The Mathematical Theory of Non-uniform Gases*]{}, (Cambridge University Press, Third edition 1990), Chap. 16. J. O. Hirschfelder, Ch. F. Curtiss, R. B. Bird, [*Molecular Theory of Gases and Liquids*]{}, Chapts. 6.4a and 9.3, (Wiley, New York 1964). R. Malfliet, Nucl. Phys. [**A 420**]{}, 621 (1984). G. Kortemeyer, F. Daffin, W. Bauer, Phys. Lett. B [**374**]{}, 25 (1996). P. Danielewicz, S. Pratt, Phys. Rev. C [**53**]{}, 249 (1996). V. Špička, P. Lipavský, K. Morawetz, Phys. Rev. B. [**55**]{}, 5084 (1997); 5095 (1997). N. N. Bogoliubov, J. Phys. (USSR) [**10**]{}, 256 (1946); transl. in [*Studies in Statistical Mechanics*]{}, Vol. 1, editors D. de Boer and G. E. Uhlenbeck (North-Holland, Amsterdam 1962). H. S. Green, [*The Molecular Theory of Fluids*]{} (North-Holland, Amsterdam 1952). Later studies of higher-order corrections (for hard spheres) have undermined a trust in any virial expansions of the kinetic equation finding divergencies which result in nonanalytic behavior of the scattering integral with the density, e.g. K. Kawasaki and I. Oppenheim, Phys. Rev. [**139**]{}, A 1763 (1965). The comparison of Enskog’s equation with experiment is, however, a sufficient reason to use this expansion in spite of its questionable theoretical support. R. Snider, J. Chem. Phys. [**32**]{}, 1051 (1960). P. J. Nacher, G. Tastevin, F. Laloë, Annalen der Physik [**48**]{}, 149 (1991); Journal de Physique I [**1**]{}, 181 (1991). M. de Haan, Physica A [**164**]{}, 373 (1990); [**165**]{}, 224 (1990); [**170**]{}, 571 (1191). K. Baerwinkel, Z. Naturforsch. [**24**]{} a, 22 and 38 (1969). P. Danielewicz, Ann. Phys. (N.Y.) [**152**]{}, 239 (1984). V. Špička, P. Lipavský, Phys. Rev. Lett. [**73**]{}, 3439 (1994); Phys. Rev. B [**52**]{}, 14615 (1995). K. Morawetz, G. Röpke, Phys. Rev. E [**51**]{}, 4246 (1995). Th. Bornath, D. Kremp, W. D. Kraeft, M. Schlanges, Phys. Rev. E [**54**]{}, 3274 (1996). H. S. Köhler, R. Malfliet, Phys. Rev. C [**48**]{}, 1034 (1993). T. Alm, G. Röpke, M. Schmidt, Phys. Rev. C [**50**]{}, 31 (1994). [^1]: The well known term $[{\rm Re}G,\Sigma^<]$. [^2]: The basic idea of the recollection can be demonstrated on the following rearrangement of the gradient approximation of a matrix product $C(1,2)=A(1,\bar 3)B(\bar 3,2)$. In the mixed representation $C=AB+{i\over 2}\left( {\partial A\over\partial\omega}{\partial B\over\partial t}- {\partial A\over\partial t}{\partial B\over\partial\omega}- {\partial A\over\partial k}{\partial B\over\partial r}+ {\partial A\over\partial r}{\partial B\over\partial k}\right)$, see [@D84; @SL95]. We denote $\varphi={i\over 2}\ln A$ and rearrange the product as $C=A\left(B+ {\partial \varphi\over\partial\omega}{\partial B\over\partial t}- {\partial \varphi\over\partial t}{\partial B\over\partial\omega}- {\partial \varphi\over\partial k}{\partial B\over\partial r}+ {\partial \varphi\over\partial r}{\partial B\over\partial k}\right)$. The gradient term in brackets can be viewed as a linear expansion of $B$ with all arguments shifted as .

ArXiv

--- abstract: 'The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with $d\gg4$ and for the spread-out model with $d>4$ and $L\gg1$, without assuming reflection positivity.' author: - 'Akira Sakai[^1]' date: 'October 26, 2005[^2]' title: Lace expansion for the Ising model --- Introduction and results ======================== Model and the motivation {#ss:model} ------------------------ The Ising model is a statistical-mechanical model that was first introduced in [@i25] as a model for magnets. Consider the $d$-dimensional integer lattice ${{\mathbb Z}^d}$, and let $\Lambda$ be a finite subset of ${{\mathbb Z}^d}$ containing the origin $o\in{{\mathbb Z}^d}$. For example, $\Lambda$ is a $d$-dimensional hypercube centered at the origin. At each site $x\in\Lambda$, there is a spin variable $\varphi_x$ that takes values either $+1$ or $-1$. The Hamiltonian represents the energy of the system, and is defined by $$\begin{aligned} {\label{eq:hamilton}} H^h_\Lambda(\varphi)=-\sum_{\{x,y\}\subset\Lambda}J_{x,y}\varphi_x\varphi_y -h\sum_{x\in\Lambda}\varphi_x,\end{aligned}$$ where $\varphi\equiv\{\varphi_x\}_{x\in\Lambda}$ is a spin configuration, $\{J_{x,y}\}_{x,y\in{{\mathbb Z}^d}}$ is a collection of spin-spin couplings, and $h\in{{\mathbb R}}$ represents the strength of an external magnetic field uniformly imposed on $\Lambda$. We say that the model is ferromagnetic if $J_{x,y}\ge0$ for all pairs $\{x,y\}$; in this case, the Hamiltonian becomes lower as more spins align. The partition function $Z_{p,h;\Lambda}$ at the inverse temperature $p\ge0$ is the expectation of the Boltzmann factor $e^{-pH^h_\Lambda(\varphi)}$ with respect to the product measure $\prod_{x\in\Lambda}(\frac12{\mathbbm{1}{\scriptstyle\{\varphi_x=+1\}}}+\frac12{\mathbbm{1}{\scriptstyle\{\varphi_x=-1\}}})$: $$\begin{aligned} {\label{eq:ZL-def}} Z_{p,h;\Lambda}=2^{-|\Lambda|}\sum_{\varphi\in\{\pm1\}^\Lambda}e^{-pH^h_\Lambda (\varphi)}.\end{aligned}$$ Then, we denote the thermal average of a function $f=f(\varphi)$ by $$\begin{aligned} {\label{eq:faver}} {{\langle f \rangle}}_{p,h;\Lambda}=\frac{2^{-|\Lambda|}}{Z_{p,h;\Lambda}}\sum_{\varphi\in \{\pm1\}^\Lambda}f(\varphi)\,e^{-pH^h_\Lambda(\varphi)}.\end{aligned}$$ Suppose that the spin-spin coupling is translation-invariant, ${{\mathbb Z}^d}$-symmetric and finite-range (i.e., there exists an $L<\infty$ such that $J_{o,x}=0$ if $\|x\|_\infty>L$) and that $J_{o,x}\ge0$ for any $x\in{{\mathbb Z}^d}$ and $h\ge0$. Then, there exist monotone infinite-volume limits of ${{\langle \varphi_x \rangle}}_{p,h;\Lambda}$ and ${{\langle \varphi_x\varphi_y \rangle}}_{p,h;\Lambda}$. Let $$\begin{aligned} M_{p,h}=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}{{\langle \varphi_o \rangle}}_{p,h;\Lambda},&& G_p(x)=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}{{\langle \varphi_o\varphi_x \rangle}}_{p,h=0; \Lambda},&& \chi_p=\sum_{x\in{{\mathbb Z}^d}}G_p(x).\end{aligned}$$ When $d\ge2$, there exists a unique critical inverse temperature ${p_\text{c}}\in(0,\infty)$ such that the spontaneous magnetization $M^+_p\equiv\lim_{h\downarrow0}M_{p,h}$ equals zero, $G_p(x)$ decays exponentially as $|x|\uparrow\infty$ (we refer, e.g., to [@civ03] for a sharper Ornstein-Zernike result) and thus the magnetic susceptibility $\chi_p$ is finite if $p<{p_\text{c}}$, while $M^+_p>0$ and $\chi_p=\infty$ if $p>{p_\text{c}}$ (see [@abf87] and references therein). We should also refer to [@b05b] for recent results on the phase transition for the Ising model. We are interested in the behavior of these observables around $p={p_\text{c}}$. The susceptibility $\chi_p$ is known to diverge as $p\uparrow{p_\text{c}}$ [@a82; @ag83]. It is generally expected that $\lim_{p\downarrow{p_\text{c}}}M^+_p=\lim_{h\downarrow0}M_{{p_\text{c}},h}=0$. We believe that there are so-called critical exponents $\gamma=\gamma(d)$, $\beta=\beta(d)$ and $\delta=\delta(d)$, which are insensitive to the precise definition of $J_{o,x}\ge0$ (universality), such that (we use below the limit notation “$\approx$” in some appropriate sense) $$\begin{aligned} M^+_p\stackrel{p\downarrow{p_\text{c}}}{\approx}(p-{p_\text{c}})^\beta,&& \chi_p\stackrel{p\uparrow{p_\text{c}}}{\approx}({p_\text{c}}-p)^{-\gamma},&& M_{{p_\text{c}},h}\stackrel{h\downarrow0}{\approx}h^{1/\delta}.\end{aligned}$$ These exponents (if they exist) are known to obey the mean-field bounds: $\beta\leq1/2$, $\gamma\ge1$ and $\delta\ge3$. For example, $\beta=1/8$, $\gamma=7/4$ and $\delta=15$ for the nearest-neighbor model on ${{\mathbb Z}}^2$ [@o44]. Our ultimate goal is to identify the values of the critical exponents in other dimensions and to understand the universality for the Ising model. There is a sufficient condition, the so-called bubble condition, for the above critical exponents to take on their respective mean-field values. Namely, the finiteness of $\sum_{x\in{{\mathbb Z}^d}}G_{{p_\text{c}}}(x)^2$ (or the finiteness of $\sum_{x\in{{\mathbb Z}^d}}G_p(x)^2$ uniformly in $p<{p_\text{c}}$) implies that $\beta=1/2$, $\gamma=1$ and $\delta=3$ [@a82; @abf87; @af86; @ag83]. It is therefore crucial to know how fast $G_{{p_\text{c}}}(x)$ (or $G_p(x)$ near $p={p_\text{c}}$) decays as $|x|\uparrow\infty$. We note that the bubble condition holds for $d>4$ if the anomalous dimension $\eta$ takes on its mean-field value $\eta=0$, where the anomalous dimension is another critical exponent formally defined as $$\begin{aligned} {\label{eq:eta-formal}} G_{{p_\text{c}}}(x)\stackrel{|x|\uparrow\infty}{\approx}|x|^{-(d-2+\eta)}.\end{aligned}$$ Let $\hat J_k=\sum_{x\in{{\mathbb Z}^d}}J_{o,x}\,e^{ik\cdot x}$ and $\hat G_p(k)=\sum_{x\in{{\mathbb Z}^d}}G_p(x)\,e^{ik\cdot x}$ for $p<{p_\text{c}}$. For a class of models that satisfy the so-called reflection positivity [@fss76], the following infrared bound[^3] holds: $$\begin{aligned} {\label{eq:IRbd-so}} 0\leq\hat G_p(k)\leq\frac{\text{const.}}{\hat J_0-\hat J_k}\qquad \text{uniformly in }p<{p_\text{c}},\end{aligned}$$ where $d$ is supposed to be large enough to ensure integrability of the upper bound. For finite-range models, $d$ has to be bigger than 2, since $\hat J_0-\hat J_k\asymp|k|^2$, where “$f\asymp g$” means that $f/g$ is bounded away from zero and infinity. By Parseval’s identity, the infrared bound [(\[eq:IRbd-so\])]{} implies the bubble condition for finite-range reflection-positive models above four dimensions, and therefore $$\begin{aligned} {\label{eq:MFbehavior}} M^+_p\stackrel{p\downarrow{p_\text{c}}}{\asymp}(p-{p_\text{c}})^{1/2},&& \chi_p\stackrel{p\uparrow{p_\text{c}}}{\asymp}({p_\text{c}}-p)^{-1},&& M_{{p_\text{c}},h}\stackrel{h\downarrow0}{\asymp}h^{1/3}.\end{aligned}$$ The class of reflection-positive models includes the nearest-neighbor model, a variant of the next-nearest-neighbor model, Yukawa potentials, power-law decaying interactions, and their combinations [@bcc05]. For the nearest-neighbor model, we further obtain the following $x$-space Gaussian bound [@s82]: for $x\ne o$, $$\begin{aligned} {\label{eq:IRbd-sokal}} G_p(x)\leq\frac{\text{const.}}{|x|^{d-2}}\qquad\text{uniformly in }p<{p_\text{c}}.\end{aligned}$$ The problem in this approach to investigate critical behavior is that, since general finite-range models do not always satisfy reflection positivity, their mean-field behavior cannot necessarily be established, even in high dimensions. If we believe in universality, we expect that finite-range models exhibit the same mean-field behavior as soon as $d>4$. Therefore, it has been desirable to have approaches that do not assume reflection positivity. The lace expansion has been used successfully to investigate mean-field behavior for self-avoiding walk, percolation, lattice trees/animals and the contact process, above the upper-critical dimension: 4, 6 (4 for oriented percolation), 8 and 4, respectively (see, e.g., [@s04]). One of the advantages in the application of the lace expansion is that we do not have to require reflection positivity to prove a Gaussian infrared bound and mean-field behavior. Another advantage is the possibility to show an asymptotic result for the decay of correlation. Our goal in this paper is to prove the lace-expansion results for the Ising model. Main results ------------ From now on, we fix $h=0$ and abbreviate, e.g., ${{\langle \varphi_o\varphi_x \rangle}}_{p,h=0;\Lambda}$ to ${{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}$. In this paper, we prove the following lace expansion for the two-point function, in which we use the notation $$\begin{aligned} \tau_{x,y}=\tanh(pJ_{x,y}).\end{aligned}$$ \[prp:Ising-lace\] For any $p\ge0$ and any $\Lambda\subset{{\mathbb Z}^d}$, there exist $\pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ and $R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)$ for $x\in\Lambda$ and $j\ge0$ such that $$\begin{aligned} {\label{eq:Ising-lace}} {{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}=\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x) +\sum_{u,v}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(u)\,\tau_{u,v}{{\langle \varphi_v\varphi_x \rangle}}_{p;\Lambda}+(-1)^{j+1}R_{p;\Lambda}^{{\scriptscriptstyle}(j +1)}(x),\end{aligned}$$ where $$\begin{aligned} {\label{eq:Pij-def}} \Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)&=\sum_{i=0}^j(-1)^i\,\pi_{p; \Lambda}^{{\scriptscriptstyle}(i)}(x).\end{aligned}$$ For the ferromagnetic case, we have the bounds $$\begin{aligned} {\label{eq:pij-Rj-naivebd}} \pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)\ge\delta_{j,0}\delta_{o,x},&& 0\leq R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)\leq\sum_{u,v}\pi_{p;\Lambda} ^{{\scriptscriptstyle}(j)}(u)\,\tau_{u,v}{{\langle \varphi_v\varphi_x \rangle}}_{p;\Lambda}.\end{aligned}$$ We defer the display of precise expressions of $\pi_{p;\Lambda}^{{\scriptscriptstyle}(i)}(x)$ and $R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)$ to Section \[sss:complexp\], since we need a certain representation to describe these functions. We introduce this representation in Section \[ss:RCrepr\] and complete the proof of Proposition \[prp:Ising-lace\] in Section \[ss:derivation\]. It is worth emphasizing that the above proposition holds independently of the properties of the spin-spin coupling: $J_{u,v}$ does not have to be translation-invariant or ${{\mathbb Z}^d}$-symmetric. In particular, the identity [(\[eq:Ising-lace\])]{} holds independently of the sign of the spin-spin coupling. A spin glass, whose spin-spin coupling is randomly negative, is an extreme example for which [(\[eq:Ising-lace\])]{} holds. Whether or not the lace expansion [(\[eq:Ising-lace\])]{} is useful depends on the possibility of good control on the expansion coefficients and the remainder. As explained below, it is indeed possible to have optimal bounds on the expansion coefficients for the nearest-neighbor interaction (i.e., $J_{o,x}={\mathbbm{1}{\scriptstyle\{\|x\|_1=1\}}}$) and for the following spread-out interaction: $$\begin{aligned} {\label{eq:J-def}} J_{o,x}=L^{-d}\mu(L^{-1}x)\qquad(1\leq L<\infty),\end{aligned}$$ where $\mu:[-1,1]^d\setminus\{o\}\mapsto[0,\infty)$ is a bounded probability distribution, which is symmetric under rotations by $\pi/2$ and reflections in coordinate hyperplanes, and piecewise continuous so that the Riemann sum $L^{-d}\sum_{x\in{{\mathbb Z}^d}}\mu(L^{-1}x)$ approximates $\int_{{{\mathbb R}^d}}d^dx\;\mu(x)\equiv1$. One of the simplest examples would be $$\begin{aligned} {\label{eq:Juniform-def}} J_{o,x}=\frac{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}}{\sum_{z\in{{\mathbb Z}^d}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|z \|_\infty\leq L\}$}}}}=O(L^{-d})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|L^{-1}x\|_\infty\leq1\}$}}}.\end{aligned}$$ \[prp:Pij-Rj-bd\] Let $\rho=2(d-4)>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, there are finite constants $\theta$ and $\lambda$ such that $$\begin{aligned} {\label{eq:prp-bds}} |\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)-\delta_{o,x}|&\leq\theta\delta_{o,x} +\frac{\lambda(1-\delta_{o,x})}{|x|^{d+2+\rho}}\quad(j\ge0),&& |R_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)|\to0\quad(j\uparrow\infty),\end{aligned}$$ for any $p\leq{p_\text{c}}$, any $\Lambda\subset{{\mathbb Z}^d}$ and any $x\in\Lambda$. The proof of Proposition \[prp:Pij-Rj-bd\] depends on certain bounds on the expansion coefficients in terms of two-point functions. These diagrammatic bounds arise from counting the number of “disjoint connections”, corresponding to applications of the BK inequality in percolation (e.g., [@bk85]). We prove these bounds in Section \[s:bounds\], and in anticipation of this, in Section \[s:reduction\] we explain how we use their implication to prove Proposition \[prp:Pij-Rj-bd\], with $\theta=O(d^{-1})$ and $\lambda=O(1)$ for the nearest-neighbor model, and $\theta=O(L^{-2+{\epsilon}})$ and $\lambda=O(\theta^2)$ with a small ${\epsilon}>0$ for the spread-out model. Let $$\begin{aligned} \tau\equiv\tau(p)=\sum_x\tau_{o,x},&& D(x)=\frac{\tau_{o,x}}{\tau},&& \sigma^2=\sum_x|x|^2D(x).\end{aligned}$$ Due to [(\[eq:prp-bds\])]{} uniformly in $\Lambda\subset{{\mathbb Z}^d}$, there is a limit $\Pi_p(x)\equiv\lim_{\Lambda\uparrow{{\mathbb Z}^d}} \lim_{j\uparrow\infty}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ such that $$\begin{aligned} {\label{eq:Ising-lace-Zdlim}} G_p(x)=\Pi_p(x)+(\Pi_p*\tau D*G_p)(x),&& |\Pi_p(x)-\delta_{o,x}|\leq\theta\delta_{o,x}+\frac{\lambda(1 -\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ for any $p\leq{p_\text{c}}$ and any $x\in{{\mathbb Z}^d}$, where $(f*g)(x)=\sum_{y\in{{\mathbb Z}^d}}f(y)\,g(x-y)$. We note that the identity in [(\[eq:Ising-lace-Zdlim\])]{} is similar to the recursion equation for the random-walk Green’s function: $$\begin{aligned} S_r(x)\equiv\sum_{i=0}^\infty r^iD^{*i}(x)=\delta_{o,x}+(rD*S_r)(x) \qquad(|r|<1),\end{aligned}$$ where $f^{*i}(x)=(f^{*(i-1)}*f)(x)$, with $f^{*0}(x)=\delta_{o,x}$ by convention. The leading asymptotics of $S_1(x)$ for $d>2$ is known as $\frac{a_d}{\sigma^2}|x|^{-(d-2)}$, where $a_d=\frac{d}2\pi^{-d/2}\Gamma(\frac{d}2-1)$ (e.g., [@h05; @hhs03]). Following the model-independent analysis of the lace expansion in [@h05; @hhs03], we obtain the following asymptotics of the critical two-point function: \[thm:x-asy\] Let $\rho=2(d-4)>0$ and fix any small ${\epsilon}>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, we have that, for $x\ne o$, $$\begin{aligned} {\label{eq:thm-asy}} G_{{p_\text{c}}}(x)=\frac{A}{\tau({p_\text{c}})}\,\frac{a_d}{\sigma^2|x|^{d-2}} \times\begin{cases} \big(1+O(|x|^{-\frac{(\rho-{\epsilon})\wedge2}d})\big)&(\text{NN model}),\\ \big(1+O(|x|^{-\rho\wedge2+{\epsilon}})\big) &(\text{SO model}), \end{cases}\end{aligned}$$ where constants in the error terms may vary depending on ${\epsilon}$, and $$\begin{aligned} {\label{eq:constants}} \tau({p_\text{c}})=\bigg(\sum_x\Pi_{{p_\text{c}}}(x)\bigg)^{-1},&& A=\bigg(1+\frac{\tau({p_\text{c}})}{\sigma^2}\sum_x|x|^2\Pi_{{p_\text{c}}}(x)\bigg)^{-1}.\end{aligned}$$ Consequently, [(\[eq:MFbehavior\])]{} holds and $\eta=0$. In this paper, we restrict ourselves to the nearest-neighbor model for $d\gg4$ and to the spread-out model for $d>4$ with $L\gg1$. However, it is strongly expected that our method can show the same asymptotics of the critical two-point function for *any* translation-invariant, ${{\mathbb Z}^d}$-symmetric finite-range model above four dimensions, by taking the coordination number sufficiently large. Organization ------------ In the rest of this paper, we focus our attention on the model-dependent ingredients: the lace expansion for the Ising model (Proposition \[prp:Ising-lace\]) and the bounds on (the alternating sum of) the expansion coefficients for the ferromagnetic models (Proposition \[prp:Pij-Rj-bd\]). In Section \[s:laceexp\], we prove Proposition \[prp:Ising-lace\]. In Section \[s:reduction\], we reduce Proposition \[prp:Pij-Rj-bd\] to a few other propositions, which are then results of the aforementioned diagrammatic bounds on the expansion coefficients. We prove these diagrammatic bounds in Section \[s:bounds\]. As soon as the composition of the diagrams in terms of two-point functions is understood, it is not so hard to establish key elements of the above reduced propositions. We will prove these elements in Section \[ss:proof-so\] for the spread-out model and in Section \[ss:proof-nn\] for the nearest-neighbor model. Lace expansion for the Ising model {#s:laceexp} ================================== The lace expansion was initiated by Brydges and Spencer [@bs85] to investigate weakly self-avoiding walk for $d>4$. Later, it was developed for various stochastic-geometrical models, such as strictly self-avoiding walk for $d>4$ (e.g., [@hs92]), lattice trees/animals for $d>8$ (e.g., [@hs90]), unoriented percolation for $d>6$ (e.g., [@hs90']), oriented percolation for $d>4$ (e.g., [@ny93]) and the contact process for $d>4$ (e.g., [@s01]). See [@s04] for an extensive list of references. This is the first lace-expansion paper for the Ising model. In this section, we prove the lace expansion [(\[eq:Ising-lace\])]{} for the Ising model. From now on, we fix $p\ge0$ and abbreviate, e.g., $\pi_{p;\Lambda}^{{\scriptscriptstyle}(i)}(x)$ to $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$. There may be several ways to derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$, using, e.g., the high-temperature expansion, the random-walk representation (e.g., [@ffs92]) or the FK random-cluster representation (e.g., [@fk72]). In this paper, we use the random-current representation (Section \[ss:RCrepr\]), which applies to models in the Griffiths-Simon class (e.g., [@a82; @ag83]). This representation is similar in philosophy to the high-temperature expansion, but it turned out to be more efficient in investigating the critical phenomena [@a82; @abf87; @af86; @ag83]. The main advantage in this representation is the source-switching lemma (Lemma \[lmm:switching\] below in Section \[sss:2ndexp\]) by which we have an identity for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda -{{\langle \varphi_o\varphi_x \rangle}}_{{\cal A}}$ with “${{\cal A}}\subset\Lambda$” (the meaning will be explained in Section \[ss:RCrepr\]). We will repeatedly apply this identity to complete the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ in Section \[sss:complexp\]. Random-current representation {#ss:RCrepr} ----------------------------- In this subsection, we describe the random-current representation and introduce some notation that will be essential in the derivation of the lace expansion. First we introduce some notions and notation. We call a pair of sites $b=\{u,v\}$ with $J_b\ne0$ a *bond*. So far we have used the notation $\Lambda\subset{{\mathbb Z}^d}$ for a site set. However, we will often abuse this notation to describe a *graph* that consists of sites of $\Lambda$ and are equipped with a certain bond set, which we denote by ${{\mathbb B}}_\Lambda$. Note that “$\{u,v\}\in{{\mathbb B}}_\Lambda$” always implies “$u,v\in\Lambda$”, but the latter does not necessarily imply the former. If we regard ${{\cal A}}$ and $\Lambda$ as graphs, then “${{\cal A}}\subset\Lambda$” means that ${{\cal A}}$ is a subset of $\Lambda$ as a site set, and that ${{\mathbb B}}_{{\cal A}}\subset{{\mathbb B}}_\Lambda$. Now we consider the partition function $Z_{{\cal A}}$ on ${{\cal A}}\subset\Lambda$. By expanding the Boltzmann factor in [(\[eq:ZL-def\])]{}, we obtain $$\begin{aligned} {\label{eq:ZA-rewr}} Z_{{\cal A}}&=2^{-|{{\cal A}}|}\sum_{\varphi\in\{\pm1\}^{{\cal A}}}\,\prod_{\{u,v\}\in{{\mathbb B}}_{{\cal A}}}\, \bigg(\sum_{n_{u,v}\in{{\mathbb Z}_+}}\frac{(p J_{u,v})^{n_{u,v}}}{n_{u,v}!} \,\varphi_u^{n_{u,v}}\varphi_v^{n_{u,v}}\bigg){\nonumber}\\ &=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}\bigg(\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b) ^{n_b}}{n_b!}\bigg)\prod_{v\in{{\cal A}}}\bigg(\frac12\sum_{\varphi_v=\pm1} \varphi_v^{\sum_{b\ni v}n_b}\bigg),\end{aligned}$$ where we call ${{\bf n}}=\{n_b\}_{b\in{{\mathbb B}}_{{\cal A}}}$ a *current configuration*. Note that the single-spin average in the last line equals 1 if $\sum_{b\ni v}n_b$ is an even integer, and 0 otherwise. Denoting by ${\partial}{{\bf n}}$ the set of *sources* $v\in\Lambda$ at which $\sum_{b\ni v}n_b$ is an *odd* integer, and defining $$\begin{aligned} {\label{eq:weight}} w_{{\cal A}}({{\bf n}})=\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b)^{n_b}}{n_b!}\qquad({{\bf n}}\in {{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}),\end{aligned}$$ we obtain $$\begin{aligned} {\label{eq:ZA-RCrepr1}} Z_{{\cal A}}=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}w_{{\cal A}}({{\bf n}})\,\prod_{v\in{{\cal A}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{ \sum_{b\ni v}n_b\text{ even}\}$}}}=\sum_{{\partial}{{\bf n}}={\varnothing}}w_{{\cal A}}({{\bf n}}).\end{aligned}$$ The partition function $Z_{{\cal A}}$ equals the partition function on $\Lambda$ with $J_b=0$ for all $b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{\cal A}}$. We can also think of $Z_{{\cal A}}$ as the sum of $w_\Lambda({{\bf n}})$ over ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ satisfying ${{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0$, where ${{\bf n}}|_{{\mathbb B}}$ is a projection of ${{\bf n}}$ over the bonds in a bond set ${{\mathbb B}}$, i.e., ${{\bf n}}|_{{\mathbb B}}=\{n_b:b\in{{\mathbb B}}\}$. By this observation, we can rewrite [(\[eq:ZA-RCrepr1\])]{} as $$\begin{aligned} {\label{eq:ZA-RCrepr2}} Z_{{\cal A}}=\sum_{\substack{{\partial}{{\bf n}}={\varnothing}\\ {{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus {{\mathbb B}}_{{{\cal A}}}} \equiv0}}w_\Lambda({{\bf n}}).\end{aligned}$$ Following the same calculation, we can rewrite $Z_{{\cal A}}{{\langle \varphi_x\varphi_y \rangle}}_{{\cal A}}$ for $x,y\in{{\cal A}}$ as $$\begin{aligned} {\label{eq:2pt-rewr}} Z_{{\cal A}}{{\langle \varphi_x\varphi_y \rangle}}_{{\cal A}}&=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}\bigg(\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b) ^{n_b}}{n_b!}\bigg)\prod_{v\in{{\cal A}}}\bigg(\frac12\sum_{\varphi_v=\pm1} \varphi_v^{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle {\scriptscriptstyle}\{v\in x{\vartriangle}y\}$}}}+\sum_{b\ni v}n_b}\bigg){\nonumber}\\ &=\sum_{{\partial}{{\bf n}}=x{\vartriangle}y}w_{{\cal A}}({{\bf n}})=\sum_{\substack{{\partial}{{\bf n}}=x{\vartriangle}y\\ {{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0}}w_\Lambda({{\bf n}}),\end{aligned}$$ where $x{\vartriangle}y$ is an abbreviation for the symmetric difference $\{x\}{\,\triangle\,}\{y\}$: $$\begin{aligned} {\label{eq:symmdiff}} x{\vartriangle}y\equiv\{x\}{\,\triangle\,}\{y\}=\begin{cases} {\varnothing}&\text{if }x=y,\\ \{x,y\}&\text{otherwise}. \end{cases}\end{aligned}$$ If $x$ or $y$ is in ${{\cal A}}{^{\rm c}}\equiv\Lambda\setminus{{\cal A}}$, then we define both sides of [(\[eq:2pt-rewr\])]{} to be zero. This is consistent with the above representation when $x\ne y$, since, for example, if $x\in{{\cal A}}{^{\rm c}}$, then the leftmost expression of [(\[eq:2pt-rewr\])]{} is a multiple of $\frac12\sum_{\varphi_x=\pm1}\varphi_x=0$, while the last expression in [(\[eq:2pt-rewr\])]{} is also zero because there is no way of connecting $x$ and $y$ on a current configuration ${{\bf n}}$ with ${{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0$. The key observation in the representation [(\[eq:2pt-rewr\])]{} is that the right-hand side is nonzero only when $x$ and $y$ are connected by a chain of bonds with *odd* currents (see Figure \[fig:RCrepr\]). ![\[fig:RCrepr\]A current configuration with sources at $x$ and $y$. The thick-solid segments represent bonds with odd currents, while the thin-solid segments represent bonds with positive even currents, which cannot be seen in the high-temperature expansion.](RCrepr) We will exploit this peculiar underlying percolation picture to derive the lace expansion for the two-point function. Derivation of the lace expansion {#ss:derivation} -------------------------------- In this subsection, we derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ using the random-current representation. In Section \[sss:1stexp\], we introduce some definitions and perform the first stage of the expansion, namely [(\[eq:Ising-lace\])]{} for $j=0$, simply using inclusion-exclusion. In Section \[sss:2ndexp\], we perform the second stage of the expansion, where the source-switching lemma (Lemma \[lmm:switching\]) plays a significant role to carry on the expansion indefinitely. Finally, in Section \[sss:complexp\], we complete the proof of Proposition \[prp:Ising-lace\]. ### The first stage of the expansion {#sss:1stexp} As mentioned in Section \[ss:RCrepr\], the underlying picture in the random-current representation is quite similar to percolation. We exploit this similarity to obtain the lace expansion. First, we introduce some notions and notation. \[defn:perc\] (i) Given ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ and ${{\cal A}}\subset\Lambda$, we say that $x$ is ${{\bf n}}$-connected to $y$ in (the graph) ${{\cal A}}$, and simply write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y$ *in* ${{\cal A}}$, if either $x=y\in{{\cal A}}$ or there is a self-avoiding path (or we simply call it a path) from $x$ to $y$ consisting of bonds $b\in{{\mathbb B}}_{{\cal A}}$ with $n_b>0$. If ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}$, we omit “in ${{\cal A}}$” and simply write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y$. We also define $$\begin{aligned} {\label{eq:incl/excl}} \{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}y\}=\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y\}\setminus\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y\text{ in } {{\cal A}}{^{\rm c}}\},\end{aligned}$$ and say that $x$ is ${{\bf n}}$-connected to $y$ *through* ${{\cal A}}$. (ii) Given an event $E$ (i.e., a set of current configurations) and a bond $b$, we define $\{E$ off $b\}$ to be the set of current configurations ${{\bf n}}\in E$ such that changing $n_b$ results in a configuration that is also in $E$. Let ${{\cal C}}_{{\bf n}}^b(x)=\{y:x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y\text{ off }b\}$. (iii) For a *directed* bond $b=(u,v)$, we write ${\underline{b}}=u$ and ${\overline{b}}=v$. We say that a directed bond $b$ is *pivotal* for $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y$ from $x$, if $\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}$ off $b\}\cap\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y$ in ${{\cal C}}_{{\bf n}}^b(x){^{\rm c}}\}$ occurs. If $\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y\}$ occurs with no pivotal bonds, we say that $x$ is *${{\bf n}}$-doubly connected to* $y$, and write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}y$. We begin with the first stage of the lace expansion. First, by using the above percolation language, the two-point function can be written as $$\begin{aligned} {\label{eq:2pt-perclang}} {{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\equiv\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}$}}}}.\end{aligned}$$ We decompose the indicator on the right-hand side into two parts depending on whether or not there is a pivotal bond for $o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $o$; if there is, we take the *first* bond among them. Then, we have $$\begin{aligned} {\label{eq:0th-ind-fact}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}+\sum_{b\in{{\mathbb B}}_\Lambda}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}} {\underline{b}}\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{\bf n}}^b(o) {^{\rm c}}\}$}}}.\end{aligned}$$ Let $$\begin{aligned} {\label{eq:pi0-def}} \pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Substituting [(\[eq:0th-ind-fact\])]{} into [(\[eq:2pt-perclang\])]{}, we obtain (see Figure \[fig:1stpiv\]) $$\raisebox{0.2pc}{\includegraphics[scale=0.17]{1stpiv1}}~~~=~~~ \raisebox{-0.5pc}{\includegraphics[scale=0.17]{1stpiv2}}~~~+~~ \sum_b~\raisebox{-1.7pc}{\includegraphics[scale=0.17]{1stpiv3}}$$ $$\begin{aligned} {\label{eq:pre-1st-exp}} {{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in {{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x \text{ in }{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}\}$}}}.\end{aligned}$$ Next, we consider the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{}. Since $b$ is pivotal for $o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $o\,(\ne x$, due to the last indicator) and ${\partial}{{\bf n}}=o{\vartriangle}x$, in fact $n_b$ is an *odd* integer. We alternate the parity of $n_b$ by changing the source constraint into $o{\vartriangle}b{\vartriangle}x\equiv\{o\}{\,\triangle\,}\{{\underline{b}},{\overline{b}}\}{\,\triangle\,}\{x\}$ and multiplying by $$\begin{aligned} \frac{\sum_{n\text{ odd}}(p J_b)^n/n!}{\sum_{n\text{ even}}(p J_b)^n/n!}=\tanh(p J_b)\equiv\tau_b.\end{aligned}$$ Then, the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{} equals $$\begin{aligned} {\label{eq:0th-summand1}} \sum_{{\partial}{{\bf n}}=o{\vartriangle}b{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}\}$}}}.\end{aligned}$$ Note that, except for $b$, there are no positive currents on the boundary bonds of ${{\cal C}}_{{\bf n}}^b(o)$. Now, we condition on ${{\cal C}}_{{\bf n}}^b(o)={{\cal A}}$ and decouple events occurring on ${{\mathbb B}}_{{{\cal A}}{^{\rm c}}}$ from events occurring on ${{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}$, by using the following notation: $$\begin{aligned} {\label{eq:tildew-def}} \tilde w_{\Lambda,{{\cal A}}}({{\bf k}})=\prod_{b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{(pJ_b)^{k_b}}{k_b!}\qquad({{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda \setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}).\end{aligned}$$ Conditioning on ${{\cal C}}_{{\bf n}}^b(o)={{\cal A}}$, multiplying $Z_{{{\cal A}}{^{\rm c}}}/Z_{{{\cal A}}{^{\rm c}}}\equiv1$ (and using the notation ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}$ and ${{\bf m}}={{\bf n}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}$) and then summing over ${{\cal A}}\subset\Lambda$, we have $$\begin{aligned} {\label{eq:0th-summand2}} {(\ref{eq:0th-summand1})}&=\sum_{{{\cal A}}\subset\Lambda}\,\sum_{\substack{{\partial}{{\bf k}}=o{\vartriangle}{\underline{b}}\\ {\partial}{{\bf m}}={\overline{b}}{\vartriangle}x}}\frac{\tilde w_{\Lambda,{{\cal A}}}({{\bf k}})\, Z_{{{\cal A}}{^{\rm c}}}}{Z_\Lambda}\,\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf k}}}^b(o)={{\cal A}}}\, \tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{k_b\text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}{\nonumber}\\ &=\sum_{{{\cal A}}\subset\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{\bf n}}^b(o)={{\cal A}}} \,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\underbrace{\sum_{{\partial}{{\bf m}}={\overline{b}}{\vartriangle}x}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}} {\overset{}{\longleftrightarrow}}}x\text{ (in }{{\cal A}}{^{\rm c}})\}$}}}}_{=\;{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}}{\nonumber}\\ &=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$ Furthermore, “off $b$” and ${\mathbbm{1}{\scriptstyle\{n_b\text{ even}\}}}$ in the last line can be omitted, since $\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\}\setminus\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}$ off $b\}$ and $\{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}\}\cap\{n_b$ odd} are subsets of $\{{\overline{b}}\in{{\cal C}}_{{\bf n}}^b(o)\}$, on which ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}=0$. As a result, $$\begin{aligned} {\label{eq:0th-summand3}} {(\ref{eq:0th-summand2})}~=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$ By [(\[eq:pre-1st-exp\])]{} and [(\[eq:0th-summand3\])]{}, we arrive at $$\begin{aligned} {\label{eq:1st-exp}} {{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in {{\mathbb B}}_\Lambda}\pi_\Lambda^{{\scriptscriptstyle}(0)}({\underline{b}})\,\tau_b\,{{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_\Lambda-R_\Lambda^{{\scriptscriptstyle}(1)}(x),\end{aligned}$$ where $$\begin{aligned} {\label{eq:R1-def}} R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_{b\in{{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}} \frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\Big( {{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{ {{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}\Big).\end{aligned}$$ This completes the proof of [(\[eq:Ising-lace\])]{} for $j=0$, with $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ being defined in [(\[eq:pi0-def\])]{} and [(\[eq:R1-def\])]{}, respectively. ### The second stage of the expansion {#sss:2ndexp} In the next stage of the lace expansion, we further expand $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ in [(\[eq:1st-exp\])]{}. To do so, we investigate the difference ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}$ in [(\[eq:R1-def\])]{}. First, we prove the following key proposition[^4]: \[prp:through\] For $v,x\in\Lambda$ and ${{\cal A}}\subset\Lambda$, we have $$\begin{aligned} {\label{eq:lmm-through}} {{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}} =\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}} ({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Therefore, ${{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\leq {{\langle \varphi_v\varphi_x \rangle}}_\Lambda$ for the ferromagnetic case. Since both sides of [(\[eq:lmm-through\])]{} are equal to ${\mathbbm{1}{\scriptstyle\{x\in{{\cal A}}\}}}$ when $v=x$ (see below [(\[eq:symmdiff\])]{}), it suffices to prove [(\[eq:lmm-through\])]{} for $v\ne x$. First, by using [(\[eq:ZA-RCrepr1\])]{}–[(\[eq:2pt-rewr\])]{}, we obtain $$\begin{aligned} {\label{eq:WZ-num}} Z_\Lambda Z_{{{\cal A}}{^{\rm c}}}\Big({{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v \varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\Big)=\sum_{{\partial}{{\bf n}}=\{v,x\}}Z_{{{\cal A}}{^{\rm c}}}\,w_\Lambda ({{\bf n}})-\sum_{{\partial}{{\bf m}}=\{v,x\}}w_{{{\cal A}}{^{\rm c}}}({{\bf m}})\,Z_\Lambda{\nonumber}\\ =\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus {{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}})-\sum_{\substack{ {\partial}{{\bf m}}=\{v,x\},\,{\partial}{{\bf n}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}} \equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}}).\end{aligned}$$ Note that the second term is equivalent to the first term if the source constraints for ${{\bf m}}$ and ${{\bf n}}$ are exchanged. Next, we consider the second term of [(\[eq:WZ-num\])]{}, whose exact expression is $$\begin{gathered} {\label{eq:2ndterm-expl}} \sum_{\substack{{\partial}{{\bf m}}=\{v,x\},\,{\partial}{{\bf n}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus {{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\bigg(\prod_{b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{(pJ_b)^{n_b}}{n_b!}\bigg)\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{ (pJ_b)^{m_b+n_b}}{m_b!\,n_b!}=\sum_{{\partial}{{\bf N}}=\{v,x\}}w_\Lambda({{\bf N}})\sum_{ \substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}} \equiv0}}\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{gathered}$$ The following is a variant of the source-switching lemma [@a82; @ghs70] and allows us to change the source constraints in [(\[eq:2ndterm-expl\])]{}. \[lmm:switching\] $$\begin{aligned} {\label{eq:switching}} \sum_{\substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {{\bf m}}|_{ {{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{aligned}$$ The idea of the proof of [(\[eq:switching\])]{} can easily be extended to more general cases, in which the source constraint in the left-hand side of [(\[eq:switching\])]{} is replaced by ${\partial}{{\bf m}}={{\cal V}}$ for some ${{\cal V}}\subset\Lambda$ and that in the right-hand side is replaced by ${\partial}{{\bf m}}={{\cal V}}{\,\triangle\,}\{v,x\}$ (e.g., [@a82]). We will explain the proof of [(\[eq:switching\])]{} after completing the proof of Proposition \[prp:through\]. We continue with the proof of Proposition \[prp:through\]. Substituting [(\[eq:switching\])]{} into [(\[eq:2ndterm-expl\])]{}, we obtain $$\begin{aligned} {\label{eq:switching-appl}} {(\ref{eq:2ndterm-expl})}&=\sum_{{\partial}{{\bf N}}=\{v,x\}}w_\Lambda({{\bf N}})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in {{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}{\nonumber}\\ &=\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda \setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}})\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}.\end{aligned}$$ Note that the source constraints for ${{\bf m}}$ and ${{\bf n}}$ in the last line are identical to those in the first term of [(\[eq:WZ-num\])]{}, under which ${\mathbbm{1}{\scriptstyle\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}}}$ is always 1. By [(\[eq:incl/excl\])]{}, we can rewrite [(\[eq:WZ-num\])]{} as $$\begin{aligned} {\label{eq:through}} {{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}} &=\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda \setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\frac{w_\Lambda({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}} \,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Using [(\[eq:ZA-RCrepr1\])]{}–[(\[eq:ZA-RCrepr2\])]{} to omit “${{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0$” and replace $w_\Lambda({{\bf m}})$ by $w_{{{\cal A}}{^{\rm c}}}({{\bf m}})$, we arrive at [(\[eq:lmm-through\])]{}. This completes the proof of Proposition \[prp:through\]. We explain the meaning of the identity [(\[eq:switching\])]{} and the idea of its proof. Given ${{\bf N}}=\{N_b\}_{b\in{{\mathbb B}}_\Lambda}$, we denote by ${{\mathbb G}}_{{\bf N}}$ the graph consisting of $N_b$ *labeled* edges between ${\underline{b}}$ and ${\overline{b}}$ for every $b\in{{\mathbb B}}_\Lambda$ (see Figure \[fig:switching\]). $$\begin{aligned} {{\bf N}}~:&\qquad\includegraphics[scale=0.33]{switching1}\\[5pt] {{\mathbb G}}_{{\bf N}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33] {switching2}}\\[1pc] {{\mathbb S}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33] {switching3}}\\[7pt] {{\mathbb S}}{\,\triangle\,}\omega~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33] {switching4}}\end{aligned}$$ For a subgraph ${{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}$, we denote by ${\partial}{{\mathbb S}}$ the set of vertices at which the number of incident edges in ${{\mathbb S}}$ is *odd*, and let ${{\mathbb S}}_{{\cal A}}={{\mathbb S}}\cap{{\mathbb G}}_{{{\bf N}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}}$. Then, the left-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}|$ of $$\begin{aligned} {\label{eq:Sbefore}} {\mathfrak{S}}=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial}{{\mathbb S}}=\{v,x\},~{{\mathbb S}}_{{\cal A}}={\varnothing}\},\end{aligned}$$ and the sum in the right-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}'|$ of $$\begin{aligned} {\mathfrak{S}}'=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial}{{\mathbb S}}={\varnothing},~{{\mathbb S}}_{{\cal A}}={\varnothing}\}.\end{aligned}$$ We note that $|{\mathfrak{S}}|$ is zero when there are no paths on ${{\mathbb G}}_{{\bf N}}$ between $v$ and $x$ consisting of edges whose endvertices are both in ${{\cal A}}{^{\rm c}}$, while $|{\mathfrak{S}}'|$ may not be zero. The identity [(\[eq:switching\])]{} reads that $|{\mathfrak{S}}|$ equals $|{\mathfrak{S}}'|$ if we compensate for this discrepancy. Suppose that there is a path (i.e., a ) $\omega$ from $v$ to $x$ consisting of edges in ${{\mathbb G}}_{{\bf N}}$ whose endvertices are both in ${{\cal A}}{^{\rm c}}$. Then, the map $$\begin{aligned} {\label{eq:bijection}} {{\mathbb S}}\in{\mathfrak{S}}~\mapsto~{{\mathbb S}}{\,\triangle\,}\omega\in{\mathfrak{S}}',\end{aligned}$$ is a bijection [@a82; @ghs70], and therefore $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$. Here and in the rest of the paper, the symmetric difference between graphs is only in terms of *edges*. For example, ${{\mathbb S}}{\,\triangle\,}\omega$ is the result of adding or deleting edges (not vertices) contained in $\omega$. This completes the proof of [(\[eq:switching\])]{}. We now start with the second stage of the expansion by using Proposition \[prp:through\] and applying inclusion-exclusion as in the first stage of the expansion in Section \[sss:1stexp\]. First, we decompose the indicator in [(\[eq:lmm-through\])]{} into two parts depending on whether or not there is a pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $v$ such that $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Let $$\begin{aligned} {\label{eq:E-def}} E_{{\bf N}}(v,x;{{\cal A}})=\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{\nexists \text{ pivotal bond }b\text{ for }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{}{\longleftrightarrow}}}x \text{ from $v$ such that }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}\}.\end{aligned}$$ On the event $\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\setminus E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$, we take the *first* pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $v$ satisfying $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Then, we have (cf., [(\[eq:0th-ind-fact\])]{}) $$\begin{aligned} {\label{eq:1st-ind-fact}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}+\sum_{b\in {{\mathbb B}}_\Lambda}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}\}$}}}.\end{aligned}$$ Let $$\begin{aligned} {\label{eq:Theta-def}} \Theta_{v,x;{{\cal A}}}[X]=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}\,X({{\bf m}}+{{\bf n}}),&& \Theta_{v,x;{{\cal A}}}=\Theta_{v,x;{{\cal A}}}[1].\end{aligned}$$ Substituting [(\[eq:1st-ind-fact\])]{} into [(\[eq:lmm-through\])]{}, we obtain (see Figure \[fig:through\]) $$\raisebox{-1.3pc}{\includegraphics[scale=0.17]{through1}}~~~=~~~ \raisebox{-1.3pc}{\includegraphics[scale=0.17]{through2}}~~~+~~ \sum_b~\raisebox{-1.3pc}{\includegraphics[scale=0.17]{through3}}$$ $$\begin{aligned} {\label{eq:2nd-ind-fact}} &{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\\ &=\Theta_{v,x;{{\cal A}}}+\sum_{b\in{{\mathbb B}}_\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b \text{ even, }n_b\text{ odd}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{ {{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}\}$}}},{\nonumber}\end{aligned}$$ where we have replaced “$m_b+n_b>0$” in [(\[eq:1st-ind-fact\])]{} by “$m_b$ even, $n_b$ odd” that is the only possible combination consistent with the source constraints and the conditions in the indicators. As in [(\[eq:0th-summand1\])]{}, we alternate the parity of $n_b$ by changing the source constraint from ${\partial}{{\bf n}}=v{\vartriangle}x$ to ${\partial}{{\bf n}}=v{\vartriangle}b{\vartriangle}x$ and multiplying by $\tau_b$. Then, the sum over ${{\bf m}}$ and ${{\bf n}}$ in [(\[eq:2nd-ind-fact\])]{} equals $$\begin{aligned} {\label{eq:3rd-ind-ppfact}} \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}b{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b \text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b (v){^{\rm c}}\}$}}}.\end{aligned}$$ Then, as in [(\[eq:0th-summand2\])]{}, we condition on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)={{\cal B}}$ and decouple events occurring on ${{\mathbb B}}_{{{\cal B}}{^{\rm c}}}$ from events occurring on ${{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}$. Let ${{\bf m}}'={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf m}}''={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf n}}'={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\bf n}}''={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$. Note that ${\partial}{{\bf m}}'={\partial}{{\bf m}}''={\varnothing}$, ${\partial}{{\bf n}}'=v{\vartriangle}{\underline{b}}$ and ${\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x$. Multiplying [(\[eq:3rd-ind-ppfact\])]{} by $(Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}/Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}})(Z_{{{\cal B}}{^{\rm c}}}/Z_{{{\cal B}}{^{\rm c}}})\equiv1$ and using the notation [(\[eq:tildew-def\])]{}, we obtain $$\begin{aligned} {\label{eq:3rd-ind-prefact}} {(\ref{eq:3rd-ind-ppfact})}&=\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}'={\varnothing}\\{\partial}{{\bf n}}'=v{\vartriangle}{\underline{b}}}}\frac{\tilde w_{{{\cal A}}{^{\rm c}},{{\cal B}}}({{\bf m}}') \,Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda, {{\cal B}}}({{\bf n}}')\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}'+{{\bf n}}'}(v,{\underline{b}};{{\cal A}}) \text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf m}}'+{{\bf n}}'}^b(v)={{\cal B}}}\,{\nonumber}\\ &\qquad\qquad\times\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m'_b,n'_b\text{ even}\}$}}}\sum_{\substack{ {\partial}{{\bf m}}''={\varnothing}\\ {\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}} ({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf n}}'')} {Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}} {\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal B}}{^{\rm c}}\}$}}}{\nonumber}\\ &=\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}} \frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)= {{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal B}}{^{\rm c}}}{\nonumber}\\ &=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}}; {{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}},\end{aligned}$$ where we have been able to perform the sum over ${{\bf m}}''$ and ${{\bf n}}''$ independently, due to the fact that ${\mathbbm{1}{\scriptstyle\{{\overline{b}}\,{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}} {\overset{}{\longleftrightarrow}}}\,x \text{ in }{{\cal B}}{^{\rm c}}\}}}\equiv1$ for any ${{\bf n}}''\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ with ${\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x$. As in the derivation of [(\[eq:0th-summand3\])]{} from [(\[eq:0th-summand2\])]{}, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b \text{ even}\}}}$ in [(\[eq:3rd-ind-prefact\])]{} using the source constraints and the fact that ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}=0$ whenever ${\overline{b}}\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$. Therefore, $$\begin{aligned} {\label{eq:3rd-ind-fact}} {(\ref{eq:3rd-ind-prefact})}~=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}} \frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{ {{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}.\end{aligned}$$ By [(\[eq:Theta-def\])]{}–[(\[eq:3rd-ind-fact\])]{}, we arrive at $$\begin{aligned} {\label{eq:2nd-exp}} {{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}= \Theta_{v,x;{{\cal A}}}&+\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\,\tau_b\, {{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda{\nonumber}\\ &-\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\Big[\tau_b\Big({{\langle \varphi_{{\overline{b}}} \varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}^b(v) {^{\rm c}}}\Big)\Big],\end{aligned}$$ where ${{\cal C}}^b(v)\equiv{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$ is a variable for the operation $\Theta_{v,{\underline{b}};{{\cal A}}}$. This completes the second stage of the expansion. ### Completion of the lace expansion {#sss:complexp} For notational convenience, we define $w_{\varnothing}({{\bf m}})/Z_{\varnothing}={\mathbbm{1}{\scriptstyle\{{{\bf m}}\equiv0\}}}$. Since $E_{{\bf n}}(o,x;\Lambda)=\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$ (cf., [(\[eq:E-def\])]{}), we can write $$\begin{aligned} {\label{eq:pi0-rewr}} \pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\Theta_{o,x;\Lambda}.\end{aligned}$$ Also, we can write $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ in [(\[eq:R1-def\])]{} as $$\begin{aligned} {\label{eq:R1-rewr}} R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_b\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\Big({{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}^b (o){^{\rm c}}}\Big)\Big].\end{aligned}$$ Using [(\[eq:2nd-exp\])]{}, we obtain $$\begin{aligned} {\label{eq:R1R2}} R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_b\bigg(&\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\, \Theta_{{\overline{b}},x;{{\cal C}}^b(o)}\Big]+\sum_{b'}\Theta_{o,{\underline{b}};\Lambda}\Big[ \tau_b\,\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b(o)}\Big]\,\tau_{b'}{{\langle \varphi_{ {{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda{\nonumber}\\ &-\sum_{b'}\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\,\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b (o)}\Big[\tau_{b'}\Big({{\langle \varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_{{{\cal C}}^{b'}({\overline{b}}){^{\rm c}}}\Big)\Big] \Big]\bigg),\end{aligned}$$ where ${{\cal C}}^b(o)\equiv{{\cal C}}_{{\bf n}}^b(o)$ is a variable for the outer operation $\Theta_{o,{\underline{b}};\Lambda}$, and ${{\cal C}}^{b'}({\overline{b}})\equiv{{\cal C}}_{{{\bf m}}'+{{\bf n}}'}^{b'}({\overline{b}})$ is a variable for the inner operation $\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b(o)}$. For $j\ge1$, we define $$\begin{aligned} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&=\sum_{b_1,\dots,b_j}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1; \Lambda}\Big[\tau_{b_1}\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0}\Big[ \cdots\tau_{b_{j-1}}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}} \Big[\tau_{b_j}\Theta^{{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]\cdots\Big] \Big],{\label{eq:pij-def}}\\ R_\Lambda^{{\scriptscriptstyle}(j)}(x)&=\sum_{b_1,\dots,b_j}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1; \Lambda}\Big[\tau_{b_1}\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0}\Big[ \cdots\tau_{b_{j-1}}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}} \Big[\tau_{b_j}\Big({{\langle \varphi_{{\overline{b}}_j}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{ {\overline{b}}_j}\varphi_x \rangle}}_{\tilde{{\cal C}}_{j-1}{^{\rm c}}}\Big)\Big]\cdots\Big]\Big], {\label{eq:Rj-def}}\end{aligned}$$ where the operation $\Theta^{{\scriptscriptstyle}(i)}$ determines the variable $\tilde{{\cal C}}_i={{\cal C}}_{{{\bf m}}_i+{{\bf n}}_i}^{b_{i+1}}({\overline{b}}_i)$ (provided that ${\overline{b}}_0=o$). Then, we can rewrite [(\[eq:R1R2\])]{} as $$\begin{aligned} {\label{eq:R1R2-rewr}} R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)+\sum_{b'}\pi_\Lambda ^{{\scriptscriptstyle}(1)}({\underline{b}}')\,\tau_{b'}{{\langle \varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda -R_\Lambda^{{\scriptscriptstyle}(2)}(x).\end{aligned}$$ As a result, $$\begin{aligned} {{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\big(\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)-\pi_\Lambda ^{{\scriptscriptstyle}(1)}(x)\big)+\sum_b\big(\pi_\Lambda^{{\scriptscriptstyle}(0)}({\underline{b}})-\pi_\Lambda^{{\scriptscriptstyle}(1)}({\underline{b}})\big)\,\tau_b\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda+R_\Lambda^{ {\scriptscriptstyle}(2)}(x).\end{aligned}$$ By repeated applications of [(\[eq:2nd-exp\])]{} to the remainder $R_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we obtain [(\[eq:Ising-lace\])]{}–[(\[eq:Pij-def\])]{} in Proposition \[prp:Ising-lace\]. For the ferromagnetic case, $\tau_b$ and $w_{{{\cal A}}}({{\bf n}})$ for any ${{\cal A}}\subset\Lambda$ and ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}$ are nonnegative. This proves the first inequality in [(\[eq:pij-Rj-naivebd\])]{} and, with the help of Proposition \[prp:through\], the nonnegativity of $R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)$ . To prove the upper bound on $R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)$, we simply ignore ${{\langle \varphi_{{\overline{b}}_j}\varphi_x \rangle}}_{\tilde{{\cal C}}_{j-1}{^{\rm c}}}$ in [(\[eq:Rj-def\])]{} and replace $j$ by $j+1$, where $b_{j+1}=\{u,v\}$. This completes the proof of Proposition \[prp:Ising-lace\]. Comparison to percolation {#ss:percolation} ------------------------- Since we have exploited the underlying percolation picture to derive the lace expansion [(\[eq:Ising-lace\])]{} for the Ising model, it is not so surprising that the expansion coefficients [(\[eq:pi0-rewr\])]{} and [(\[eq:pij-def\])]{} (also recall [(\[eq:Theta-def\])]{}) are quite similar to the lace-expansion coefficients for unoriented bond-percolation (cf., [@hs90']): $$\begin{aligned} {\label{eq:pij-perc}} \pi_p^{{\scriptscriptstyle}(j)}(x)= \begin{cases} ~{\displaystyle}{{\mathbb E}}_p^{{\scriptscriptstyle}(0)}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}_0$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}\big]\equiv{{\mathbb P}}_p(o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\Longleftrightarrow}}}x) &(j=0),\\[1pc] {\displaystyle}\sum_{b_1,\dots,b_j}{{\mathbb E}}_p^{{\scriptscriptstyle}(0)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}_0$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\}$}}}\, p_{b_1}{{\mathbb E}}_p^{{\scriptscriptstyle}(1)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf n}}_1}({\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0)$}}} \cdots p_{b_j}{{\mathbb E}}_p^{{\scriptscriptstyle}(j)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf n}}_j}({\overline{b}}_j,x;\tilde {{\cal C}}_{j-1})$}}}\Big]\cdots\Big]\Big]&(j\ge1), \end{cases}\end{aligned}$$ where $p\equiv\sum_xp_{o,x}$ is the bond-occupation parameter, and each ${{\mathbb E}}_p^{{\scriptscriptstyle}(i)}$ denotes the expectation with respect to the product measure $\prod_b(p_b{\mathbbm{1}{\scriptstyle\{{{\bf n}}_i|_b=1\}}}+(1-p_b){\mathbbm{1}{\scriptstyle\{{{\bf n}}_i|_b=0\}}})$. In particular, the events involved in [(\[eq:pi0-rewr\])]{} and [(\[eq:pij-def\])]{} are identical to those in [(\[eq:pij-perc\])]{}. Hoever, there are significant differences between these two models. The major differences are the following: (a) Each current configuration must satisfy not only the conditions in the indicators, but also its source constraint that is absent in percolation. (b) An operation $\Theta$ is not an expectation, since the source constraints in the numerator and denominator of $\Theta$ in [(\[eq:Theta-def\])]{} are different. (c) In each $\Theta^{{\scriptscriptstyle}(i)}$ for $i\ge1$, the sum ${{\bf m}}_i+{{\bf n}}_i$ of two current configurations is coupled with ${{\bf m}}_{i-1}+{{\bf n}}_{i-1}$ via the cluster $\tilde{{\cal C}}_{i-1}$ determined by ${{\bf m}}_{i-1}+{{\bf n}}_{i-1}$. By contrast, in each ${{\mathbb E}}_p^{{\scriptscriptstyle}(i)}$ in [(\[eq:pij-perc\])]{}, a single percolation configuration ${{\bf n}}_i$ is coupled with ${{\bf n}}_{i-1}$ via $\tilde{{\cal C}}_{i-1}={{\cal C}}_{{{\bf n}}_{i-1}}^{b_i}({\overline{b}}_{i-1})$. In addition, ${{\bf m}}_i$ is nonzero only on bonds in ${{\mathbb B}}_{\tilde{{\cal C}}_{i-1}{^{\rm c}}}$, while the current configuration ${{\bf n}}_i$ has no such restriction. These elements are responsible for the difference in the method of bounding diagrams for the expansion coefficients. Take the $0^\text{th}$-expansion coefficient for example. For percolation, the BK inequality simply tells us that $$\begin{aligned} {\label{eq:pi0perc-comp}} \pi_p^{{\scriptscriptstyle}(0)}(x)\leq{{\mathbb P}}_p(o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}x)^2.\end{aligned}$$ For the ferromagnetic Ising model, on the other hand, we first recall [(\[eq:pi0-def\])]{}, i.e., $$\begin{aligned} \pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}},\end{aligned}$$ where $w_\Lambda({{\bf n}})/Z_\Lambda\ge0$. Due to the indicator, every current configuration ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ that gives nonzero contribution has at least *two bond-disjoint* paths $\zeta_1,\zeta_2$ from $o$ to $x$ such that $n_b>0$ for all $b\in\zeta_1{\:\Dot{\cup}\:}\zeta_2$. Also, due to the source constraint, there should be at least one path $\zeta$ from $o$ to $x$ such that $n_b$ is odd for all $b\in\zeta$. Suppose, for example, that $\zeta=\zeta_1$ and that $n_b$ for $b\in\zeta_2$ are all positive-even. Since a positive-even integer can split into two odd integers, on the labeled graph ${{\mathbb G}}_{{\bf n}}$ with ${\partial}{{\mathbb G}}_{{\bf n}}=o{\vartriangle}x$ (recall the notation introduced above [(\[eq:Sbefore\])]{}) there are at least *three edge-disjoint* paths from $o$ to $x$. This observation naturally leads us to expect that $$\begin{aligned} {\label{eq:pi0-comp}} \pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\end{aligned}$$ holds for the ferromagnetic Ising model. This naive argument to justify [(\[eq:pi0-comp\])]{} will be made rigorous in Section \[s:bounds\] by taking account of partition functions. The higher-order expansion coefficients are more involved, due to the above item (c). This will also be explained in detail in Section \[s:bounds\]. Bounds on $\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for the ferromagnetic models {#s:reduction} ================================================================================= From now on, we restrict ourselves to the ferromagnetic models. In this section, we explain how to prove Proposition \[prp:Pij-Rj-bd\] assuming a few other propositions (Propositions \[prp:GimpliesPix\]–\[prp:exp-bootstrap\] below). These propositions are results of diagrammatic bounds on the expansion coefficients in terms of two-point functions. We will show these diagrammatic bounds in Section \[s:bounds\]. The strategy to prove Proposition \[prp:Pij-Rj-bd\] is model-independent, and we follow the strategy in [@h05] for the nearest-neighbor model and that in [@hhs03] for the spread-out model. Since the latter is simpler, we first explain the strategy for the spread-out model. In the rest of this paper, we will frequently use the notation $$\begin{aligned} {\vbx{|\!|\!|}}=|x|\vee1.\end{aligned}$$ We also emphasize that constants in the $O$-notation used below (e.g., $O(\theta_0)$ in [(\[eq:pi-bd\])]{}) are independent of $\Lambda\subset{{\mathbb Z}^d}$. Strategy for the spread-out model --------------------------------- Using the diagrammatic bounds below in Section \[s:bounds\], we will prove in detail in Section \[ss:proof-so\] that the following proposition holds for the spread-out model: \[prp:GimpliesPix\] Let $J_{o,x}$ be the spread-out interaction. Suppose that $$\begin{aligned} {\label{eq:IR-xbd}} \tau\leq2,&& G(x)\leq\delta_{o,x}+\theta_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$ and $q\in(\frac{d}2,d)$. Then, for sufficiently small $\theta_0$ (with $\theta_0L^{d-q}$ being bounded away from zero) and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned} {\label{eq:pi-bd}} \pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq \begin{cases} O(\theta_0)^i\delta_{o,x}+O(\theta_0^3){\vbx{|\!|\!|}}^{-3q}&(i=0,1),\\ O(\theta_0)^i{\vbx{|\!|\!|}}^{-3q}&(i\ge2). \end{cases}\end{aligned}$$ The exact value of the assumed upper bound on $\tau$ in [(\[eq:IR-xbd\])]{} is unimportant and can be any finite number, as long as it is independent of $\theta_0$ and bigger than the mean-field critical point 1. We note that the exponent $3q$ in [(\[eq:pi-bd\])]{} is due to [(\[eq:pi0-comp\])]{} (and diagrammatic bounds on the higher-expansion coefficients), and is replaced by $2q$ with $q\in(\frac{2d}3,d)$ for percolation, due to, e.g., [(\[eq:pi0perc-comp\])]{}. We will show below that, at $p={p_\text{c}}$, $$\begin{aligned} {\label{eq:IR-xbd-so}} \tau\leq2,&& G(x)\leq\delta_{o,x}+O(L^{-2+\epsilon}){\vbx{|\!|\!|}}^{-(d-2)},\end{aligned}$$ for some small ${\epsilon}>0$. Since $\tau$ and $G(x)$ are nondecreasing and continuous in $p\leq{p_\text{c}}$ for the ferromagnetic models, these bounds imply [(\[eq:IR-xbd\])]{} for all $p\leq{p_\text{c}}$, with $\theta_0=cL^{-2+{\epsilon}}>0$ and $q=d-2$, where $q\in(\frac{d}2,d)$ if $d>4$ and $\theta_0L^{d-q}=cL^{\epsilon}>0$. Then, by Proposition \[prp:GimpliesPix\], the bound [(\[eq:pi-bd\])]{} with $\theta_0=O(L^{-2+{\epsilon}})$ and $q=d-2$ holds for $d>4$ and $\theta_0\ll1$ (thus $L\gg1$). Therefore, by [(\[eq:pij-Rj-naivebd\])]{} with ${{\langle \varphi_v\varphi_x \rangle}}_\Lambda\leq1$, $$\begin{aligned} {\label{eq:Rj-optSO}} 0\leq R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)\leq\tau\sum_u\pi_\Lambda^{{\scriptscriptstyle}(j)}(u) =O(\theta_0)^j\to0\qquad(j\uparrow\infty),\end{aligned}$$ and by [(\[eq:Pij-def\])]{} for $j\ge0$, $$\begin{aligned} {\label{eq:Pij-optSO}} |\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)-\delta_{o,x}|\leq O(\theta_0)\delta_{o,x} +\frac{O(\theta_0^2)}{{\vbx{|\!|\!|}}^{3(d-2)}}=O(\theta_0)\delta_{o,x}+ \frac{O(\theta_0^2)(1-\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ where $\rho=2(d-4)$. This completes the proof of Proposition \[prp:Pij-Rj-bd\] for the spread-out model, assuming [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$. It thus remains to show the bounds in [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$. These bounds are proved by adapting the model-independent bootstrapping argument in [@hhs03] (see the proof of [@hhs03 Proposition 2.2] for self-avoiding walk and percolation), together with the fact that $G(x)$ decays exponentially as $|x|\uparrow\infty$ for every $p<{p_\text{c}}$ [@l80; @s80] so that $\sup_xG(x)$ is continuous in $p<{p_\text{c}}$ [@s05]. We complete the proof. Strategy for the nearest-neighbor model --------------------------------------- Since $\sigma^2=O(1)$ for short-range models, we cannot expect that $\theta_0$ in [(\[eq:IR-xbd\])]{} is small, or that Proposition \[prp:GimpliesPix\] is applicable to bound the expansion coefficients in this setting. Under this circumstance, we follow the strategy in [@h05]. The following is the key proposition, whose proof will be explained in Section \[ss:proof-nn\]: \[prp:GimpliesPik\] Let $J_{o,x}$ be the nearest-neighbor or spread-out interaction, and suppose that $$\begin{aligned} {\label{eq:IR-kbd}} \tau-1\leq\theta_0,&& \sup_x(D*G^{*2})(x)\leq\theta_0,&& \sup_{\substack{x\equiv(x_1,\dots,x_d)\ne o\\ l=1,\dots,d}} \bigg(\frac{x_l^2}{\sigma^2}\vee1\bigg)G(x)\leq\theta_0\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$. Then, for sufficiently small $\theta_0$ and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned} {\label{eq:pi-sumbd}} \sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq \begin{cases} 1+O(\theta_0^2)&(i=0),\\ O(\theta_0)^i&(i\ge1), \end{cases}&& \sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq d\sigma^2(i+1)^2O(\theta_0)^{i \vee2}.\end{aligned}$$ Furthermore, in addition to [(\[eq:IR-kbd\])]{} with $\theta_0\ll1$, if $$\begin{aligned} {\label{eq:IR-xbdNN}} G(x)\leq\lambda_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ holds for some $\lambda_0\in[1,\infty)$ and $q\in(0,d)$, then we have for $i\ge0$ $$\begin{aligned} {\label{eq:pi-kbd}} \pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq O(\theta_0)^i\delta_{o,x}+\frac{\lambda_0^3 (i+1)^{3q+2}O(\theta_0)^{(i-2)\vee0}}{|x|^{3q}}(1-\delta_{o,x}).\end{aligned}$$    First we claim that the assumed bounds in [(\[eq:IR-kbd\])]{} indeed hold for any $p\leq{p_\text{c}}$ if $d>4$ and $\theta_0\ll1$, where $\theta_0=O(d^{-1})$ for the nearest-neighbor model and $\theta_0=O(L^{-d})$ for the spread-out model. The proof is based on the orthodox model-independent bootstrapping argument in, e.g., [@ms93] (see also [@hs02] for improved random-walk estimates; bootstrapping assumptions that are different from, but philosophically similar to, [(\[eq:IR-kbd\])]{} are used in [@hhs?]). Therefore, [(\[eq:pi-sumbd\])]{} holds for $p\leq{p_\text{c}}$ and hence ensures the existence of an infinite-volume limit $\Pi(x)=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}\lim_{j\uparrow\infty}\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ that satisfies $$\begin{aligned} {\label{eq:Pi-bdNN}} \sum_x|\Pi(x)|=1+O(\theta_0),&& \sum_x|x|^2|\Pi(x)|=d\sigma^2O(\theta_0^2).\end{aligned}$$ As a byproduct, we obtain the identity in [(\[eq:constants\])]{} for $\tau({p_\text{c}})$ for both models. Suppose that $$\begin{aligned} {\label{eq:IR-xbd-nn}} G(x)\leq\lambda_0{\vbx{|\!|\!|}}^{-(d-2)}\end{aligned}$$ holds at $p={p_\text{c}}$. Then, by Proposition \[prp:GimpliesPik\], we obtain [(\[eq:pi-kbd\])]{} with $q=d-2$. Using this in [(\[eq:Rj-optSO\])]{}–[(\[eq:Pij-optSO\])]{}, we can prove Proposition \[prp:Pij-Rj-bd\]. To complete the proof, it thus remains to show [(\[eq:IR-xbd-nn\])]{} at $p={p_\text{c}}$. To show this, we use the following proposition: \[prp:exp-bootstrap\] Let $$\begin{aligned} {\label{eq:GbarWbar}} \bar G^{{\scriptscriptstyle}(s)}=\sup_x|x|^sG(x),&& \bar W^{{\scriptscriptstyle}(t)}=\sup_x\sum_y|y|^tG(y)\,G(x-y),\end{aligned}$$ and suppose that the bounds in [(\[eq:IR-kbd\])]{} hold with $\theta_0\ll1$. (i) If $\sum_x\Pi(x)=\tau^{-1}$ and $|\Pi(x)|\leq O({\vbx{|\!|\!|}}^{-(d+2)})$, then we have $$\begin{aligned} {\label{eq:prp-asy}} G(x)\sim\frac{\sum_x\Pi(x)}{\tau\sum_x|x|^2(D*\Pi)(x)}\,\frac{a_d} {|x|^{d-2}}\qquad\text{as }|x|\uparrow\infty.\end{aligned}$$ (ii) If $\sum_x|x|^r|\Pi(x)|<\infty$ for some $r>0$, then, for $s,t>0$ which are not odd integers, we have $$\begin{aligned} \begin{cases} \bar G^{{\scriptscriptstyle}(s)}<\infty&\text{if}~~~s\leq r~~\text{and}~~s<d-2,\\ \bar W^{{\scriptscriptstyle}(t)}<\infty&\text{if}~~~t\leq\lfloor r\rfloor~~\text{and} ~~t<d-4. \end{cases}\end{aligned}$$ (iii) If $\bar W^{{\scriptscriptstyle}(t)}<\infty$ for some $t\ge0$, then $\sum_x|x|^{t+2}|\Pi(x)|<\infty$. The above proposition is a summary of key elements in [@h05 Proposition 1.3 and Lemmas 1.5–1.6] that are sufficient to prove [(\[eq:IR-xbd-nn\])]{} in the current setting. The proofs of Propositions \[prp:exp-bootstrap\](i) and \[prp:exp-bootstrap\](ii) are model-independent and can be found in [@h05 Sections 2 and 4], respectively. The proof of Proposition \[prp:exp-bootstrap\](iii) is similar to that of the first statement of Proposition \[prp:GimpliesPik\]: [(\[eq:IR-kbd\])]{} implies [(\[eq:pi-sumbd\])]{}. We will explain this in Section \[ss:proof-nn\]. Now we continue with the proof of [(\[eq:IR-xbd-nn\])]{}. Fix $p={p_\text{c}}$. Since the asymptotic behavior [(\[eq:prp-asy\])]{} is good enough for the bound [(\[eq:IR-xbd-nn\])]{}, it suffices to check the assumptions of Proposition \[prp:exp-bootstrap\](i). The first assumption on the sum of $\Pi(x)$ is satisfied at $p={p_\text{c}}$, as mentioned below [(\[eq:Pi-bdNN\])]{}. The second assumption is also satisfied if $\bar G^{({{\scriptscriptstyle}\frac{d+2}3})}<\infty$, because of the second statement of Proposition \[prp:GimpliesPik\]: [(\[eq:IR-xbdNN\])]{} implies [(\[eq:pi-kbd\])]{}. By Proposition \[prp:exp-bootstrap\](ii), it thus suffices to show that $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$ is finite if $d>4$. To show this, we let $$\begin{aligned} r_0=2,&& r_{i+1}=\Big((d-2)\wedge\big(\lfloor r_i\rfloor+2\big) \Big)-{\epsilon},\end{aligned}$$ where $0<{\epsilon}\leq\frac{2}3(d-4)$. Note that, by this definition, $r_i$ for $i\ge1$ equals $((d-2)\wedge(i+3))-{\epsilon}$ and increases until it reaches $d-2-{\epsilon}$. We prove below by induction that $\sum_x|x|^{r_i}|\Pi(x)|$ is finite for all $i\ge0$. This is sufficient for the finiteness of $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$, since $$\begin{aligned} \lim_{i\uparrow\infty}r_i=d-2-{\epsilon}\ge d-2-\tfrac{2}3(d-4) =\tfrac{d+2}3.\end{aligned}$$ Note that, by [(\[eq:Pi-bdNN\])]{}, $\sum_x|x|^{r_0}|\Pi(x)|<\infty$. Suppose $\sum_x|x|^{r_i}|\Pi(x)|<\infty$ for some $i\ge0$. Then, by Proposition \[prp:exp-bootstrap\](ii), $\bar W^{{\scriptscriptstyle}(t)}$ is finite for $t\in(0,\lfloor r_i\rfloor]\cap(0,d-4)$. Since $\lfloor r_0\rfloor=2$ and $\lfloor r_i\rfloor=(d-3)\wedge(i+2)$ for $i\ge1$, $\bar W^{{\scriptscriptstyle}(T)}$ with $T=(i+2)\wedge(d-4-{\epsilon})$ is finite. Then, by Proposition \[prp:exp-bootstrap\](iii), $\sum_x|x|^{T+2}|\Pi(x)|$ is finite. Since $$\begin{aligned} T+2=(i+4)\wedge(d-2-{\epsilon})\ge\big((d-2)\wedge(i+4)\big)-{\epsilon}=r_{i+1},\end{aligned}$$ we obtain that $\sum_x|x|^{r_{i+1}}|\Pi(x)|<\infty$. This completes the induction and the proof of [(\[eq:IR-xbd-nn\])]{}. The proof of Proposition \[prp:Pij-Rj-bd\] is now completed. Diagrammatic bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ {#s:bounds} ================================================================= In this section, we prove diagrammatic bounds on the expansion coefficients. In Section \[ss:diagram\], we construct diagrams in terms of two-point functions and state the bounds. In Section \[ss:pi0bd\], we prove a key lemma for the diagrammatic bounds and show how to apply this lemma to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$. In Section \[ss:pijbd\], we prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$. Construction of diagrams {#ss:diagram} ------------------------ To state bounds on the expansion coefficients (as in Proposition \[prp:diagram-bd\] below), we first define diagrammatic functions consisting of two-point functions. Let $$\begin{aligned} {\label{eq:tildeG-def}} \tilde G_\Lambda(y,x) =\sum_{b:{\overline{b}}=x}{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b,\end{aligned}$$ which satisfies[^5] $$\begin{aligned} {\label{eq:G-delta-bd}} {{\langle \varphi_y\varphi_x \rangle}}_\Lambda\leq\delta_{y,x}+\sum_{b:{\overline{b}}=x}\, \sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ n_b\text{ odd}}}\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}=\delta_{y,x}+\sum_{b:{\overline{b}}=x}\tau_b\sum_{\substack{ {\partial}{{\bf n}}=y{\vartriangle}{\underline{b}}\\ n_b\text{ even}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \leq\delta_{y,x}+\tilde G_\Lambda(y,x).\end{aligned}$$ Let $$\begin{aligned} {\label{eq:psi-def}} \psi_\Lambda(y,x)=\sum_{j=0}^\infty\big(\tilde G_\Lambda^2\big)^{*j}(y,x) &\equiv\delta_{y,x}+\sum_{j=1}^\infty\sum_{\substack{u_0,\dots,u_j\\ u_0= y,\;u_j=x}}\prod_{l=1}^j\tilde G_\Lambda(u_{l-1},u_l)^2,\end{aligned}$$ and define (see the first line in Figure \[fig:P-def\]) $$\begin{aligned} P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)&=2\big(\psi_\Lambda(v_1,v'_1)- \delta_{v_1,v'_1}\big)\,{{\langle \varphi_{v_1}\varphi_{v'_1} \rangle}}_\Lambda, {\label{eq:P1-def}}\\[5pt] P_\Lambda^{{\scriptscriptstyle}(j)}(v_1,v'_j)&=\sum_{\substack{v_2,\dots,v_j\\ v'_1,\dots,v'_{j-1}}}\bigg(\prod_{i=1}^j\big(\psi_\Lambda(v_i, v'_i)-\delta_{v_i,v'_i}\big)\bigg){{\langle \varphi_{v_1}\varphi_{ v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{\nonumber}\\ &\qquad\qquad\times\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{v'_{i -1}}\varphi_{v_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{v_{i+1}}\varphi_{ v'_i} \rangle}}_\Lambda\bigg){{\langle \varphi_{v'_{j-1}}\varphi_{v'_j} \rangle}} _\Lambda\qquad(j\ge2),{\label{eq:Pj-def}}\end{aligned}$$ where the empty product for $j=2$ is regarded as 1. $$\begin{gathered} P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics[scale=.1] {P0}}\qquad P_\Lambda^{{\scriptscriptstyle}(2)}(v_1,v'_2)=\raisebox{-9pt}{\includegraphics[scale=.1] {P1}}\qquad P_\Lambda^{{\scriptscriptstyle}(3)}(v_1,v'_3)=\raisebox{-15pt}{\includegraphics[scale=.1] {P2}}\\[5pt] P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics [scale=.1]{P0p}}\hspace{5pc} P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-18pt} {\includegraphics[scale=.1]{P0pp}}+~ \raisebox{-18pt}{\includegraphics[scale=.1]{P0pp2}}\\[5pt] P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-12pt}{\includegraphics [scale=.1]{P0prime}}\hspace{5pc} P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-18pt} {\includegraphics[scale=.1]{P0primeprime}}\end{gathered}$$ Next, we define $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ by replacing one of the $2j-1$ two-point functions on the right-hand side of [(\[eq:P1-def\])]{}–[(\[eq:Pj-def\])]{} by the product of *two* two-point functions, such as replacing ${{\langle \varphi_z\varphi_{z'} \rangle}}_\Lambda$ by ${{\langle \varphi_z\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z'} \rangle}}_\Lambda$, and then summing over all $2j-1$ choices of this replacement. For example, we define (see the second line in Figure \[fig:P-def\]) $$\begin{aligned} {\label{eq:P'1-def}} P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=2\big(\psi_\Lambda(v_1,v'_1)- \delta_{v_1,v'_1}\big){{\langle \varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u \varphi_{v'_1} \rangle}}_\Lambda,\end{aligned}$$ and $$\begin{aligned} P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(2)}}(v_1,v'_2)=\sum_{v_2,v'_1}\bigg(\prod_{i =1}^2\big(\psi_\Lambda(v_i,v'_i)-\delta_{v_i,v'_i}\big)\bigg)\Big({{\langle \varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{v_2} \rangle}}_\Lambda {{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}\varphi_{ v'_2} \rangle}}_\Lambda&{\nonumber}\\ +{{\langle \varphi_{v_1}\varphi_{v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1} \varphi_{v'_2} \rangle}}_\Lambda&{\nonumber}\\[7pt] +{{\langle \varphi_{v_1}\varphi_{v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_{ v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u \varphi_{v'_2} \rangle}}_\Lambda&\Big).\end{aligned}$$ We define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ similarly as follows. First we take *two* two-point functions in $P_\Lambda^{{\scriptscriptstyle}(j)}(v_1,v'_j)$, one of which (say, ${{\langle \varphi_{z_1}\varphi_{z'_1} \rangle}}_\Lambda$ for some $z_1,z'_1$) is among the aforementioned $2j-1$ two-point functions, and the other (say, $\tilde G_\Lambda(z_2,z'_2)$ for some $z_2,z'_2$) is among those of which $\psi_\Lambda(v_i,v'_i)-\delta_{v_i,v'_i}$ for $i=1,\dots,j$ are composed. The product ${{\langle \varphi_{z_1}\varphi_{z'_1} \rangle}}_\Lambda\tilde G_\Lambda(z_2,z'_2)$ is then replaced by $$\begin{aligned} &\bigg(\sum_{v'}{{\langle \varphi_{z_1}\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'} \varphi_{z'_1} \rangle}}_\Lambda\,\psi_\Lambda(v',v)\bigg)\Big({{\langle \varphi_{z_2} \varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,z'_2)+\tilde G_\Lambda(z_2,z'_2) \,\delta_{u,z'_2}\Big){\nonumber}\\ &+{{\langle \varphi_{z_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z'_1} \rangle}}_\Lambda\sum_{v'}\Big({{\langle \varphi_{z_2}\varphi_{v'} \rangle}}_\Lambda\tilde G_\Lambda(v',z'_2)+\tilde G_\Lambda(z_2,z'_2)\,\delta_{v',z'_2}\Big) \,\psi_\Lambda(v',v).\end{aligned}$$ Finally, we define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ by taking account of all possible combinations of ${{\langle \varphi_{z_1} \varphi_{z'_1} \rangle}}_\Lambda$ and $\tilde G_\Lambda(z_2,z'_2)$. For example, we define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)$ as (see Figure \[fig:P-def\]) $$\begin{aligned} {\label{eq:P''1-def}} P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1){\nonumber}\\ =\sum_{u',u'',v'}\bigg(&2\psi_\Lambda(v_1,u')\,\tilde G_\Lambda(u',u'') \Big({{\langle \varphi_{u'}\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,u'')+\tilde G_\Lambda(u',u'')\,\delta_{u,u''}\Big)\,\psi_\Lambda(u'',v'_1){\nonumber}\\ &\times{{\langle \varphi_{v_1}\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}\varphi_{ v'_1} \rangle}}_\Lambda\psi_\Lambda(v',v)+(\text{permutation of $u$ and }v') \bigg),\end{aligned}$$ where the permutation term corresponds to the second term for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)$ in Figure \[fig:P-def\]. In addition to the above quantities, we define (see the third line in Figure \[fig:P-def\]) $$\begin{aligned} P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}}_\Lambda^2 {{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda, {\label{eq:P'0-def}}\\[5pt] P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}} _\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}} _\Lambda\sum_{v'}{{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'} \varphi_x \rangle}}_\Lambda\,\psi_\Lambda(v',v),{\label{eq:P''0-def}}\end{aligned}$$ and let $$\begin{aligned} {\label{eq:P'P''-def}} P'_{\Lambda;u}(y,x)=\sum_{j\ge0}P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x),&& P''_{\Lambda;u,v}(y,x)&=\sum_{j\ge0}P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x),\end{aligned}$$ where $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)$ are the leading contributions to $P'_{\Lambda;u}(y,x)$ and $P''_{\Lambda;u,v}(y,x)$, respectively. Finally, we define $$\begin{aligned} Q'_{\Lambda;u}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big) P'_{\Lambda;u}(z,x),{\label{eq:Q'-def}}\\ Q''_{\Lambda;u,v}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z) \big)P''_{\Lambda;u,v}(z,x){\nonumber}\\ &\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\tilde G_\Lambda(v',z)\,P'_{\Lambda;u}(z,x)\,\psi_\Lambda(v',v).{\label{eq:Q''-def}}\end{aligned}$$ The following are the diagrammatic bounds on the expansion coefficients (see Figure \[fig:piN-bd\]): \[prp:diagram-bd\] For the ferromagnetic Ising model, we have $$\begin{aligned} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq \begin{cases}{\label{eq:piNbd}} P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(o,x)\equiv{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3 &(j=0),\\[5pt] {\displaystyle}\sum_{\substack{b_1,\dots,b_j\\ v_1,\dots,v_j}}P_{\Lambda;v_1}^{\prime {{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\,\bigg(\prod_{i=1}^{j-1}\tau_{b_i}Q''_{\Lambda;v_i,v_{ i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})\bigg)\,\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x)&(j\ge1), \end{cases}\end{aligned}$$ where, as well as in the rest of the paper, the empty product is regarded as 1 by convention. $$\begin{aligned} \pi^{{\scriptscriptstyle}(1)}_\Lambda(x)\lesssim\raisebox{-11pt}{\includegraphics[scale= 0.18]{pi1}}\qquad \pi^{{\scriptscriptstyle}(2)}_\Lambda(x)\lesssim\raisebox{-20pt}{\includegraphics[scale= 0.18]{pi21}}+\raisebox{-20pt}{\includegraphics[scale=0.18]{pi22}}\end{aligned}$$ Bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ {#ss:pi0bd} --------------------------------------------------- The key ingredient of the proof of Proposition \[prp:diagram-bd\] is Lemma \[lmm:GHS-BK\] below, which is an extension of the GHS idea used in the proof of Lemma \[lmm:switching\]. In this subsection, we demonstrate how this extension works to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and the inequality $$\begin{aligned} {\label{eq:pi0'-bd}} \sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}} x\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y}\leq P_{\Lambda;y}^{\prime{{\scriptscriptstyle}(0)}}(o,x),\end{aligned}$$ which will be used in Section \[ss:pijbd\] to obtain the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$. Since the inequality is trivial if $x=o$, we restrict our attention to the case of $x\ne o$. First we note that, for each current configuration ${{\bf n}}$ with ${\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}=1$, there are at least *three edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ between $o$ and $x$. See, for example, the first term on the right-hand side in Figure \[fig:1stpiv\]. Suppose that the thick line in that picture, referred to as $\zeta_1$ and split into $\zeta_{11}{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_{13}$ from $o$ to $x$, consists of bonds $b$ with $n_b=1$, and that the thin lines, referred to as $\zeta_2$ and $\zeta_3$ that terminate at $o$ and $x$ respectively, consist of bonds $b'$ with $n_{b'}=2$. Let $\zeta'_i$, for $i=2,3$, be the duplication of $\zeta_i$. Then, the three paths $\zeta_2{\:\Dot{\cup}\:}\zeta_{13}$, $\zeta'_2{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_3$ and $\zeta_{11}{\:\Dot{\cup}\:}\zeta'_3$ are edge-disjoint. Then, by multiplying $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ by *two* dummies $(Z_\Lambda/Z_\Lambda)^2\,(\equiv1)$, we obtain $$\begin{aligned} {\label{eq:pi0*Z2}} \pi_\Lambda^{{\scriptscriptstyle}(0)}(x)&=\sum_{\substack{{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}'' ={\varnothing}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}}')}{Z_\Lambda} \,\frac{w_\Lambda({{\bf m}}'')}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}{\nonumber}\\ &=\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^3}\sum_{\substack{ {\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}''={\varnothing}\\ {{\bf N}}\equiv{{\bf n}}+{{\bf m}}'+{{\bf m}}''}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}\prod_b\frac{N_b!}{n_b!\;m'_b!\;m''_b!},\end{aligned}$$ where the sum over ${{\bf n}},{{\bf m}}',{{\bf m}}''$ in the second line equals the cardinality of the following set of partitions: $$\begin{aligned} {\label{eq:Ssubset}} {\mathfrak{S}}_0=\bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0,1,2}{{\mathbb S}}_i,\; {\partial}{{\mathbb S}}_0=\{o,x\},\;{\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2={\varnothing},\;o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\Longleftrightarrow}}}x\text{ in } {{\mathbb S}}_0\bigg\},\end{aligned}$$ where “$o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\Longleftrightarrow}}}x$ in ${{\mathbb S}}_0$” means that there are at least two *bond*-disjoint paths in ${{\mathbb S}}_0$. We will show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$, where $$\begin{aligned} {\label{eq:Ssupset}} {\mathfrak{S}}'_0=\bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0,1,2}{{\mathbb S}}_i,\; {\partial}{{\mathbb S}}_0={\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2=\{o,x\}\bigg\}.\end{aligned}$$ This implies [(\[eq:piNbd\])]{} for $j=0$, because $$\begin{aligned} |{\mathfrak{S}}'_0|=\sum_{\substack{{\partial}{{\bf n}}={\partial}{{\bf m}}'={\partial}{{\bf m}}''=\{o,x\}\\ {{\bf N}}\equiv {{\bf n}}+{{\bf m}}'+{{\bf m}}''}}\prod_b\frac{N_b!}{n_b!\,m'_b!\, m''_b!},\end{aligned}$$ and $$\begin{aligned} \sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^3}\sum_{\substack{ {\partial}{{\bf n}}={\partial}{{\bf m}}'={\partial}{{\bf m}}''=\{o,x\}\\ {{\bf N}}\equiv{{\bf n}}+{{\bf m}}'+{{\bf m}}''}}\prod_b \frac{N_b!}{n_b!\;m'_b!\;m''_b!}=\bigg(\sum_{{\partial}{{\bf n}}=\{ o,x\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\bigg)^3.\end{aligned}$$ It remains to show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. To do so, we use the following lemma, in which we denote by $\Omega_{z\to z'}^{{{\bf N}}}$ the set of paths on ${{\mathbb G}}_{{\bf N}}$ from $z$ to $z'$ and write $\omega\cap\omega'={\varnothing}$ to mean that $\omega$ and $\omega'$ are *edge*-disjoint (not necessarily *bond*-disjoint). \[lmm:GHS-BK\] Given a current configuration ${{\bf N}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$, $k\ge1$, ${{\cal V}}\subset\Lambda$ and $z_i\ne z'_i\in\Lambda$ for $i=1,\dots,k$, we let $$\begin{aligned} {\label{eq:fS-gen}} {\mathfrak{S}}=\left\{({{\mathbb S}}_0,{{\mathbb S}}_1,\dots,{{\mathbb S}}_k): \begin{array}{r} {{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{\raisebox{-3pt}{$\scriptstyle k$}} {{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0={{\cal V}},\;{\partial}{{\mathbb S}}_i={\varnothing}~(i=1,\dots,k),\;\\ {{}^\exists}\omega_i\in\Omega^{{\bf N}}_{z_i\to z'_i}~(i=1,\dots,k)~ \text{\rm such that }\omega_i\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_i\\ \text{\rm and }\omega_i\cap\omega_j={\varnothing}~(i\ne j) \end{array}\right\},\end{aligned}$$ and define ${\mathfrak{S}}'$ to be the right-hand side of [(\[eq:fS-gen\])]{} with “${\partial}{{\mathbb S}}_0={{\cal V}}$, ${\partial}{{\mathbb S}}_i={\varnothing}$” being replaced by “${\partial}{{\mathbb S}}_0={{\cal V}}{\,\triangle\,}\{z_1,z'_1\}{\,\triangle\,}\cdots{\,\triangle\,}\{z_k,z'_k\}$, ${\partial}{{\mathbb S}}_i=\{z_i,z'_i\}$”. Then, $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$. We will prove this lemma at the end of this subsection. Now we use Lemma \[lmm:GHS-BK\] with $k=2$ and ${{\cal V}}=\{z_1,z'_1\}=\{z_2,z'_2\}=\{o,x\}$. Note that ${\mathfrak{S}}_0$ in [(\[eq:Ssubset\])]{} is a subset of ${\mathfrak{S}}$, since ${\mathfrak{S}}$ includes partitions $({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2)$ in which there does not exist two *bond*-disjoint paths on ${{\mathbb S}}_0$. In addition, ${\mathfrak{S}}'$ is trivially a subset of ${\mathfrak{S}}'_0$ in [(\[eq:Ssupset\])]{}. Therefore, we have $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. This completes the proof of [(\[eq:piNbd\])]{} for $j=0$. Here, we summarize the basic steps that we have followed to bound $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and which we generalize to prove [(\[eq:pi0’-bd\])]{} below and the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ in Section \[sss:dbconn\]. (i) Count the (minimum) number, say, $k+1$, of *edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ that satisfy the source constraint (as well as other additional conditions, if there are) of the considered function $f(x)$. For example, $k=2$ for $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\equiv\frac1{Z_\Lambda}\sum_{{\partial}{{\bf n}}=\{o,x\}} w_\Lambda({{\bf n}})\,{\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$. (ii) Multiply $f(x)$ by $(\frac{Z_\Lambda}{Z_\Lambda})^k=\prod_{i=1}^k (\frac1{Z_\Lambda}\sum_{{\partial}{{\bf m}}_i={\varnothing}}w_\Lambda({{\bf m}}_i)) \,(\equiv1)$ and then overlap the $k$ dummies ${{\bf m}}_1,\dots,{{\bf m}}_k$ on the original current configuration ${{\bf n}}$. Choose $k$ paths $\omega_1,\dots,\omega_k$ among $k+1$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf n}}+\sum_{i=1}^k{{\bf m}}_i}$. (iii) Use Lemma \[lmm:GHS-BK\] to exchange the occupation status of edges on $\omega_i$ between ${{\mathbb G}}_{{\bf n}}$ and ${{\mathbb G}}_{{{\bf m}}_i}$ for every $i=1,\dots,k$. The current configurations after the mapping, denoted by $\tilde{{\bf n}},\tilde{{\bf m}}_1,\dots,\tilde{{\bf m}}_k$, satisfy ${\partial}\tilde{{\bf n}}={\partial}{{\bf n}}{\vartriangle}{\partial}\omega_1{\vartriangle}\cdots{\vartriangle}{\partial}\omega_k$ and ${\partial}\tilde{{\bf m}}_i={\partial}\omega_i$ for $i=1,\dots,k$. If $y=o$ or $x$, then [(\[eq:pi0’-bd\])]{} is reduced to the inequality for $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$. Also, if $y\ne o=x$, then the left-hand side of [(\[eq:pi0’-bd\])]{} multiplied by $Z_\Lambda/Z_\Lambda=\sum_{{\partial}{{\bf m}}={\varnothing}}w_\Lambda({{\bf m}})/Z_\Lambda\equiv1$ equals $$\begin{aligned} {\label{eq:dbbd}} \sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}={\varnothing}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda ({{\bf m}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y\}$}}}&\leq\sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}={\varnothing}}\frac{ w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}+{{\bf m}}$}} {\overset{}{\longleftrightarrow}}}y\}$}}}{\nonumber}\\ &=\sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}=\{o,y\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{ w_\Lambda({{\bf m}})}{Z_\Lambda}\;={{\langle \varphi_o\varphi_y \rangle}}_\Lambda^2,\end{aligned}$$ where the first equality is due to Lemma \[lmm:switching\]. Therefore, we can assume $o\ne x\ne y\ne o$. We follow the three steps described above. \(i) Since $y\notin{\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y}=1$, it is not hard to see that there is an *edge*-disjoint cycle (closed path) $o\to y\to x\to o$. Since a cycle does not have a source, there must be another edge-disjoint connection from $o$ to $x$, due to the source constraint ${\partial}{{\bf n}}=\{o,x\}$. Therefore, there are at least $4\,(=k+1)$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$: one is between $o$ and $y$, another is between $y$ and $x$, and the other two are between $o$ and $x$. \(ii) Multiplying both sides of [(\[eq:pi0’-bd\])]{} by $(Z_\Lambda/Z_\Lambda)^3$ is equivalent to $$\begin{aligned} {\label{eq:pi0'-equiv}} &\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^4}\sum_{\substack{ {\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}_i={\varnothing}~{{}^\forall}i=1,2,3\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^3 {{\bf m}}_i}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}y}\prod_b \frac{N_b!}{n_b!\;m^{{\scriptscriptstyle}(1)}_b!\;m^{{\scriptscriptstyle}(2)}_b!\;m^{{\scriptscriptstyle}(3)}_b!}{\nonumber}\\ &\quad\leq\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^4}\sum_{ \substack{{\partial}{{\bf n}}={\partial}{{\bf m}}_3=\{o,x\}\\ {\partial}{{\bf m}}_1=\{o,y\},~{\partial}{{\bf m}}_2=\{y,x\}\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^3{{\bf m}}_i}}\prod_b\frac{N_b!}{n_b!\; m^{{\scriptscriptstyle}(1)}_b!\;m^{{\scriptscriptstyle}(2)}_b!\;m^{{\scriptscriptstyle}(3)}_b!},\end{aligned}$$ where we have used the notation $m_b^{{\scriptscriptstyle}(i)}={{\bf m}}_i|_b$. Note that the second sum on the left-hand side equals the cardinality of $$\begin{aligned} {\label{eq:S03sub}} \bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2,{{\mathbb S}}_3): \begin{array}{c} {{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{\raisebox{-3pt}{$\scriptstyle3$}} {{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0=\{o,x\},\;{\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2={\partial}{{\mathbb S}}_3={\varnothing}\\ o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\Longleftrightarrow}}}x\text{ in }{{\mathbb S}}_0,~o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}y\text{ in }{{\mathbb S}}_0 \end{array}\bigg\},\end{aligned}$$ and the second sum on the right-hand side of [(\[eq:pi0’-equiv\])]{} equals the cardinality of $$\begin{aligned} {\label{eq:S03sup}} \textstyle\Big\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2,{{\mathbb S}}_3):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{ \raisebox{-3pt}{$\scriptstyle3$}}{{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0={\partial}{{\mathbb S}}_3=\{o,x\},\;{\partial}{{\mathbb S}}_1=\{o,y\},\;{\partial}{{\mathbb S}}_2=\{y,x\}\Big\}.\end{aligned}$$ Therefore, to prove [(\[eq:pi0’-equiv\])]{}, it is sufficient to show that the cardinality of [(\[eq:S03sub\])]{} is not bigger than that of [(\[eq:S03sup\])]{}. \(iii) Now we use Lemma \[lmm:GHS-BK\] with $k=3$ and ${{\cal V}}=\{z_3,z'_3\}=\{o,x\}$, $\{z_1,z'_1\}=\{o,y\}$ and $\{z_2,z'_2\}=\{y,x\}$. Since [(\[eq:S03sub\])]{} is a subset of ${\mathfrak{S}}$ in the current setting, while ${\mathfrak{S}}'$ is a subset of [(\[eq:S03sup\])]{}, we obtain [(\[eq:pi0’-equiv\])]{}. This completes the proof of [(\[eq:pi0’-bd\])]{}. We prove Lemma \[lmm:GHS-BK\] by decomposing ${\mathfrak{S}}^{(\prime)}$ into ${\mathop{\Dot{\bigcup}}}_{\vec\omega_k}{\mathfrak{S}}_{\vec\omega_k}^{(\prime)}$ (described in detail below) and then constructing a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$ for every $\vec\omega_k$. To do so, we first introduce some notation. 1. For every $i=1,\dots,k$, we introduce an arbitrarily fixed order among elements in $\Omega_{z_i\to z'_i}^{{\bf N}}$. For $\omega,\omega'\in\Omega_{z_i\to z'_i}^{{\bf N}}$, we write $\omega\prec\omega'$ if $\omega$ is earlier than $\omega'$ in this order. Let $\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ be the set of paths $\zeta\in\Omega_{z_1\to z'_1}^{{\bf N}}$ such that there are $k-1$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf N}}}\setminus\zeta$ (= the resulting graph by removing the edges in $\zeta$) each of which connects $z_i$ and $z'_i$ for every $i=2,\dots,k$. 2. Then, for $\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$, we define $\Xi_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ to be the set of paths $\zeta\in\Omega_{z_2\to z'_2}^{{\bf N}}$ on ${{\mathbb G}}_{{\bf N}}\setminus\omega_1$ such that $\zeta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ earlier than $\omega_1$. Then, we define $\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ to be the set of paths $\zeta\in\Xi_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ such that there are $k-2$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf N}}}\setminus(\omega_1{\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=3,\dots,k$. 3. More generally, for $l<k$ and $\vec\omega_l=(\omega_1, \dots,\omega_l)$ with $\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$, $\omega_2\in\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1},\dots$, $\omega_l\in\tilde\Omega_{z_l\to z'_l}^{{{\bf N}};\vec\omega_{l-1}}$, we define $\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Omega_{z_{l+1}\to z'_{l+1}}^{{\bf N}}$ on ${{\mathbb G}}_{{\bf N}}\setminus{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle l$}} \omega_i$ such that $\zeta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ earlier than $\omega_i$, for every $i=1,\dots,l$. Then, we define $\tilde\Omega_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ such that there are $k-(l+1)$ edge-disjoint paths on ${{\mathbb G}}_{ {{\bf N}}}\setminus({\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle l$}}\omega_i {\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=l+2,\dots,k$. 4. If $l=k-1$, then we simply define $\tilde\Omega_{z_k\to z'_k}^{{{\bf N}};\vec \omega_{k-1}}=\Xi_{z_k\to z'_k}^{{{\bf N}};\vec\omega_{k-1}}$. We will also abuse the notation to denote $\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ by $\tilde\Omega_{z_1\to z'_1}^{{{\bf N}};\vec\omega_0}$. Using the above notation, we can decompose ${\mathfrak{S}}^{(\prime)}$ disjointly as follows. For a collection $\omega_i\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ for $i=1,\dots,k$, we denote by ${\mathfrak{S}}_{\vec\omega_k}^{(\prime)}$ the set of partitions $\vec{{\mathbb S}}_k\equiv({{\mathbb S}}_0,{{\mathbb S}}_1,\dots,{{\mathbb S}}_k)\in{\mathfrak{S}}^{(\prime)}$ such that, for every $i=1,\dots,k$, the earliest element of $\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ contained in ${{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_i$ is $\omega_i$. Then, ${\mathfrak{S}}^{(\prime)}$ is decomposed as $$\begin{aligned} {\label{eq:SS'-dec}} {\mathfrak{S}}^{(\prime)}={\mathop{\Dot{\bigcup}}}_{\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}}\, {\mathop{\Dot{\bigcup}}}_{\omega_2\in\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1}}\cdots {\mathop{\Dot{\bigcup}}}_{\omega_k\in\tilde\Omega_{z_k\to z'_k}^{{{\bf N}};\vec\omega_{k-1}}} {\mathfrak{S}}_{\vec\omega_k}^{(\prime)}.\end{aligned}$$ To complete the proof of Lemma \[lmm:GHS-BK\], it suffices to construct a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$ for every $\vec\omega_k$. For $\vec{{\mathbb S}}_k\in{\mathfrak{S}}_{\vec\omega_k}$, we define $$\begin{aligned} {\label{eq:Fdef}} \textstyle\vec F_{\vec\omega_k}(\vec{{\mathbb S}}_k)\equiv\big(F_{\vec\omega_k}^{ {\scriptscriptstyle}(0)}({{\mathbb S}}_0),\dots,F_{\vec\omega_k}^{{\scriptscriptstyle}(k)}({{\mathbb S}}_k)\big)=\Big({{\mathbb S}}_0 {\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle k$}}\omega_i,\,{{\mathbb S}}_1 {\,\triangle\,}\omega_1,\dots,{{\mathbb S}}_k{\,\triangle\,}\omega_k\Big),\end{aligned}$$ where ${\partial}F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0)={{\cal V}}{\,\triangle\,}\{z_1,z'_1\} {\,\triangle\,}\cdots{\,\triangle\,}\{z_k,z'_k\}$ and ${\partial}F_{\vec\omega_k}^{{\scriptscriptstyle}(i)}({{\mathbb S}}_i)= \{z_i,z'_i\}$ for $i=1,\dots,k$. Note that, by definition using symmetric difference, we have $\vec F_{ \vec\omega_k}(\vec F_{\vec\omega_k}(\vec{{\mathbb S}}_k))=\vec{{\mathbb S}}_k$. Also, by simple combinatorics using $\omega_i\cap\omega_j={{\mathbb S}}_i\cap{{\mathbb S}}_j={\varnothing}$ and $\omega_j\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_j$ for $1\leq j\leq k$ and $i\ne j$, we have $$\begin{aligned} {\label{eq:F0DcupFi}} F_{\vec\omega_k}^{{\scriptscriptstyle}(i)}({{\mathbb S}}_i)\cap F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j)= {\varnothing},&& F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)} ({{\mathbb S}}_j)\textstyle=\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i\ne j}\omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j.\end{aligned}$$ Since $\omega_j\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_j$ and $\omega_j\cap{\mathop{\Dot{\bigcup}}}_{i\ne j}\omega_i={\varnothing}$, we have $\omega_j\subset F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)} ({{\mathbb S}}_j)$. It remains to show that $\omega_j$ is the earliest element of $\tilde\Omega_{z_j\to z'_j}^{{{\bf N}};\vec\omega_{j-1}}$ in $F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)} ({{\mathbb S}}_j)$. To see this, we first recall that $\tilde\Omega_{z_j\to z'_j}^{{{\bf N}};\vec \omega_{j-1}}$ is a set of paths on ${{\mathbb G}}_{{\bf N}}\setminus{\mathop{\Dot{\bigcup}}}_{i<j} \omega_i$, so that its earliest element contained in $({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i<j}\omega_i){\:\Dot{\cup}\:}{{\mathbb S}}_j$ is still $\omega_j$. Furthermore, since each $\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec \omega_{i-1}}$ for $i>j$ is a set of paths that do not fully contain $\omega_j$ or any earlier element of $\tilde\Omega_{z_j\to z'_j}^{{{\bf N}};\vec\omega_{j-1}}$ as a subset, $\omega_j$ is still the earliest element of $$\begin{aligned} \bigg(\textstyle\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i<j}\omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j\bigg) {\,\triangle\,}\Big({\mathop{\Dot{\bigcup}}}_{i>j}\omega_i\Big)=\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i\ne j} \omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j\equiv F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j).\end{aligned}$$ Therefore, $\vec F_{\vec\omega_k}$ is a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$. This completes the proof of Lemma \[lmm:GHS-BK\]. Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ {#ss:pijbd} ---------------------------------------------------------------- First we prove [(\[eq:piNbd\])]{} for $j\ge1$ assuming the following two lemmas, in which we recall [(\[eq:Theta-def\])]{} and use $$\begin{aligned} &E'_{{\bf N}}(z,x;{{\cal A}})=\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{}{\Longleftrightarrow}}}x\},& &E''_{{\bf N}}(z,x,v;{{\cal A}})=E'_{{\bf N}}(z,x;{{\cal A}})\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}} {\overset{}{\longleftrightarrow}}}v\},{\label{eq:E'E''-def}}\\ &\Theta'_{z,x;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\! \frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}(z,x;{{\cal A}})$}}},\quad& &\Theta''_{z,x,v;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\! \frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}(z,x,v;{{\cal A}})$}}}.{\label{eq:Theta'Theta''-def}}\end{aligned}$$ \[lmm:Thetabds\] For the ferromagnetic Ising model, we have $$\begin{aligned} \Theta_{y,x;{{\cal A}}}&\leq\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big) \,\Theta'_{z,x;{{\cal A}}},{\label{eq:Theta[1]-bd}}\\[5pt] \Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}\big]&\leq\sum_z\big(\delta_{y,z} +\tilde G_\Lambda(y,z)\big)\,\Theta''_{z,x,v;{{\cal A}}}{\nonumber}\\ &\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\tilde G_\Lambda(v',z)\,\Theta'_{z,x; {{\cal A}}}\,\psi_\Lambda(v',v).{\label{eq:Theta[I]-bd}}\end{aligned}$$ \[lmm:Theta’Theta”bd\] For the ferromagnetic Ising model, we have $$\begin{aligned} {\label{eq:Theta'Theta''bd}} \Theta'_{y,x;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P'_{\Lambda;u}(y,x),&& \Theta''_{y,x,v;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P''_{\Lambda;u,v}(y,x).\end{aligned}$$ We prove Lemma \[lmm:Thetabds\] in Section \[sss:chopping-off\], and Lemma \[lmm:Theta’Theta”bd\] in Section \[sss:dbconn\]. Recall [(\[eq:pij-def\])]{}. By [(\[eq:Theta\[1\]-bd\])]{}, [(\[eq:Theta’Theta”bd\])]{} and [(\[eq:Q’-def\])]{}, we obtain $$\begin{aligned} {\label{eq:nest-diagbd}} \Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\Big[\tau_{b_j} \Theta^{{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]&\leq\Theta^{ {\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\bigg[\sum_z\tau_{b_j}\big( \delta_{{\overline{b}}_j,z}+\tilde G_\Lambda({\overline{b}}_j,z)\big)\sum_{v_j\in \tilde{{\cal C}}_{j-1}}P'_{\Lambda;v_j}(z,x)\bigg]{\nonumber}\\ &\leq\sum_{v_j}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}} \big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}_{j-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v_j\}$}}}\big]\,\tau_{b_j}Q'_{\Lambda;v_j} ({\overline{b}}_j,x).\end{aligned}$$ For $j=1$, we use [(\[eq:pi0’-bd\])]{} and [(\[eq:nest-diagbd\])]{} to obtain $$\begin{aligned} {\label{eq:pi0'-bd-appl}} &\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)\equiv\sum_{b_1}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;\Lambda} \Big[\tau_{b_1}\,\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,x;\tilde{{\cal C}}_0}\Big]\leq\sum_{b_1, v_1}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;\Lambda}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v_1\}$}}}\big]\,\tau_{ b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x)\\ &~=\sum_{b_1,v_1}\bigg(\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}_1}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v_1}\bigg)\tau_{ b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x)\leq\sum_{b_1,v_1}P_{\Lambda;v_1}^{\prime{ {\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\,\tau_{b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x).{\nonumber}\end{aligned}$$ For $j\ge2$, we use [(\[eq:Theta\[I\]-bd\])]{}–[(\[eq:Theta’Theta”bd\])]{} and then [(\[eq:Q’-def\])]{}–[(\[eq:Q”-def\])]{} to obtain $$\begin{aligned} {\label{eq:Theta-bd-appl}} &\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\Big[\tau_{b_j}\Theta^{ {\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]\leq\sum_{v_j}\Theta^{{\scriptscriptstyle}(j-1)}_{ {\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}_{j-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v_j\}$}}}\big]\,\tau_{ b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x){\nonumber}\\ &\quad\leq\sum_{v_j}\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x)\Bigg(\sum_z\big( \delta_{{\overline{b}}_{j-1},z}+\tilde G_\Lambda({\overline{b}}_{j-1},z)\big)\sum_{v_{j-1}\in \tilde{{\cal C}}_{j-2}}P''_{\Lambda;v_{j-1},v_j}(z,{\underline{b}}_j){\nonumber}\\ &\qquad\qquad+\sum_{v',z}\big(\delta_{{\overline{b}}_{j-1},v'}+\tilde G_\Lambda({\overline{b}}_{ j-1},v')\big)\,\tilde G_\Lambda(v',z)\,\sum_{v_{j-1}\in\tilde{{\cal C}}_{j-2}} P'_{\Lambda;v_{j-1}}(z,{\underline{b}}_j)\,\psi_\Lambda(v',v_j)\Bigg){\nonumber}\\ &\quad\leq\sum_{v_{j-1},v_j}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{j-1}\in\tilde{{\cal C}}_{j-2}\}$}}}\,Q''_{\Lambda; v_{j-1},v_j}({\overline{b}}_{j-1},{\underline{b}}_j)\,\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x).\end{aligned}$$ We repeatedly use [(\[eq:Theta\[I\]-bd\])]{}–[(\[eq:Theta’Theta”bd\])]{} to bound $\Theta^{{\scriptscriptstyle}(i)}_{{\overline{b}}_i,{\underline{b}}_{i+1};\tilde{{\cal C}}_{i-1}} [{\mathbbm{1}{\scriptstyle\{{\overline{b}}_i {\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v_{i+1}\}}}]$ for $i=j-2,\dots,1$ as in [(\[eq:Theta-bd-appl\])]{}, and then at the end we apply [(\[eq:pi0’-bd\])]{} as in [(\[eq:pi0’-bd-appl\])]{} to obtain [(\[eq:piNbd\])]{}. This completes the proof. ### Proof of Lemma \[lmm:Thetabds\] {#sss:chopping-off} Recall [(\[eq:Theta-def\])]{} and [(\[eq:Theta’Theta”-def\])]{}. Then, to prove [(\[eq:Theta\[1\]-bd\])]{}, it suffices to bound the contribution from ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})$}}}$ by $\sum_z\tilde G_\Lambda(y,z)\,\Theta'_{z,x;{{\cal A}}}$. First we recall [(\[eq:E-def\])]{} and [(\[eq:E’E”-def\])]{}. Then, we have $$\begin{aligned} E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})=E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}}) \cap\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\big\}.\end{aligned}$$ On $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$, there is at least one pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $y$. Let $b$ be the last pivotal bond among them. Then, we have ${\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\text{ off }b$, $m_b+n_b>0$, and $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}$ in ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}$. Moreover, on the event $E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})$, we have that $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}$ in ${{\cal A}}{^{\rm c}}$ and ${\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x$. Since $\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\text{ off }b\}\cap\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}=\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$ on the event that $b$ is pivotal for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $y$, we have $$\begin{aligned} {\label{eq:EE'-dec}} &E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}}){\nonumber}\\ &={\mathop{\Dot{\bigcup}}}_b\Big\{\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}\cap\{m_b+n_b>0 \}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x) {^{\rm c}}\big\}\Big\}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned} {\label{eq:Theta[1]-rewr}} &\Theta_{y,x;{{\cal A}}}-\Theta'_{y,x;{{\cal A}}}{\nonumber}\\ &=\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}\}$}}}.\end{aligned}$$ It remains to bound the right-hand side of [(\[eq:Theta\[1\]-rewr\])]{}, which is nonzero only if $m_b$ is even and $n_b$ is odd, due to the source constraints and the conditions in the indicators. First, as in [(\[eq:2nd-ind-fact\])]{}, we alternate the parity of $n_b$ by changing the source constraint into ${\partial}{{\bf n}}=y{\vartriangle}b{\vartriangle}x$ and multiplying by $\tau_b$. Then, by conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)$ as in [(\[eq:3rd-ind-prefact\])]{} (i.e., conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)={{\cal B}}$, letting ${{\bf m}}'={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf m}}''={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf n}}'={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\bf n}}''={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, and then summing over ${{\cal B}}\subset\Lambda$), we obtain $$\begin{aligned} {\label{eq:2ndind-contr}} \sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}'={\varnothing}\\ {\partial}{{\bf n}}'= {\overline{b}}{\vartriangle}x}}\frac{\tilde w_{{{\cal A}}{^{\rm c}},{{\cal B}}}({{\bf m}}')\,Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda,{{\cal B}}} ({{\bf n}}')\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}'+{{\bf n}}'}({\overline{b}},x; {{\cal A}})\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}^b_{{{\bf m}}'+{{\bf n}}'}(x)={{\cal B}}}{\nonumber}\\ \times\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m'_b,n'_b\text{ even}\}$}}}\underbrace{\sum_{ \substack{{\partial}{{\bf m}}''={\varnothing}\\ {\partial}{{\bf n}}''=y{\vartriangle}{\underline{b}}}}\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\, \frac{w_{{{\cal B}}{^{\rm c}}}({{\bf n}}'')}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+ {{\bf n}}''$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}~\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}\}$}}}}_{\stackrel{ \because{(\ref{eq:switching-appl})}\;}={{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}{\nonumber}\\ =\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\, \tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}.\end{aligned}$$ Since ${{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}=0$ on $E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\setminus\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$ off $b\}\subset\{{\underline{b}}\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)\}$ and on the event that $m_b$ or $n_b$ is odd (see below [(\[eq:0th-summand2\])]{} or above [(\[eq:3rd-ind-fact\])]{}), we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b\text{ even}\}}}$ in [(\[eq:2ndind-contr\])]{}. Since ${{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+ {{\bf n}}}^b(x) {^{\rm c}}}\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda$ due to Proposition \[prp:through\], we have $$\begin{aligned} {\label{eq:2ndind-contrbd}} {(\ref{eq:2ndind-contr})}\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}}={{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\,\Theta'_{{\overline{b}},x;{{\cal A}}}.\end{aligned}$$ Therefore, [(\[eq:Theta\[1\]-rewr\])]{} is bounded by $\sum_b{{\langle \varphi_y \varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\,\Theta'_{{\overline{b}},x;{{\cal A}}}\equiv\sum_z\tilde G_\Lambda(y,z)\,\Theta'_{z,x;{{\cal A}}}$. This completes the proof of [(\[eq:Theta\[1\]-bd\])]{}. Recall [(\[eq:Theta-def\])]{} and [(\[eq:Theta’Theta”-def\])]{}. To prove [(\[eq:Theta\[I\]-bd\])]{}, we investigate $$\begin{aligned} L&\equiv\big\{E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}\big\} \setminus E''_{{{\bf m}}+{{\bf n}}}(y,x,v;{{\cal A}}){\nonumber}\\ &=\{E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\}\cap \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\},\end{aligned}$$ where $\Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle L$}}}]=\Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}} v\}}}]-\Theta''_{y,x,v;{{\cal A}}}$. First we recall [(\[eq:EE’-dec\])]{}, in which $b$ is the last pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$ from $y$, and define $$\begin{aligned} R_1(b)&=\{E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})\text{ off }b\}\cap\{m_b +n_b>0\}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap {{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}\big\},{\label{eq:R1b-def}}\\ R_2(b)&=\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\} \cap\{m_b+n_b>0\}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}},\;y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\big\}, {\label{eq:R2b-def}}\end{aligned}$$ where $v\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)$ on $R_1(b)$, while $v\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(y)$ on $R_2(b)$. Since $$\begin{aligned} {\label{eq:EE'E''-predec}} L={\mathop{\Dot{\bigcup}}}_b\{R_1(b){\:\Dot{\cup}\:}R_2(b)\},\end{aligned}$$ we have $$\begin{aligned} {\label{eq:EE'E''predec2}} \Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}} v\}}}]-\Theta''_{y,x,v;{{\cal A}}}=\sum_b \Big(\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_1(b)$}}}\big]+\Theta_{y,x;{{\cal A}}} \big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]\Big).\end{aligned}$$ Following the same argument as in [(\[eq:2ndind-contr\])]{}–[(\[eq:2ndind-contrbd\])]{}, we easily obtain $$\begin{aligned} {\label{eq:EE'E''predec3}} \Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_1(b)$}}}\big]&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{ {{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b \text{ even}\}$}}}\,{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\, {{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}{\nonumber}\\ &\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\sum_{\substack{ {\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{ {{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})$}}}={{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b \,\Theta''_{{\overline{b}},x,v;{{\cal A}}}.\end{aligned}$$ Similarly, we have $$\begin{aligned} {\label{eq:EE'E''decpre3}} \Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]&=\sum_{{{\cal B}}\subset \Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}} \frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off } b\\}$}}}\,\cap\,\{{{\cal C}}^b_{{{\bf m}}+{{\bf n}}}(x)={{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b \text{ even}\}$}}}{\nonumber}\\ &\qquad\qquad\times\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=y{\vartriangle}{\underline{b}}}} \frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}},~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\text{ (in }{{\cal B}}{^{\rm c}})\}$}}}{\nonumber}\\ &\leq\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\, \tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)},\end{aligned}$$ where $$\begin{aligned} {\label{eq:Psi-def}} \Psi_{y,z,v;{{\cal A}},{{\cal B}}}=\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}= y{\vartriangle}z}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ We note that, by ignoring the indicator in [(\[eq:Psi-def\])]{}, we have $0\leq\Psi_{y,z,v;{{\cal A}},{{\cal B}}}\leq{{\langle \varphi_y\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}$, which is zero whenever $z\in{{\cal B}}$. Therefore, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b\text{ even}\}}}$ in [(\[eq:EE’E”decpre3\])]{} to obtain $$\begin{aligned} {\label{eq:EE'E''dec3}} \Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]\leq\sum_{\substack{{\partial}{{\bf m}}= {\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}} \,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}} \,\tau_b\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)}.\end{aligned}$$ Substituting [(\[eq:EE’E”predec3\])]{} and [(\[eq:EE’E”dec3\])]{} to [(\[eq:EE’E”predec2\])]{}, we arrive at $$\begin{aligned} {\label{eq:EE'E''dec2}} \Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}\big]&\leq\sum_z\big(\delta_{y, z}+\tilde G_\Lambda(y,z)\big)\,\Theta''_{z,x,v;{{\cal A}}}{\nonumber}\\ &\quad+\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{ w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}}\,\tau_b\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{ {{\bf m}}+{{\bf n}}}^b(x)}.\end{aligned}$$ The proof of [(\[eq:Theta\[I\]-bd\])]{} is completed by using $$\begin{aligned} {\label{eq:Psi-bd}} \Psi_{y,z,v;{{\cal A}},{{\cal B}}}\leq\sum_{v'}{{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda {{\langle \varphi_{v'}\varphi_z \rangle}}_\Lambda\,\psi_\Lambda(v',v),\end{aligned}$$ and replacing ${{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda$ in [(\[eq:Psi-bd\])]{} by $\delta_{y,v'}+\tilde G_\Lambda(y,v')$, due to [(\[eq:G-delta-bd\])]{}. To complete the proof of [(\[eq:Theta\[I\]-bd\])]{}, it thus remains to show [(\[eq:Psi-bd\])]{}. First we note that, if ${{\cal A}}\subset{{\cal B}}$, then by Lemma \[lmm:switching\] we have $$\begin{aligned} \Psi_{y,z,v;{{\cal A}},{{\cal B}}}=\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=y{\vartriangle}z}} \frac{w_{{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})} {Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}={{\langle \varphi_y\varphi_v \rangle}}_{{{\cal B}}{^{\rm c}}}{{\langle \varphi_v\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}\leq{{\langle \varphi_y \varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_z \rangle}}_\Lambda.\end{aligned}$$ However, to prove [(\[eq:Psi-bd\])]{} for a general ${{\cal A}}$ that does not necessarily satisfy ${{\cal A}}\subset{{\cal B}}$, we use $$\begin{aligned} {\label{eq:Psi-ind-dec}} \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}{\:\Dot{\cup}\:}\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\} \setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}\big\},\end{aligned}$$ and consider the two events on the right-hand side separately. The contribution to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ from $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$ is easily bounded, similarly to [(\[eq:dbbd\])]{}, as $$\begin{aligned} {\label{eq:psi-delta}} \sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}\leq\sum_{\substack{{\partial}{{\bf k}}=y{\vartriangle}z\\ {\partial}{{\bf k}}'={\varnothing}}}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}}}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}+{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}v\}$}}}&={{\langle \varphi_y\varphi_v \rangle}}_{{{\cal B}}{^{\rm c}}}{{\langle \varphi_v\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}{\nonumber}\\ &\leq{{\langle \varphi_y\varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_z \rangle}}_\Lambda.\end{aligned}$$ Next we consider the contribution to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ from $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$ in [(\[eq:Psi-ind-dec\])]{}. We denote by ${{\cal C}}_{{\bf k}}(y)$ the set of sites ${{\bf k}}$-connected from $y$. Since $v\in{{\cal C}}_{{{\bf h}}+{{\bf k}}}(y)\setminus{{\cal C}}_{{\bf k}}(y)$, there is a *nonzero* alternating chain of mutually-disjoint ${{\bf h}}$-connected clusters and mutually-disjoint ${{\bf k}}$-connected clusters, from some $u_0\in{{\cal C}}_{{\bf k}}(y)$ to $v$. Therefore, we have $$\begin{aligned} {\label{eq:ind-bd}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\\}$}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v}\leq\sum_{j=1}^\infty \sum_{\substack{u_0,\dots,u_j\\ u_l\ne u_{l'}\,{{}^\forall}l\ne l'\\ u_j =v}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_0\}$}}}\bigg(\prod_{l\ge0}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}} {\overset{}{\longleftrightarrow}}}u_{2l +1}\}$}}}\bigg)\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg){\nonumber}\\ \times\bigg(\prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf h}}(u_{2l}) \,\cap\,{{\cal C}}_{{\bf h}}(u_{2l'})={\varnothing}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg),\end{aligned}$$ where we regard an empty product as 1. Using this bound, we can perform the sums over ${{\bf h}}$ and ${{\bf k}}$ in [(\[eq:Psi-def\])]{} independently. For $j=1$ and given $u_0\ne u_1=v$, the summand of [(\[eq:ind-bd\])]{} equals ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_0\}}}{\mathbbm{1}{\scriptstyle\{u_0{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$, which is simply equal to ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$ if $u_0=y$. Then, by [(\[eq:psi-delta\])]{} and [(\[eq:G-delta-bd\])]{}, the contribution from this to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ is $$\begin{aligned} {\label{eq:psi-delta-G2}} \sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_0\}$}}}\sum_{{\partial}{{\bf h}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}({{\bf h}})} {Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_0{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}\leq{{\langle \varphi_y \varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z \rangle}}_\Lambda\,\tilde G_\Lambda(u_0,v)^2.\end{aligned}$$ Fix $j\ge2$ and a sequence of distinct sites $u_0,\dots,u_j\,(=v)$, and first consider the contribution to the sum over ${{\bf k}}$ in [(\[eq:Psi-def\])]{} from the relevant indicators in the right-hand side of [(\[eq:ind-bd\])]{}, which is $$\begin{aligned} {\label{eq:nsum-0thbd}} &\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_0\}$}}}\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg) \prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\, {{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\\ &=\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}} \bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg)\bigg( \prod_{\substack{l,l'\ge1\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\, {{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_0\\}$}}}\,\cap\,\{{{\cal C}}_{{\bf k}}(u_0)\,\cap\,{{\cal U}}_{{{\bf k}};1}={\varnothing}}{\nonumber},\end{aligned}$$ where ${{\cal U}}_{{{\bf k}};1}={\mathop{\Dot{\bigcup}}}_{l\ge1}{{\cal C}}_{{\bf k}}(u_{2l})$. Conditioning on ${{\cal U}}_{{{\bf k}};1}$, we obtain that $$\begin{gathered} {(\ref{eq:nsum-0thbd})}=\sum_{{\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})} {Z_{{{\cal B}}{^{\rm c}}}}\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_{2l}\}$}}} \bigg)\bigg(\prod_{\substack{l,l'\ge1\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\ \times\underbrace{\sum_{{\partial}{{\bf k}}'=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}\cap\, {{\cal U}}_{{{\bf k}};1}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};1}{^{\rm c}}}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}u_0\}$}}}}_{\stackrel{\because{(\ref{eq:psi-delta})}\;}\leq {{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z \rangle}}_\Lambda}.{\label{eq:nsum-1stbd}}\end{gathered}$$ Then, by conditioning on ${{\cal U}}_{{{\bf k}};2}\equiv{\mathop{\Dot{\bigcup}}}_{l\ge2}{{\cal C}}_{{\bf k}}(u_{2l})$, following the same computation as above and using [(\[eq:G-delta-bd\])]{}, we further obtain that $$\begin{gathered} {(\ref{eq:nsum-0thbd})}\leq{{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z \rangle}}_\Lambda\sum_{{\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}} ({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\bigg(\prod_{l\ge2}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}u_{2 l}\}$}}}\bigg)\bigg(\prod_{\substack{l,l'\ge2\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\ \times\underbrace{\sum_{{\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}u_2\}$}}}}_{\leq\;\tilde G_\Lambda(u_1,u_2)^2}.{\label{eq:nsum-2ndbd}}\end{gathered}$$ We repeat this computation until all indicators for ${{\bf k}}$ are used up. We also apply the same argument to the sum over ${{\bf h}}$ in [(\[eq:Psi-def\])]{}. Summarizing these bounds with [(\[eq:psi-delta\])]{} and [(\[eq:psi-delta-G2\])]{}, and replacing $u_0$ in [(\[eq:ind-bd\])]{}–[(\[eq:nsum-1stbd\])]{} by $v'$, we obtain [(\[eq:Psi-bd\])]{}. This completes the proof of [(\[eq:Theta\[I\]-bd\])]{}. ### Proof of Lemma \[lmm:Theta’Theta”bd\] {#sss:dbconn} We note that the common factor ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$ in $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ can be decomposed as $$\begin{aligned} {\label{eq:Theta'-evdec}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}$}}} \setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ We estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$ to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ in the following paragraphs (a) and (b), respectively. Then, in the paragraphs (c) and (d) below, we will estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{} to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$, respectively. **(a)** First we investigate the contribution to $\Theta'_{y,x;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$: $$\begin{aligned} {\label{eq:contr-(a)}} \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}} ({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ For a set of events $E_1,\dots,E_N$, we define $E_1\circ\cdots\circ E_N$ to be the event that $E_1,\dots,E_N$ occur *bond*-disjointly. Then, we have $$\begin{aligned} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}\leq\sum_{u\in{{\cal A}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u \\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x},\end{aligned}$$ where the right-hand side does not depend on ${{\bf m}}$. Therefore, the contribution to $\Theta'_{y,x;{{\cal A}}}$ is bounded by $$\begin{aligned} {\label{eq:Theta'-bd1stbd}} {(\ref{eq:contr-(a)})}\leq\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{ w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x}\leq\sum_{u\in{{\cal A}}}P_{\Lambda; u}^{\prime{{\scriptscriptstyle}(0)}}(y,x),\end{aligned}$$ where we have applied the same argument as in the proof of [(\[eq:pi0’-bd\])]{}, which is around [(\[eq:dbbd\])]{}–[(\[eq:S03sup\])]{}. **(b)** Next we investigate the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-evdec\])]{}: $$\begin{aligned} {\label{eq:contr-(b)}} \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}} ({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+ {{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ Note that, by using [(\[eq:Psi-ind-dec\])]{} and ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}\leq{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$, we have $$\begin{aligned} {\label{eq:Theta''-1stindbd}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}} v}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}\,\Big({\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}} v\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\\}$}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}\Big).\end{aligned}$$ We investigate the contributions from the two indicators in the parentheses separately. We begin with the contribution from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$, which is independent of ${{\bf m}}$. Since $$\begin{aligned} \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}&\subset\{y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\circ\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\},\\ \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}&\subset\bigcup_{u\in{{\cal A}}}\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\}\circ\{u {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\},{\label{eq:Theta''-1stind1st-pcontr}}\end{aligned}$$ the contribution to [(\[eq:contr-(b)\])]{} from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$ in [(\[eq:Theta”-1stindbd\])]{} is bounded by $$\begin{aligned} {\label{eq:Theta''-1stind1stcontr}} \sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ We follow Steps (i)–(iii) described above [(\[eq:dbbd\])]{} in Section \[ss:pi0bd\]. Without loss of generality, we can assume that $y,u,x$ and $v$ are all different; otherwise, the following argument can be simplified. (i) Since $y$ and $x$ are sources, but $u$ and $v$ are not, there is an edge-disjoint cycle $y\to u\to x\to v\to y$, with an extra edge-disjoint path from $y$ to $x$. Therefore, we have in total at least $5\,(=4+1)$ edge-disjoint paths. (ii) Multiplying by $(Z_\Lambda/Z_\Lambda)^4$, we have $$\begin{aligned} {\label{eq:Theta''-bd1stprebd}} {(\ref{eq:Theta''-1stind1stcontr})}=\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})} {Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_i={\varnothing}~ {{}^\forall}i=1,\dots,4\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}\prod_b\frac{N_b!}{n_b!\prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!},\end{aligned}$$ where we have used the notation $m_b^{{\scriptscriptstyle}(i)}={{\bf m}}_i|_b$. (iii) The sum over ${{\bf n}},{{\bf m}}_1,\dots,{{\bf m}}_4$ in [(\[eq:Theta”-bd1stprebd\])]{} is bounded by the cardinality of ${\mathfrak{S}}$ in Lemma \[lmm:GHS-BK\] with $k=4$, ${{\cal V}}=\{y,x\}$, $\{z_1,z'_1\}=\{y,u\}$, $\{z_2,z'_2\}=\{u,x\}$, $\{z_3,z'_3\}=\{y,v\}$ and $\{z_4,z'_4\}=\{v,x\}$. Bounding the cardinality of ${\mathfrak{S}}'$ in Lemma \[lmm:GHS-BK\] for this setting, we obtain $$\begin{aligned} {\label{eq:Theta''-bd1stbd1}} {(\ref{eq:Theta''-bd1stprebd})}&\leq\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_1=y{\vartriangle}u,~{\partial}{{\bf m}}_2=u{\vartriangle}x\\ {\partial}{{\bf m}}_3=y{\vartriangle}v,~{\partial}{{\bf m}}_4=v{\vartriangle}x\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}\prod_b\frac{N_b!}{n_b! \prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!}{\nonumber}\\ &\leq\sum_{u\in{{\cal A}}}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y \varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y \varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_x \rangle}}_\Lambda.\end{aligned}$$ Next we investigate the contribution to [(\[eq:contr-(b)\])]{} from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}$ in [(\[eq:Theta”-1stindbd\])]{}. On the event $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\cap\{\{ y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}\}$, there exists a $v_0\ne v$ such that $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\circ\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v_0\}$ occurs and that $v_0$ and $v$ are connected via a nonzero alternating chain of mutually-disjoint ${{\bf m}}$-connected clusters and mutually-disjoint ${{\bf n}}$-connected clusters. Therefore, by [(\[eq:ind-bd\])]{} and [(\[eq:Theta”-1stind1st-pcontr\])]{} (see also [(\[eq:Theta”-1stind1stcontr\])]{}), we obtain $$\begin{aligned} {\label{eq:Theta''-1stindbd2}} &{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v \}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}}{\nonumber}\\ &\leq\sum_{u\in{{\cal A}}}\,\sum_{j\ge1}\sum_{\substack{v_0,\dots,v_j\\ v_l\ne v_{l'}\,{{}^\forall}l\ne l'\\ v_j=v}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}} x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v_0}\,\bigg(\prod_{l\ge0}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2 l}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}} {\overset{}{\longleftrightarrow}}}v_{2l+1}\}$}}}\bigg){\nonumber}\\ &\qquad\times\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v_{2l}\}$}}}\bigg)\bigg( \prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf m}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf m}}(v_{2l'})={\varnothing}\}$}}}\;{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf n}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf n}}(v_{2l'})={\varnothing}\}$}}}\bigg).\end{aligned}$$ For the three products of indicators, we repeate the same argument as in [(\[eq:psi-delta-G2\])]{}–[(\[eq:nsum-2ndbd\])]{} to derive the factor $\psi_\Lambda(v_0,v)-\delta_{v_0,v}$. As a result, we have $$\begin{aligned} {\label{eq:Theta''-prebd1stbd2}} &\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}} \,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v\}}{\nonumber}\\ &~\leq\sum_{v_0}\big(\psi_\Lambda(v_0,v)-\delta_{v_0,v}\big)\sum_{u\in{{\cal A}}} \,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}} u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v_0}.\end{aligned}$$ Following the same argument as in [(\[eq:Theta”-1stind1stcontr\])]{}–[(\[eq:Theta”-bd1stbd1\])]{}, we obtain $$\begin{aligned} {\label{eq:Theta''-bd1stbd2}} {(\ref{eq:Theta''-prebd1stbd2})}&\leq\sum_{u\in{{\cal A}},\;v_0}\big(\psi_\Lambda(v_0, v)-\delta_{v_0,v}\big)\,{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y \varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y \varphi_{v_0} \rangle}}_\Lambda{{\langle \varphi_{v_0}\varphi_x \rangle}}_\Lambda{\nonumber}\\ &\leq\sum_{u\in{{\cal A}}}\Big(P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)- {{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda {{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_v \rangle}}_\Lambda {{\langle \varphi_v\varphi_x \rangle}}_\Lambda\Big).\end{aligned}$$ Summarizing [(\[eq:Theta”-1stindbd\])]{}, [(\[eq:Theta”-bd1stbd1\])]{} and [(\[eq:Theta”-bd1stbd2\])]{}, we arrive at $$\begin{aligned} {\label{eq:Theta''-0bdfin}} {(\ref{eq:contr-(b)})}\leq\sum_{u\in{{\cal A}}}P_{\Lambda;u,v}^{\prime\prime {{\scriptscriptstyle}(0)}}(y,x).\end{aligned}$$ This completes the bound on the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-evdec\])]{}. **(c)** The contribution to $\Theta'_{y,x;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{} equals $$\begin{aligned} {\label{eq:contr-(c)}} \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}}.\end{aligned}$$ Note that, if ${\mathbbm{1}{\scriptstyle\{{\partial}{{\bf n}}=y{\vartriangle}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}=1$, then $y$ is ${{\bf n}}$-connected, but not ${{\bf n}}$-doubly connected, to $x$, and therefore there exists at least one pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$. Given an ordered set of bonds $\vec b_T=(b_1,\dots,b_T)$, we define $$\begin{aligned} {\label{eq:H-def}} H_{{{\bf n}};\vec b_T}(y,x)=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\}\cap\bigcap_{i=1}^T\Big\{\{{\overline{b}}_i {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}{\underline{b}}_{i+1}\}\cap\big\{n_{b_i}>0,~b_i\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\big\}\Big\},\end{aligned}$$ where, by convention, ${\underline{b}}_{T+1}=x$. Then, by ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}$, we obtain $$\begin{aligned} {\label{eq:Theta'-2ndindbd2}} {(\ref{eq:contr-(c)})}&=\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x) \,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}{\nonumber}\\ &\leq\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y {\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)\,\cap\, \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ On the event $H_{{{\bf n}};\vec b_T}(y,x)$, we denote the ${{\bf n}}$-double connections between the pivotal bonds $b_1,\dots,b_T$ by $$\begin{aligned} {{\cal D}}_{{{\bf n}};i}=\begin{cases} {{\cal C}}_{{\bf n}}^{b_1}(y)&(i=0),\\ {{\cal C}}_{{\bf n}}^{b_{i+1}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_i}(y)&(i=1,\dots,T-1),\\ {{\cal C}}_{{\bf n}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_T}(y)&(i=T). \end{cases}\end{aligned}$$ As in Figure \[fig:lace-edges\], we can think of ${{\cal C}}_{{\bf n}}(y)$ as the interval $[0,T]$, where each integer $i\in[0,T]$ corresponds to ${{\cal D}}_{{{\bf n}};i}$ and the unit interval $(i-1,i)\subset[0,T]$ corresponds to the pivotal bond $b_i$. Since $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x$, we see that, for every $b_i$, there must be an $({{\bf m}}+{{\bf n}})$-bypath (i.e., an $({{\bf m}}+{{\bf n}})$-connection that does not go through $b_i$) from some $z\in{{\cal D}}_{{{\bf n}};s}$ with $s<i$ to some $z'\in{{\cal D}}_{{{\bf n}};t}$ with $t\ge i$. We abbreviate $\{s,t\}$ to $st$ if there is no confusion. Let ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(2)}=\{\{0t_1,s_2T\}:0<s_2\leq t_1<T\}$ and generally for $j\leq T$ (see Figure \[fig:lace-edges\]), ![\[fig:lace-edges\]An element in ${{\cal L}}_{[0,8]}^{{\scriptscriptstyle}(4)}$, which consists of $s_1t_1=\{0,3\}$, $s_2t_2=\{2,4\}$, $s_3t_3=\{4,6\}$ and $s_4t_4=\{5,8\}$.](lace-edges) $$\begin{aligned} {{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}=\big\{\{s_it_i\}_{i=1}^j:0=s_1<s_2\leq t_1<s_3 \leq\cdots\leq t_{j-2}<s_j\leq t_{j-1}<t_j=T\big\}.\end{aligned}$$ For every $j\in\{1,\dots,T\}$, we have $\bigcup_{st\in\Gamma}[s,t]= [0,T]$ for any $\Gamma\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}$, which implies double connection. Conditioning on ${{\cal C}}_{{\bf n}}(y)\equiv\bigcup_{i=0}^{ \raisebox{-3pt}{$\scriptstyle T$}}{{\cal D}}_{{{\bf n}};i}={{\cal B}}$ (and denoting ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, ${{\bf h}}={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\cal D}}_{{{\bf n}};i}\equiv{{\cal D}}_{{{\bf h}};i}={{\cal B}}_i$) and multiplying by $Z_{{{\cal B}}{^{\rm c}}}/Z_{{{\cal B}}{^{\rm c}}}$, we obtain $$\begin{aligned} {\label{eq:Theta'-2ndindbd2.2}} {(\ref{eq:Theta'-2ndindbd2})}=\sum_{{{\cal B}}\subset\Lambda}\,\sum_{T\ge1} \sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}\\ {\partial}{{\bf h}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda, {{\cal B}}}({{\bf h}})\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})} {Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf h}};\vec b_T}(y,x) \,\cap\,\{{{\cal C}}_{{\bf h}}(y)={{\cal B}}}{\nonumber}\\ \times\sum_{j=1}^T\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}} \,\sum_{\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal B}}_{s_i},~z'_i\in{{\cal B}}_{t_i}\\}$}}}\,\cap\,\{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}} z'_i}\bigg)\prod_{i\ne l}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_l)={\varnothing}\}$}}}.\end{aligned}$$ Reorganizing this expression and then summing over ${{\cal B}}\subset{{\cal A}}$, we obtain $$\begin{aligned} {\label{eq:Theta'-2ndindbd3}} {(\ref{eq:Theta'-2ndindbd2.2})}&=\sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\,\cap\,H_{{{\bf n}};\vec b_T} (y,x)$}}}{\nonumber}\\ &\quad\times\sum_{j=1}^T\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\, \sum_{\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};s_i},~z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg){\nonumber}\\ &\quad\times\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}} \bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\prod_{i\ne l} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}},\end{aligned}$$ where we have denoted ${{\cal C}}_{{\bf n}}(y)$ by $\tilde{{\cal D}}$. In the rightmost expression, the first line determines $\tilde{{\cal D}}$ that contains vertices $z_i,z'_i$ for all $i=1,\dots,j$ in a specific manner, while the second line determines the bypaths ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ joining $z_i$ and $z'_i$ for every $i=1,\dots,j$. We first derive ${{\bf n}}$-independent bounds on these bypaths in the following paragraph (c-1). Then, in (c-2) below, we will bound the first two lines of the rightmost expression in [(\[eq:Theta’-2ndindbd3\])]{}. **(c-1)** For $j=1$, the last line of the rightmost expression in [(\[eq:Theta’-2ndindbd3\])]{} simply equals $$\begin{aligned} {\label{eq:Theta'-2ndindbd3:j=1}} \sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1 {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_1\}$}}}.\end{aligned}$$ Since $z_1,z'_1\in\tilde{{\cal D}}$ and $z_1\ne z'_1$, these two vertices are connected via a nonzero alternating chain of mutually-disjoint ${{\bf m}}$-connected clusters and mutually-disjoint ${{\bf k}}$-connected clusters. Moreover, since $z_1,z'_1\in\tilde{{\cal D}}$ and ${{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$, this chain of bubbles starts and ends with ${{\bf m}}$-connected clusters (possibly with a single ${{\bf m}}$-connected cluster), not with ${{\bf k}}$-connected clusters. Therefore, by following the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, we can easily show $$\begin{aligned} {\label{eq:Theta'-2ndindbd3:j=1bd}} {(\ref{eq:Theta'-2ndindbd3:j=1})}\leq\sum_{l\ge1}\big(\tilde G_\Lambda^2 \big)^{*(2l-1)}(z_1,z'_1).\end{aligned}$$ For $j\ge2$, since ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for $i=1,\dots,j$ are mutually-disjoint due to the last product of the indicators in [(\[eq:Theta’-2ndindbd3\])]{}, we can treat each bypath separately by the conditioning-on-clusters argument. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last line in the rightmost expression of [(\[eq:Theta’-2ndindbd3\])]{} equals $$\begin{gathered} \sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i= 2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,l\ge 2\\ i\ne l}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}} \bigg){\nonumber}\\ \times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+ {{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,\frac{ w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}z'_1\}$}}}. {\label{eq:lace-edges}}\end{gathered}$$ By using [(\[eq:Theta’-2ndindbd3:j=1bd\])]{} (and replacing ${{\cal A}}{^{\rm c}}$ and $\tilde{{\cal D}}{^{\rm c}}$ in [(\[eq:Theta’-2ndindbd3:j=1\])]{} by ${{\cal A}}{^{\rm c}}\cap{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}$ and $\tilde{{\cal D}}{^{\rm c}}\cap{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}$, respectively), the second line of [(\[eq:lace-edges\])]{} is bounded by $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. Repeating the same argument until the remaining products of the indicators are used up, we obtain $$\begin{aligned} {\label{eq:lace-edgesbd}} {(\ref{eq:lace-edges})}&\leq\prod_{i=1}^j\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i).\end{aligned}$$ We have proved that $$\begin{aligned} {\label{eq:Theta'-2ndindbd4}} {(\ref{eq:Theta'-2ndindbd3})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots, z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i)\bigg)\sum_{{\partial}{{\bf n}}=y{\vartriangle}x} \frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}{\nonumber}\\ \times\sum_{T\ge j}\sum_{\vec b_T}\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0, T]}^{(j)}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{ {{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}.\end{aligned}$$ **(c-2)** Since [(\[eq:Theta’-2ndindbd4\])]{} depends only on a single current configuration, we may use Lemma \[lmm:GHS-BK\] to obtain an upper bound. To do so, we first simplify the second line of [(\[eq:Theta’-2ndindbd4\])]{}, which is, by definition, equal to the indicator of the disjoint union $$\begin{aligned} {\label{eq:fin-ind}} &{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T}{\mathop{\Dot{\bigcup}}}_{\{s_it_i\}_{i=1}^j\in {{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap\bigcap_{i=1}^j \big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\}\bigg\}\\ &={\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\Bigg\{{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T} {\mathop{\Dot{\bigcup}}}_{\substack{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}\\ b_{t_i +1}=e_{i+1}\;{{}^\forall}i=0,\dots,j-1}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap \bigcap_{i=1}^j\big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\} \bigg\}\Bigg\},{\nonumber}\end{aligned}$$ where $t_0=0$ by convention. On the left-hand side of [(\[eq:fin-ind\])]{}, the first two unions identify the number and location of the pivotal bonds for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$, and the third union identifies the indices of double connections associated with the bypaths between $z_i$ and $z'_i$, for every $i=1,\dots,j$. The union over $e_1,\dots,e_j$ on the right-hand side identifies some of the pivotal bonds $b_1,\dots,b_T$ that are essential to decompose the chain of double connections $H_{{{\bf n}};\vec b_T}(y,x)$ into the following building blocks (see Figure \[fig:I-def\]): $$\begin{gathered} I_1(y,z,x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I1}}\hspace{7pc} I_2(y,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I2}}\\[1pc] I_3(y,z,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I31}}\quad~ \cup\quad~\raisebox{-12pt}{\includegraphics[scale=0.12]{I32}}\end{gathered}$$ $$\begin{gathered} I_1(y,z,x)=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}z\},\qquad I_2(y,z',x)=\bigcup_u\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\}\circ I_1(u,z',x)\big\}, {\label{eq:I12-def}}\\ I_3(y,z,z',x)=\bigcup_u\Big\{\{I_2(y,z,u)\circ I_2(u,z',x)\}\cup\big\{ \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}u\}\circ\{I_1(u,z,x)\cap I_1(u,z',x)\}\big\}\Big\}. {\label{eq:I3-def}}\end{gathered}$$ For example, since ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, we have $$\begin{aligned} {\label{eq:fin-ind:=1}} &({(\ref{eq:fin-ind})}\text{ for }j=1)={\mathop{\Dot{\bigcup}}}_{e_1}{\mathop{\Dot{\bigcup}}}_{T\ge1}\, {\mathop{\Dot{\bigcup}}}_{\vec b_T:b_1=e_1}\Big\{H_{{{\bf n}};\vec b_T}(y,x)\cap\big\{ z_1\in{{\cal D}}_{{{\bf n}};0},\,z'_1\in{{\cal D}}_{{{\bf n}};T}\big\}\Big\}{\nonumber}\\ &\qquad\subset{\mathop{\Dot{\bigcup}}}_{e_1}\Big\{\big\{I_1(y,z_1,{\underline{e}}_1)\circ I_2 ({\overline{e}}_1,z'_1,x)\big\}\cap\big\{n_{e_1}>0,~e_1\text{ is pivotal for } y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\big\}\Big\}.\end{aligned}$$ It is not hard to see in general that $$\begin{aligned} {\label{eq:fin-ind:geq2}} &({(\ref{eq:fin-ind})}\text{ for }j\ge2){\nonumber}\\ &\quad\subset{\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\bigg\{\Big\{I_1(y,z_1,{\underline{e}}_1) \circ I_3({\overline{e}}_1,z_2,z'_1,{\underline{e}}_2)\circ\cdots\circ I_3({\overline{e}}_{j-1}, z_j,z'_{j-1},{\underline{e}}_j)\circ I_2({\overline{e}}_j,z'_j,x)\Big\}{\nonumber}\\ &\hspace{5pc}\cap\bigcap_{i=1}^j\big\{n_{e_i}>0,~e_i \text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x\big\}\bigg\}.\end{aligned}$$ To bound [(\[eq:Theta’-2ndindbd4\])]{} using Lemma \[lmm:GHS-BK\], we further consider an event that includes [(\[eq:fin-ind:=1\])]{}–[(\[eq:fin-ind:geq2\])]{} as subsets. Without losing generality, we can assume that $y\ne{\underline{e}}_1$, ${\overline{e}}_{i-1}\ne{\underline{e}}_i$ for $i=2,\dots,j$, and ${\overline{e}}_j\ne x$; otherwise, the following argument can be simplified. We consider each event $I_i$ in [(\[eq:fin-ind:=1\])]{}–[(\[eq:fin-ind:geq2\])]{} individually, and to do so, we assume that $y$ and ${\underline{e}}_1$ are the only sources for $I_1(y,z_1,{\underline{e}}_1)$, that ${\overline{e}}_{i-1}$ and ${\underline{e}}_i$ are the only sources for $I_3({\overline{e}}_{i-1},z_i,z'_{i-1},{\underline{e}}_i)$ for every $i=2,\dots,j$, and that ${\overline{e}}_j$ and $x$ are the only sources for $I_2({\overline{e}}_j,z'_j,x)$. This is because $y$ and $x$ are the only sources for the entire event [(\[eq:fin-ind:geq2\])]{}, and every $e_i$ is pivotal for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}x$. On $I_1(y,z,x)$ with $y,x$ being the only sources, according to the observation in Step (i) described below [(\[eq:dbbd\])]{}, we have two edge-disjoint connections from $y$ to $z$, one of which may go through $x$, and another edge-disjoint connection from $y$ to $x$ (cf., $I_1(y,z,x)$ in Figure \[fig:I-def\]). Therefore, $$\begin{aligned} {\label{eq:I1-supset}} I_1(y,z,x)\subset\big\{{{}^\exists}\omega_1,\omega_2\in\Omega_{y\to z}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{y\to x}^{{\bf n}}\text{ such that }~\omega_i \cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$ Similarly, for $I_2(y,z',x)$ with $y,x$ being the only sources (cf., $I_2(y,z',x)$ in Figure \[fig:I-def\]), $$\begin{aligned} {\label{eq:I2-supset}} I_2(y,z',x)\subset\big\{{{}^\exists}\omega_1,\omega_2\in\Omega_{x\to z'}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{y\to x}^{{\bf n}}\text{ such that } ~\omega_i\cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$ On $I_3(y,z,z',x)$ with $y,x$ being the only sources, there are at least three edge-disjoint paths, one from $y$ to $z$, another one from $z$ to $z'$, and another one from $z'$ to $x$. It is not hard to see this from $\bigcup_u\{I_2(y,z,u)\circ I_2(u,z',x)\}$ in [(\[eq:I3-def\])]{}, which corresponds to the first event depicted in Figure \[fig:I-def\]. It is also possible to extract such three edge-disjoint paths from the remaining event in [(\[eq:I3-def\])]{}. See the second event depicted in Figure \[fig:I-def\] for one of the worst topological situations. Since there are at least three edge-disjoint paths between $u$ and $x$, say, $\zeta_1,\zeta_2$ and $\zeta_3$, we can go from $y$ to $z$ via $\zeta_1$ and a part of $\zeta_2$, and go from $z$ to $z'$ via the middle part of $\zeta_2$, and then go from $z'$ to $x$ via the remaining part of $\zeta_2$ and $\zeta_3$. The other cases can be dealt with similarly. As a result, we have $$\begin{aligned} {\label{eq:I3-supset}} I_3(y,z,z',x)\subset\big\{{{}^\exists}\omega_1\in\Omega_{y\to z}^{{\bf n}}\,{{}^\exists}\omega_2\in\Omega_{z\to z'}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{z'\to x}^{{\bf n}}\text{ such that }\omega_i\cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$ Since $$\begin{aligned} \bigcup_e\Big\{\big\{\{{{}^\exists}\omega\in\Omega_{z\to{\underline{e}}}^{{\bf n}}\}\circ\{{{}^\exists}\omega\in\Omega_{{\overline{e}}\to z'}^{{\bf n}}\}\big\}\cap\{n_e>0\}\Big\}\subset\{{{}^\exists}\omega\in\Omega_{z\to z'}^{{\bf n}}\},\end{aligned}$$ we see that [(\[eq:fin-ind:=1\])]{} is a subset of $$\begin{aligned} {\label{eq:tildeI-def:=1}} \tilde I_{z_1,z'_1}^{{\scriptscriptstyle}(1)}(y,x)=\left\{\! \begin{array}{c} {{}^\exists}\omega_1,\omega_2\in\Omega_{z_1\to y}^{{\bf n}}\;{{}^\exists}\omega_3\in \Omega_{y\to x}^{{\bf n}}\;{{}^\exists}\omega_4,\omega_5\in\Omega_{x\to z'_1}^{{\bf n}}\\ \text{such that }~\omega_i\cap\omega_l={\varnothing}~(i\ne l) \end{array}\!\right\},\end{aligned}$$ and that [(\[eq:fin-ind:geq2\])]{} is a subset of (see Figure \[fig:eventI\]) $$\begin{aligned} {\label{eq:tildeI-def:geq2}} \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)=\left\{\! \begin{array}{c} {{}^\exists}\omega_1,\omega_2\in\Omega_{z_1\to y}^{{\bf n}}\;{{}^\exists}\omega_3\in \Omega_{y\to z_2}^{{\bf n}}\;{{}^\exists}\omega_4\in\Omega_{z_2\to z'_1}^{{\bf n}}\; {{}^\exists}\omega_5\in\Omega_{z'_1\to z_3}^{{\bf n}}\cdots\\ \cdots{{}^\exists}\omega_{2j}\in\Omega_{z_j\to z'_{j-1}}^{{\bf n}}\,{{}^\exists}\omega_{ 2j+1}\in\Omega_{z'_{j-1}\to x}^{{\bf n}}\;{{}^\exists}\omega_{2j+2},\omega_{2j+3} \in\Omega_{x\to z'_j}^{{\bf n}}\\ \text{such that }~\omega_i\cap\omega_l={\varnothing}~(i\ne l) \end{array}\!\right\},\end{aligned}$$ ![\[fig:eventI\]A schematic representation of $\tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$ for $j\ge2$ consisting of $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$.](eventI) where $\vec z_j^{(\prime)}=(z_1^{(\prime)},\dots,z_j^{(\prime)})$. Therefore, $$\begin{aligned} {\label{eq:Theta'-2ndindbd5}} {(\ref{eq:Theta'-2ndindbd4})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j\sum_{l\ge1}\big(\tilde G_\Lambda^2 \big)^{*(2l-1)}(z_i,z'_i)\bigg)\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j, \vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}.\end{aligned}$$ Now we apply Lemma \[lmm:GHS-BK\] to bound [(\[eq:Theta’-2ndindbd5\])]{}. To clearly understand how it is applied, for now we ignore ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-2ndindbd5\])]{} and only consider the contribution from ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j, \vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}$. Without losing generality, we assume that $y,x,z_i,z'_i$ for $i=1,\dots,j$ are all different. Since there are $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$ as in [(\[eq:tildeI-def:=1\])]{}–[(\[eq:tildeI-def:geq2\])]{} (see also Figure \[fig:eventI\]), we multiply [(\[eq:Theta’-2ndindbd5\])]{} by $(Z_\Lambda/Z_\Lambda)^{2j+2}$, following Step (ii) of the strategy described in Section \[ss:pi0bd\]. Overlapping the $2j+3$ current configurations and using Lemma \[lmm:GHS-BK\] with ${{\cal V}}=\{y,x\}$ and $k=2j+2$, we obtain $$\begin{aligned} {\label{eq:Theta'-2ndindbd6}} \sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}&\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}\leq{{\langle \varphi_{z_1}\varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x\varphi_{z'_j} \rangle}}_\Lambda^2\\ &\times\begin{cases} {\displaystyle}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda&(j=1),\\ {\displaystyle}{{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda{{\langle \varphi_{z_2}\varphi_{ z'_1} \rangle}}_\Lambda\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{z'_{i-1}}\varphi_{ z_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{z_{i+1}}\varphi_{z'_i} \rangle}}_\Lambda\bigg) {{\langle \varphi_{z'_{j-1}}\varphi_x \rangle}}_\Lambda&(j\ge2). \end{cases}{\nonumber}\end{aligned}$$ Note that, by [(\[eq:G-delta-bd\])]{}, we have $$\begin{aligned} \left.\begin{array}{r} \sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(y,x)\\[5pt] \sum_z{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z,x)\\[5pt] \sum_{z'}{{\langle \varphi_x\varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1} (\tilde G_\Lambda^2)^{*(2l-1)}(y,z') \end{array}\right\}&\leq\psi_\Lambda(y,x)-\delta_{y,x},\\[5pt] \sum_{z,z'}{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x \varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1}\big(\tilde G_\Lambda^2 \big)^{*(2l-1)}(z,z')&\leq2\big(\psi_\Lambda(y,x)-\delta_{y,x}\big).\end{aligned}$$ Therefore, [(\[eq:Theta’-2ndindbd5\])]{} without ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ is bounded by $$\begin{aligned} {\label{eq:Theta'-2ndindbd7}} &{{\langle \varphi_y\varphi_x \rangle}}_\Lambda\sum_{z_1,z'_1}{{\langle \varphi_{z_1} \varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x\varphi_{z'_1} \rangle}}_\Lambda^2\sum_{l \ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_1,z'_1){\nonumber}\\ &+\sum_{j\ge2}\sum_{\substack{z_2,\dots,z_j\\ z'_1,\dots,z'_{j-1}}} \bigg(\prod_{i=2}^{j-1}\big(\psi_\Lambda(z_i,z'_i)-\delta_{z_i,z'_i} \big)\bigg)\bigg(\sum_{z_1}{{\langle \varphi_y\varphi_{z_1} \rangle}}_\Lambda^2 \sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_1,z'_1)\bigg) {\nonumber}\\ &\hspace{4pc}\times\bigg(\sum_{z'_j}{{\langle \varphi_x\varphi_{z'_j} \rangle}}_\Lambda^2\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_j, z'_j)\bigg){{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda{{\langle \varphi_{z_2} \varphi_{z'_1} \rangle}}_\Lambda{\nonumber}\\ &\hspace{4pc}\times\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{z'_{i-1}} \varphi_{z_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{z_{i+1}}\varphi_{z'_i} \rangle}}_\Lambda \bigg){{\langle \varphi_{z'_{j-1}}\varphi_x \rangle}}_\Lambda\leq\sum_{j\ge1} P_\Lambda^{{\scriptscriptstyle}(j)}(y,x).\end{aligned}$$ If ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ is present in the above argument, then at least one of the paths $\omega_i$ for $i=3,\dots,2j+1$ has to go through ${{\cal A}}$. For example, if $\omega_3~(\in\Omega_{y\to z_2}^{{\bf n}})$ goes through ${{\cal A}}$, then we can split it into two edge-disjoint paths at some $u\in{{\cal A}}$, such as $\omega'_3\in\Omega_{y\to u}^{{\bf n}}$ and $\omega''_3\in\Omega_{u\to z_2}^{{\bf n}}$. The contribution from this case is bounded, by following the same argument as above, by [(\[eq:Theta’-2ndindbd6\])]{} with ${{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda$ being replaced by $\sum_{u\in{{\cal A}}} {{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z_2} \rangle}}_\Lambda$. Bounding the other $2j-2$ cases similarly and summing these bounds over $j\ge1$, we obtain $$\begin{aligned} {\label{eq:Theta'-2ndindbd8}} {(\ref{eq:Theta'-2ndindbd5})}\leq\sum_{u\in{{\cal A}}}\sum_{j\ge1}P_{\Lambda; u}^{\prime{{\scriptscriptstyle}(j)}}(y,x).\end{aligned}$$ This together with [(\[eq:Theta’-bd1stbd\])]{} in the above paragraph (a) complete the proof of the bound on $\Theta'_{y,x;{{\cal A}}}$ in [(\[eq:Theta’Theta”bd\])]{}. **(d)** Finally, we investigate the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\\}}}\setminus \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{}: $$\begin{aligned} {\label{eq:contr-(d)}} \sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\}\,\cap \,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ Using $H_{{{\bf n}};\vec b_T}(y,x)$ defined in [(\[eq:H-def\])]{}, we can write [(\[eq:contr-(d)\])]{} as (cf., [(\[eq:Theta’-2ndindbd2\])]{}) $$\begin{aligned} {\label{eq:Theta''-2ndindrewr}} {(\ref{eq:contr-(d)})}=\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda ({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x) \,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ To bound this, we will also use a similar expression to [(\[eq:Theta’-2ndindbd3\])]{}, in which ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$ with $\tilde{{\cal D}}={{\cal C}}_{{\bf n}}^b(y)$. We investigate [(\[eq:Theta”-2ndindrewr\])]{} separately (in the following paragraphs (d-1) and (d-2)) depending on whether or not there is a bypath ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for some $i\in\{1,\dots,j\}$ containing $v$. **(d-1)** If there is such a bypath, then we use ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}\leq{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ as in [(\[eq:Theta’-2ndindbd2\])]{} to bound the contribution from this case to [(\[eq:Theta”-2ndindrewr\])]{} by $$\begin{aligned} {\label{eq:Theta''-2ndindbd1}} \sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)$}}} \sum_{j=1}^T\,\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\,\sum_{ \substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i \in{{\cal D}}_{{{\bf n}};s_i},~z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg){\nonumber}\\ \times\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf k}}={\varnothing}}}\frac{w_{{{\cal A}}{^{\rm c}}} ({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg( \prod_{i\ne l}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}} \bigg)\sum_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\}$}}}.\end{aligned}$$ Note that the last sum of the indicators is the only difference from [(\[eq:Theta’-2ndindbd3\])]{}. When $j=1$, the second line of [(\[eq:Theta”-2ndindbd1\])]{} equals $$\begin{aligned} {\label{eq:Theta''-2ndindbd1:j=1}} \sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1 {\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_1\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ As described in [(\[eq:Theta’-2ndindbd3:j=1\])]{}–[(\[eq:Theta’-2ndindbd3:j=1bd\])]{}, we can bound [(\[eq:Theta”-2ndindbd1:j=1\])]{} without ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$ by a chain of bubbles $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. If ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}}}=1$, then, by the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, one of the bubbles has an extra vertex $v'$ that is further connected to $v$ with another chain of bubbles $\psi_\Lambda(v',v)$. That is, the effect of ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$ is to replace one of the $\tilde G_\Lambda$’s in the chain of bubbles, say, $\tilde G_\Lambda(a,a')$, by $\sum_{v'}({{\langle \varphi_{a} \varphi_{v'} \rangle}}_\Lambda\tilde G_\Lambda(v',a')+\tilde G_\Lambda (a,a')\delta_{v',a'})\,\psi_\Lambda(v',v)$. Let $$\begin{aligned} {\label{eq:g-def}} g_{\Lambda;y}(z,z')=\sum_{l\ge1}\sum_{i=1}^{2l-1}\sum_{a,a'}\big( \tilde G_\Lambda^2\big)^{*(i-1)}(z,a)\,\tilde G_\Lambda(a,a')\, \big(\tilde G_\Lambda^2\big)^{*(2l-1-i)}(a',z'){\nonumber}\\ \times\Big({{\langle \varphi_a\varphi_y \rangle}}_\Lambda\tilde G_\Lambda(y,a') +\tilde G_\Lambda(a,a')\,\delta_{y,a'}\Big).\end{aligned}$$ Then, we have $$\begin{aligned} {\label{eq:Theta''-2ndindbd1:j=1bd}} {(\ref{eq:Theta''-2ndindbd1:j=1})}\leq\sum_{v'}g_{\Lambda;v'}(z_1,z'_1) \,\psi_\Lambda(v',v).\end{aligned}$$ Let $j\ge2$ and consider the contribution to [(\[eq:Theta”-2ndindbd1\])]{} from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_1)\}}}$; the contribution from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\}}}$ with $i\ne1$ can be estimated in the same way. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ as in [(\[eq:lace-edges\])]{}, the contribution to the second line of [(\[eq:Theta”-2ndindbd1\])]{} from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_1)\}}}\equiv{\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}}}$ equals $$\begin{aligned} {\label{eq:Theta''-2ndindbd2}} &\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i= 2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,i'\ge 2\\ i\ne i'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})= {\varnothing}\}$}}}\bigg){\nonumber}\\ &\qquad\times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{ {{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde {{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}z'_1\}$}}} \,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}} {\overset{}{\longleftrightarrow}}}v\}$}}},\end{aligned}$$ where the second line is bounded by [(\[eq:Theta”-2ndindbd1:j=1bd\])]{} for $j=1$, and then the first line is bounded by $\prod_{i=2}^j\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_i,z'_i)$, due to [(\[eq:lace-edges\])]{}–[(\[eq:lace-edgesbd\])]{}. Summarizing the above bounds, we have (cf., [(\[eq:Theta’-2ndindbd5\])]{}) $$\begin{aligned} {\label{eq:Theta''-2ndindbd2.2}} {(\ref{eq:Theta''-2ndindbd1})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots, z_j\\ z'_1,\dots,z'_j}}\bigg(&\sum_{h=1}^j\sum_{v'}g_{\Lambda;v'} (z_h,z'_h)\,\psi_\Lambda(v',v)\prod_{i\ne h}\sum_{l\ge1}\big( \tilde G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i)\bigg){\nonumber}\\ &\times\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{ y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y, x)$}}},\end{aligned}$$ to which we can apply the bound discussed between [(\[eq:Theta’-2ndindbd2\])]{} and [(\[eq:Theta’-2ndindbd8\])]{}. **(d-2)** If $v\notin{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for any $i=1,\dots,j$, then there exists a $v'\in{{\cal D}}_{{{\bf n}};l}$ for some $l\in\{0,\dots,T\}$ such that $v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v$ and ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\cap{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)={\varnothing}$ for any $i$. In addition, since all connections from $y$ to $x$ on the graph $\tilde{{\cal D}}\cup{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-2pt} {$\scriptstyle j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)$ have to go through ${{\cal A}}$, there is an $h\in\{1,\dots,j\}$ such that $z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h$. Therefore, the contribution from this case to [(\[eq:Theta”-2ndindrewr\])]{} is bounded by $$\begin{gathered} \sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\!\frac{w_\Lambda({{\bf n}})} {Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\sum_{j=1}^T\sum_{\{s_it_i\}_{i =1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\sum_{\substack{v',z_1,\dots,z_j\\ z'_1,\dots, z'_j}}\!\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};s_i},\;z'_i\in{{\cal D}}_{{{\bf n}}; t_i}\}$}}}\bigg)\sum_{l=0}^T{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'\in{{\cal D}}_{{{\bf n}};l}\}$}}}{\nonumber}\\ \times\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf k}}={\varnothing}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})} {Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}} \bigg(\sum_{h=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}} \prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{i\ne i'} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\ \times{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}v\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\,\cap\, {{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)={\varnothing}\}$}}},{\label{eq:Theta''-2ndindbd3}}\end{gathered}$$ where, by conditioning on ${{\cal S}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox {-2pt}{$\scriptstyle j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last two lines are (see below [(\[eq:lace-edges\])]{}) $$\begin{gathered} \sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\sum_{h=1}^j {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}} \bigg)\bigg(\prod_{i\ne i'}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\ \times\underbrace{\sum_{{\partial}{{\bf m}}''={\partial}{{\bf k}}''={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\, {{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}'')}{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf k}}''$}} {\overset{}{\longleftrightarrow}}}v\}$}}}}_{\leq\; \psi_\Lambda(v',v)}.{\label{eq:Theta''-2ndindbd3-l2,3}}\end{gathered}$$ When $j=1$, we have $$\begin{aligned} {\label{eq:Theta''-2ndindbd3-l2,3:j=1}} ({(\ref{eq:Theta''-2ndindbd3-l2,3})}\text{ for }j=1)\leq\psi_\Lambda (v',v)\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\, {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}$}}}.\end{aligned}$$ If we ignore the “through ${{\cal A}}$”-condition in the last indicator, then the sum is bounded, as in [(\[eq:Theta’-2ndindbd3:j=1bd\])]{}, by a chain of bubbles $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. However, because of this condition, one of the $\tilde G_\Lambda$’s in the bound, say, $\tilde G_\Lambda(a,a')$, is replaced by $\sum_{u\in{{\cal A}}}({{\langle \varphi_{a}\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,a') +\tilde G_\Lambda(a,a')\delta_{u,a'})$. Using [(\[eq:g-def\])]{}, we have $$\begin{aligned} {\label{eq:Theta''-2ndindbd3-l2,3:j=1bd}} {(\ref{eq:Theta''-2ndindbd3-l2,3:j=1})}\leq\psi_\Lambda(v',v)\sum_{y\in{{\cal A}}} g_{\Lambda;y}(z_1,z'_1).\end{aligned}$$ Let $j\ge2$ and consider the contribution to [(\[eq:Theta”-2ndindbd3-l2,3\])]{} from ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}}}$; the contributions from ${\mathbbm{1}{\scriptstyle\{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}}}$ with $h\ne1$ can be estimated similarly. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the contribution to [(\[eq:Theta”-2ndindbd3-l2,3\])]{} from ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}}}$ equals $$\begin{aligned} {\label{eq:Theta''-2ndindbd4}} \sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}&\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\, \frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i= 2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}} {\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,i'\ge 2\\ i\ne i'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})= {\varnothing}\}$}}}\bigg){\nonumber}\\ &\times\psi_\Lambda(v',v)\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}} {^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')} {Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+ {{\bf k}}'$}} {\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}$}}},\end{aligned}$$ where the second line is bounded by [(\[eq:Theta”-2ndindbd3-l2,3:j=1bd\])]{} for $j=1$, and then the first line is bounded by $\prod_{i=2}^j\sum_{l\ge1}(\tilde G_\Lambda^2)^{ *(2l-1)}(z_i,z'_i)$, as described below [(\[eq:Theta”-2ndindbd2\])]{}. As a result, [(\[eq:Theta”-2ndindbd3\])]{} is bounded by $$\begin{aligned} {\label{eq:Theta''-2ndindbd5}} &\sum_{j\ge1}\sum_{\substack{v'\!,z_1,\dots,z_j\\ z'_1,\dots,z'_j}} \psi_\Lambda(v',v)\bigg(\sum_{h=1}^j\sum_{y\in{{\cal A}}}g_{\Lambda;y}(z_h, z'_h)\prod_{i\ne h}\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l- 1)}(z_i,z'_i)\bigg){\nonumber}\\ &\times\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\sum_{T \ge j}\sum_{\vec b_T}\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}} {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}}; s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg)\sum_{l=0}^T{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'\in{{\cal D}}_{{{\bf n}};l}\}$}}}.\end{aligned}$$ The second line can be bounded by following the argument between [(\[eq:Theta’-2ndindbd4\])]{} and [(\[eq:Theta’-2ndindbd7\])]{}; note that the sum of the indicators in [(\[eq:Theta”-2ndindbd5\])]{}, except for the last factor $\sum_{l=0}^T{\mathbbm{1}{\scriptstyle\{v'\in{{\cal D}}_{{{\bf n}};l}\}}}$, is identical to that in [(\[eq:Theta’-2ndindbd4\])]{}. First, we rewrite the sum of the indicators in [(\[eq:Theta”-2ndindbd5\])]{} as a single indicator of an event ${{\cal E}}$ similar to [(\[eq:fin-ind\])]{}. Then, we construct another event similar to $\tilde I^{{\scriptscriptstyle}(j)}_{\vec z_j,\vec z'_j}(y,x)$ in [(\[eq:tildeI-def:=1\])]{}–[(\[eq:tildeI-def:geq2\])]{}, of which ${{\cal E}}$ is a subset. Due to $\sum_{l=0}^T{\mathbbm{1}{\scriptstyle\{v'\in{{\cal D}}_{{{\bf n}};l}\}}}$ in [(\[eq:Theta”-2ndindbd5\])]{}, one of the paths in the definition of $\tilde I^{{\scriptscriptstyle}(j)}_{\vec z_j,\vec z'_j}(y,x)$, say, $\omega_i\in\Omega^{{{\bf n}}}_{a\to a'}$ for some $a,a'$ (depending on $i$) is split into two edge-disjoint paths $\omega'_i\in\Omega^{{{\bf n}}}_{a\to v'}$ and $\omega''_i\in\Omega^{{{\bf n}}}_{v'\to a'}$, followed by the summation over $i=3,\dots,2j+1$ (cf., Figure \[fig:eventI\]). Finally, we apply Lemma \[lmm:GHS-BK\] to obtain the desired bound on the last line of [(\[eq:Theta”-2ndindbd5\])]{}. Summarizing the above (d-1) and (d-2), we obtain $$\begin{aligned} {(\ref{eq:Theta''-2ndindrewr})}\leq\sum_{j\ge1}\sum_{u\in{{\cal A}}}P_{ \Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x).\end{aligned}$$ This together with [(\[eq:Theta”-0bdfin\])]{} in the above paragraph (b) complete the proof of the bound on $\Theta''_{y,x,v;{{\cal A}}}$ in [(\[eq:Theta’Theta”bd\])]{}. Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ assuming the decay of $G(x)$ ================================================================================= Using the diagrammatic bounds proved in the previous section, we prove Proposition \[prp:GimpliesPix\] in Section \[ss:proof-so\], and Propositions \[prp:GimpliesPik\] and \[prp:exp-bootstrap\](iii) in Section \[ss:proof-nn\]. Bounds for the spread-out model {#ss:proof-so} ------------------------------- We prove Proposition \[prp:GimpliesPix\] for the spread-out model using the following convolution bounds: \[prp:conv-star\] (i) Let $a\ge b>0$ and $a+b>d$. There is a $C=C(a,b,d)$ such that $$\begin{aligned} {\label{eq:conv}} \sum_y\frac1{{\vby-v{|\!|\!|}}^a}\,\frac1{{\vbx-y{|\!|\!|}}^b}\leq\frac{C} {{\vbx-v{|\!|\!|}}^{(a\wedge d+b)-d}}.\end{aligned}$$ (ii) Let $q\in(\frac{d}2,d)$. There is a $C'=C'(d,q)$ such that $$\begin{aligned} \sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\,\frac1{{\vbx'-z{|\!|\!|}}^q}\,\frac1{{\vbz-y{|\!|\!|}}^q}\, \frac1{{\vbz-y'{|\!|\!|}}^q}\leq\frac{C'}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}.{\label{eq:star}}\end{aligned}$$ The inequality [(\[eq:conv\])]{} is identical to [@hhs03 Proposition 1.7(i)]. We use this to prove [(\[eq:star\])]{}. By the triangle inequality, we have $\frac12{\vbx-y{|\!|\!|}}\leq{\vbx-z{|\!|\!|}}\vee{\vbz-y{|\!|\!|}}$ and $\frac12{\vbx'-y'{|\!|\!|}}\leq{\vbx'-z{|\!|\!|}}\vee{\vbz-y'{|\!|\!|}}$. Suppose that ${\vbx-z{|\!|\!|}}\leq{\vbz-y{|\!|\!|}}$ and ${\vbx'-z{|\!|\!|}}\leq{\vbz-y'{|\!|\!|}}$. Then, by [(\[eq:conv\])]{} with $a=b=q$, the contribution from this case is bounded by $$\begin{aligned} \frac{2^{2q}}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}\sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\, \frac1{{\vbx'-z{|\!|\!|}}^q}\leq\frac{2^{2q}c{\vbx-x'{|\!|\!|}}^{d-2q}}{{\vbx-y{|\!|\!|}}^q {\vbx'-y'{|\!|\!|}}^q},\end{aligned}$$ for some $c<\infty$, where we note that ${\vbx-x'{|\!|\!|}}^{d-2q}\leq1$ because of $\frac12d<q$. The other three possible cases can be estimated similarly (see Figure \[fig:star\](a)). This completes the proof of Proposition \[prp:conv-star\]. $$\begin{aligned} \begin{array}{cc} \text{(a)}&{\displaystyle}\sum_z~~\raisebox{-1.4pc}{\includegraphics[scale=0.2] {star1}}~~~~\lesssim~~~~\raisebox{-1.4pc}{\includegraphics[scale=0.2] {star2}}\\[2pc] \text{(b)}&\qquad{\displaystyle}\sum_{u_j,v_j}~~\raisebox{-21pt}{\includegraphics [scale=0.2]{fish1}}~~~~\lesssim~~~~\sum_{v_j}~~\raisebox{-14pt}{ \includegraphics[scale=0.2]{fish2}}~~~~\lesssim~~~\raisebox{-14pt}{ \includegraphics[scale=0.2]{fish3}} \end{array}\end{aligned}$$ Before going into the proof of Proposition \[prp:GimpliesPix\], we summarize prerequisites. Recall that [(\[eq:Q’-def\])]{}–[(\[eq:Q”-def\])]{} involve $\tilde G_\Lambda$, and note that, by [(\[eq:G-delta-bd\])]{}, $$\begin{aligned} {\label{eq:pi0-1stbd}} {{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\leq\delta_{o,x}+\tilde G_\Lambda(o,x)^3.\end{aligned}$$ We first show that $$\begin{aligned} {\label{eq:tildeG-bd}} \tilde G_\Lambda(o,x)\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q},&& \sum_{b:{\underline{b}}=o}\tau_b\big(\delta_{{\overline{b}},x}+\tilde G_\Lambda({\overline{b}},x)\big) \leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}\end{aligned}$$ hold assuming the bounds in [(\[eq:IR-xbd\])]{}. By the assumed bound $\tau\leq2$ in [(\[eq:IR-xbd\])]{}, we have $$\begin{aligned} {\label{eq:tildeG-1stbd}} \tilde G_\Lambda(o,x)=\tau D(x)+\sum_{y\ne x}\tau D(y)\,{{\langle \varphi_y \varphi_x \rangle}}_\Lambda\leq2D(x)+\sum_{y\ne x}2D(y)\,G(x-y),\end{aligned}$$ where, and from now on without stating explicitly, we use the translation invariance of $G(x)$ and the fact that $G(x-y)$ is an increasing limit of ${{\langle \varphi_y\varphi_x \rangle}}_\Lambda$ as $\Lambda\uparrow{{\mathbb Z}^d}$. By [(\[eq:J-def\])]{} and the assumption in Proposition \[prp:GimpliesPix\] that $\theta_0L^{d-q}$, with $q<d$, is bounded away from zero, we obtain $$\begin{aligned} {\label{eq:Dbd}} D(x)\leq O(L^{-d}){\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}\leq\frac{O(L^{-d+q})} {{\vbx{|\!|\!|}}^q}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$ For the last term in [(\[eq:tildeG-1stbd\])]{}, we consider the cases for $|x|\leq2\sqrt{d}L$ and $|x|\ge2\sqrt{d}L$ separately. When $|x|\leq2\sqrt{d}L$, we use [(\[eq:Dbd\])]{}, [(\[eq:IR-xbd\])]{} and [(\[eq:conv\])]{} with $\frac12d<q<d$ to obtain $$\begin{aligned} \sum_{y\ne x}D(y)\,G(x-y)\leq\sum_y\frac{O(L^{-d+q})}{{\vby{|\!|\!|}}^q}\, \frac{\theta_0}{{\vbx-y{|\!|\!|}}^q}\leq\frac{O(\theta_0L^{-d+q})} {{\vbx{|\!|\!|}}^{2q-d}}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$ When $|x|\ge2\sqrt{d}L$, we use the triangle inequality $|x-y|\ge|x|-|y|$ and the fact that $D(y)$ is nonzero only when $0<\|y\|_\infty\leq L$ (so that $|y|\leq\sqrt{d}\|y\|_\infty\leq\sqrt{d}L\leq\frac12|x|$). Then, we obtain $$\begin{aligned} \sum_{y\ne x}D(y)\,G(x-y)\leq\sum_yD(y)\,\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q} =\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$ This completes the proof of the first inequality in [(\[eq:tildeG-bd\])]{}. The second inequality can be proved similarly. By repeated use of [(\[eq:tildeG-bd\])]{} and Proposition \[prp:conv-star\](i) with $a=b=2q$ (or Proposition \[prp:conv-star\](ii) with $x=x'$ and $y=y'$), we obtain $$\begin{aligned} {\label{eq:psi-bd}} \psi_\Lambda(v',v)\leq\delta_{v',v}+\frac{O(\theta_0^2)}{{\vbv-v'{|\!|\!|}}^{2q}}.\end{aligned}$$ Together with the naive bound $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ (cf., [(\[eq:IR-xbd\])]{}) as well as Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we also obtain $$\begin{aligned} {\label{eq:GGpsi-bd}} \sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)&\leq G(v-y)\,G(z-v)+ \sum_{v'}\frac{O(\theta_0^2)}{{\vbv'-y{|\!|\!|}}^q{\vbz-v'{|\!|\!|}}^q{\vbv-v'{|\!|\!|}}^{2q}} {\nonumber}\\ &\leq\frac{O(1)}{{\vbv-y{|\!|\!|}}^q{\vbz-v{|\!|\!|}}^q}.\end{aligned}$$ The $O(1)$ term in the right-hand side is replaced by $O(\theta_0)$ or $O(\theta_0^2)$ depending on the number of $G$’s on the left being replaced by $\tilde G_\Lambda$’s. Since [(\[eq:pi0-1stbd\])]{}–[(\[eq:tildeG-bd\])]{} immediately imply the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it suffices to prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$ for $i\ge1$. To do so, we first estimate the building blocks of the diagrammatic bound [(\[eq:piNbd\])]{}: $\sum_{b:{\underline{b}}=y}\tau_b\,Q'_{\Lambda;u}({\overline{b}},x)$ and $\sum_{b:{\underline{b}}=y}\tau_b\,Q''_{\Lambda;u,v}({\overline{b}},x)$. Recall [(\[eq:P’0-def\])]{}–[(\[eq:Q”-def\])]{}. First, by using $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ and [(\[eq:GGpsi-bd\])]{}, we obtain $$\begin{aligned} P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx-y{|\!|\!|}}^{2q} {\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'0-bd}}\\ P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx -y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.{\label{eq:P''0-bd}}\end{aligned}$$ We will show at the end of this subsection that, for $j\ge1$, $$\begin{aligned} P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j)\,O(\theta_0^2)^j} {{\vbx-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'j-bd}}\\ P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j^2)\,O (\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q {\vbx-v{|\!|\!|}}^q}.{\label{eq:P''j-bd}}\end{aligned}$$ As a result, $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)$ (resp., $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)$) is the leading term of $P'_{\Lambda;u}(y,x)$ (resp., $P''_{\Lambda;u,v}(y,x)$), which thus obeys the same bound as in [(\[eq:P’0-bd\])]{} (resp., [(\[eq:P”0-bd\])]{}), with a different constant in $O(1)$. Combining these bounds with [(\[eq:tildeG-bd\])]{} and [(\[eq:GGpsi-bd\])]{} (with both $G$ in the left-hand side being replace by $\tilde G_\Lambda$) and then using Proposition \[prp:conv-star\](ii), we obtain $$\begin{aligned} {\label{eq:bb1-bd}} \sum_{b:{\underline{b}}=y}\tau_b\,Q'_{\Lambda;u}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)} {{\vbz-y{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^{2q}{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}\leq \frac{O(\theta_0)}{{\vbx-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^{2q}},\end{aligned}$$ and $$\begin{aligned} {\label{eq:bb2-bd}} \sum_{b:{\underline{b}}=y}\tau_b\,Q''_{\Lambda;u,v}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)} {{\vbz-y{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^q{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-z{|\!|\!|}}^q {\vbx-v{|\!|\!|}}^q}{\nonumber}\\ &\quad+\sum_z\frac{O(\theta_0)}{{\vbv-y{|\!|\!|}}^q}\,\frac{O(\theta_0)}{{\vbz- v{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^{2q}{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}{\nonumber}\\ &\leq\frac{O(\theta_0)}{{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^{2q}}.\end{aligned}$$ This completes bounding the building blocks. Now we prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$. For the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge2$, we simply apply [(\[eq:P’0-bd\])]{} and [(\[eq:bb1-bd\])]{}–[(\[eq:bb2-bd\])]{} to the diagrammatic bound [(\[eq:piNbd\])]{}. Then, we obtain $$\begin{aligned} {\label{eq:piNgeq2-prebd}} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{\substack{u_1,\dots,u_j\\ v_1,\dots, v_j}}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}&\bigg( \prod_{i=1}^{j-1}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i+1}- v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\ &\times\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\qquad(j\ge2).\end{aligned}$$ First, we consider the sum over $u_j$ and $v_j$. By successive applications of Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we obtain (see Figure \[fig:star\](b)) $$\begin{aligned} {\label{eq:succ-appl}} &\sum_{v_j}\sum_{u_j}\frac{O(\theta_0)}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbu_j -v_j{|\!|\!|}}^q{\vbu_j-v_{j-1}{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q {\vbx-v_j{|\!|\!|}}^{2q}}\\ &\leq\sum_{v_j}\frac{O(\theta_0)^2}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbv_{j-1} -v_j{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^2} {{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}},{\nonumber}\end{aligned}$$ and thus $$\begin{aligned} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{\substack{u_1,\dots,u_{j-1}\\ v_1, \dots,v_{j-1}}}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q} &\bigg(\prod_{i=1}^{j-2}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i +1}-v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\ &\times\frac{O(\theta_0)^2}{{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}}.\end{aligned}$$ Repeating the application of Proposition \[prp:conv-star\](ii) as in [(\[eq:succ-appl\])]{}, we end up with $$\begin{aligned} {\label{eq:piNgeq2-bd}} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&\leq\sum_{u_1,v_1}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2 q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}\,\frac{O(\theta_0)^j}{{\vbx-u_1{|\!|\!|}}^q {\vbx-v_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^j}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$ For the bound on $\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)$, we use the following bound, instead of [(\[eq:P’0-bd\])]{}: $$\begin{aligned} {\label{eq:P'0-dec}} P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)=\delta_{o,u}\delta_{o,v}+(1-\delta_{ o,u}\delta_{o,v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)\leq\delta_{o,u} \delta_{o,v}+\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q{\vbu-v{|\!|\!|}}^q}.\end{aligned}$$ In addition, instead of using [(\[eq:bb1-bd\])]{}, we use $$\begin{aligned} {\label{eq:bb1-dec}} \sum_{b:{\underline{b}}=u}\tau_b\,Q'_{\Lambda;v}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)} {{\vbz-u{|\!|\!|}}^q}\bigg(\delta_{z,v}\delta_{z,x}+(1-\delta_{z,x}\delta_{z, v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z,x)+\sum_{j\ge1}P_{\Lambda;v}^{ \prime{{\scriptscriptstyle}(j)}}(z,x)\bigg){\nonumber}\\ &\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\sum_z\frac{O( \theta_0^3)}{{\vbz-u{|\!|\!|}}^q{\vbx-z{|\!|\!|}}^{2q}{\vbv-z{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}{\nonumber}\\ &\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\frac{O(\theta_0^3)} {{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}},\end{aligned}$$ due to [(\[eq:tildeG-bd\])]{}, [(\[eq:P’j-bd\])]{} and [(\[eq:P’0-dec\])]{}. Applying [(\[eq:P’0-dec\])]{}–[(\[eq:bb1-dec\])]{} to [(\[eq:piNbd\])]{} for $j=1$ and then using Proposition \[prp:conv-star\](ii), we end up with $$\begin{aligned} \pi_\Lambda^{{\scriptscriptstyle}(1)}(x)&\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)} {{\vbx{|\!|\!|}}^{3q}}+\sum_{u,v}\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q {\vbu-v{|\!|\!|}}^q}\bigg(\frac{O(\theta_0)\,\delta_{v,x}}{{\vbx-u{|\!|\!|}}^q}+\frac{ O(\theta_0^3)}{{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}}\bigg){\nonumber}\\ &\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$ To complete the proof of Proposition \[prp:GimpliesPix\], it thus remains to show [(\[eq:P’j-bd\])]{}–[(\[eq:P”j-bd\])]{}. The inequality [(\[eq:P’j-bd\])]{} for $j=1$ immediately follows from the definition [(\[eq:P’1-def\])]{} of $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (see also Figure \[fig:P-def\]) and the bound [(\[eq:psi-bd\])]{} on $\psi_\Lambda-\delta$. To prove [(\[eq:P”j-bd\])]{} for $j=1$, we first recall the definition [(\[eq:P”1-def\])]{} of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (and Figure \[fig:P-def\]). Note that, by [(\[eq:GGpsi-bd\])]{}, $\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)$ obeys the same bound on $\sum_{v'}G(v'-y)\,G(z-v')$ (with a different $O(1)$ term). That is, the effect of an additional $\psi_\Lambda$ is not significant. Therefore, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ is identical, with a possible modification of the $O(1)$ multiple, to the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (or $P_{\Lambda;v}^{\prime{\scriptscriptstyle}(1)}$) with $v$ (resp., $u$) “being embedded” in one of the bubbles consisting of $\psi_\Lambda-\delta$. By [(\[eq:psi-bd\])]{}, $\psi_\Lambda(y,x)-\delta_{y,x}$ with $v$ being embedded in one of its bubbles is bounded as $$\begin{aligned} {\label{eq:psipsi-bd}} &\sum_{k=1}^\infty\sum_{l=1}^k\sum_{y',x'}\big(\tilde G_\Lambda^2 \big)^{*(l-1)}(y,y')\,\tilde G_\Lambda(y',x')\Big({{\langle \varphi_{y'} \varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde G_\Lambda(y',x')\, \delta_{v,x'}\Big)\big(\tilde G_\Lambda^2\big)^{*(k-l)} (x',x){\nonumber}\\ &=\sum_{y',x'}\psi_\Lambda(y,y')\,\tilde G_\Lambda(y',x')\Big( {{\langle \varphi_{y'}\varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde G_\Lambda(y',x')\,\delta_{v,x'}\Big)\psi_\Lambda(x',x){\nonumber}\\ &\leq\sum_{y',x'}\frac{O(1)}{{\vby'-y{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)} {{\vbx'-y'{|\!|\!|}}^q}\,\frac{O(\theta_0)}{{\vbv-y'{|\!|\!|}}^q{\vbx'-v{|\!|\!|}}^q}\, \frac{O(1)}{{\vbx-x'{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)} {{\vbx-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ By this observation and using [(\[eq:IR-xbd\])]{} to bound the remaining two two-point functions consisting of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (recall [(\[eq:P”1-def\])]{}), we obtain [(\[eq:P”j-bd\])]{} for $j=1$. For [(\[eq:P’j-bd\])]{}–[(\[eq:P”j-bd\])]{} with $j\ge2$, we first note that, by applying [(\[eq:IR-xbd\])]{} and [(\[eq:psi-bd\])]{} to the definition [(\[eq:Pj-def\])]{} of $P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)$, we have $$\begin{gathered} P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)\leq\sum_{\substack{v_2,\dots,v_j\\ v'_1,\dots, v'_{j-1}}}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbv_2-y{|\!|\!|}}^q{\vbv'_1 -v_2{|\!|\!|}}^q}\prod_{i=2}^{j-1}\frac{O(\theta_0^2)}{{\vbv'_i-v_i{|\!|\!|}}^{2q}{{|\!|\!|}v_{i+1}-v'_{i-1}{|\!|\!|}}^q{\vbv'_i-v_{i+1}{|\!|\!|}}^q}{\nonumber}\\ \times\frac{O(\theta_0^2)}{{\vbx-v_j{|\!|\!|}}^{2q}{\vbx-v'_{j-1}{|\!|\!|}}^q}.{\label{eq:Pj-bd}}\end{gathered}$$ By definition, the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(j)}(y,x)$ is obtained by “embedding $u$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ (not ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$) and then summing over all these $2j-1$ choices. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ is replaced by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ is bounded, similarly to [(\[eq:piNgeq2-bd\])]{}, by $$\begin{aligned} &\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbv_2 -u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j-1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx -v_2{|\!|\!|}}^{2q}}{\nonumber}\\ &\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{{|\!|\!|}x-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^{2q}{\vbu -y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}.\end{aligned}$$ The other $2j-2$ contributions can be estimated in a similar way, with the same form of the bound. This completes the proof of [(\[eq:P’j-bd\])]{}. By [(\[eq:psipsi-bd\])]{}, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(j)}(y,x)$ is also obtained by “embedding $u$ and $v$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ and one of the $j$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])]{}, and then summing over all these combinations. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])]{} are replaced, respectively, by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1-v{|\!|\!|}}^q$, is bounded by $$\begin{aligned} &\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1 -v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j -1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx-v_2{|\!|\!|}}^{2q}}{\nonumber}\\ &\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1 -v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}{\nonumber}\\ &\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q {\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ The other $(2j-1)j-1$ contributions can be estimated similarly, with the same form of the bound. This completes the proof of [(\[eq:P”j-bd\])]{} and thus Proposition \[prp:GimpliesPix\]. Bounds for finite-range models {#ss:proof-nn} ------------------------------ First, we prove [(\[eq:pi-sumbd\])]{} and Proposition \[prp:exp-bootstrap\](iii) assuming [(\[eq:IR-kbd\])]{}. Then, we prove [(\[eq:pi-kbd\])]{} assuming [(\[eq:IR-kbd\])]{} and [(\[eq:IR-xbdNN\])]{} to complete the proof of Propositions \[prp:GimpliesPik\]. By applying [(\[eq:G-delta-bd\])]{} to the bound [(\[eq:piNbd\])]{} on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it is easy to show that, for $r=0,2$, $$\begin{aligned} {\label{eq:pi0-rthmombd}} \sum_x|x|^r\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq\delta_{r,0}+\sum_{x\ne o} |x|^r{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3&\leq\delta_{r,0}+\Big( \sup_{x\ne o}|x|^rG(x)\Big)\sum_{x\ne o}(\tau D*G)(x)\,G(x){\nonumber}\\ &\leq\delta_{r,0}+(d\sigma^2)^{\delta_{r,2}}O(\theta_0)^2.\end{aligned}$$ For $i\ge1$, by using the diagrammatic bound [(\[eq:piNbd\])]{} and translation invariance, we have $$\begin{aligned} {\label{eq:dec-bd}} \sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq\bigg(\sum_{v,x}P_{\Lambda;v} ^{\prime{{\scriptscriptstyle}(0)}}(o,x)\bigg)\bigg(\sup_y\sum_{z,v,x}\tau_{y,z} Q''_{\Lambda;o,v}(z,x)\bigg)^{i-1}\bigg(\sup_y\sum_{z,x}\tau_{y, z}Q'_{\Lambda;o}(z,x)\bigg).\end{aligned}$$ The proof of the bound on $\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$ for $i\ge1$ is completed by showing that $$\begin{aligned} {\label{eq:block-sumbd}} \bigg(\sum_{v,x}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,x)-1\bigg)\vee \bigg(\sup_y\sum_{z,v,x}\tau_{y,z}Q''_{\Lambda;o,v}(z,x)\bigg) \vee\bigg(\sup_y\sum_{z,x}\tau_{y,z}Q'_{\Lambda;o}(z,x)\bigg) =O(\theta_0).\end{aligned}$$ The key idea to obtain this estimate is that the bounding diagrams for the Ising model are similar to those for self-avoiding walk (cf., Figure \[fig:piN-bd\]). The diagrams for self-avoiding walk are known to be bounded by products of bubble diagrams (see, e.g., [@ms93]), and we can apply the same method to bound the diagrams for the Ising model by products of bubbles. For example, consider $$\begin{aligned} {\label{eq:tau*Q'-rewr}} \sum_{z,x}\tau_{y,z}Q'_{\Lambda;o}(z,x)=\sum_{z',x}\bigg(\sum_z \tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(z,z')\big)\bigg) P'_{\Lambda;o}(z',x).\end{aligned}$$ The factor of $\theta_0$ is due to the nonzero line segment $\sum_z\tau_{y,z}(\delta_{z,z'}+\tilde G_\Lambda(z,z'))$, because $$\begin{gathered} \sum_z\tau_{o,z}\big(\delta_{z,x}+\tilde G_\Lambda(z,x)\big)=\tau D(x)+\tau\sum_zD(z)\,\tilde G_\Lambda(z,x)\leq O(\theta_0)+\tau \sup_x\tilde G_\Lambda(o,x),{\label{eq:tau*delta+G-bd}}\\ \tilde G_\Lambda(o,x)\leq\tau D(x)+\tau\sum_{y\ne o}G(y)\,D(x-y)\leq O(\theta_0)+\tau\sup_{y\ne o}G(y)=O(\theta_0),{\label{eq:tildeG-bdnn}}\end{gathered}$$ where we have used translation invariance, [(\[eq:IR-kbd\])]{} and $\sup_xD(x)=O(\theta_0)$. By [(\[eq:P’P”-def\])]{}, $$\begin{aligned} {\label{eq:tau*Q'-rewrbd}} {(\ref{eq:tau*Q'-rewr})}\leq O(\theta_0)\sum_{z',x}P'_{\Lambda;o}(z',x) =O(\theta_0)\sum_{z',x}\bigg(P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x) +\sum_{j\ge1}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)\bigg).\end{aligned}$$ Similarly to [(\[eq:pi0-rthmombd\])]{} for $r=0$, the sum of $P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z',x)$ is easily estimated as $1+O(\theta_0)$. We claim that the sum of $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ for $j\geq1$ is $(2j-1)\,O(\theta_0)^j$, since $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ is a sum of $2j-1$ terms, each of which contains $j$ chains of nonzero bubbles; each chain is $\psi_\Lambda(v,v')-\delta_{v,v'}$ for some $v,v'$ and satisfies $$\begin{aligned} \sum_{v'}\big(\psi_\Lambda(v,v')-\delta_{v,v'}\big)\leq\sum_{l\ge1} \Big(\tau^2\big(D*(D*G^{*2})\big)(o)\Big)^l=\sum_{l\ge1} O(\theta_0)^l=O(\theta_0).\end{aligned}$$ For example, $$\begin{aligned} \sum_{z',x}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(4)}}(z',x)&=\raisebox{-1.2pc} {\includegraphics[scale=0.15]{Pprime4}}~+6\text{ other possibilities},\end{aligned}$$ which can be estimated, by translation invariance, as $$\begin{aligned} \raisebox{-1.5pc}{\includegraphics[scale=0.15]{Pprime4}}&\leq ~\raisebox{-1.8pc}{\includegraphics[scale=0.15]{Pprime4dec}}{\nonumber}\\[5pt] &\leq\bigg(\sum_y\big(\psi_\Lambda(o,y)-\delta_{o,y}\big)\bigg)^4\big( \bar W^{{\scriptscriptstyle}(0)}\big)^4=O(\theta_0)^4,\end{aligned}$$ where $\bar W^{{\scriptscriptstyle}(t)}$ is given by [(\[eq:GbarWbar\])]{}. The sum of $\tau_{y,z}Q''_{\Lambda;o,v}(z,x)$ in [(\[eq:block-sumbd\])]{} is estimated similarly [@sNN]. We complete the proof of the bound on $\sum_x\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$. To estimate $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$, we recall that, in each bounding diagram, there are at least three distinct paths between $o$ and $x$: the uppermost path (i.e., $o\to b_1\to v_2\to b_3\to\cdots\to x$ in [(\[eq:piNbd\])]{}; see also Figure \[fig:piN-bd\]), the lowermost path (i.e., $o\to v_1\to b_2\to v_3\to\cdots\to x$) and a middle zigzag path. We use the lowermost path to bound $|x|^2$ as $$\begin{aligned} {\label{eq:x2-bd}} |x|^2=\sum_{n=0}^j|a_n|^2+2\sum_{0\leq m<n\leq j}a_m\cdot a_n \leq(j+1)\sum_{n=0}^j|a_n|^2,\end{aligned}$$ where $a_0=v_1$, $a_1={\underline{b}}_2-v_1$ ,$a_2=v_3-{\underline{b}}_2,\dots$, and $a_j=x-v_j$ or $x-{\underline{b}}_j$ depending on the parity of $j$. $$\begin{gathered} \includegraphics[scale=0.16]{pi3dec}\\[1pc] \text{(i)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12] {pi3dec4}}\qquad\qquad \text{(ii)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12] {pi3dec1}}\\[5pt] \text{(iii)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12] {pi3dec2}}\qquad~~~\&~~\qquad\raisebox{-1.2pc}{\includegraphics [scale=0.12]{pi3dec3}}\end{gathered}$$ We discuss the contributions to $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from (i) $|a_j|^2$, (ii) $|a_0|^2$ and (iii) $|a_n|^2$ for $n\ne0,j$, separately (cf., Figure \[fig:pi3-dec\]). \(i) The contribution from $|a_j|^2$ is bounded by $$\begin{aligned} {\label{eq:2nddec-bd:n=j}} &\bigg(\sum_{v,y}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,y)\bigg)\bigg(\sup_y \sum_{\substack{b,v,z\\ {\underline{b}}=y}}\tau_bQ''_{\Lambda;o,v}({\overline{b}},z)\bigg)^{j -1}{\nonumber}\\ &\qquad\qquad\times\bigg(\sup_y\sum_{\substack{b,x\\ {\underline{b}}=y}}\Big(|x|^2 {\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ odd}\}$}}}+|x-{\underline{b}}|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ even}\}$}}}\Big)\,\tau_b Q'_{\Lambda;o}({\overline{b}},x)\bigg){\nonumber}\\ &\leq O(\theta_0)^{j-1}\sup_y\sum_{z,z',x}\Big(|x|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ odd}\}$}}} +|x-y|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ even}\}$}}}\Big)\,\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(z,z')\big)P'_{\Lambda;o}(z',x).\end{aligned}$$ By [(\[eq:P’0-def\])]{}, the leading contribution from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x)$ for an odd $j$ can be estimated as $$\begin{aligned} {\label{eq:2nddec-bd:n=jbd}} &\sup_y\sum_{z,z',x}|x|^2\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(z,z')\big)P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x){\nonumber}\\ &=\sup_y\sum_{z,z',x}\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda( z,z')\big)\,{{\langle \varphi_{z'}\varphi_o \rangle}}_\Lambda\,{{\langle \varphi_{z'} \varphi_x \rangle}}_\Lambda^2\,|x|^2{{\langle \varphi_o\varphi_x \rangle}}_\Lambda{\nonumber}\\ &\leq\sup_y\Big((\tau D*G)(y)+(\tau D*G)^{*2}(y)\Big)\,G^{*2}(o)\, \bar G^{{\scriptscriptstyle}(2)}=d\sigma^2O(\theta_0)^2,\end{aligned}$$ where $\bar G^{{\scriptscriptstyle}(s)}$ is given by [(\[eq:GbarWbar\])]{}. The other contributions from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(i)}}(z',x)$ for $i\ge1$ and from the even-$j$ case can be estimated similarly; if $j$ is even, then, by using $|x-y|^2\leq2|z'-y|^2+2|x-z'|^2$ and estimating the contributions from $|z'-y|^2$ and $|x-z'|^2$ separately, we obtain that the supremum in [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)$. Consequently, [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}$. \(ii) To bound the contributions to $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from $|a_n|^2$ for $n<j$, we define (cf., Figure \[fig:tildeQ”\]) $$\begin{aligned} {\label{eq:tildeQ''-def}} \tilde Q''_{\Lambda;u,v}(y,x)=\sum_b\bigg(P''_{\Lambda;u,v}(y, {\underline{b}})+\sum_{y'}\tilde G_\Lambda(y,y')\,P'_{\Lambda;u}(y',{\underline{b}}) \,\psi_\Lambda(y,v)\bigg)\,\tau_b\big(\delta_{{\overline{b}},x}+\tilde G_\Lambda({\overline{b}},x)\big).\end{aligned}$$ By translation invariance and a similar argument to show [(\[eq:block-sumbd\])]{}, we can easily prove $$\begin{aligned} {\label{eq:tildeQ''-bd}} \sup_z\sum_{y,v}\tilde Q''_{\Lambda;o,v}(y,v+z)=\sum_{y,v}\tilde Q''_{\Lambda;v,o}(y,z)=O(\theta_0).\end{aligned}$$ Therefore, the contribution from $|a_0|^2$ to $\sum_x|x|^2 \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ is bounded by $$\begin{aligned} {\label{eq:2nddec-bd:n=0}} &\bigg(\sup_y\sum_{v,b}|v|^2P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(v,{\underline{b}})\, \tau_b\big(\delta_{{\overline{b}},y}+\tilde G_\Lambda({\overline{b}},y)\big)\bigg)\bigg( \sup_z\sum_{y,v}\tilde Q''_{\Lambda;v,o}(y,z)\bigg)^{j-1}\bigg( \sum_{z,x}P'_{\Lambda;o}(z,x)\bigg){\nonumber}\\ &\quad\leq d\sigma^2O(\theta_0)^{j+1}.\end{aligned}$$ ![\[fig:tildeQ”\]The leading diagrams of $\tilde Q''_{\Lambda;u,v}(y,x)$, due to $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,{\underline{b}})$ and $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y',{\underline{b}})$ in [(\[eq:tildeQ”-def\])]{}, respectively.](tildeQpp1 "fig:") ![\[fig:tildeQ”\]The leading diagrams of $\tilde Q''_{\Lambda;u,v}(y,x)$, due to $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,{\underline{b}})$ and $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y',{\underline{b}})$ in [(\[eq:tildeQ”-def\])]{}, respectively.](tildeQpp2 "fig:") \(iii) By translation invariance and [(\[eq:tildeQ”-def\])]{}–[(\[eq:tildeQ”-bd\])]{}, the contribution from $|a_n|^2$ for an $n\ne0,j$ is bounded by $$\begin{aligned} {\label{eq:2nddec-bd:0<n<j}} &\bigg(\sum_{v,y}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,y)\bigg)\bigg(\sup_y \sum_{\substack{b,v,z\\ {\underline{b}}=y}}\tau_bQ''_{\Lambda;o,v}({\overline{b}},z)\bigg)^{n -1}\bigg(\sup_z\sum_{y,v}\tilde Q''_{\Lambda;v,o}(y,z)\bigg)^{j-1-n} \bigg(\sum_{z,x}P'_{\Lambda;o}(z,x)\bigg){\nonumber}\\ &\times\bigg(\sup_{y,z}\sum_{\substack{b,b',v\\ {\underline{b}}=y}}\Big(|{\underline{b}}' |^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n\text{ odd}\}$}}}+|v-{\underline{b}}|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n\text{ even}\}$}}}\Big)\,\tau_b Q''_{\Lambda;o,v}({\overline{b}},{\underline{b}}')\,\tau_{b'}\big(\delta_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}},v+z} +\tilde G_\Lambda({{\overline{b}}^{\raisebox{-2pt}{$\scriptstyle\prime$}}},v+z)\big)\bigg),\end{aligned}$$ where the first line is $O(\theta_0)^{j-2}$. The leading contribution to the second line from $P_{\Lambda;o,v}^{\prime\prime{{\scriptscriptstyle}(0)}}$ and $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}$ in $Q''_{\Lambda;o,v}$ for an odd $n$ is bounded, due to translation invariance, by $$\begin{aligned} {\label{eq:2nddec-bd:0<n<jbd}} &\bar G^{{\scriptscriptstyle}(2)}\sup_{y,z}\Bigg(~\raisebox{-1.4pc}{\includegraphics [scale=0.14]{Pprime0Wdec24}}~+\raisebox{-1.4pc}{\includegraphics [scale=0.14]{Pprime0Wdec25}}~\Bigg){\nonumber}\\ &\leq d\sigma^2O(\theta_0)^2\sup_z\Bigg(~\raisebox{-1.4pc} {\includegraphics[scale=0.14]{Pprime0Wdec26}}~+~\raisebox{-1.4pc} {\includegraphics[scale=0.14]{Pprime0Wdec27}}~\Bigg)\leq d\sigma^2 O(\theta_0)^3.\end{aligned}$$ The other contributions from $P_{\Lambda;o,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(i)}}$ for $i\ge1$ and from the even-$n$ case can be estimated similarly; if $n$ is even, then the second supremum in [(\[eq:2nddec-bd:0&lt;n&lt;jbd\])]{} is $O(\theta_0)$. Therefore, [(\[eq:2nddec-bd:0&lt;n&lt;j\])]{} is $d\sigma^2O(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}$. Summarizing the above (i)–(iii) and using $2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor\geq j\vee2$ for $j\ge1$, we have $$\begin{aligned} \frac1{j+1}\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq d\sigma^2\Big( jO(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}+O(\theta_0)^{j+1} \Big)\leq d\sigma^2(j+1)\,O(\theta_0)^{j\vee2}.\end{aligned}$$ This together with [(\[eq:pi0-rthmombd\])]{} complete the proof of [(\[eq:pi-sumbd\])]{}. It is easy to see that $$\begin{aligned} {\label{eq:pi0-t+2ndmombd}} \sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq\sum_x|x|^{t+2}G(x)^3\leq \bar G^{{\scriptscriptstyle}(2)}\sum_x|x|^tG(x)^2\leq d\sigma^2\theta_0\bar W^{{\scriptscriptstyle}(t)}.\end{aligned}$$ We show below that, for $j\ge1$, $$\begin{aligned} {\label{eq:pij-t+2ndmombd}} \sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq d\sigma^2\bar W^{{\scriptscriptstyle}(t)} (j+1)^{t+3}O(\theta_0)^{j\vee2-1},\end{aligned}$$ where the bound is independent of $\Lambda$. Due to these uniform bounds, we conclude that the sum of $|x|^{t+2}|\Pi(x)|$ is finite if $\theta_0\ll1$. Now we explain the main idea of the proof of [(\[eq:pij-t+2ndmombd\])]{}. First we recall that, in the proof of the bound on $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we distribute $|x|^2$ along the lowermost path of each bounding diagram. To bound $\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we again use the lowermost path in the same way to distribute $|x|^2$, and use the uppermost path to distribute the remaining $|x|^t$. More precisely, we use $$\begin{aligned} {\label{eq:|x|-max}} |x|\leq(j+1)\max_{n=0,1,\dots,j}|a'_n|,\end{aligned}$$ where $a'_0,a'_1,\dots,a'_j$ are the displacements along the uppermost path: $a'_0={\underline{b}}_1$, $a'_1=v_2-{\underline{b}}_1$, $a'_2={\underline{b}}_3-v_2,\dots$, and $a'_j=x-v_j$ or $x-{\underline{b}}_j$ depending on the parity of $j$. Let $m$ be such that $|a'_m|=\max_n|a'_n|$. For the contribution to $\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from $|a_n|^2$ in [(\[eq:x2-bd\])]{} for $n\ne m$, we simply follow the same strategy as explained above in the paragraphs (i)–(iii) to prove the bound on $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$. The only difference is that one of the bubbles $\bar W^{{\scriptscriptstyle}(0)}$ contained in the bound on the $m^\text{th}$ block is now replaced by $\bar W^{{\scriptscriptstyle}(t)}$. The contribution from $|a_m|^2$ in [(\[eq:x2-bd\])]{} can be estimated in a similar way, except for a few complicated cases, due to $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ for $i\ge1$ contained in the $m^\text{th}$ block. For example, let $j$ be even and let $m=j$ (cf., the second line of [(\[eq:2nddec-bd:n=j\])]{}). The following are two possibile diagrams in the contribution from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(4)}}(f,x)$ to $\sum_{z,x}|x-y|^2|x|^t\tau_{y,z}Q'_{\Lambda;o}(z,x)$: $$\begin{aligned} {\label{eq:IRSchwarz}} \text{(i)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14] {IRnonSchwarz}}\hspace{5pc} \text{(ii)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14] {IRSchwarz}}\end{aligned}$$ where, for simplicity, $\psi_\Lambda(f,g)-\delta_{f,g}$ and $\psi_\Lambda(u,z)-\delta_{u,z}$ are reduced to $\tilde G_\Lambda(f,g)^2$ and $\tilde G_\Lambda(f,g)^2$, respectively. We suppose that $|v|$ is bigger than $|w-v|$ and $|x-w|$ along the lowermost path from $o$ to $x$ through $v$ and $w$, so that $|x|^t$ is bounded by $3^t|v|^t$. We also suppose that $|z-u|$ in (\[eq:IRSchwarz\].i) (resp., $|g-f|$ in (\[eq:IRSchwarz\].ii)) is bigger than the end-to-end distance of any of the other four segments along the uppermost path from $y$ to $x$ through $f,g,u$ and $z$. Therefore, we can bound $|x-y|^2$ by $5^2|z-u|^2$ in (\[eq:IRSchwarz\].i) (resp., $5^2|g-f|^2$ in (\[eq:IRSchwarz\].ii)) and bound the weighted arc between $u$ and $z$ (resp., between $f$ and $g$) by $5^2\bar G^{{\scriptscriptstyle}(2)}$. By translation invariance, the remaining diagram of (\[eq:IRSchwarz\].i) is easily bounded as $$\begin{aligned} {\label{eq:IRSchwarz-bdi}} \sum_{f',g,u',v}\!\raisebox{-1.4pc}{\includegraphics[scale=0.14] {IRnonSchwarzdec1}}\;=\sup_{f',g,u'}\,\raisebox{-1.4pc} {\includegraphics[scale=0.14]{IRnonSchwarzdec2}}\leq\bar W^{{\scriptscriptstyle}(t)}\,O(\theta_0)^4,\end{aligned}$$ where the power 4 (not 3) is due to the fact that the segment from $u'$ in the last block is nonzero. To bound the remaining diagram of (\[eq:IRSchwarz\].ii) is a little trickier. We note that at least one of $|u|,|z-u|,|w-z|$ and $|v-w|$ along the path from $o$ to $v$ through $u,z,w$ is bigger than $\frac14|v|$. Suppose $|v-w|\ge\frac14|v|$, so that $|v|^t\leq2^t|v-w|^{t/2}|v|^{t/2}$. Then, by using the Schwarz inequality, we obtain $$\begin{aligned} {\label{eq:IRSchwarz-bdii}} \raisebox{-1.3pc}{\includegraphics[scale=0.13]{IRSchwarzdec1}}~~\leq~~ \Bigg(~~\raisebox{-2pc}{\includegraphics[scale=0.12]{IRSchwarzdec2}} ~~\Bigg)^{1/2}~\Bigg(~~\raisebox{-1.5pc}{\includegraphics[scale=0.12] {IRSchwarzdec3}}~~\Bigg)^{1/2},\end{aligned}$$ where the two weighted arcs between $o$ and $v$ in the second term is $|v|^tG(v)^2\equiv(|v|^{t/2}G(v))^2$. By translation invariance and the fact that the north-east and north-west segments from $g$ in the first term are nonzero, we obtain $$\begin{aligned} \raisebox{-2pc}{\includegraphics[scale=0.13]{IRSchwarzdec2}}~ &\leq\bigg(\sup_z\raisebox{-0.9pc}{\includegraphics[scale=0.12] {IRSchwarzdec4}}~\bigg)\Big(\sup_{g'}\tau(D*G^{*2})(g')\Big)^2\bar W^{{\scriptscriptstyle}(0)}\bigg(\sum_v\big(\psi_\lambda(o,v)-\delta_{o,v}\big) \bigg)^2{\nonumber}\\ &\leq O(\theta_0)^5.\end{aligned}$$ With the help of $(\bar W^{{\scriptscriptstyle}(t/2)})^2\leq\bar W^{{\scriptscriptstyle}(0)}\bar W^{{\scriptscriptstyle}(t)}$ (due to the Schwarz inequality), we also obtain $$\begin{aligned} \raisebox{-1.8pc}{\includegraphics[scale=0.13]{IRSchwarzdec3}}~~\leq \,\bar W^{{\scriptscriptstyle}(0)}\,\bar W^{{\scriptscriptstyle}(t)}\Bigg(\sup_v\raisebox{-1.1pc} {\includegraphics[scale=0.13]{IRSchwarzdec5}}~\Bigg)^2\leq\big(\bar W^{{\scriptscriptstyle}(t)}\big)^2O(\theta_0)^4.\end{aligned}$$ Therefore, [(\[eq:IRSchwarz-bdii\])]{} is bounded by $\bar W^{{\scriptscriptstyle}(t)}O(\theta_0)^{9/2}$. The other cases can be estimated similarly [@sNN]. As a result, we obtain $$\begin{aligned} \sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{m=0}^jd\sigma^2 \bar W^{{\scriptscriptstyle}(t)}(j+1)^{t+2}O(\theta_0)^{j\vee2-1},\end{aligned}$$ which implies [(\[eq:pij-t+2ndmombd\])]{}. This completes the proof of Proposition \[prp:exp-bootstrap\](iii). If $x=o$, then we simply use the bound on the sum in [(\[eq:pi-sumbd\])]{} to obtain $\pi_\Lambda^{{\scriptscriptstyle}(i)}(o)\leq O(\theta_0)^i$ for any $i\ge0$. It is also easy to see that $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ with $x\ne o$ obeys [(\[eq:pi-kbd\])]{}, due to [(\[eq:IR-xbdNN\])]{} and the diagrammatic bound [(\[eq:piNbd\])]{}. It thus remains to show [(\[eq:pi-kbd\])]{} for $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ with $x\ne o$ and $j\ge1$. The idea of the proof is somewhat similar to that of Proposition \[prp:exp-bootstrap\](iii) explained above. First, we take $|a_m|\equiv\max_n|a_n|$ from the lowermost path and $|a'_l|\equiv\max_n|a'_n|$ from the uppermost path of a bounding diagram. Note that, by [(\[eq:|x|-max\])]{}, $|a_m|$ and $|a'_l|$ are both bigger than $\frac1{j+1}|x|$. That is, $|a_m|^{-q}$ and $|a'_l|^{-q}$ are both bounded from above by $(j+1)^q|x|^{-q}$. If the path corresponding to $a_m$ in the $m^\text{th}$ block consists of $N$ segments, we take the “longest” segment whose end-to-end distance is therefore bigger than $\frac1{N(j+1)}|x|$. That is, the corresponding two-point function is bounded by $\lambda_0 N^q(j+1)^q|x|^{-q}$. Here, $N$ depends on the parity of $m$, as well as on $i\ge0$ for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ (or $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ if $m=0$ or $j$) and the location of $u,v$ in each diagram, and is at most $N\leq O(i+1)$. However, the number of nonzero chains of bubbles contained in each diagram of $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ is $O(i)$, and hence their contribution would be $O(\theta_0)^{O(i)}$. This compensates the growing factor of $N^q$, and therefore we will not have to take the effect of $N$ seriously. The same is true for $a'_l$, and we refrain from repeating the same argument. Next, we take the “longest” segment, denoted $a''$, among those which together with $a'_l$ (or a part of it) form a “loop”; a similar observation was used to obtain [(\[eq:IRSchwarz-bdii\])]{}. The loop consists of segments contained in the $l^\text{th}$ block and possibly in the $(l-1)^\text{st}$ block, and hence the number of choices for $a''$ is at most $O(i_{l-1}+i_l+1)$, where $i_l$ is the index of $P_{\Lambda}^{\prime{{\scriptscriptstyle}(i_l)}}$ or $P_{\Lambda}^{\prime\prime{{\scriptscriptstyle}(i_l)}}$ in the $l^\text{th}$ block ($i_{-1}=0$ by convention). By [(\[eq:|x|-max\])]{}, we have $|a''|\ge O(i_{l-1}+i_l+1)^{-1}|a'_l|$, and the corresponding two-point function is bounded by $\lambda_0O(i_{l-1}+i_l+1)^q (j+1)^q|x|^{-q}$. As explained above, the effect of $O(i_{l-1}+i_l+1)^q$ would not be significant after summing over $i_{l-1}$ and $i_l$. We have explained how to extract three “long” segments from each bounding diagram, which provide the factor $\lambda_0^3(j+1)^{3q}|x|^{-3q}$ in [(\[eq:pi-kbd\])]{}; the extra factor of $(j+1)^2$ in [(\[eq:pi-kbd\])]{} is due to the number of choices of $m,l\in\{0,1,\dots,j\}$. Therefore, the remaining task is to control the rest of the diagram. Suppose, for example, $0<m<l<j$ (so that $j\ge3$). Using $\tilde Q''_\Lambda$ defined in [(\[eq:tildeQ”-def\])]{}, we can reorganize the diagrammatic bound [(\[eq:piNbd\])]{} on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ as (cf., [(\[eq:2nddec-bd:0&lt;n&lt;j\])]{}) $$\begin{aligned} {\label{eq:diagbd-reorg}} \pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&\leq\sum_{\substack{b_m,v_m\\ y_{l+1}, v_{l+1}}}\bigg(\sum_{\substack{b_1,\dots,b_{m-1}\\ v_1,\dots, v_{m-1}}}P_{\Lambda;v_1}^{\prime{{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\prod_{i= 1}^{m-1}\tau_{b_i}Q''_{\Lambda;v_i,v_{i+1}}({\overline{b}}_i,{\underline{b}}_{i+1}) \bigg){\nonumber}\\ &\qquad\times\sum_{b_{l+1}}\bigg(\sum_{\substack{b_{m+1},\dots, b_l\\ v_{m+1},\dots,v_l}}\prod_{i=m}^l\tau_{b_i}Q''_{\Lambda; v_i,v_{i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})\bigg)\tau_{b_{l+1}}\big(\delta_{ {\overline{b}}_{l+1},y_{l+1}}+\tilde G_\Lambda({\overline{b}}_{l+1},y_{l+1})\big){\nonumber}\\ &\qquad\times\Bigg(\sum_{\substack{y_{l+2},\dots,y_j\\ v_{l+2}, \dots,v_j}}\bigg(\prod_{i=l+1}^{j-1}\tilde Q''_{\Lambda;v_i, v_{i+1}}(y_i,y_{i+1})\bigg)P'_{\Lambda;v_j}(y_j,x)\Bigg).\end{aligned}$$ As explained above, we bound three “long” two-point functions contained in the second line of [(\[eq:diagbd-reorg\])]{}; let $Y_{m,l}$ be the supremum of what remains in the second line over $b_m,v_m,y_{l+1},v_{l+1}$. Then we can perform the sum of the first line over $b_m,v_m$ and the sum of the third line over $y_{l+1},v_{l+1}$ independently; the former is $O(\theta_0)^{m-1}$ and the latter is $O(\theta_0)^{j-1-l}$, due to [(\[eq:block-sumbd\])]{} and [(\[eq:tildeQ”-bd\])]{}, respectively. Finally, we can bound $Y_{m,l}$ using the Schwarz inequality by $O(\theta_0)^{l-m}$, where $l-m$ is the number of nonzero segments in the second line of [(\[eq:diagbd-reorg\])]{} (i.e., $\sum_{b_i}\tau_{b_i}(\delta_{{\overline{b}}_i,y_i}+\tilde G_\Lambda({\overline{b}}_i,y_i))$ for some $y_m,\dots,y_{l+1}$) minus 2 (= the maximum number of those along the uppermost and lowermost paths that are extracted to obtain the aformentioned $|x|$-decaying term). For example, one of the leading contributions to $Y_{m,m+4}$ is bounded, by using translation invariance and the Schwarz inequality, as $$\begin{aligned} \sup_{u,v,y}\raisebox{-1pc}{\includegraphics[scale=0.14]{Yml1}}~\leq O(\theta_0)~\sup_{u,z}\raisebox{-1pc}{\includegraphics[scale=0.14] {Yml2}}\\ \leq O(\theta_0)^{3/2}\left(\raisebox{-1.9pc}{\includegraphics[scale =0.14]{Yml3}}\right)^{1/2}\leq O(\theta_0)^2~\sup_{s'}\raisebox{-1pc} {\includegraphics[scale=0.14]{Yml4}}~&\leq O(\theta_0)^4.{\nonumber}\end{aligned}$$ The other cases can be estimated similarly [@sNN]. This completes the proof of [(\[eq:pi-kbd\])]{}. Acknowledgements {#acknowledgements .unnumbered} ================ First of all, I am grateful to Masao Ohno for having drawn my attention to the subject of this paper. I would like to thank Takashi Hara for stimulating discussions and his hospitality during my visit to Kyushu University in December 2004 and April 2005. I would also like to thank Aernout van Enter for useful discussions on reflection positivity. Special thanks go to Mark Holmes and John Imbrie for continual encouragement and valuable comments to the former versions of the manuscript, and Remco van der Hofstad for his constant support in various aspects. This work was supported in part by the Postdoctoral Fellowship of EURANDOM, and in part by the Netherlands Organization for Scientific Research (NWO). [99]{} M. Aizenman. Geometric analysis of $\phi^4$ fields and Ising models. *Commun. Math. Phys.* [**86**]{} (1982): 1–48. M. Aizenman, D.J. Barsky and R. Fernández. The phase transition in a general class of Ising-type models is sharp. *J. Statist. Phys.* [**47**]{} (1987): 343–374. M. Aizenman and R. Fernández. On the critical behavior of the magnetization in high-dimensional Ising models. *J. Statist. Phys.* [**44**]{} (1986): 393–454. M. Aizenman and R. Graham. On the renormalized coupling constant and the susceptibility in $\phi_4^4$ field theory and the Ising model in four dimensions. *Nucl. Phys. B* [**225**]{} (1983): 261–288. J. van den Berg and H. Kesten. Inequalities with applications to percolation and reliability. (1985): 556–569. M. Biskup, L. Chayes and N. Crawford. Mean-field driven first-order phase transitions in systems with long-range interactions. *Preprint* (2005). T. Bodineau. Translation invariant Gibbs states for the Ising model. *Preprint* (2005). To appear in *Probab. Th. Rel. Fields*. D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. *Commum. Math. Phys.* [**97**]{} (1985): 125–148. M. Campanino, D. Ioffe and Y. Velenik. Ornstein-Zernike theory for finite range Ising models above $T_c$. *Probab. Th. Rel. Fields* [**125**]{} (2003): 305–349. R. Fernández, J. Fröhlich and A.D. Sokal. *Random walks, critical phenomena, and triviality in quantum field theory*. Springer, Berlin (1992). C.M. Fortuin and P.W. Kasteleyn. On the random-cluster model I. Introduction and relation to other models. *Physica* [**57**]{} (1972): 536–564. J. Fröhlich, B. Simon and T. Spencer. Infrared bounds, phase transitions and continuous symmetry breaking. *Commun. Math. Phys.* [**50**]{} (1976): 79–95. R.B. Griffiths, C.A. Hurst and S. Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external field. *J. Math. Phys.* [**11**]{} (1970): 790–795. T. Hara. Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. *Preprint* (2005). T. Hara, R. van der Hofstad and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. *Ann. Probab.* [**31**]{} (2003): 349–408. T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. *J. Statist. Phys.* [**59**]{} (1990): 1469–1510. T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. *Commun. Math. Phys.* [**128**]{} (1990): 333–391. T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. *Commun. Math. Phys.* [**147**]{} (1992): 101–136. T. Hara and G. Slade. Mean-field behaviour and the lace expansion. *Probability and Phase Transition* (ed., G. R. Grimmett). Kluwer, Dordrecht (1994): 87–122. M. Heydenreich, R. van der Hofstad and A. Sakai. Mean-field behavior for long- and finite-range Ising model, percolation and self-avoiding walk. *In preparation*. R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. *Probab. Th. Rel. Fields* [**122**]{} (2002): 389–430. E. Ising. Beitrag zur Theorie des Ferromagnetismus. *Zeitschrift für Physik* [**31**]{} (1925): 253–258. E.H. Lieb. A refinement of Simon’s correlation inequality. (1980): 127–135. N. Madras and G. Slade. *The Self-Avoiding Walk*. Birkhäuser, Boston (1993). B.G. Nguyen and W-S. Yang. Triangle condition for oriented percolation in high dimensions. *Ann. Probab.* [**21**]{} (1993): 1809–1844. L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. (1944): 117–149. A. Sakai. Mean-field critical behavior for the contact process. *J. Statist. Phys.* [**104**]{} (2001): 111–143. A. Sakai. Asymptotic behavior of the critical two-point function for spread-out Ising ferromagnets above four dimensions. *Unpublished manuscript* (2005). A. Sakai. Diagrammatic estimates of the lace-expansion coefficients for finite-range Ising ferromagnets. *Unpublished notes* (2006). B. Simon. Correlation inequalities and the decay of correlations in ferromagnets. (1980): 111–126. G. Slade. The lace expansion and its applications. Springer Lecture Notes in Mathematics [**1879**]{} (2006). A.D. Sokal. An alternate constructive approach to the $\varphi_3^4$ quantum field theory, and a possible destructive approach to $\varphi_4^4$. *Ann. Inst. Henri Poincaré Phys. Théorique* [**37**]{} (1982): 317–398. [^1]: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. [[email protected]]{} [^2]: Updated: November 13, 2006 [^3]: In [(\[eq:IRbd-so\])]{} and [(\[eq:IRbd-sokal\])]{}, we also use the fact that, for $p<{p_\text{c}}$, our $G_p$ (i.e., the infinite-volume limit of the two-point function under the free-boundary condition) is equal to the infinite-volume limit of the two-point function under the periodic-boundary condition. [^4]: The mean-field results in [@a82; @abf87; @af86; @ag83] are based on a couple of differential inequalities for $M_{p,h}$ and $\chi_p$ (under the periodic-boundary condition) using a certain random-walk representation. We can simplify the proof of the same differential inequalities (under the free-boundary condition as well) using Proposition \[prp:through\]. [^5]: Repeated applications of [(\[eq:G-delta-bd\])]{} to the translation-invariant models result in the random-walk bound: ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda\leq S_\tau(x)$ for $\Lambda\subset{{\mathbb Z}^d}$ and $\tau\leq1$.

ArXiv

--- author: - | Dehua Cheng\ USC - | Yu Cheng\ USC - | Yan Liu[^1]\ USC - | Richard Peng\ MIT - | Shang-Hua Teng[^2]\ USC bibliography: - 'isr.bib' - 'parallelSampling.bib' title: Scalable Parallel Factorizations of SDD Matrices and Efficient Sampling for Gaussian Graphical Models --- [^1]: Supported in part by NSF research grants IIS-1134990, IIS-1254206 and U.S. Defense Advanced Research Projects Agency (DARPA) under Social Media in Strategic Communication (SMISC) program, Agreement Number W911NF-12-1-0034. [^2]: Supported in part by NSF grants CCF-1111270 and CCF-096448 and by the Simons Investigator Award from the Simons Foundation.

ArXiv

--- abstract: '*We present an algorithm for classification tasks on big data. Experiments conducted as part of this study indicate that the algorithm can be as accurate as ensemble methods such as random forests or gradient boosted trees. Unlike ensemble methods, the models produced by the algorithm can be easily interpreted. The algorithm is based on a divide and conquer strategy and consists of two steps. The first step consists of using a decision tree to segment the large dataset. By construction, decision trees attempt to create homogeneous class distributions in their leaf nodes. However, non-homogeneous leaf nodes are usually produced. The second step of the algorithm consists of using a suitable classifier to determine the class labels for the non-homogeneous leaf nodes. The decision tree segment provides a coarse segment profile while the leaf level classifier can provide information about the attributes that affect the label within a segment.*' author: - Rajiv Sambasivan - Sourish Das bibliography: - 'ldcct.bib' title: Big Data Classification Using Augmented Decision Trees --- Introduction and Motivation =========================== Classification tasks are arguably the most common applications of machine learning. Over the years, several sophisticated techniques have been developed for classification. The size of the data set has started becoming an important consideration today in picking a method for classification. Solving the problem for linearly separable decision boundaries was an important first step [@zhang2004solving]. Linear decision boundaries may offer an adequate solution for some datasets but many real world classification problems are characterized by non-linear decision boundaries. Kernel methods [@bosern] are useful in these situations. However applying kernel methods to large datasets also has many challenges. On moderate size datasets, evaluating multiple kernels on the data and then subsequently picking hyper-parameters using a technique like grid search is a tractable approach. However with large datasets, this approach may be impractical because each experimental evaluation may be computationally expensive. Sometimes, such an iterative approach to kernel selection may not yield kernels that perform well and we may need to resort to multiple kernel learning [@bach2004multiple] to arrive at a suitable kernel for the problem. Since developing a single complex model for the entire dataset is a difficult task, a natural line of inquiry would be a divide and conquer strategy. This would entail developing models on segments of the data. Though ideas such as Hierarchical Generalized Linear Models [@lee1996hierarchical] have been developed, the method to determine the segments is a critical aspect of such an approach. Recently we reported a method to perform big data regression using a Classification and Regression Tree (CART) [@breiman1984classification] to perform this segmentation [@sambasivan2017big] . The effectiveness of this approach with regression problems suggested that this technique could be applied to classification tasks as well.\ Experiments reported in this study suggest that this approach could be effective for classification tasks as well. The approach is characterized by two steps. The first step uses a CART decision tree to segment the large dataset. The leaves of the decision tree represent the segments. Decision trees minimize an impurity measure like the misclassification error, gini-index [@gini1971variability] or the cross-entropy at the leaves. While some leaves may be almost homogeneous with respect to the class distribution, in a large dataset a decision tree that generalizes well may have many leaves where the class distribution is not homogeneous. These nodes may require a classifier that can determine complex decision boundaries or these leaves may represent noisy regions of the data . Accordingly, the second step of the algorithm fits a classifier to those nodes where the class distribution is non-homogeneous. In the experiments reported in this study we found that it was possible to increase classification accuracies in some cases. When this strategy fails we observed that this was because all classifiers perform poorly at certain leaf nodes. This suggests that these nodes are either noisy or may require additional features to achieve good classification performance. In this study, we observed this behavior with the census income dataset (see section \[sec:datasets\] for details of the dataset). The classification task for this dataset is to predict the income level for an individual given socio-economic features. In segments of poor performance we found records of individuals working a high number of hours per week with the state government jobs, but reporting a low income. These records seem to be of dubious quality, since even with minimum-wage, these instances should belong to the higher income category. When these noisy segments were removed, we are able to enhance accuracy. Therefore this algorithm either achieves good accuracies or it helps us identify potentially noisy or difficult regions of our dataset. An attractive feature of this algorithm is the ease with which the resulting models can be interpreted. For any data instance, the decision tree model yields the aggregate properties associated with that instance. The leaf level model obtained from the second step can then be interpreted to yield insights into factors that affect the decision for that leaf. In the experiments conducted as part of this study we found that the accuracy of the proposed approach matches what is obtained with ensemble methods like gradient boosted trees [@chen2016xgboost] or random forests [@breiman2001random]. Models produced by ensemble methods are difficult to interpret in contrast to the models produced by the proposed method. Therefore the proposed method can produce models that are both interpretable and accurate. This is highly desirable.\ Problem Context {#sec:pc} =============== We are given a dataset $\mathcal{D}$, let $x_i$ represent the predictor variables and $y_i$ represent the label associated with the instance $i$. Observations are ordered pairs $(x_i, y_i),\ i = 1,2,\hdots, \mathit{N}$. Class labels $y_i$ are represented by $\{0,1,\hdots, K-1\}$. Classification trees partition the predictor space into $m$ regions, $\mathit{R_1},\mathit{R_2}, \hdots, \mathit{R_m}$. Consider a $\mathit{K}$ class classification problem. For a leaf node $m$, representing region $\mathit{R_m}$ with $\mathit{N_m}$ observations, the proportion of observations belonging to class $\mathit{k}$ is defined as: $$\mathit{\hat{p}_{mk}} = \frac{1}{\mathit{N_m}} \sum_{x_i \in \mathit{R_m}} \mathit{I}(y_i = k),$$ $\mathit{I}(y_i = k) = \begin{cases*} 1\quad if\ y_i = k\\ 0,\ otherwise. \end{cases*} $ CART labels the instances in leaf node $m$ with label $$k(m) = \underset{ k \in \mathit{K}} \operatorname*{arg\,max}\quad \mathit{\hat{p}_{mk}},$$ see [@friedman2001elements]\[Chapter 9, section 9.2\]. During tree development, CART tries to produce leaf nodes that are as homogeneous as possible. Typically not all leaf nodes are homogeneous. Leaves that are non-homogeneous with respect to the class distribution are data regions where we can enhance the performance of the decision tree. Section \[sec:dt\_for\_seg\] provides the details of the how this is achieved. Decision Trees For Segmentation {#sec:dt_for_seg} =============================== $\leftarrow$ The first step of the algorithm is to segment the dataset with a CART decision tree. The second step of the algorithm is to augment the performance of the decision tree classifier in segments or leaves where the class distribution is non-homogeneous. This is achieved by using a suitable leaf level classifier. A pool of classifiers is developed for these segments and the best performing classifier, as indicated by the cross-validated training error is used as the leaf classifier for the segment. Algorithm \[algo:dt\_tree\_seg\_reg\] summarizes these ideas. The number of instances at the leaf or equivalently the height of the decision tree is an important parameter. The following factors need to considered in picking this parameter: 1. Generalization error of the decision tree: We need to avoid over-fitting the decision tree model. The decision rules produced by the tree should be valid for the test set and produce a test error that is not very different from the training error. 2. Total generalization error of the algorithm: We want our algorithm to be as accurate as possible. The leaf size at which the best decision tree error is obtained may be different from the leaf size at which the lowest overall error is obtained for the algorithm. We need to ensure that the composite model generalizes well. These ideas are discussed and illustrated in section \[sec:experiments\]. Leaf Classifiers {#sec:leaf_classifiers} ================ The key idea with the algorithm presented in this work is to augment the performance of decision tree nodes where the class distribution is non-homogeneous. Using a suitable classifier, we may be able to determine decision boundaries in these segments that result in better classification accuracy than what is produced with the plain decision tree. This strategy works very well for some datasets. Sometimes however we do encounter nodes where all classifiers perform poorly. This typically happens for a small proportion of the segments. These segments are probably noisy or require additional features for achieving good classification performance. Strategies to deal with these segments are discussed in section \[sec:dor\]. [@kohavi1996scaling] presents an algorithm called NBTree that is similar to the idea presented in this work. [@kohavi1996scaling] work uses a Naive Bayes classifier for the leaf nodes. The tree algorithm used in [@kohavi1996scaling] is C4.5 [@quinlan2014c4]. In this work we used the CART [@breiman2001random] algorithm for the decision tree. The accuracies obtained with a decision tree based on the CART algorithm is higher than what is reported with NBTree in [@kohavi1996scaling], see section \[sec:accuracy\] for details. In [@kohavi1996scaling], the Naive Bayes classifier is the leaf classifier for every leaf whereas this algorithm permits flexibility with this decision. We can pick any classifier for the leaf node. Further, we can fit a pool of classifiers and then pick the best classifier for a node based on cross-validation scores observed during training. It is also important to note that leaf nodes may homogeneous with respect to the class distribution, in which case there is no need to fit a classifier. We can accept the decision tree results for that leaf. The implementation of Algorithm \[algo:dt\_tree\_seg\_reg\] that was used for this study implements these features. Feature Selection {#sec:eol} ================= The datasets used for this study were high dimensional. These public datasets have also been used research and in kaggle competitions [@kaggle]. Feature selection performed using the extremely randomized trees algorithm [@geurts2006extremely] helps in removing noisy features for some of the datasets used in this study. Extremely randomized trees is a tree based algorithm that is similar to random forests, but with some important differences. Unlike in random forests where the splits are determined on the basis of an impurity measure like gini [@gini1971variability] or cross-entropy, the splits in extremely randomized trees are randomly determined. Methodology {#sec:methodology} =========== The datasets used in this study have been featured in kaggle competitions. A review of the competition forums reveal that feature selection and ensemble tree based algorithms are the key components for good performance on these datasets. The methodology used to evaluate the effectiveness of the algorithm proposed in this study is as follows. Feature selection is performed to determine the relevant features for a dataset. If feature selection did not help improve accuracy, we retain all original features. We use a CART decision tree on the datasets used for this study and obtain a performance measurement. Next we apply the random forest and gradient boosted trees algorithm on the datasets and obtain a performance measurement. Finally we apply Algorithm \[algo:dt\_tree\_seg\_reg\] and obtain the performance measurement for the proposed algorithm. We can then evaluate how the accuracy obtained with the proposed algorithm compares with ensemble methods such as random forests or gradient boosted trees. We can also evaluate the accuracy gain obtained with Algorithm \[algo:dt\_tree\_seg\_reg\] over a plain CART decision tree. It should be noted that the leaf size is an important parameter in applying Algorithm \[algo:dt\_tree\_seg\_reg\]. The leaf size used was one that produced good accuracy and good generalization. In general, the decision tree generalizes well at this leaf size. Experiments that illustrate the effect of the leaf size parameter are discussed in section \[sec:experiments\]. Experiments {#sec:experiments} =========== Datasets {#sec:datasets} -------- The datasets for this study all came from the UCI data repository [@Lichman] and the US Department of Transportation website [@RITA_Delay_Data_Download]. These datasets also figure in the kaggle playground category competitions. Playground category competitions are organized for the purpose of testing out machine learning ideas [@no_free_hunch_2016]. There is no prize money involved. The following datasets were used for this study: 1. **Forest Cover Type**: This dataset is used to predict the type of forest cover given cartographic information. The data covers four wilderness areas in the Roosevelt National Forest of northern Colorado. The dataset consists of 581012 instances with 54 attributes. There are no missing values for attributes. 2. **Airline Delay**: This dataset is used to predict airline travel delays. The dataset is obtained from the US Department of Transportation website [@RITA_Delay_Data_Download]. The data consists of flight on time arrival performance for the months of January and February of 2017. A flight is considered delayed if it is associated with an arrival delay of fifteen minutes or more. Thirteen flight information attributes are extracted. The dataset has over eight hundred thousand records. 3. **Census Income**: This dataset contains data extracted from the 1994 census database. The prediction task associated with this dataset is to predict if the annual income of an individual is over 50,000 dollars or not. The dataset has missing attributes. This dataset has been studied in a machine learning context in [@kohavi1996scaling]. As done in [@kohavi1996scaling], we have ignored records with missing values. Unlike the previous datasets, feature selection did not improve accuracy with this dataset and all original attributes were retained for analysis. This dataset has 14 attributes and 45220 complete instances (rows with no missing values). . Experimental Evaluation of Leaf Size ------------------------------------ As discussed in section \[sec:dt\_for\_seg\] and section \[sec:methodology\], the leaf size (or equivalently the tree height) is an important parameter for the algorithm presented in this work. The leaf size can affect: 1. The generalization of the decision tree used to segment the data. 2. The generalization of the overall model. Therefore we need two experiments. The first experiment illustrates the effect of the leaf size on the generalization of the decision tree model. The second experiment illustrates the effect of the leaf size on the overall model (Algorithm \[algo:dt\_tree\_seg\_reg\]). For both these experiments, 70 percent of the data was used for training and 30 percent of the data was used for the test set. For both these experiments, the classification accuracy was used as the metric for error. All modeling for this study was done in `Python` with the `scikit-learn`([@scikit-learn]) library. Discussion of Experimental Results {#sec:dor} ================================== The key idea with the proposed algorithm is that we can enhance decision tree classification performance in leaf nodes where the decision tree is unable to produce homogeneous class distributions. It is possible that other classification techniques like kernel methods or K-Nearest Neighbors may be able to identify good decision boundaries in these nodes. In the experiments reported in this study we found one of the following two types of behavior: 1. Leaf Classifiers can enhance classification performance: This was the case with the forest cover type identification dataset and the airline delay dataset. We could enhance classification accuracy at the leaf nodes by using another classifier. 2. Leaf Classifiers are unable to enhance classification performance: In this case all classifiers perform poorly in certain regions of the dataset. This kind of behavior was noted with the census income dataset. ![Segment Accuracy Enhancement - Airline Delay[]{data-label="fig:EXP_SAE_AD"}](AD_DT_Vs_Aug_DT){width="\linewidth"} An illustration of the effectiveness of leaf level classifiers is provided in Figure \[fig:EXP\_SAE\_FC\] through Figure \[fig:EXP\_SAE\_CI\]. These plots illustrate the increase in accuracy of test set prediction when using a leaf level classifier. The accuracies obtained with a plain decision tree are illustrated in red while the accuracies obtained when a leaf level classifier is used is illustrated in blue. The leaf size for these experiments are the values at which both good accuracy and good generalization are observed . With the forest cover identification dataset, the leaf level classifiers are able to enhance the accuracy in almost every segment of the dataset. The leaf level classifier that was effective in almost all leaf segments of the forest cover dataset was the KNN (K Nearest Neighbors) classifier with a window size of 3 neighbors. In this experiment we used a neighborhood size of 3 for all leaves. It is possible that this neighborhood size is non-optimal for some segments. So we could possibly enhance the accuracy reported in section \[sec:accuracy\] by tuning this parameter in segments in the lower end of the accuracy range.\ The response for the airline delay dataset indicates if a flight is going to delayed over 15 minutes. The response for this dataset is skewed. The proportion of delays for the test set is shown in Figure \[fig:SEG\_DELAYED\_PROP\_AD\]. As is evident, flight delays are fairly uncommon for most segments. However there are a small proportion of segments characterized by higher delays. The segment ID’s for these higher delay segments range from 100 through 140. An analysis of Figure \[fig:EXP\_SAE\_AD\] shows that leaf level classifiers help enhance accuracies in these segments. It appears that there are more blue points than red points in Figure \[fig:EXP\_SAE\_AD\]. Many segments have very low proportion of delays. In these segments, there is really no advantage in using a leaf level classifier. The plain CART decision tree does well in regions where the response is fairly homogeneous (very low delays). The accuracies of the leaf level classifier and the plain decision tree overlap in these regions. The leaf level classifier accuracy is plotted second, so there is more blue evident. In the higher delay segments the leaf level classifiers perform better than the plain decision tree and therefore the difference is evident in such regions. For the higher delay segments there is no classifier that performs best for all the segments. For some segments a logistic regression classifier performed best, while for others an SVM classifier or a KNN classifier performed best. As with the forest cover dataset, there is scope for improvement of the accuracy reported in section \[sec:accuracy\] by fine tuning the classifier hyper-parameters in the higher delay segments.\ The census income dataset provides an example of where leaf level classifiers do not help in enhancing accuracy. An analysis of Figure \[fig:EXP\_SAE\_CI\] shows that there are many segments in the segment ID range 100 through 200 where the accuracy of prediction is low. For example, there are many segments where the accuracy of prediction is less than 60%. An analysis of the classifier accuracies in these regions revealed that all classifiers perform poorly in these problematic segments. The accuracy obtained with the plain decision tree is the same as the accuracy obtained with leaf level classifiers in these segments. This suggests that these regions are noisy or that we may require a better set of features for these regions. This is discussed in section \[sec:accuracy\]. As with the airline delay dataset, there is a lot of overlap in the accuracies produced by the plain decision tree and the leaf level classifiers. However, the performance is characterized by two regions. A region where the decision tree and the augmented decision tree perform well and a region where the the decision tree and the augmented decision tree perform poorly. In summary, we are either able to enhance performance or we are able to identify problematic regions of our dataset when we use Algorithm \[algo:dt\_tree\_seg\_reg\]. Problematic regions are those where all classifiers perform poorly. This data could be isolated for further analysis. Removing these problematic regions enhances accuracy (see section\[sec:accuracy\]). Effect of Leaf Size on Decision Tree Generalization {#sec:eff_ldct} --------------------------------------------------- These experiments illustrate the effect of the leaf size parameter on the generalization error of the decision tree. Tree growth along a particular path in the tree is stopped when the number of instances in the node falls below a threshold level. The training error and the test set error associated with the leaf size setting is noted. This procedure is repeated for various values of the threshold level of the leaf size parameter. The results are shown in Figure \[fig:EXP\_LSDT\_AD\] through Figure \[fig:EXP\_LSDT\_FC\]. ![Airline Delay Decision Tree Generalization[]{data-label="fig:EXP_LSDT_AD"}](AD_ls_vs_DT_Accuracy){width="\linewidth"} As is evident, the generalization error of the decision tree is good over the entire range of leaf sizes. The single exception is the case of using a leaf size of one for the forest cover and airline delay datasets. As expected, there is significant over-fitting for this case. Effect of Leaf Size on Model Generalization ------------------------------------------- These experiments illustrate the effect of the leaf size on the error of Algorithm \[algo:dt\_tree\_seg\_reg\]. For each leaf size, the training and test error are noted. The results of these experiments are shown in Figure \[fig:EXP\_LSMODEL\_AD\] through \[fig:EXP\_LSMODEL\_FC\]. ![Airline Delay Overall Model Generalization[]{data-label="fig:EXP_LSMODEL_AD"}](AD_ls_vs_Total_Accuracy){width="\linewidth"} The optimal leaf size is one where we achieve good accuracy and good generalization. For the airline delay dataset this optimal value is about 6000 (see Figure \[fig:EXP\_LSMODEL\_AD\]). For the forest cover dataset, the optimal value is around 1500 (see Figure \[fig:EXP\_LSMODEL\_FC\]). There is very little accuracy gain with increasing the leaf size beyond 1500 for the forest cover dataset. For both the forest cover and the airline delay dataset, the decision tree generalizes well at the optimal settings (see Figure \[fig:EXP\_LSDT\_AD\] and Figure \[fig:EXP\_LSDT\_FC\]). Accuracy {#sec:accuracy} -------- As discussed in section \[sec:methodology\], we can evaluate the effectiveness of the algorithm by comparing the accuracy obtained with Algorithm \[algo:dt\_tree\_seg\_reg\] with those obtained from ensemble methods like random forests or gradient boosted trees. We can also evaluate the improvement in accuracy over using a plain decision tree. The accuracies obtained with a plain decision tree are shown in Table \[tab:baseline\_accuracy\]. The leaf size used with the plain decision tree is one where the best accuracy was observed. The sizes of the datasets are provided in section \[sec:datasets\]. For all experiments, 70% of the dataset was used for training and 30% was used as the test set. Dataset Leaf.Size Accuracy --- --------------- ----------- ---------- 1 Forest Cover 1 0.916 2 Airline 1000 0.937 3 Census Income 100 0.853 : Baseline Accuracies - CART Decision Tree Algorithm[]{data-label="tab:baseline_accuracy"} Algorithm \[algo:dt\_tree\_seg\_reg\] can enhance the baseline accuracies reported in Table \[tab:baseline\_accuracy\] for the forest cover and the airline delay datasets. The improvement in accuracies are reported in Table \[tab:algorithm\_accuracy\]. Ensemble methods also achieve high accuracies for these datasets, however the models they produce are not interpretable. Algorithm \[algo:dt\_tree\_seg\_reg\] produces models that are very easily interpretable. Dataset Method Accuracy --- --------------- --------------- ---------- 1 Airline Delay XGBoost 0.944 2 Airline Delay Random Forest 0.946 3 Airline Delay 0.945 4 Forest Cover XGBoost 0.936 5 Forest Cover Random Forest 0.945 6 Forest Cover 0.957 : Accuracies obtained with Algorithm \[algo:dt\_tree\_seg\_reg\][]{data-label="tab:algorithm_accuracy"} With the census income dataset we observed that we have a small proportion (13.58% of the data set instances) of decision tree nodes where all classifiers perform poorly. A preliminary analysis of these records indicates that there are possible data errors with this segment. For example there are records of people working with the state government working over 70 hours a week but reporting less than 50 thousand dollars in income. These records seem dubious since even at minimum wage, such employees should make over fifty thousand dollars. In any case these set of records may require further analysis to determine if they are either noisy or require additional features to obtain better classification performance.\ When these records are removed from the dataset, we obtain an accuracy of 90.04% (see Figure \[fig:EXP\_LSDT\_CI\]). Algorithm \[algo:dt\_tree\_seg\_reg\] provides us a method to identify such problematic regions of our dataset. [@xiong2006enhancing] provides techniques to remove noise from datasets. Finding the noisy regions in large datasets and separating them from regions of good data quality is a time consuming task. Algorithm \[algo:dt\_tree\_seg\_reg\] can help identify these regions. Noise removal techniques, such as those discussed in [@xiong2006enhancing] can then be applied to see if these can help improve classification accuracy. Therefore, there is scope for improving the accuracy with the census income dataset as well. [@kohavi1996scaling] report an accuracy of 84.47% with the NBTree algorithm. Training and test sizes used in [@kohavi1996scaling] are similar to those used in this work. A review of Table \[tab:baseline\_accuracy\] shows that the baseline accuracy with a CART decision tree is 85.3%. [@kohavi1996scaling] reports an accuracy of 81.91% for a C 4.5 decision tree. This suggests that the choice of the decision tree (C 4.5 versus CART) can affect the accuracy. Interpreting the Model {#sec:interpretation} ---------------------- Models produced by Algorithm \[algo:dt\_tree\_seg\_reg\] have a simple interpretation. A data instance can be associated with two models - the segment model and the leaf classification model. The segment model provides an aggregate profile for the data instance while the leaf classification model can yield insights into the factors that affect the label for an instance within the segment. Therefore we can interpret the model at coarse and fine granularities. A sample of the segment profiles for the forest cover dataset is shown in Table \[tab:sample\_forest\_cover\_seg\]. The columns provide the relative proportion of the different types of tree cover in that segment. It is clear that each segment is characterized by a particular set of tree cover. Seg. ID CT\_1 CT\_2 CT\_3 CT\_4 CT\_5 CT\_6 CT\_7 --- --------- ------- ------- ------- ------- ------- ------- ------- 1 3 0.003 0.085 0.262 0.361 0.003 0.287 0.000 2 10 0.000 0.015 0.785 0.002 0.000 0.197 0.000 3 11 0.000 0.032 0.653 0.033 0.000 0.281 0.000 4 12 0.000 0.026 0.885 0.011 0.000 0.078 0.000 5 15 0.000 0.000 0.437 0.017 0.000 0.546 0.000 : Sample of Segment Profiles - Forest Cover Dataset[]{data-label="tab:sample_forest_cover_seg"} Similarly, Figure \[fig:SEG\_DELAYED\_PROP\_AD\] shows the proportion of flights delayed by segment ID. It is evident that some segments are associated with a higher proportion of delays while many segments have very low proportion of delays. Most decision tree implementations provide a feature to generate decision rules or tree visualizations. The decision tree visualization for the airline delay dataset is shown in Figure \[fig:DT\_VISUAL\_AD\]. The leaves are color coded to indicate the majority class for that node. The blue nodes indicate the nodes associated with delays. This provides an easy way to generate the coarse grained profile for a segment. We can then interpret the leaf level model for an instance, for example with the airline delay dataset, a logistic regression model, to determine the factors that affect flight delays for a particular segment. In summary the models provided by the algorithm reported in this work can be easily interpreted. This is in contrast to ensemble models like Random Forests or Gradient Boosted Trees. While these can provide accurate predictions, the models they produce are not interpretable. Analysis of Segment Classifiers =============================== Generalization of Segment Classifiers ------------------------------------- The effect of the leaf size on the generalization of the decision tree and Algorithm 1 was evaluated experimentally. A related concern is the generalization of a particular segment classifier. We can analyze the generalization of the classifier for a particular segment using concentration inequalities. The segment classifier is a function $f:x\rightarrow y$. Here $x\in \mathcal{X}=\mathbb{R}^d$ represents the predictor variables and $y\in \mathcal{Y}=\{0,1,...,k-1\}$ represents the label. The ‘0-1’ loss function $l$ is defined as $$l(f(x),y)=\bigg\{\begin{array}{cc} 1 & \text{, if } f(x)\neq y,\\ 0 & \text{, if } f(x) = y. \end{array}$$ Ideally, we want to learn the function by minimizing the risk of misclassification, where the misclassification risk is the expected value of the loss function over the joint density of the data. The joint density of the data, $\mathbb{P}(x,y)$ is defined over $\mathcal{X}\times\mathcal{Y}$. The statistical misclassification risk for the classifier $f$ is defined as $$R(f)=\mathbb{E}_{\mathbb{P}}[l(f(x),y)]=\int l(f(x),y)d\mathbb{P}(x,y).$$ The joint density of the data for a segment , $\mathbb{P}(x,y)$, is however not known in practice. What we have access to is the training data. Therefore in practice the loss of the classifier is evaluated over the training data. This yields the empirical misclassification risk. The empirical misclassification risk for the classifier $f$ is defined as $$\hat{R}_n(f)=\mathbb{E}_{\hat{\mathbb{P}}}[l(f(x),y)]=\frac{1}{n}\sum_{i=1}^nl(f(x_i),y_i)$$ \[lemma:sample\_size\] For a given $n \geq \frac{1}{2\epsilon^2}\log(\frac{2}{\delta})$ and $0<\delta<1$, $$\mathbb{P}(|\hat{R}_n(f)-R(f)|<\epsilon)>1-\delta.$$ Hoeffding’s inequality [@hoeffding_63] states that > If $Z_1, Z_2,\hdots,Z_n$ are independent with $\mathbb{P}(a\leq Z_i \leq b)=1$ and have a common mean $\mu$ then $$\mathbb{P}(|\bar{Z}-\mu|>\epsilon)<\delta$$ where $\bar{Z}=\frac{1}{n}\sum_{i=1}^nZ_i$ and $\delta=2\exp\{-\frac{2n\epsilon^2}{(b-a)^2}\}$. In our case, we define $Z_i=l(f(x_i),y_i)$, which is bounded with probability one ($a = 0\ and\ b = 1$) and have common mean $R(f)$. When we set $n=\frac{1}{2\epsilon^2}\log\big(\frac{2}{\delta}\big)$, we have $$\begin{aligned} \mathbb{P}[|\hat{R}_n(f)-R(f)|\geq\epsilon]&\leq& 2\exp\{-2n\epsilon^2\},\\ &=&2 \exp\{-2\frac{1}{2\epsilon^2}\log(\frac{2}{\delta})\epsilon^2\},\\ &=&2e^{-\log\big(\frac{2}{\delta}\big)},\\ &=&\delta.\end{aligned}$$ Hence the result. Lemma \[lemma:sample\_size\] provides a method to determine the sample size needed to keep the difference between the misclassification risk and the empirical misclassification risk to a small value, $\epsilon$, with high probability $(1-\delta)$. Bayes Error Rate ---------------- A review of the results of applying Algorithm \[algo:dt\_tree\_seg\_reg\] to various datasets used in the study reveals that the divide and conquer approach has some very useful implications in analyzing large datasets. It is evident that most problems are characterized by many regions where we achieve good success in predicting the class label and a few regions where predicting the class label is challenging. This characteristic is very useful because it points out the difficult regions of the dataset in terms of the classification task. Some questions that are of interest in the problematic regions of the dataset are the following: What is the best possible accuracy in these problematic regions? Are the features useful for the classification task in the problematic regions? The census income dataset is an example of where such questions are very relevant. The Bayes Error is a very useful theoretical idea to answer these questions. (see [@devroye96]\[Chapter 2\]). The optimal classifier $f^*(x)$ associated with a classification task is the Bayes classifier. For a binary classification problem, as is the case with the census income dataset, the Bayes classifier assigns class labels using the following rule: $$f^*(x) = \begin{cases*} 1\quad if\ \mathbb{P}\left[y = 1| x\right] > \frac{1}{2}\\ 0,\ otherwise. \end{cases*}$$ If we can estimate $\mathbb{P}\left[y = 1| x\right]$, we can estimate the performance of the Bayes classifier (see [@tumer2003bayes]). Density estimation is a computationally expensive task (see [@friedman2001elements]\[Chapter 6, section 6.9\]). Estimating the density for the entire dataset is computationally intractable. However, we are interested in evaluating this only for segments where we have poor classification accuracy. Since segments sizes are small and we want to perform this for a few segments only, this is computationally tractable. If it turns out that the Bayes classifier performs also poorly at these segments, then we know that the features are not useful for that segment and we need better features to improve classification accuracy. Applying these ideas to evaluate poorly performing segments is an area of future work. The intent here is to point out the localizing the problematic areas enables us to apply theoretical tools like the Bayes error rate to a small subset of our data. This makes such analysis more tractable than applying this to the entire dataset. Conclusion {#sec:conclusion} ========== We presented an an algorithm to perform classification tasks on large datasets. The algorithm uses a divide and conquer strategy to scale classification tasks. A decision tree is used to segment the dataset. By construction, many of the decision tree leaves are relatively homogeneous in terms of class distribution. Suitable classifiers can be used on the non-homogeneous leaves to determine class labels for the leaf instances. We demonstrated the effectiveness of this algorithm on large datasets. The algorithm achieves one of the following outcomes: 1. We achieve good classification accuracy. The levels of accuracy obtained is higher than what is achievable with a simple decision tree and can match the accuracy obtained using ensemble techniques like random forests or gradient boosted trees. Further the model produced by the proposed algorithm is easy to interpret and can yield insights related to the learning task. In contrast, though ensemble methods can be accurate, the models they produce are not very interpretable. 2. We are able to identify problematic or noisy regions of the dataset. This situation is characterized by decision tree nodes where all classifiers perform poorly. Typically this is a small portion of the dataset. This algorithm can be used to identify such regions of the dataset. These segments can then be isolated for further analysis. After removing these problematic segments, we are able to achieve high classification accuracy. In summary, the proposed algorithm can produce models that are accurate and interpretable. This is highly desirable.

ArXiv

--- abstract: 'The impact of the incoherent electron-positron pairs from beamstrahlung on the occupancy of the vertex detector (VXD) for the International Large Detector concept (ILD) has been studied, based on the standard ILD simulation tools. The occupancy was evaluated for two substantially different sensor technology in order to estimate the importance of the latter. The influence of an anti-DID field removing backscattered electrons has also been studied.' author: - | Rita De Masi and Marc Winter\ Institut Pluridisciplinaire Hubert Curien (IPHC)\ 23 rue du Loess - BP28- F67037 Strasbourg (France) title: Improved Estimate of the Occupancy by Beamstrahlung Electrons in the ILD Vertex Detector --- Introduction ============ The incoherent production of electron-positron pairs resulting from the beam-beam interaction is the main source of background for the ILD vertex detector, and it is most constraining for its innermost layer. These electrons and positrons are produced with a longitudinal momentum up to few hundreds GeV and a transverse momentum of few tens of MeV on average. Due to their low $p_T$, they spiralize in the solenoidal magnetic field, whose field lines are parallel to the beam line, thus several of them can traverse repeatedly the same VXD layer. Those primary electrons and positrons may also hit elements of the detector further down the beam line, originating low energy particles traveling backward (secondaries), which may reach the VXD. The rate of secondaries reaching the VXD depends strongly on the presence of an additional dipole field located further down the beam line, as shown in Section \[aD\]; thus primaries and secondaries will be analized separately in the following. The time when the hit has taken place, will be used to distinguish them. Namely, will be considered as generated by primaries all hits with a hit time shorter than 20 ns and by secondaries those with a hit time larger than 20 ns. A detailed description of this analysis can be found in [@bkgnote]. Analysis ======== 100 bunch crossings (BX) generated with the GuineaPig [@GP] generator have been studied. The standard simulation and reconstruction tools for the ILD detector concept have been used (i.e. Mokka [@Mokka] and Marlin [@Marlin] respectively). The model of detector used in this study takes properly into account the angle of 14 mrad between the beam directions. A preliminary description of the calorimeters along the beam line is also included [@Mokka]. Hit density {#sec:ht} ----------- The number of hits in the first layer of the VXD as function of the coordinate along the beam line $z$ and the polar angle $\phi$ is shown in Fig. \[Fig::occl1\]. Besides a change of the absolute hit rate, analogous distributions can be observed for the remaining layers. The $\phi$ distribution shows a significant increase of the number of hits in the region $|\phi|<50^\circ$, due to the particles with large hit time which are not produced symmetrically around the $z$ axis. The “spikes” in the $\phi$ distributions are due to particle crossing the overlapping regions of neighbouring ladders. Occupancy --------- [r]{}[0.5]{} {width="0.5\columnwidth"} In order to calculate the occupancy, the effective path length of the particles inside the sensitive volume of the detector ougth to be accounted for. It may reach up to several millimeters, especially for backscattered particles which were produced at small polar angle in order to reach the VXD.\ The occupancy depends on the characteristics of the VXD, namely pixel size, integration time, number of hit pixels per impact, effective thickness of the sensitive volume. In absence of choice of the sensor technology, a set of those parameters has been agreed upon in the ILD vertex community as reference and they have been used to estimate the occupancy. As a comparison, the occupancy has been also estimated in the framework of a specific technology (CMOS [@cmos]). The parameters describing both options are shown in Tab. \[Tab::sC\]). ------- ---------------- --------------------------- ---------------- --------------------------- layer pitch ($\mu$m) integration time ($\mu$s) pitch ($\mu$m) integration time ($\mu$s) 1 25 50 20 25 2 25 200 25 50 3 25 200 33 100 4 25 200 33 100 5 25 200 33 100 ------- ---------------- --------------------------- ---------------- --------------------------- : Parameters of the VXD layers for the standard and CMOS configuration. 50 $\mu$m and 15 $\mu$m sensitive thickness, 3 and 5 hit pixels in average for straight impact respectively.[]{data-label="Tab::sC"} The results for the occupancy in each layer are shown for the two configurations in Tab. \[Tab::occupancy\]. ------- -------- ---------------- ---------------- -------- ---------------- ---------------- layer total large hit time short hit time total large hit time short hit time 1 0.0790 0.0347 0.0443 0.0183 0.0080 0.0103 2 0.0381 0.0164 0.0217 0.0062 0.0026 0.0035 3 0.0105 0.0049 0.0056 0.0054 0.0025 0.0029 4 0.0041 0.0020 0.0021 0.0021 0.0010 0.0011 5 0.0016 0.0006 0.0010 0.0008 0.0003 0.0005 ------- -------- ---------------- ---------------- -------- ---------------- ---------------- : Occupancy for each layer in absence of an anti-DID for the standard and CMOS configurations. The large and small hit time components are shown, as well as their sum.[]{data-label="Tab::occupancy"} The values are averaged over $\phi$. In fact, due to the $\phi$ dependence shown in Figure \[Fig::occl1\], the local occupancy in a $\phi$ sector can be twice as high as the mean. In average, one can conclude that the large hit time contribution to the occupancy is more than $40\%$ of the total rate. Anti-DID magnetic field {#aD} ----------------------- A Detector Integrated Dipole (anti-DID), aligning the outgoing beam with the experimental magnetic field, can be used to reduce the beam size growth due to synchrotron radiation. The anti-DID impacts also the hit rate on the VXD due to beamstrahlung electrons, by reducing the number of backscattered electrons travelling backwards from further along the beam line. [r]{}[0.5]{} {width="0.5\columnwidth"} The anti-DID reduces by roughly 30% the number of hits on the VXD, in particular the large hit time component, as can be seen in Figure \[Fig::occl1aD\]. This leads to a more homogeneous local distribution in $\phi$. The occupancy of the ILD vertex detector, which is a driving parameter of its requirements, has been evaluated with the latest version of the experimental apparatus, assuming a five-layer VXD geometry with 15 mm inner radius and a 3.5 T magnetic field. The evaluation was performed for two different sets of pixel characteristics, representative of the most mature sensor technologies under consideration. Both sets assume a continous read-out during the train. They differ by their read-out time, pixel pitch, cluster multiplicity and sensitive volume thickness. Conclusion ========== Occupancies of $\sim2\%$ and $\sim7\%$ were found in the innermost layer for the two sets. The average occupancy would be about 30% lower in presence of anti-DID, with a 50% decrease in one azimuthal sector. Accounting for the uncertainties on these predictions translates into upper limits on the occupancy in the innermost layer in the range 5-15%, depending on the sensor characteristics. These high rates plead for additional R&D on the sensors equipping this layer, in particular for shortening the read-out time significantly below 50 $\mu s$. [99]{} Presentation:\ `http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=221&sessionId=21&confId=2628` R.De Masi [*et al.*]{}, ILC-note in preparation. D. Schulte, PhD Thesis, University of Hamburg, (1996). P.M. de Freitas, MOKKA, `http://mokka.in2p3.fr`. ILC software `http://ilcsoft.desy.de`. `http://iphc.in2p3.fr/-CMOS-ILC-.html`.

ArXiv

--- abstract: 'In this article, we study the high order term of the fidelity of the Heisenberg chain with next-nearest-neighbor interaction and analyze its connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless type happened in the system. We calculate the fidelity susceptibility of the system and find that although the phase transition point can’t be well characterized by the fidelity susceptibility, it can be effectively picked out by the higher order of the ground-state fidelity for finite-size systems.' author: - Li Wang - 'Shi-Jian Gu' - Shu Chen title: 'High-order fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbor interaction' --- Introduction ============ Quantum phase transitions (QPTs) of a quantum many-body system have been attracting the persistent interest of physical researchers in recent years. Due to the diversity of quantum phases and QPTs, finding universal ways or methods to characterize QPTs is very meaningful and urgent. From the viewpoint of Landau-Ginzburg theory which has been widely accepted and known in condensed matter physics [@sachdev], QPT is connected with the corresponding order parameter and symmetry breaking. However, there are also some QPTs which cannot be well understood under the Landau-Ginzburg paradigm, such as the topological phase transitions [@xgwen] and Beresinskii-Kosterlitz-Thouless (BKT) phase transitions [beresinskii,kosterlitz]{}. Recently, an increasing research effort has been focused on the role of ground-state fidelity in characterizing QPTs[Gu\_review,htquan,zanardi06, hqzhou,YouWL07,zanardi07PRl,schen07pre,schen08pra,buonsante,mfyang,ZhouPRL]{}. As a basic concept in quantum information science, the fidelity measures the similarity between two states and is simply defined as modulus of their overlap [@zanardi06]. The fidelity approach provides us a novel way to understand QPTs from the viewpoint of quantum information theory. So far QPTs in various quantum many-body systems [@YouWL07; @hqzhou; @mfyang; @qhchen; @mfyang08; @WangXG08; @schen07pre; @schen08pra; @Paunkovic; @Venuti; @buonsante; @AHamma07; @abasto; @YangS; @zanardi07PRl; @WangXG; @Zhou09; @Zhou0803; @ZhouPRL] have been shown to be well characterized by the ground-state fidelity or fidelity susceptibility which is the leading term of the fidelity [YouWL07,zanardi07PRl]{}. Generally, one may expect that the structure of the ground states at the different phases is basically different and should reveal itself by some sort of singular behavior in the ground state fidelity or the fidelity susceptibility at the transition point [@zanardi06; @hqzhou]. Despite its great successes of application in various systems, this intuitive idea turns out to be not complete [@schen07pre; @mfyang; @schen08pra; @YouWL07; @mfyang08]. Although the fidelity and the fidelity susceptibility can be used to describe first- and second-order QPTs[@schen08pra], as well as the topological QPTs [@AHamma07; @abasto; @YangS; @Zhou0803] successfully, nevertheless there are also some ambiguous cases for that both the two methods mentioned above do not work very effectively [@schen07pre; @schen08pra; @YouWL07; @mfyang08]. Very recently, the controversial issue of BKT phase transition and ground state fidelity has been studied in Ref. [@Zhou09] from a perspective of matrix product states which essentially depend on a classical simulations of quantum lattice systems [@ZhouPRL]. In case that the leading term of the fidelity (fidelity susceptibility) works not very effectively, the higher order term in the fidelity may be worth studying. Up to now, there is still lack of literature concerning this part of the fidelity. Here, in this paper, we make an attempt on investigating the effect of higher order term of the fidelity on the characterization of the BKT-type phase transition happened in the Heisenberg chain with next-nearest-neighbor (NNN) interaction [@Haldane]. We will show that although the fidelity and fidelity susceptibility cannot effectively characterize the BKT-type phase transition point for the Heisenberg chain with NNN interaction, the higher order term of the fidelity gives a good attempt on detecting such a transition. Our paper is organized as follows. In Sec. II, we display the formulism of the higher order term of the fidelity. The subsequent section is devoted to the calculation of the higher order term of the fidelity for the model of Heisenberg chain with NNN interaction and show its connection to the quantum phase transition of the system. A brief summary is given in Sec. [sec:sum]{}. Higher order of the fidelity {#sec:highorder} ============================ As usual, the ground state fidelity is defined as the modulus of the overlap between $|\Psi_0(\lambda) \rangle$ and $| \Psi_0(\lambda+\delta\lambda) \rangle$, i.e. $$F(\lambda, \delta\lambda) =\left| f(\lambda, \lambda+\delta\lambda) \right | = \left | \langle \Psi_0(\lambda)| \Psi_0(\lambda+\delta\lambda) \rangle \right | , \tag{1} \label{eqF}$$ where $\Psi_0(\lambda)$ is the ground-state wavefunction of Hamiltonian $% H=H_0 + \lambda H_I$, $\lambda$ is the driving parameter and $\delta \lambda$ is a small deviation in the parameter space of $\lambda$. The fidelity susceptibility denotes only the leading term of the fidelity. Straightforwardly, one can get the higher order term of the fidelity following similar expansion in deriving the fidelity susceptibility [YouWL07]{}. By using the Taylor expansion, the overlap between two wavefunction $|\Psi _{0}(\lambda )\rangle $ and $|\Psi _{0}(\lambda +\delta \lambda )\rangle $ can be expanded to an arbitrary order of $\delta\lambda$, i.e. $$f(\lambda ,\lambda +\delta \lambda )=1+\sum_{n=1}^{\infty }\frac{(\delta \lambda )^{n}}{n!}\left\langle \Psi _{0}(\lambda )\left\vert \frac{\partial ^{n}}{\partial \lambda ^{n}}\Psi _{0}(\lambda )\right. \right\rangle . \tag{2} \label{eqf}$$ Therefore, the fidelity becomes $$\begin{aligned} F^{2}=&1+\sum_{n=1}^{\infty }\frac{(\delta \lambda )^{n}}{n!}\left\langle \Psi _{0}\left\vert \frac{\partial ^{n}}{\partial \lambda ^{n}}\Psi _{0}\right. \right\rangle+ \notag \\ &\sum_{n=1}^{\infty }\frac{(\delta \lambda )^{n}}{n!}\left\langle \left. \frac{\partial ^{n}}{\partial \lambda ^{n}}\Psi _{0}\right\vert \Psi _{0}\right\rangle + \notag \\ &\sum_{m,n=1}^{\infty }\frac{(\delta \lambda )^{m+n}}{m!n!}\left\langle \Psi _{0}\left\vert \frac{\partial ^{n}}{\partial \lambda ^{n}}\Psi _{0}\right. \right\rangle \left\langle \left. \frac{\partial ^{m}}{\partial \lambda ^{m}}% \Psi _{0}\right\vert \Psi _{0}\right\rangle. \tag{3} \label{eqF2}\end{aligned}$$ We note that $\frac {\partial^{n}} {\partial \lambda^{n}} \langle \Psi_0(\lambda)| \Psi_0(\lambda) \rangle =0$ and use the relation for a given $n$ $$\sum_{m=0}^{n}\frac{n!}{m!(n-m)!}\left\langle \left. \frac{\partial ^{m}}{% \partial \lambda ^{m}}\Psi _{0}\right\vert \frac{\partial ^{n-m}}{\partial \lambda ^{n-m}}\Psi _{0}\right\rangle =0 , \tag{4} \label{relation}$$ then we can simplify the expression of (\[eqF2\]) into $$F^{2}=1-\sum_{l=1}^{\infty }(\delta \lambda )^{l}\chi _{F}^{(l)} \tag{5} \label{F2simple}$$where $$\chi _{F}^{(l)}=\sum_{l=m+n}\frac{1}{m!n!}\left\langle \left. \frac{\partial ^{m}}{\partial \lambda ^{m}}\Psi _{0}\right\vert \hat {P} \left\vert \frac{% \partial ^{n}}{\partial \lambda ^{n}}\Psi _{0}\right. \right\rangle , \tag{6} \label{eq:higherorderdiff}$$ with the projection operator $\hat {P}$ defined as $\hat {P}=1- |\Psi_0 \rangle \langle \Psi_0 |$. It is easy to check that $\chi _{F}^{(1)}$ is zero and $\chi _{F}^{(2)}$ the fidelity susceptibility [@YouWL07]. Next we shall consider the third order fidelity $\chi _{F}^{(3)}$ and apply it to judge the phase transition in the spin chain model with NNN exchanges. Alternatively, one can directly derive the expression of $\chi _{F}^{(3)}$ from the perturbation expansion of the GS wavefunction. According the perturbation theory, the GS wavefunction, up to the second order, is $$\begin{aligned} |\Psi _{0}(\lambda +\delta \lambda )\rangle =& |\Psi _{0}\rangle +\delta \lambda \sum_{n\neq 0}\frac{H_{I}^{n0}|\Psi _{n}\rangle }{E_{0}-E_{n}} \\ & +\left( \delta \lambda \right) ^{2}\sum_{m,n\neq 0}\frac{% H_{I}^{nm}H_{I}^{m0}|\Psi _{n}\rangle }{(E_{0}-E_{m})(E_{0}-E_{n})} \\ & -\left( \delta \lambda \right) ^{2}\sum_{n\neq 0}\frac{% H_{I}^{00}H_{I}^{n0}|\Psi _{n}\rangle }{(E_{0}-E_{n})^{2}} \\ & -\frac{\left( \delta \lambda \right) ^{2}}{2}\sum_{n\neq 0}\frac{% H_{I}^{0n}H_{I}^{n0}|\Psi _{0}\rangle }{(E_{0}-E_{n})^{2}}.\end{aligned}$$The 3rd order term $\chi _{F}^{(3)}$, which is proportional to the 3rd order derivative of GS fidelity, can be then directly extracted from eq. (\[F2simple\]): $$\chi _{F}^{(3)}=\sum_{m,n\neq 0}\frac{2H_{I}^{0m}H_{I}^{mn}H_{I}^{n0}}{% (E_{0}-E_{m})(E_{0}-E_{n})^{2}}-\sum_{n\neq 0}\frac{2H_{I}^{00}\left\vert H_{I}^{n0}\right\vert ^{2}}{(E_{0}-E_{n})^{3}}. \tag{8} \label{eq:higheroderperturb}$$ Eqs. (\[eq:higherorderdiff\]) and (\[eq:higheroderperturb\]) present the main formulism of the higher order expansion of the fidelity. So far the explicit physical meaning of the high order term in the fidelity is still not clear. The expression of 3rd fidelity bears the similarity to its correspondence of the 3rd derivative of GS energy which has the following form $$\frac {\partial^3 E} {\partial \lambda^3} = \sum_{m,n\neq0}\frac{6 H_I^{0n}H_I^{nm}H_I^{m0}}{(E_0-E_m)(E_0-E_n)} -\sum_{n\neq0}\frac{6 H_I^{00}\left|H_I^{n0}\right|^2}{(E_0-E_n)^2}. \tag{9}$$ Obviously, the 3rd fidelity is more divergent than the 3rd derivative of GS energy. Similar connection between the fidelity susceptibility and 2nd derivative of GS energy has been unveiled [@schen08pra]. Generally the $n $-th order fidelity is much more divergent than its counterpart of $n$-th order derivative of GS energy, therefore an $n$-th order QPT can be certainly detected by the $n$-th order fidelity. However, this conclusion does not exclude the possibility that $n$-th order fidelity can detect a even higher order or infinite order QPT. A concrete example has been given in Ref. [mfyang08]{}, where a QPT of higher than second order was singled out unambiguously by using the fidelity susceptibility despite the corresponding second derivative of the ground-state energy density showing no signal of divergence. So far no example of BKT-type QPT unambiguously detected by fidelity susceptibility has been given. Next we shall attempt to apply the third-order fidelity to study the BKT-type transition in a the spin chain model with NNN exchanges. The model and the calculation of 3rd order fidelity {#sec:model} =================================================== Now we turn to the one-dimensional Heisenberg chain with the NNN coupling described by the Hamiltonian $$H(\lambda )=\sum_{j=1}^{L}\left( \hat{s}_{j}\hat{s}_{j+1}+\lambda \hat{s}_{j}% \hat{s}_{j+2}\right) , \tag{10} \label{Ham}$$where $\hat{s}_{j}$ denotes the spin-1/2 operator at the $j\,$th site, $L$ denotes the total number of sites. The driving parameter $\lambda $ represents the ratio between the NNN coupling and the nearest-neighbor (NN) coupling. The GS properties of the model (\[Ham\]) has been widely studied by both analytical method [@Haldane; @Giamarchi] and numerical method [Okamoto,Castilla,RChitra,SRWhite96]{}. The QPT driven by $\lambda $ is well understood. The driving term due to $\lambda $ is irrelevant when $\lambda <\lambda _{c}(\simeq 0.2411)$, and the system flows to a spin fluid or Luttinger liquid with massless spinon excitations. As $\lambda >\lambda _{c}$, the frustration term is relevant and the ground state flows to the dimerized phase with a spin gap open [@Haldane; @Giamarchi]. The transition from spin fluid to dimerized phase is known to be of BKT type [@Haldane; @Giamarchi], for which the transition point was hard to be determined numerically due to the problem of logarithmic correction [@Affleck]. The critical value of $\lambda _{c}=0.2411\pm 0.0001$ has been accurately determined by various numerical methods [@Okamoto; @Castilla; @RChitra; @SRWhite96]. ![The GS fidelity susceptibility of the heisenberg chain with next-nearest-neighbor interaction for finite system size from 14 sites to 26 sites. Obviously, there is no expected peaks can be observed.[]{data-label="Figure1"}](Fig1.eps){width="9cm"} The GS fidelity for the model (\[Ham\]) has been studied in Ref. [schen07pre]{} and also in Ref. [@WangXG] in terms of operator fidelity. No singularities in the GS fidelity or operator fidelity around $\lambda_c$ have been detected for the system with different sizes, which implies that the GS fidelity may be not an effective characterization of the BKT-type QPT in this model. The BKT-type QPT is a infinite order phase transition where the $n$-th order derivative of GS energy is continuous. ![The third order term of the GS fidelity of the spin chain with next-nearest-neighbor interaction for the finite system size from 14 sites to 26 sites. Explicit peaks can be observed in this figure. As the system size increases, the position of the peak gets closer to the BKT-transition point.[]{data-label="Figure2"}](Fig2.eps){width="9cm"} ![Finite-size scaling of the extrema of the third term of the GS fidelity. A linear fit is made. According to this fit, when it comes to the point $N\rightarrow \infty$, $\protect\lambda_c=0.238 \pm0.006$. []{data-label="Figure3"}](Fig3.eps){width="9cm"} In light of the higher-order fidelity being more powerful than its energy judgement, we study the possibility for detecting the infinite-order BKT-type QPT via the 3rd order fidelity and focus on the QPT in the spin chain with NNN interactions as a concrete example. We first calculate the GS wave functions by using the numerical exact diagonalization method for finite size system, and thus the fidelity susceptibility and the 3rd order fidelity can be extracted from the overlap of neighboring GS wave functions. In Fig.1, we display the fidelity susceptibility for systems with different sizes. We observe that no an obvious peak for the fidelity susceptibility is detected in a wide range of the parameter $0<\lambda<0.5$. This result suggests that the transition point for the BKT-type QPTs cannot be very effectively characterized by the fidelity susceptibility either for a finite-size system. The BKT-type phase transition generally is an infinite order phase transition for which the infinite order derivatives of the ground-state energy is continuous. A good example with exact proof is the BKT-type transition happened in the antiferromagnetic XXZ spin chain model [YangCN]{}. In the BKT-type transition point, it has been proven analytically that all the $n$-th order derivatives of ground state energy is continuous [@YangCN]. Since the $n$-th order fidelity is much divergent than its correspondence of derivative of the ground-state energy, one might expect that there exists the possibility that the $n$-th order fidelity is divergent even its $n$-th order energy derivative is continuous. To see whether a higher order fidelity works better than fidelity susceptibility in detecting the BKT-type QPT happened in this model, we calculated the 3rd order fidelity versus the driving parameter as shown in Fig. 2. It is clear that a peak is developed in the 3rd order fidelity and the location of peaks tends to get close to the side of transition point $% \lambda _{c}$ with the increase of lattice size. To extrapolate the $\lambda _{c}$ in the infinite size limit, we analyze the finite size scaling of position of peak in the Fig. 3. When the system size comes to infinity, the extrapolated value of the phase transition point is $\lambda _{c}=0.238\pm 0.006$, which, within the scope of fitting error, agrees well with $\lambda _{c}=0.2411\pm 0.0001$ obtained by highly accurate numerical methods [Okamoto,Castilla,RChitra,SRWhite96]{}. Summary {#sec:sum} ======= We have shown the formulism for the high order of the fidelity in detail and applied it to a concrete model, *i.e.*, the one dimensional Heisenberg chain with NNN interaction. We first calculate the ground-state wavefunction of the system by exact diagnolization method, and then extract fidelity susceptibility and the third order of the GS fidelity. We find that despite the GS fidelity and the fidelity susceptibility being not a very effective detector, the BKT-type phase transition happened in this spin chain model might be effectively detected by the 3rd order term of the GS fidelity for finite-size system. Although the physical meaning of the higher order term of the GS fidelity hasn’t been deeply understood, we wish that our observation would stimulate further studies on this issue. This work is supported by NSF of China under Grant No. 10821403, programs of Chinese Academy of Sciences, National Program for Basic Research of MOST, China and the Earmarked Grant Research from the Research Grants Council of HKSAR, China (Project No. CUHK 400807). [99]{} S. Sachdev, *Quantum Phase Transitions*(Cambridge University Press, Cambridge, England, 1999) X. G. Wen, *Quantum Field Theory of Many-Body Systems* (Oxford University, New York, 2004) V. L. Beresiskii, Sov. Phys. JETP 32, 493 (1971). J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181(1973); J. M. Kosterlitz, *ibid* 7, 1046 (1974). S. J. Gu, e-print arXiv: 0811.3127. H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96, 140604 (2006). P. Zanardi and N. Paunkovi$\acute{c}$, Phys. Rev. E 74, 031123 (2006). H. Q. Zhou and J. P. Barjaktarevic, J. Phys. A: Math. Theor. 41 412001 (2008); H. Q. Zhou, J. H. Zhao, H. L. Wang, B. Li, arXiv: 0711.4651. P. Buonsante and A. Vezzani, Phys. Rev. Lett. 98, 110601 (2007). S. Chen, L. Wang, S. J. Gu, and Y. Wang, Phys. Rev. E 76, 061108 (2007). S. Chen, L. Wang, Y. Hao, and Y. Wang, Phys. Rev. A **77**, 032111 (2008). M. F. Yang, Phys. Rev. B 76, 180403(R)(2007); Y. -C. Tzeng and M. F. Yang, Phys. Rev. A 77, 012311 (2008); J. O. Fjaerestad, J. Stat. Mech. P07011 (2008). W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. E 76, 022101 (2007); S. J. Gu, H. M. Kwok, W. Q. Ning, and H. Q. Lin, Phys. Rev. B **77**, 245109 (2008). P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. **99**, 100603 (2007). H. Q. Zhou, R. Orus, and G. Vidal, Phys. Rev. Lett. **100**, 080601 (2008). Y. C. Tzeng, H. H. Hung, Y. C. Chen, and M. F. Yang, Phys. Rev. A **77**, 062321 (2008). H. L. Wang, J. H. Zhao, B. Li, and H. Q. Zhou, arXiv:0902.1670. L. C. Venuti, M. Cozzini, P. Buonsante, F. Massel, N. Bray-Ali, and P. Zanardi, Phys. Rev. B **78**, 115410 (2008). A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, Phys. Rev. B **77**, 155111 (2008). D. F. Abasto, A. Hamma, and P. Zanardi, Phys. Rev. A **78**, 010301 (2008). S. Yang, S. J. Gu, C. P. Sun, and H. Q. Lin, Phys. Rev. A **78**, 012304 (2008). J. H. Zhao, H. Q. Zhou, arXiv:0803.0814. X. M. Lu, Z. Sun, X. G. Wang, and P. Zanardi, Phys. Rev. A **78**, 032309 (2008); J. Ma, L. Xu, H. N. Xiong, and X. G. Wang, Phys. Rev. E **78**, 051126 (2008). N. Paunkovic *et al.*, Phys. Rev. A 77, 052302 (2008). K. W. Sun, Y. Y. Zhang, Q. H. Chen, Phys. Rev. B **79**, 104429 (2009); T. Liu, Y. Y. Zhang, Q. H. Chen, K. L. Wang, arXiv:0812.0321; L. Gong and P. Q. Tong, Phys. Rev. B **78**, 115114 (2008). X. G. Wang, Z. Sun and Z. D. Wang, Phys. Rev. A **79** , 012105 (2009). F. D. M. Haldane, Phys. Rev. B **25**, 4925 (1982). K. Okamto and K. Nomura, Phys. Lett. A **169**, 433 (1992). G. Castilla, S. Chakravarty, and V.J. Emery, Phys. Rev. Lett. **75**, 1823 (1995). T. Giamarchi, *Quantum Physics in One Dimension* (Oxford University Press, Oxford, England, 2004). R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, and S. Ramasesha, Phys. Rev. B **52**, 6581 (1995). S. R. White, and I. Affleck, Phys. Rev. B **54**, 9862 (1996). I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys. A 22, 511 (1989). C. N. Yang and C. P. Yang, Phys. Rev. 150, 321 (1966); J. D. Cloizeaux and M. Gaudin, J. Math. Phys, 7, 1387 (1966).

ArXiv

--- abstract: 'We calculate model-independently the impact of the critical point on higher order baryon susceptibilities $\chi_n$, showing how they depend on fluctuations of the order parameter. Including all tree level diagrams, we find new contributions to $\chi_4$ equally important to the kurtosis of the order parameter fluctuations, and we characterize the kurtosis and other nonguassian moments as functions on the phase diagram. Important features of this analysis are then confirmed by a Gross-Neveu model study with good agreement with other model studies as well as lattice and experimental data. This suggests the universality of the characteristic peak in baryon susceptibilities as a signal of the critical point. We discuss leveraging measurements of different $\chi_n$ to extrapolate the location of the critical point.' author: - 'Jiunn-Wei Chen$^{1,2}$, Jian Deng$^{3}$ and Lance Labun$^2$' date: '18 October, 2014' title: 'Baryon susceptibilities, nongaussian moments and the QCD critical point' --- A major goal of QCD theory and heavy-ion collision (HIC) experiment is to locate the critical end point in the chemical potential–temperature ($\mu\!-\!T$) plane[@QCDphases]. It is the target of the beam energy scan at RHIC and the future FAIR experiment, which are designed to create and measure QCD matter at high temperature and density. Lattice simulations are also developing methods to calculate properties of QCD matter at $\mu \neq 0$[@altlattice; @Gavai:2010zn], which cannot be reached directly due to the sign problem. The critical point itself is a second-order transition, characterized by diverging correlation length $\xi $, due to vanishing mass of the order parameter field $\sigma $. This fact, $m_{\sigma }^{-1}=\xi \rightarrow \infty $, is a statement about the two-point correlation function of the $\sigma $ field, and we can use low energy effective field theory to relate other correlation functions to the critical point and phase structure. $\sigma $ correlations influence observables such as baryon number fluctuations because the $\sigma $ couples like a mass term for the baryons, meaning that the presence of $\sigma $ changes the baryon energy[@Stephanov:1998dy]. Thus our aim is to establish the theory connection from the phase structure through $\sigma $ dynamics to observables, here proton number fluctuations, which can be compared to event-by-event fluctuations in HICs[@Aggarwal:2010wy; @Adamczyk:2013dal] and to lattice simulations[@Gavai:2010zn]. It is important to keep in mind that the QCD matter created in HICs is dynamic. The measured data in general integrate properties from the initial state and expansion dynamics, and they may not represent equilibrium properties of QCD matter at the freeze-out $\mu ,T$, especially if the fireball has passed near the critical point [@slowing]. Assuming the departure from equilibrium is small, we interpret the freeze-out data as approximate measurements of the phase diagram, which can be compared with theory and lattice predictions to help locate the critical point. The fluctuation observables compared between HICs and lattice simulations are ratios of baryon susceptibilities $$m_{1}=\frac{T\chi _{3}}{\chi _{2}},\quad m_{2}=\frac{T^{2}\chi _{4}}{\chi _{2}},\quad \chi _{n}=\frac{\partial ^{n}\ln \mathcal{Z}}{\partial \mu ^{n}} \label{chin}$$with the volume dependence eliminated in the ratios. More precisely, HICs measure proton fluctuations, which are shown to directly reflect the baryon fluctuations, because the order parameter field, the scalar $\sigma$, is an isospin singlet [@Hatta:2003wn]. From here, one approach is model independent, considering the partition function as a path integral over $\sigma $, $\mathcal{Z}=\int \!\mathcal{D}\sigma \ e^{-\Omega \lbrack \sigma ]/T}$, and the effective potential of the Landau theory $\Omega[\sigma]$ contains the phase structure in its coefficients. However those parameters are not determined by the theory. Previously this has been used to search for dominant contributions to $\chi _{n}$ close to the critical point [@Stephanov:2008qz; @Stephanov:2011pb]. Another approach is to evaluate $\ln \mathcal{Z}$ in a QCD-like model, such as NJL [@Asakawa:2009aj], to gain predictive power of $\chi _{n}$ as functions on the phase diagram. We pursue both approaches to put the model independent results into the context of the global phase diagram. We analyze a general polynomial form of the effective potential $\Omega[\sigma]$. We derive the $\chi _{n}$ as functions of the $\sigma $ fluctuation moments $\langle \delta \sigma ^{k}\rangle $, extracting new, equally important contributions to $m_{2}$ in additional to the $\sigma $ field kurtosis $\kappa_{4}$, studied by [@Stephanov:2011pb]. We show that negative $\kappa_{4}$ is restricted to the normal phase, and thus these new contributions are necessary to understand recent HIC and lattice results for $m_{2}$. Our model independent results are corroborated with quantitative study of the 1+1 dimensional Gross-Neveu (GN) model, revealing remarkably good qualitative agreement with both other model studies [@Asakawa:2009aj] as well as the experimental data. This consistency suggests that those features of our findings are model-independent. We begin with the effective potential for the order parameter field, $$\Omega \lbrack \sigma ]=\int d^{3}x\left( -\!J\sigma +\frac{g_{2}}{2}\sigma ^{2}+\frac{g_{4}}{4}\sigma ^{4}+\frac{g_{6}}{6}\sigma ^{6}+\cdots \right) \label{Veffglobal}$$with coefficients $g_{2n}$ functions of temperature and chemical potential, determining the phase diagram. Focusing on long range correlations, we consider only the zero momentum $\vec{k}=0$ mode, and so do not write the kinetic energy term $(\vec{\nabla}\sigma )^{2}$ here [@Stephanov:2008qz]. With the explicit symmetry breaking parameter $J\rightarrow 0$, the point where $g_{2}=g_{4}=0$ is the tricritical point (TCP), separating the second order transition line for $g_{4}>0$ from the first order line for $g_{4}<0$. When $J\neq 0$, the second order line disappears into a crossover transition through which the $\sigma $ minimum $\langle \sigma \rangle \equiv v$ changes smoothly as a function of temperature, and the TCP becomes a critical end point (CEP). Fluctuations of the order parameter field obey an effective potential obtained by first minimizing the potential Eq.(\[Veffglobal\]) and then Taylor expanding around $v$, yielding $$\Omega \lbrack \delta \sigma ]-\Omega _{0}=\int d^{3}x\left( \frac{m_{\sigma }^{2}}{2}\delta \sigma ^{2}+\frac{\lambda _{3}}{3}\delta \sigma ^{3}+\frac{\lambda _{4}}{4}\delta \sigma ^{4}+\cdots \right) \label{Veffflucns}$$with $\delta \sigma (x)=\sigma (x)-v$. The constant $\Omega _{0}\equiv\Omega[\sigma \!=\!v]$ does not influence the fluctuations, but does appear in the observables corresponding to the mean field contribution. The vev $v$ satisfies the gap equation $v(g_{2}+g_{4}v^{2}+g_{6}v^{4})=J$, and depends on $\mu ,T$ through the $g_{2n}$. Calculating $\mu $-derivatives of the partition function gives an explicit relation between susceptibilities $\chi _{n}$ and $\delta \sigma $ fluctuations. Starting with the second order, $$T^{2}\chi _{2}=T^{2}\frac{\partial ^{2}\ln \mathcal{Z}}{\partial \mu ^{2}}=-T\langle \Omega ^{\prime \prime }\rangle +\langle (\Omega ^{\prime })^{2}\rangle -\langle \Omega ^{\prime }\rangle ^{2} \label{dPdmu2}$$where $\langle f\rangle =\mathcal{Z}^{-1}\int \mathcal{D}\sigma \ f\ e^{-\Omega /T}$ is the expectation value of the function $f$ including $\sigma $ fluctuations. The prime indicates differentiation with respect to $\mu $, $$\label{aijexp} \frac{\partial ^{k}\Omega }{\partial \mu ^{k}}=\int d^{3}x\left( a_{k0}+a_{k1}\delta \sigma +a_{k2}\delta \sigma ^{2}+\cdots \right) .$$The first term $a_{k0}$ is the mean-field contribution from differentiating $\Omega_0$. The linear term arises from the $\mu$-dependence of the vev $v$. Plugging these derivatives into Eq.(\[dPdmu2\]), we keep all tree-level contributions, where the power of the correlator is less than or equal to order of the $\mu$-derivative. This means that the expectation value of a product of correlators at different points is equal to the product of expectation values of correlators formed by making all possible contractions of $\delta\sigma $ at different points. The combination $\langle (\Omega ^{\prime})^{2}\rangle -\langle \Omega ^{\prime }\rangle ^{2}$ cancels disconnected diagrams. Applying these rules, $$T^{2}\chi _{2}=-VTa_{20}+V^2a_{11}^{2}\langle\delta\sigma^{2}\rangle \label{dPdmu2flucns}$$A diagrammatic method helps to organize these calculations and distinguish loops arising from contractions. So far [Eq.(\[dPdmu2flucns\])]{} is just the usual second moment of particle number, here expanded in terms of the fluctuations of the $\delta\sigma$ field. Applying this procedure, the higher order susceptibilities are $$\begin{aligned} \label{dPdmu3sigma} T^{3}\chi _{3}=& -VT^{2}a_{30}+3V^2Ta_{11}a_{21}\langle \delta \sigma^{2}\rangle \\ \notag & -V^3a_{11}^{3}\langle \delta \sigma^{3}\rangle -6V^{3}a_{11}^{2}a_{12}\langle \delta \sigma^{2}\rangle ^{2} \end{aligned}$$ and $$\begin{aligned} \label{dPdmu4sigma} T^{4}\chi _{4}=& -VT^{3}a_{40}+V^2T^{2}(4a_{31}a_{11}+3a_{21}^{2})\langle \delta\sigma^{2}\rangle \\ & -6V^3Ta_{21}a_{11}^{2}\langle\delta\sigma^{3}\rangle +V^4a_{11}^{4}\big(\langle \delta \sigma^{4}\rangle -3\langle \delta \sigma^{2}\rangle ^{2}\big) \notag \\ & -12V^{3}T(a_{22}a_{11}^{2}+2a_{21}a_{11}a_{12})\langle \delta \sigma^{2}\rangle ^{2} \notag \\ & +24V^{4}(2a_{11}^{2}a_{12}^{2}+a_{11}^{3}a_{13})\langle \delta \sigma^{2}\rangle ^{3} \notag \\ & +24V^{4}a_{11}^{3}a_{12}\langle \delta \sigma^{3}\rangle \langle\delta \sigma^{2}\rangle \notag\end{aligned}$$ Each factor of $V$ comes from the $d^3x$ integration in [Eq.(\[aijexp\])]{}, and after inserting the expressions for $\langle\delta\sigma^{k}\rangle $, each $\chi_2,\chi_3,\chi_4\propto V$. The fluctuation moments $\langle\delta\sigma^{k}\rangle $ are derived by functional differention of Eq.(\[Veffflucns\]), $$\begin{aligned} \kappa _{2}& =\langle \delta \sigma ^{2}\rangle =\frac{T}{V}\xi ^{2},\quad \kappa _{3}=\langle \delta \sigma ^{3}\rangle =-2\lambda _{3}\frac{T^{2}}{V^{2}}\xi ^{6} \label{kappa4} \\ \kappa _{4}& =\langle \delta \sigma ^{4}\rangle -3\langle \delta \sigma ^{2}\rangle ^{2}=6\frac{T^{3}}{V^{3}}\big(2(\lambda _{3}\xi )^{2}-\lambda _{4}\big)\xi ^{8}\end{aligned}$$ The point is that the $a_{jk}$ coefficients weight how the $\delta\sigma$ correlations contribute to the higher order susceptibilities, $\chi_2,\chi_3...$. Moveover, the $a_{jk}$ have their own $\xi$ dependence, which can be estimated analytically and model-independently, as well as compared with model studies. For example, we find that $a_{11}=m^2\partial v/\partial\mu$ scales $\sim\xi^{-1}$ near the critical point. To compare to a given solvable model (such as the GN model below), the coupling constants $m_\sigma^2,\lambda_3,\lambda_4...$ are calculated from the model’s effective potential and then their $\mu$-derivatives evaluated yielding $a_{kj}$ coefficients. The third moment $\chi_{3}$ has been studied in the NJL model and found to be negative around the phase boundary [@Asakawa:2009aj]. In agreement with power-counting $\xi $ only in the $\delta\sigma$ correlators [@Stephanov:2008qz], the behavior of $m_{1}$ near the critical point can be explained by focusing on $\langle\delta\sigma^{3}\rangle $ and hence the function $\kappa_{3}(\mu ,T)$: In this case, estimating the $\xi$ dependence of the $a_{jk}$ coefficients in [Eq.(\[dPdmu3sigma\])]{} reveals that the $a_{11}^3\kappa_3$ term scales with the largest positive power of $\xi$. However, for $\chi_{4}$ there are many terms of the same (tree-level) order in the perturbation theory. Taking into account the $\xi$ dependence of the coefficients, several contributions, including those represented by the diagrams in Fig. \[fig:diags\] scale with the same power of $\xi$ as the $\kappa_{4}$ term. Although fewer $\sigma$ propagators are visible in some of these diagrams, the coefficient functions $a_{11},a_{12},$ and $a_{13}$ all have important $m_{\sigma}$ dependence. Looking at $\xi$-scaling, we find all three terms $\langle\delta\sigma^2\rangle^2$, $\langle\delta\sigma^2\rangle^3$, and $\langle\delta\sigma^2\rangle\langle\delta\sigma^3\rangle$ are approximately equally relevant as the $\kappa_4$ term. These analyses are supported by separately evaluating these terms in the GN model. (200,210) (30,150)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi3lambda3.1 "fig:"){width="70pt"}]{} (55,140)[(1a)]{} (100,150)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi3xi2.1 "fig:"){width="70pt"}]{} (125,140)[(1b)]{} (30,80)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi4lambda4.1 "fig:"){width="70pt"}]{} (55,70)[(2a)]{} (100,80)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi4lambda3sq.1 "fig:"){width="70pt"}]{} (125,70)[(2b)]{} (0,10)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi4xi31.1 "fig:"){width="70pt"}]{} (25,0)[(2c)]{} (70,10)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi4xi32.1 "fig:"){width="70pt"}]{} (95,0)[(2d)]{} (140,10)[![Diagrams 1a,1b give leading contributions to $\chi_3$. Diagrams 2(a-e) are some of the leading contributions to $\chi_4$. Omitted are diagrams involving multiple $\mu$ derivaties at the same point. []{data-label="fig:diags"}](chi4lambda3xi.1 "fig:"){width="70pt"}]{} (165,0)[(2e)]{} Next to see how the $\sigma $ fluctuations are impacted by the CEP, we investigate $\kappa _{3}(\mu ,T)$ and $\kappa _{4}(\mu ,T)$ as functions on the phase diagram. With $J\rightarrow 0$, the unbroken phase is where $\langle \sigma \rangle =0$, and in this case $\lambda _{2n}=g_{2n}$. Odd terms are zero, in particular $\lambda _{3}\equiv 0$ in the unbroken phase, and negative $\kappa _{4}$ exists when $\lambda _{4}=g_{4}>0$ above the second order phase transition line. In the symmetry broken phase, $$\begin{aligned} 2(\lambda _{3}\xi )^{2}-\lambda _{4}& =\frac{4}{\sqrt{D}}\left( (g_{4}-2\sqrt{D})^{2}+D\right) , \label{kappa4broken} \\ & D=g_{4}^{2}-4g_{2}g_{6}>0\quad (J=0) \notag\end{aligned}$$$D$ is the algebraic discriminant obtained when solving the gap question for the extrema, and it is positive in the broken phase, corresponding to real, nontrivial ($\sigma \neq 0$) solutions. Therefore, with $J=0$, $\kappa_{4}$ is positive definite in the broken phase and the $\kappa_4<0$ region is defined by the conditions $g_{2}>0$ and $g_{4}>0$ occuring only in the unbroken phase. For concreteness, this is illustrated in the GN model, Figure \[fig:GNphases\]. Turning on $J\neq 0$ produces a continuous change in the $\lambda_i$. In particular, the $\kappa_4=0$ lines, bounding the $\kappa_4<0$ region, move continuously away from their $J=0$ limits, and continue to obey the constraint “remembered” from $J=0$ theory. To see this, first recall that the tricritical point anchors one corner of the $\kappa_{4}<0$ region, and in the $J\neq 0$ theory, the critical end point continues to do so[@Stephanov:2011pb]. The reason is that $\kappa_{4}$ Eq.(\[kappa4\]) has a local minimum where $\lambda_{3}=0$. For $J=0$, $\lambda_{3}=0$ holds throughout the unbroken phase, but for any fixed $J\neq 0$, the relation $\lambda_{3}(g_{2},g_{4},...)=0$ is an equation whose solution defines a line in the $\mu -T$ plane. The $\lambda_{3}=0$ line must pass through the critical end point. Differing trajectories of the phase boundary and $\lambda_{3}=0$ line are seen in the GN model, Figure \[fig:GNphases\]. The critical end point is located by the conditions $m_{\sigma}^{2}=\lambda_{3}=0$, which means the coefficients $g_{2n}$ satisfy [@Stephanov:1998dy] $$g_{2}=5g_{6}v^{4},~~g_{4}=-\frac{10}{3}g_{6}v^{2},~~ v^{5}=\frac{3}{8}\frac{J}{g_{6}}\quad @\,\mathrm{CEP} \label{CEPg2n}$$The vev $v=\langle \sigma \rangle $ is nonzero, as expected, and as the symmetry breaking is turned off $J\rightarrow 0$ these equations return to their $J=0$ limits. Since $v^{2}>0$, the CEP always shifts to the southeast, into the fourth quadrant relative to the TCP of the $J=0$ theory at $g_{2}=g_{4}=0$. To locate the $\lambda_{3}=0$ line, relax condition on $m_{\sigma }^{2}$ to find that $\lambda_{3}=0$ is the set of points satisfying $$g_{2}=\frac{7}{3}g_{6}v^{4}+\frac{J}{v},\quad g_{4}=-\frac{10}{3}g_{6}v^{2} \label{lambda3zero}$$The $\lambda_{3}=0$ line leaves the CEP parallel to the first order line, and hence proceeds in the direction of decreasing $g_{4}$. With $g_{2}>0$ and $g_{4}<0$ near the critical point (Eq.(\[CEPg2n\])), the relation Eq.(\[lambda3zero\]) requires that $v$ decreases along the $\lambda_{3}=0$ line. In the high $T$ limit, $v\rightarrow 0$, so that the $\lambda_{3}=0$ line asymptotes to $g_{4}=0$ from below. Thus, from Eq.(\[lambda3zero\]) we deduce that $\lambda_{3}=0$ typically cannot proceed close to the $\mu =0$ axis, since that would require that the tricritical point of the $J=0$ theory is near the $\mu =0$ axis. The $\kappa_{4}<0$-region must migrate toward higher $T$ and $\mu $ with the critical end point. In the high $T$, low $\mu$ behaviour of $\kappa_4$ is given by expanding for small $v$: $2(\lambda_3\xi)^2-\lambda_4= -g_4+2\big(\frac{(3g_4)^2}{g_2}-5g_6\big)v^2+ \mathcal{O}(v^4)$ which is valid where $g_2,g_4>0$, far away from the lines where $g_2,g_4$ vanish. Approaching from high $T$, $\kappa_4$ starts out negative just as in the $J=0$ theory, and becomes positive just where the vev $v$ becomes large enough that the second term starts to win over the first. Therefore, as the magnitude of explicit breaking increases enhancing the order parameter, the $\kappa_4=0$ line and $\kappa_4<0$ region move farther from the phase boundary. We demonstrate the features derived above in the phase diagram and susceptibilities of the GN model. The fermion number susceptibilites behave very similarly to other models such as PNJL [@Skokov:2011rq]. The GN model comprises $N$ fermions in 1 spatial dimension with bare mass $m_{0}$ and a four-fermion interaction $\propto g^{2}$, and in the large $N$ limit has a rich phase structure [@Schnetz:2005ih]. The physical mass $m$ is given by $m\gamma =(\pi/Ng^{2})m_{0}$ where $\gamma =\pi /(Ng^{2})-\ln \Lambda /m$ is the parameter controlling the magnitude of explicit symmetry breaking. At small $\mu ,T$, there is a chiral condensate $\langle \bar{\psi}\psi \rangle $ and the order parameter is the effective mass $M=m_{0}-g^{2}N\langle \bar{\psi}\psi \rangle $. The effective potential is a function of $M$, and we focus on the region above and on the low $\mu $ side of the critical point [@Schnetz:2005ih] ![Phase diagram of the GN model, with phase boundaries and TCP of the $\protect\gamma=0$ theory and the CEP of the $\protect\gamma=0.1$ theory. The $\protect\kappa_4<0$ region of the $\protect\gamma=0$ is above the second order (green) line and left of the dashed (blue) line that joins the boundary at the TCP. The $\protect\kappa_4<0$ region of the $\protect\gamma=0.1$ theory is delineated by the dot-dashed (red) line, and $\protect\lambda_3=0$ the solid (red) line inside this region. []{data-label="fig:GNphases"}](lam3-kappa4-cross-0-new.eps){width="45.00000%"} The phase diagram behaves as described model-independently: For $\gamma\rightarrow 0$, there is a tricritical point and second order line extending to the $\mu =0$ axis. For $\gamma \neq 0$, the second order line vanishes into a crossover and the critical end point shifts increasingly to the southeast away from the former tricritical point. Figure \[fig:GNphases\] compares the phase diagrams of the GN model for $\gamma =0$ and $\gamma =0.1$. For $\gamma \neq 0$ the phase boundary is determined as the peak in the chiral susceptibility, $$\chi _{M}=\frac{\partial \langle \bar{\psi}\psi \rangle }{\partial m}=\frac{1}{m}\left( M-T\frac{\partial M}{\partial T}-\mu \frac{\partial M}{\partial \mu }\right) \label{chiM}$$as is used in lattice QCD studies [@chiMlattice]. The phase boundary stays near the critical line of the $\gamma =0$ theory, which is robust for different values of $\gamma $. All our results are shown in units of $m=1$. For $\gamma=0$, the ${\kappa_4}<0$ region is delineated by the second-order line and the $g_4=0$ line. For $\gamma=0.1$, it is delineated by the dot-dashed line with a cusp at the CEP. Varying $\gamma$, we see that the ${\kappa_4}<0$ region evolves continuously as a function of $\gamma$ from its $\gamma\to 0$ limit. The $\lambda_3=0$ line leaves the CEP parallel to the first order line, and the $\kappa_4<0$ region is approximately symmetric around it very near the CEP. However, the $\lambda_3=0$ line then asymptotes to the $g_4=0$ line, which pulls the $\kappa_4<0$ region away from the phase boundary. ![Upper frame: Density plot of $m_2$ in the $\protect\mu-T$ plane with $\protect\gamma=0.1$. The white lines indicate where $m_2=0$ and in the (red) wedge between these lines $m_2<0$. The first order line is the solid heavy line, and the crossover line is the dotted line, determined by the max of Eq.(\[chiM\]). The dashed lines are hypothetical freeze-out curves, color-coded to correspond to the lines in the lower frame. []{data-label="fig:c2"}](m2_muT_folines.eps "fig:"){width="45.00000%"}![Upper frame: Density plot of $m_2$ in the $\protect\mu-T$ plane with $\protect\gamma=0.1$. The white lines indicate where $m_2=0$ and in the (red) wedge between these lines $m_2<0$. The first order line is the solid heavy line, and the crossover line is the dotted line, determined by the max of Eq.(\[chiM\]). The dashed lines are hypothetical freeze-out curves, color-coded to correspond to the lines in the lower frame. []{data-label="fig:c2"}](m2_T.eps "fig:"){width="45.00000%"} We plot $m_2$ on the phase diagram in Figure \[fig:c2\]. The negative $m_2$ region forms a wedge opening up from the CEP and extends deeper across the phase boundary than the $\kappa_4<0$ region. Negative $m_2$ could be accessible to freeze-out at $\mu<\mu_{\mathrm{CEP}}$, and the signature would be a minimum followed by a rapid increase to a positive peak, as seen in the (green) freeze-out curve closest to the phase boundary. Moving freeze-out progressively away from the phase boundary, both the minimum and maximum of $m_2$ decrease in magnitude. Thus it is possible $m_2$ is only positive along the freeze-out curve (for example the lowest curve). Its maximum provides a residual signal of proximity to the CEP, seeing that the height of the peak decreases rapidly away from the phase boundary. Comparing upper and lower frames of Fig. \[fig:c2\], we see that the peak in $m_2$ is always at a temperature higher (or $\mu$ lower) than the CEP. Strikingly, the black line is in good qualitative agreement with lattice and HIC results. However, non-monotonic behaviour of $m_{2}$ along a single freeze-out line is insufficient to establish proximity to the CEP. Many possible freeze-out curves can be drawn that cross several contours of constant $m_{2}$ twice, and each will display a local maximum of $m_{2}$ as a function of $\mu $ or the collision energy. For this reason, it will be important to combine several probes of the phase diagram, and one way to start is to compare $m_{2}$ and $m_{1}$ ![$m_1$ along the hypothetical freeze-out lines given in the upper frame of Fig.\[fig:c2\].\ []{data-label="fig:c1"}](m1_T.eps){width="45.00000%"} In Figure \[fig:c1\], we plot $m_{1}$ along the same hypothetical freeze-out curves. Like $m_{2}$ it displays a positive peak close the CEP, and the magnitude of the maximum decreases for freeze-out lines farther away from the phase boundary. Again, the peak is at higher temperature (lower $\mu $) than the CEP. This fact appears to be universal, as it is seen an Ising-model evaluation of $\kappa _{3}$ and $\kappa _{4}$ similar to [@Stephanov:2011pb]. Despite many similarities, the topography of the peaks in $m_{1}$ and $m_{2}$ differ in detail. Combining measurements of these two observables along the freeze-out curve, we may be able to extract more information about the CEP location. To conclude, we have studied the fermion susceptibilities $\chi_2,\chi_3,\chi_4$ analytically using a low energy effective theory for the order parameter field and numerically using the Gross-Neveu model as an example system. The model-independent analysis shows that larger quark mass pushes the critical end point to higher $\mu$, and there are constraints on the position of the CEP relative to the tricritical point of the zero quark mass theory. In agreement with previous work, nonmonotonic behaviour of $m_{1}$ and $m_{2} $ appears as a signal of the critical region in the phase diagram. [ Consistent with experimental data, we find $m_2$ first decreases as a function of chemical potential $\mu$, which is a remnant of the $m_2<0$ region above the critical point. Seeing a large peak in $m_2$ at larger $\mu$/smaller $\sqrt{s}$ would support this explanation of the data. However, it is necessary to accumulate as much corroborating evidence as possible to preclude false positive, and we note in this same region, $m_1$ is also expected to peak and decrease again. The peaks in $m_{1},m_{2}$ are typically not the point of closest approach, and the temperature of the peaks are ordered $T_{\mathrm{max},m_{1}}>T_{\mathrm{max},m_{2}}>T_{\mathrm{CEP}}$, a fact which might be leveraged to indicate the location of the critical point]{}. To the extent that the fireball is near thermodynamic equilibrium at freeze-out, the model independent features we find can be compared to experiment. It may be possible to refine the predictions by taking into account expansion dynamics[@slowing]. More information may be extracted from the experimental data by combining measurements of $m_{1}$ and $m_{2}$ along the single available freeze-out curve (and possibly other curves available from lattice). 0.2cm *Acknowledgments*: JWC is supported in part by the MOST, NTU-CTS, and the NTU-CASTS of R.O.C. J.D. is supported in part by the Major State Basic Research Development Program in China (Contract No. 2014CB845406), National Natural Science Foundation of China (Projects No. 11105082). [99]{} M. A. Stephanov, PoS LAT **2006**, 024 (2006) \[hep-lat/0701002\]; K. Fukushima and C. Sasaki, Prog. Part. Nucl. Phys. **72**, 99 (2013) \[arXiv:1301.6377 \[hep-ph\]\]; S. Gupta, X. Luo, B. Mohanty, H. G. Ritter and N. Xu, Science [**332**]{}, 1525 (2011) \[arXiv:1105.3934 \[hep-ph\]\]; and references therein. P. de Forcrand and O. Philipsen, Phys. Rev. Lett. **105**, 152001 (2010) \[arXiv:1004.3144 \[hep-lat\]\]; A. Li, A. Alexandru and K. F. Liu, Phys. Rev. D **84**, 071503 (2011) \[arXiv:1103.3045 \[hep-ph\]\]. R. V. Gavai and S. Gupta, Phys. Lett. B **696**, 459 (2011) \[arXiv:1001.3796 \[hep-lat\]\]. M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. Lett. **81**, 4816 (1998) \[hep-ph/9806219\]. M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D **60**, 114028 (1999) \[hep-ph/9903292\]. M. M. Aggarwal *et al.* \[STAR Collaboration\], Phys. Rev. Lett. **105**, 022302 (2010) \[arXiv:1004.4959 \[nucl-ex\]\]. L. Adamczyk *et al.* \[STAR Collaboration\], Phys. Rev. Lett. **112**, 032302 (2014) \[arXiv:1309.5681 \[nucl-ex\]\]. B. Berdnikov and K. Rajagopal, Phys. Rev. D **61**, 105017 (2000) \[hep-ph/9912274\]. C. Nonaka and M. Asakawa, Phys. Rev. C **71**, 044904 (2005) \[nucl-th/0410078\]. C. Athanasiou, K. Rajagopal and M. Stephanov, Phys. Rev. D **82**, 074008 (2010) \[arXiv:1006.4636 \[hep-ph\]\]. Y. Hatta and M. A. Stephanov, Phys. Rev. Lett.  [**91**]{}, 102003 (2003) \[Erratum-ibid.  [**91**]{}, 129901 (2003)\] \[hep-ph/0302002\]. M. A. Stephanov, Phys. Rev. Lett. **102**, 032301 (2009) \[arXiv:0809.3450 \[hep-ph\]\]. M. A. Stephanov, Phys. Rev. Lett. **107**, 052301 (2011) \[arXiv:1104.1627 \[hep-ph\]\]. M. Asakawa, S. Ejiri and M. Kitazawa, Phys. Rev. Lett. **103**, 262301 (2009) \[arXiv:0904.2089 \[nucl-th\]\]. V. Skokov, B. Friman and K. Redlich, Phys. Lett. B **708**, 179 (2012) \[arXiv:1108.3231 \[hep-ph\]\]. O. Schnetz, M. Thies and K. Urlichs, Annals Phys. **321**, 2604 (2006) \[hep-th/0511206\]. Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B **643**, 46 (2006) \[hep-lat/0609068\]. A. Bazavov, T. Bhattacharya, M. Cheng, *et al.* Phys. Rev. D **85**, 054503 (2012) \[arXiv:1111.1710 \[hep-lat\]\]. L. Levkova, PoS LATTICE **2011**, 011 (2011) \[arXiv:1201.1516 \[hep-lat\]\].

ArXiv

--- abstract: '[Multisymplectic systems, partial differential equations, fluid dynamics, conservation laws, potential vorticity]{} We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map – the “back-to-labels” map – gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton’s principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle-relabelling symmetry and leading to Kelvin’s circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.' author: - 'C. J. Cotter, D. D. Holm and P. E. Hydon' title: Multisymplectic formulation of fluid dynamics using the inverse map --- \[firstpage\] Introduction ============ A system of partial differential equations (PDEs) is said to be *multisymplectic* if it is of the form $$K^{\alpha}_{ij}(\MM{z})z^j_{,\alpha} = {\frac{\partial H}{\partial z^i}},$$ where each of the two-forms $$\kappa^\alpha = \frac{1}{2}K_{ij}^\alpha(\MM{z})\, {\mathrm{d}}z^i\wedge{\mathrm{d}}z^j$$ is closed. Here $\MM{z}$ is an ordered set of dependent variables, total differentiation with respect to each independent variable $q^\alpha$ is denoted by the subscript $\alpha$ after a comma, and the Einstein summation convention is used. The closed two-form $\kappa^\alpha$ is associated with the independent variable $q^\alpha$; it is analogous to the symplectic two-form for a Hamiltonian ordinary differential equation. Hence there is a symplectic structure associated with each independent variable. In the first of a series of papers, Bridges (1997) pioneered the development of multisymplectic systems, showing that the rich geometric structure that is endowed by the symplectic two-forms can be used to understand the interaction and stability of nonlinear waves. For many important PDEs, the multisymplectic formulation has revealed hidden features that are important in stability analysis. In order to preserve at least some of these features in numerical simulations, Bridges & Reich (2001) introduced multisymplectic integrators, which generalise the symplectic methods that have been widely used in numerical Hamiltonian dynamics. Hydon (2005) showed that multisymplectic systems of PDEs may be derived from Hamilton’s principle whenever the Lagrangian is affine in the first-order derivatives and contains no higher-order derivatives. This can usually be achieved by introducing auxiliary variables to eliminate the derivatives. The aim of this paper is to provide a unified approach to producing multisymplectic formulations of fluid dynamics, based on the inverse map. Our approach covers all fluid dynamical equations that are written in Euler-Poincaré form (Holm *et al.*, 1998), *i.e.* all equations which arise due to the advection of fluid material. First we use the inverse map to form a canonical Euler-Lagrange equation (following the Clebsch representation given in Holm & Kupershmidt, 1983). Then the Lagrangian is made affine in the space and time derivatives by using constraints that introduce additional variables. Following Hydon (2005), we obtain a one-form quasi-conservation law which, when it is pulled back to the space and time coordinates, gives conservation laws for momentum and energy. We also obtain a two-form conservation law that represents conservation of symplecticity; when this is pulled back to the spatial coordinates, it leads to a conservation law for vorticity. The multisymplectic version of Noether’s Theorem yields an infinite space of conservation laws from the particle-relabelling symmetry for fluid dynamics; these conservation laws imply Kelvin’s circulation theorem. The conserved momentum that is canonically conjugate to the back-to-labels map plays a key role in the derivation of the conservation laws. The corresponding velocity is the convective velocity, whose geometric properties are discussed in Holm *et al.* (1986). In this paper we show how the above constructions are made in general, illustrating this with examples. We also discuss how multisymplectic integrators can be constructed using these methods. Sections \[review\] and \[inverse map sec\] review the relations among multisymplectic structures, the Clebsch representation and the momentum map associated with particle relabelling. Section \[inverse map EPDiff\] shows how to construct a multisymplectic formulation of the Euler-Poincaré equation for the diffeomorphism group (EPDiff), and derives the corresponding conservation laws, including the infinite set of conservation laws that yield Kelvin’s circulation theorem. Section \[advected\] extends this formulation to the Euler-Poincaré equation with advected quantities. This is illustrated by the incompressible Euler equation, showing how the circulation theorem arises in the multisymplectic formulation. Section \[numerics\] sketches numerical issues in the multisymplectic framework. Finally, Section \[summary\] summarises and outlines directions for future research. Review of multisymplectic structures {#review} ==================================== This section reviews the formulation of multisymplectic systems and their conservation laws, following Hydon (2005). A system of partial differential equation (PDEs) is multisymplectic provided that it can be represented as a variational problem with a Lagrangian that is affine in the first derivatives of the dependent variables: $$\label{mslag} L = L_j^\alpha(\MM{z})z^j_{,\alpha} - H(\MM{z}).$$ The Euler-Lagrange equations are then $$\label{Eul-Lag-eqns}K^{\alpha}_{ij}(\MM{z})z^j_{,\alpha} = {\frac{\partial H}{\partial z^i}},$$ where the functions $$\label{Msymp-struct-matrix} K^{\alpha}_{ij}(\MM{z}) = {\frac{\partial L^\alpha_j}{\partial z^i}}-{\frac{\partial L^\alpha_i}{\partial z^j}}$$ are coefficients of the multisymplectic structure matrix. We define the (closed) symplectic two-forms $$\label{kappa} \kappa^\alpha = \frac{1}{2}K_{ij}^\alpha(\MM{z})\, {\mathrm{d}}z^i\wedge{\mathrm{d}}z^j,$$ and obtain the structural conservation law (Bridges, 1997). $$\label{kappa law} \kappa^\alpha_{,\alpha} = 0.$$ Hydon showed that the Poincaré Lemma leads to a one-form quasi-conservation law $$(L^{\alpha}_jdz^j)_{,\alpha} = {\mathrm{d}}(L^\alpha_jz^j_{,\alpha}-H(\MM{z}))= {\mathrm{d}}{L}, \label{ofcl}$$ whose exterior derivative is (\[kappa law\]). Every one-parameter Lie group of point symmetries of the multisymplectic system (\[Eul-Lag-eqns\]) is generated by a differential operator of the form $$\label{X} X = Q^i(\MM{q},\MM{z}){\frac{\partial }{\partial z^i}} + (Q^i(\MM{q},\MM{z}))_{,\alpha} {\frac{\partial }{\partial z^i_{,\alpha}}}.$$ Noether’s Theorem implies that if $X$ generates variational symmetries, that is, if $$\label{varsym} XL = B^\alpha_{,\alpha}$$ for some functions $B^\alpha$, then the interior product of $X$ with the one-form quasi-conservation law yields the conservation law $$\label{noethm} (L_j^{\alpha}Q^j-B^{\alpha})_{,\alpha}=0.$$ This is the multisymplectic form of Noether’s theorem. Every multisymplectic system is invariant under translations in the independent variables $\MM{q}$. For each of these symmetries, Noether’s theorem yields a conservation law $$(L_j^\alpha z^j_{,\beta}-L\delta^\alpha_\beta)_{,\alpha}=0.$$ Such conservation laws can equally well be obtained by pulling back the quasi-conservation law (\[ofcl\]) to the base space of independent variables. Commonly, the independent variables are spatial position $\MM{x}$ and time $t$. Pulling back (\[ofcl\]) to these base coordinates yields the energy conservation law from the ${\mathrm{d}}{t}$ component, and the momentum conservation law from the remaining components. We shall see the form of these conservation laws for fluid dynamics in later sections. The inverse map and Clebsch representation {#inverse map sec} ========================================== Lagrangian fluid dynamics and the inverse map --------------------------------------------- Lagrangian fluid dynamics provides evolution equations for particles moving with a fluid flow. This is typically done by writing down a flow map $\Phi$ from some reference configuration to the fluid domain $\Omega$ at each instance in time. As the fluid particles cannot cavitate, superimpose or jump, this map must be a diffeomorphism. For an $n$-dimensional fluid flow, the flow map $\Phi:\,\mathbb{R}^n\times\mathbb{R}\mapsto\mathbb{R}^n$ given by $\MM{x}=\Phi(\MM{l},t)$ specifies the spatial position at time $t$ of the fluid particle that has *label* $\MM{l}=\Phi(\MM{x},0)$. The *inverse map* $\Phi^{-1}$ gives the label of the particle that occupies position $\MM{x}$ at time $t$ as the function $\MM{l}=\Phi^{-1}(\MM{x},t)$. The Eulerian velocity field $\MM{u}(\MM{x},t)$ gives the velocity of the fluid particle that occupies position $\MM{x}$ at time $t$ as follows: $$\MM{\dot{x}}(\MM{l},t)=\MM{u}(\MM{x}(\MM{l},t),t).$$ Each label component $l_k(\MM{x},t)$ satisfies the advection law $$\label{label eqn} l_{k,t} + u_il_{k,i} = 0.$$ Here ${}_{,t}$ and ${}_{,i}$ denote differentiation with respect to $t$ and $x_i$ respectively. We use Cartesian coordinates and the Euclidean inner product[^1], so we shall not generally distinguish between ‘up’ and ‘down’ indices; summation from 1 to $n$ is implied whenever an index is repeated. Clebsch representation using the inverse map -------------------------------------------- A canonical variational principle for fluid dynamics may be formulated by following the standard Clebsch procedure using the inverse map (Seliger & Whitham (1968), Holm & Kupershmidt, 1983). The Clebsch procedure begins with a functional $\ell[\MM{u}]$ of the Eulerian fluid velocity $\MM{u}$, which is known as the *reduced Lagrangian* in the context of Euler-Poincaré reduction (Holm *et al.*, 1998). One then enforces stationarity of the action $S=\int\ell[\MM{u}]{\mathrm{d}}t$ under the constraint that equation (\[label eqn\]) is satisfied by using a vector of $n$ Lagrange multipliers, which is denoted as $\MM{\pi}$. These Lagrange multipliers are the conjugate momenta to $\MM{l}$ in the course of the Legendre transformation to the Hamiltonian formulation. One may choose $\ell[\MM{u}]$ to be solely the kinetic energy, which depends only on $\MM{u}$. More generally, $\ell$ will also depend on thermodynamic Eulerian variables such as density, whose evolution may also be accommodated by introducing constraints. These constraints are often called the “Lin constraints” (Serrin, 1959). This idea was also used in reformulating London’s variational principle for superfluids (Lin, 1963). \[clebsch\] The Clebsch variational principle using the inverse map is $$\delta \int_{t_0}^{t_1} \ell[\MM{u}] + \int_{\Omega} \MM{\pi}\cdot(\MM{l}_t+\MM{u}\cdot\nabla\MM{l}){\mathrm{d}}V(\MM{x}) {\mathrm{d}}{t}=0,$$ where $\MM{\pi}(\MM{x},t)$ are Lagrange multipliers which enforce the constraint that particle labels $\MM{l}(\MM{x},t)$ are advected by the flow. Taking the indicated variations leads to the following equations: $$\begin{aligned} \label{momentum map} \delta \MM{u}:&& {\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\MM{l})^T\cdot\MM{\pi} = 0, \\ \nonumber \delta \MM{\pi}:&& \MM{l}_t+(\MM{u}\cdot\nabla)\MM{l} = 0, \\ \nonumber \delta \MM{l}:&& \MM{\pi}_t+\nabla\cdot(\MM{u}\MM{\pi}) = 0,\end{aligned}$$ where $$\begin{aligned} \label{notation} \left((\nabla\MM{l})^T\cdot\MM{\pi}\right)_i := \pi_kl_{k,i}, \qquad (\nabla\cdot(\MM{u}\MM{\pi}))_k := (u_j\pi_k)_{,j},\end{aligned}$$ and the variational derivative $\delta{\ell}/\delta{\MM{u}}$ is defined by $$\ell[\MM{u}+\epsilon\MM{u}'] = \ell[\MM{u}] + \epsilon\int_{\Omega} {\frac{\delta \ell}{\delta \MM{u}}}\cdot\MM{u}' \,{\mathrm{d}}V(\MM{x}) + \mathcal{O}(\epsilon^2) \,.$$ In the language of fluid mechanics, the expression (\[momentum map\]) for the spatial momentum $\MM{m}=\delta{\ell}/\delta\MM{u}$ in terms of canonically conjugate variables $(\MM{l}, \MM{\pi})$ is an example of a “Clebsch representation,” which expresses the solution of the EPDiff equations (see below) in terms of canonical variables that evolve by standard canonical Hamilton equations. This has been known in the case of fluid mechanics for more than 100 years. For modern discussions of the Clebsch representation for ideal fluids, see, for example, Holm & Kupershmidt (1983) and Marsden & Weinstein (1983). In the language of geometric mechanics, the Clebsch representation is a momentum map. Particle relabelling -------------------- As the physics of fluids should be independent of the labelling of particles, one may relabel the particles (by a diffeomorphism of the flow domain) without changing the dynamics. This is called the *particle relabelling symmetry*; Noether’s theorem applied to this symmetry leads to the Kelvin circulation theorem. See Holm *et al.* (1998) for a modern description. Clebsch momentum map -------------------- A [[******]{}momentum map]{} is a map $\mathbf{J}: T^{\ast}Q \rightarrow \mathfrak{g}^\ast$ from the cotangent bundle $T^*Q$ of the configuration manifold $Q$ to the dual $\mathfrak{g}^\ast$ of the Lie algebra $\mathfrak{g}$ of a Lie group $G$ that acts on $Q$. The momentum map is defined by the formula, $$\label{momentummapdef} \mathbf{J} (\nu_q) \cdot \xi = \left\langle \nu_q, \xi_Q (q) \right\rangle,$$ where $\nu_q \in T ^{\ast} _q Q $ and $\xi \in \mathfrak{g}$. In this formula $\xi _Q $ is the infinitesimal generator of the action of ${G}$ on $Q$ associated with the Lie algebra element $\xi$, and $\left\langle \nu_q, \xi_Q (q) \right\rangle$ is the natural pairing of an element of $T ^{\ast}_q Q $ with an element of $T _q Q $. The Clebsch relation (\[momentum map\]) defines a momentum map for the right action $\operatorname{Diff} (\Omega)$ of the diffeomorphisms of the domain $\Omega$ on the back-to-labels map $\MM{l}$. [^2] The spatial momentum in equation (\[momentum map\]) may be rewritten as a map $\MM{J}_\Omega:\,T^*\Omega\mapsto\mathfrak{X}^*(\Omega)$ from the cotangent bundle of $\Omega$ to the dual $\mathfrak{X}^*(\Omega)$ of the vector fields $\mathfrak{X}(\Omega)$ given by $$\begin{aligned} \label{rightmommap} \MM{J}_\Omega:\,\MM{m}\cdot {\mathrm{d}}\MM{x} = - \Big( (\nabla\MM{l})^T\cdot\MM{\pi} \Big)\cdot {\mathrm{d}}\MM{x} = - \,\MM{\pi} \cdot {\mathrm{d}}\MM{l} =: - \,\pi_k {\mathrm{d}}l_k \,.\end{aligned}$$ That is, $\MM{J}_\Omega$ maps the space of labels and their conjugate momenta $(\MM{l},\MM{\pi})\in T^*\Omega$ to the space of one-form densities $\MM{m}\in\mathfrak{X}^*(\Omega)$ on $\Omega$. The map (\[rightmommap\]) may be associated with the [*right action*]{} $\MM{l}\cdot\eta$ of smooth invertible maps (diffeomorphisms) $\eta$ of the back-to-labels maps $\MM{l}$ by composition of functions, as follows, $$\label{rightDiff} \operatorname{Diff}(\Omega):\ \MM{l}\cdot\eta=\MM{l}\circ\eta \,.$$ The infinitesimal generator of this right action is obtained from its definition, as $$\label{infgen-right} X_{\Omega}(\MM{l}) := \frac{d}{ds}\Big|_{s=0}\Big(\MM{l}\circ\eta(s)\Big) = T\MM{l} \circ X \,,$$ in which the vector field $X \in \mathfrak{X}(\Omega)$ is tangent to the curve of diffeomorphisms $\eta _s$ at the identity $s = 0$. Thus, pairing the map $\MM{J}_\Omega$ with the vector field $X \in \mathfrak{X}(\Omega)$ yields $$\begin{aligned} \left\langle \MM{J}_\Omega (\MM{l}, \MM{\pi} ), X \right\rangle & = -\,\langle\, \MM{\pi} \cdot {\mathrm{d}}\MM{l} \,, X \,\rangle \\& = -\,\int_S \pi_kl_{k,j}X_j (\MM{x}) \,{\mathrm{d}}V(\MM{x}) \\& = -\,\left\langle (\MM{l}, \MM{\pi}), T\MM{l}\cdot X \right\rangle \\& = -\,\left\langle (\MM{l}, \MM{\pi}), X_{\Omega}(\MM{l}) \right\rangle \,,\end{aligned}$$ where $\left\langle \,\cdot\,, \,\cdot\,\right\rangle:\,T ^{\ast}_{\MM{l}}\Omega\times T_{\MM{l}}\Omega\mapsto\mathbb{R}$ is the $L^2$ pairing of an element of $T ^{\ast}_{\MM{l}}\Omega $ (a one-form density) with an element of $T_{\MM{l}}\Omega $ (a vector field).\ Consequently, the Clebsch map (\[momentum map\]) satisfies the defining relation (\[momentummapdef\]) to be a momentum map, $$\label{momentummap-JOmega} \MM{J}(\MM{l}, \MM{\pi}) = -\,\MM{\pi} \cdot {\mathrm{d}}\MM{l} \,,$$ with the $L^2$ pairing of the one-form density $-\,\MM{\pi} \cdot {\mathrm{d}}\MM{l}$ with the vector field $X$. Being the cotangent lift of the action of $\operatorname{Diff} (\Omega)$, the momentum map $\mathbf{J}_\Omega$ in (\[rightmommap\]) is equivariant and Poisson. That is, substituting the canonical Poisson bracket into relation yields the Lie-Poisson bracket on the space of $\MM{m}$’s. See, for example, Holm & Kupershmidt (1983) and Marsden & Weinstein (1983) for more explanation, discussion and applications. The momentum map property of the Clebsch representation guarantees that the canonically conjugate variables $(\MM{l},\MM{\pi})$ may be eliminated in favour of the spatial momentum $\MM{m}$. Before its momentum map property was understood, the use of the Clebsch representation to eliminate the canonical variables in favour of Eulerian fluid variables was a tantalising mystery (Seliger & Whitham, 1968). Note that the right action of $\operatorname{Diff}(\Omega)$ on the inverse map is not a symmetry. In fact, as we shall see, the right action of $\operatorname{Diff}(\Omega)$ on the inverse map generates the fluid motion itself. Elimination theorem ------------------- Eliminating the canonically conjugate variables $(\MM{l}, \MM{\pi})$ produces an equation of motion for $\MM{m}=\delta{\ell}/\delta\MM{u}$, which is constructed in the proof of the following theorem: The labels $\MM{l}$ and their conjugate momenta $\MM{\pi}$ may be eliminated from the equations arising from the variational principle (\[clebsch\]) to obtain the weak form of the following equation of motion for $\delta{\ell}/\delta\MM{u}$: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delta \MM{u}}} + \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} =0,$$ where $$\operatorname{ad}^*_{\MM{u}}\MM{m} = \nabla\cdot(\MM{u}\MM{m}) + (\nabla\MM{u})^T\cdot \MM{m}$$ is defined by $$\langle \operatorname{ad}^*_{\MM{u}}\MM{m},\MM{w}\rangle = -\langle \MM{m},\operatorname{ad}_{\MM{u}}\MM{w}\rangle = \langle \MM{m},(\MM{u}\cdot\nabla)\MM{w} - (\MM{w}\cdot\nabla)\MM{u}\rangle,$$ and $\langle\cdot,\cdot\rangle$ is the $L^2$ inner-product. This equation of motion for ${\delta\ell}/{\delta\MM{u}}$ is the Euler-Poincaré equation for the diffeomorphism group (EPDiff) (Holm *et al.*, 1998). Take the time-derivative of the inner product of ${\delta\ell}/{\delta\MM{u}}$ with a time-independent vector field $\MM{w}$: $$\begin{aligned} {\frac{d }{d t}}\Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle & = & {\frac{d }{d t}}\Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} , \MM{w} \Bigg\rangle = {\frac{d }{d t}}\Bigg\langle -\MM{\pi},(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle, \\ & = & \Bigg\langle \nabla\cdot(\MM{u}\MM{\pi}), (\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle \MM{\pi}, (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle , \\ & = & \Bigg\langle \MM{\pi}, -(\MM{u}\cdot\nabla)(\MM{w}\cdot\nabla)\MM{l}+ (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle = \Bigg\langle \MM{\pi},-\left(\operatorname{ad}_{\MM{u}}\MM{w}\cdot\nabla\right)\MM{l} \Bigg\rangle , \\ &=& \Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi},\operatorname{ad}_{\MM{u}}\MM{w} \Bigg\rangle = \Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\operatorname{ad}_{\MM{u}}\MM{w}\Bigg\rangle = -\,\Bigg\langle \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle,\end{aligned}$$ which is the (weak form of the) EPDiff equation. Example: EPDiff($H^1$) ---------------------- To give a concrete example, consider EPDiff with $\ell[\MM{u}]$ being the $H^1_\lambda$-norm for $\MM{u}$. This is the $n$-dimensional Camassa-Holm (CH) equation (Camassa & Holm, 1993; Holm *et al.*, 1998; Holm & Marsden, 2004), which has applications in computational anatomy (Holm *et al.*, 2004; Miller *et al.*, 2002). This system has the reduced Lagrangian $$\ell[\MM{u}] = \int_{\Omega} \frac{1}{2}\,\left(|\MM{u}|^2+\lambda^2|\nabla\MM{u}|^2\right){\mathrm{d}}V(\MM{x}) = \int_{\Omega} \frac{1}{2}\,\left(u_iu_i+\lambda^2u_{i,j}u_{i,j}\right){\mathrm{d}}V(\MM{x}) = \frac{1}{2}\|\MM{u}\|^2_{H^1_\lambda}.$$ The EPDiff equation amounts to $${\frac{\partial \MM{m}}{\partial t}} + (\MM{u}\cdot\nabla)\MM{m} + (\nabla\MM{u})^T\cdot\MM{m} + \MM{m}\nabla\cdot\MM{u} = 0, \qquad \MM{m} = (1-\lambda^2\nabla^2)\MM{u} \,.$$ When $n=1$, these reduce to the Camassa-Holm (CH) equation, $$m_t + um_x + 2mu_x = 0,\qquad m = u - \lambda^2u_{xx} \,.$$ Advected quantities ------------------- To construct more general fluid equations we shall include advected quantities $a$ whose flow-rules are defined by $$a_t + \mathcal{L}_{\MM{u}}a = 0,$$ where $\mathcal{L}_{\MM{u}}$ is the Lie derivative. Such advected variables typically arise in the potential energy or the thermodynamic internal energy of an ideal fluid. For example, advected scalars $s$ (as in salinity) satisfy $${\frac{\partial }{\partial t}}s + \mathcal{L}_{\MM{u}}s=0, \quad \textrm{\emph{i.e.}} \quad s_t + (\MM{u}\cdot\nabla)s = 0,$$ and advected densities $\rho\,{\mathrm{d}}V$ satisfy, $${\frac{\partial }{\partial t}}(\rho\,{\mathrm{d}}V) + \mathcal{L}_{\MM{u}}(\rho\,{\mathrm{d}}V)=0, \quad \textrm{\emph{i.e.}} \quad \rho_t+\nabla\cdot(\rho\MM{u}) = 0. \,. \label{advecD}$$ A more extensive list of different types of advected quantity is given in Holm *et al.* (1998). We write the reduced Lagrangian $\ell$ as a functional of the Eulerian fluid variables $\MM{u}$ and $a$, and add further constraints to the action $S$ to account for their advection relations, $$\label{principle with advected qs} S = \int \ell[\MM{u},a]\,{\mathrm{d}}t + \int {\mathrm{d}}t\int_{\Omega}\MM{\pi}\cdot(\MM{l}_t+ (\MM{u}\cdot\nabla)\MM{l}) + \phi(a_t+\mathcal{L}_{\MM{u}}a) \,{\mathrm{d}}V(\MM{x}).$$ The Euler-Lagrange equations, which follow from the stationarity condition $\delta S=0$, are $$\begin{aligned} \delta\MM{u}:&& {\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\MM{l})^T\cdot\MM{\pi} + \phi\diamond a = 0, \label{EL advected 1}\\ \delta\MM{\pi}:&& \MM{l}_t + (\MM{u}\cdot\nabla)\MM{l} = 0, \nonumber\\ \delta\MM{l}:&& -\MM{\pi}_t - \nabla\cdot(\MM{u}\MM{\pi}) = 0, \label{EL advected 3}\\ \delta\phi:&& a_t + \mathcal{L}_{\MM{u}}a = 0, \nonumber \\ \delta{a}:&& -\phi_t-\mathcal{L}_{\MM{u}}\phi + {\frac{\delta \ell}{\delta a}} = 0, \label{EL advected 5}\end{aligned}$$ where the diamond operator ($\diamond$) is defined as the dual of the Lie derivative operation $\mathcal{L}_{\MM{u}}$ with respect to the $L^2$ pairing. Explicitly, under integration by parts, $$\int_\Omega (\phi\diamond a)\cdot\MM{u}\,{\mathrm{d}}{V}(\MM{x}) = -\int_\Omega (\phi\mathcal{L}_{\MM{u}}a)\,{\mathrm{d}}{V}(\MM{x}).$$ The map to the spatial momentum in equation (\[EL advected 1\]) $${\frac{\delta \ell}{\delta \MM{u}}} =: \MM{m} = -\,\pi_A\nabla l^A -\, \phi\diamond a \,,$$ is again a momentum map, this time for the semidirect-product action of the diffeomorphisms on $\Omega\times V^*$. Again the momentum map property allows the canonical variables to be eliminated in favour of the Eulerian quantities. As a result, eliminating the variables $\MM{l}$, $\MM{\pi}$ and $\phi$ leads to the Euler-Poincaré equation with advected quantities $a$. The labels $\MM{l}$, their conjugate momenta $\MM{\pi}$ and the conjugate momentum $(\phi)$ to the advected quantities $(a)$ may be eliminated from equations (\[EL advected 1\]-\[EL advected 5\]) to obtain the weak form of the Euler-Poincaré equation with advected quantities: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delta \MM{u}}} + \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} = a\diamond{\frac{\delta \ell}{\delta a}}, \qquad a_t + \mathcal{L}_{\MM{u}}a = 0.$$ Take the time-derivative of the inner product of ${\delta\ell}/{\delta\MM{u}}$ with a function of $\MM{w}$: $$\begin{aligned} {\frac{d }{d t}}\Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle & = & {\frac{d }{d t}}\Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} - \phi\diamond a , \MM{w} \Bigg\rangle = {\frac{d }{d t}}\Bigg\langle -\MM{\pi},(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + {\frac{d }{d t}}\Bigg\langle \phi,\mathcal{L}_{\MM{w}}a\Bigg\rangle \\ & = & \Bigg\langle \nabla\cdot(\MM{u}\MM{\pi}), (\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle \MM{\pi}, (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle\\ & & \quad + \Bigg\langle -{\frac{\delta \ell}{\delta a}}-\mathcal{L}_{\MM{u}}\phi, \mathcal{L}_{\MM{w}}a\Bigg\rangle + \Bigg\langle \phi,-\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a\Bigg\rangle , \\ & = & \Bigg\langle \MM{\pi}, -(\MM{u}\cdot\nabla)(\MM{w}\cdot\nabla)\MM{l}+ (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle \\ & & \quad +\Bigg\langle\phi,\mathcal{L}_{\MM{u}}\mathcal{L}_{\MM{w}}a -\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a \Bigg\rangle , \\ & =&\Bigg\langle \MM{\pi},-\left(\operatorname{ad}_{\MM{u}}\MM{w}\cdot\nabla\right)\MM{l} \Bigg\rangle + \Bigg\langle \phi,\mathcal{L}_{\operatorname{ad}_{\MM{u}}\MM{w}}a \Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle, \\ &=& \Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} -\phi\diamond a,\operatorname{ad}_{\MM{u}}\MM{w} \Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle , \\ & = & \Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\operatorname{ad}_{\MM{u}}\MM{w}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle = \Bigg\langle -\operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} + {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle.\end{aligned}$$ These Euler-Poincaré equations with advected quantities cover all conservative fluid equations which describe the advection of material. For a large collection of examples, see Holm *et al.* (1998). Example: Incompressible Euler equations --------------------------------------- As an example, consider the reduced Lagrangian for the incompressible Euler equations $$\ell[\MM{u},\rho,p] = \int_\Omega \frac{\rho|\MM{u}|^2}{2} + p(1-\rho)\,{\mathrm{d}}V(\MM{x}) \,.$$ Here $\rho(\MM{x},t)$ is the ratio of the local fluid density to the average density over $\Omega$; this is governed by the continuity equation (\[advecD\]). The pressure $p$ is a Lagrange multiplier that fixes the incompressibility constraint $\rho=1$. The variational derivatives in this case are $${\frac{\delta \ell}{\delta \MM{u}}} = \rho\MM{u}, \qquad {\frac{\delta \ell}{\delta \rho}} = \frac{|\MM{u}|^2}{2}-p,\qquad {\frac{\delta \ell}{\delta p}} =1-\rho\,.$$ Consequently, the Euler-Poincaré equations become $$\begin{aligned} (\rho\MM{u})_t + (\MM{u}\cdot\nabla)(\rho\MM{u}) + \rho\MM{u}(\nabla\cdot\MM{u}) + \rho(\nabla\MM{u})^T\cdot \MM{u} &=& \rho\nabla\left(\frac{|\MM{u}|^2}{2} -p\right), \\ \rho_t+\nabla\cdot(\rho\MM{u}) &=& 0, \\ \qquad \rho&=&1,\end{aligned}$$ and rearrangement gives the Euler fluid equations, $$\MM{u}_t + (\MM{u}\cdot\nabla)\MM{u} = -\nabla p, \qquad \nabla\cdot\MM{u}=0.$$ Inverse map multisymplectic formulation for EPDiff($H^1$) {#inverse map EPDiff} ========================================================= As we now have a canonical variational principle for fluid dynamics *via* the inverse map, one may obtain its multisymplectic formulation by extending the phase space so that the Lagrangian is affine in the space and time derivatives. In this section we show how to do this for EPDiff($H^1$) as discussed in the previous section. Affine Lagrangian for EPDiff($H^1$) ----------------------------------- After introducing the inverse map constraint, the Lagrangian becomes $$L = \frac{1}{2}u_iu_i + \frac{\lambda^2}{2}u_{i,j}u_{i,j} + \pi_k\left(l_{k,t} + u_jl_{k,j}\right).$$ Any high-order derivatives and nonlinear functions of first-order derivatives must now be removed from the Lagrangian to make it affine. We introduce a tensor variable $$W_{ij} = u_{i,j}\,;$$ this relationship may be enforced by using Lagrange multipliers. However, it turns out that the multipliers can be eliminated and the Lagrangian becomes $$\label{epmslag} L = \frac{1}{2}u_iu_i - \frac{\lambda^2}{2} W_{ij}W_{ij} +\lambda^2W_{ij}u_{i,j}+ \pi_k\left(l_{k,t} + u_jl_{k,j}\right),$$ which is now affine in the space and time derivatives of $\MM{u}$, $W$, $\MM{l}$ and $\MM{\pi}$. Multisymplectic structure ------------------------- The Euler-Lagrange equations for the affine Lagrangian (\[epmslag\]) are $$\begin{aligned} \delta u_i:&& u_i - \lambda^2W_{ij,j}+ \pi_k l_{k,i} = 0, \\ \delta l_k:&& -\pi_{k,t} - (\pi_k u_j)_{,j} = 0. \\ \delta \pi_k:&& l_{k,t} + u_j l_{k,j} = 0, \\ \delta W_{ij}:&& -\lambda^2 W_{ij} + \lambda^2 u_{i,j} = 0 .\end{aligned}$$ These equations possess the following multisymplectic structure as in equation (\[Eul-Lag-eqns\]): $$\begin{pmatrix} 0 & \pi_k\partial_i & & -\lambda^2\partial_j \\ -\pi_k\partial_i & 0 & -\partial_t-u_j\partial_j & 0 \\ 0 & \partial_t+u_j\partial_j & 0 & 0 \\ \lambda^2\partial_j & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} u_i \\ l_k \\ \pi_k \\ W_{ij}\\ \end{pmatrix} = \nabla H,$$ where $\partial_t=\partial/\partial t,\ \partial_i=\partial/\partial x_i$, and $$H = -\left(\frac{1}{2}u_iu_i -\frac{\lambda^2}{2}W_{ij}W_{ij}\right) = -\left(\frac{1}{2}|\MM{u}|^2 -\frac{\lambda^2}{2}|W|^2\right).$$ One-form quasi-conservation law ------------------------------- For our multisymplectic formulation of EPDiff($H^1$), the independent variables are $$q^j = x_j, \quad j=1,\ldots n, \qquad q^{n+1} = t,$$ and the dependent variables are $$\begin{aligned} z^i = u_i, \qquad z^{n + k} = l_k, \qquad z^{2n + k} = \pi_k,\qquad z^{(i+2)n+j}=W_{ij}\,,\end{aligned}$$ where $i,j$ and $k$ range from $1$ to $n$. Comparing (\[epmslag\]) with (\[mslag\]) gives the following non-zero components $L^{\alpha}_j$: $$L^j_i = \lambda^2W_{ij}, \qquad L_{n+k}^j = \pi_ku_j, \qquad L_{n+k}^{n+1} = \pi_k, \qquad i,j,k=1,\ldots,n.$$ Therefore the one-form quasi-conservation law amounts to $$\label{epqcl} \left(\pi_k{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2W_{ij}{\mathrm{d}}u_i+\pi_ku_j{\mathrm{d}}l_k\right)_{,j}={\mathrm{d}}L.$$ The exterior derivative of this expression yields the structural conservation law $$\label{epsympcl} \left({\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2{\mathrm{d}}W_{ij}\wedge{\mathrm{d}}u_i+u_j{\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k+\pi_k{\mathrm{d}}u_j\wedge{\mathrm{d}}l_k\right)_{,j}=0.$$ Conservation of energy ---------------------- For EPDiff($H^1$), the ${\mathrm{d}}t$-component of the pullback of the one-form conservation law (\[epqcl\]) gives $$\left(\pi_kl_{k,t} - L\right)_{,t} +\left( \lambda^2W_{ij}u_{i,t} + \pi_ku_jl_{k,t} \right)_{,j} = 0.$$ In terms of $\MM{u}$ and its derivatives, this amounts to $$\left(u_im_i-\frac{1}{2}u_iu_i-\frac{\lambda^2}{2}u_{i,j}u_{i,j}\right)_{,t} +\left(\lambda^2u_{i,j}u_{i,t}+u_iu_jm_i\right)_{,j} = 0,$$ where $$m_i=u_i-\lambda^2u_{i,kk}.$$ This is the energy conservation law for EPDiff($H^1$). Conservation of momentum ------------------------ Similarly, the conservation law that is associated with translations in the $x_i$-direction is $$\left(\pi_kl_{k,i}\right)_{,t} +\left( \lambda^2 W_{kj}u_{k,i} + \pi_ku_jl_{k,i} - \delta_{ij}L \right)_{,j}=0,$$ which amounts to the momentum conservation law $$m_{i,t}+\left(\lambda^2u_{k,i}u_{k,j}-u_jm_i-\delta_{ij}\left(\frac{1}{2}u_ku_k+\frac{\lambda^2}{2}u_{k,l}u_{k,l}\right)\right)_{,j}.$$ Conservation of vorticity ------------------------- Next, consider the coefficient of each ${\mathrm{d}}x_r\wedge{\mathrm{d}}x_s$ in the pull-back of the structural (two-form) conservation law (\[epsympcl\]). This is $$\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right)_{,t} +\left(\lambda^2\left(W_{ij,r}u_{i,s}-W_{ij,s}u_{i,r}\right) +u_j\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right) +\pi_k\left(u_{j,r}l_{k,s}-u_{j,s}l_{k,r}\right)\right)_{,j}=0,$$ which amounts to $$\left(m_{r,s}-m_{s,r}\right)_{,t}+\left(\lambda^2\left(u_{i,s}u_{i,jr}-u_{i,r}u_{i,js}\right)+(u_jm_r)_{,s}-(u_jm_s)_{,r}\right)_{,j}=0.$$ One can regard this as a vorticity conservation law for EPDiff($H^1$); it is a differential consequence of the momentum conservation law. Particle relabelling symmetry ----------------------------- As we discussed in Section \[inverse map sec\], fluid equations in general, and EPDiff in particular, are invariant under relabelling of particles. In the context of the inverse map variables, relabelling is accomplished by the action of the diffeomorphism group $\operatorname{Diff}(\Omega)$ defined by $$\MM{l}\mapsto \eta\circ\MM{l}\equiv\eta(\MM{l}), \qquad \eta\in\operatorname{Diff}(\Omega).$$ The corresponding infinitesimal action of the vector fields $\mathfrak{X}(\Omega)$ is then $$\MM{l} \mapsto \MM{\xi}\circ\MM{l}\equiv\MM{\xi}(\MM{l}), \qquad \MM{\xi}\in\mathfrak{X}(\Omega),$$ and the cotangent lift of this action is $$(\MM{\pi},\MM{l}) \mapsto \left(-(\nabla\MM{\xi}(\MM{l}))^T\cdot\MM{\pi} ,\MM{\xi}(\MM{l})\right).$$ To obtain the symmetry generator (\[X\]), we extend the above action to first derivatives as follows: $$\begin{aligned} X &=& \xi_k(\MM{l}){\frac{\partial }{\partial l_k}} + (\xi_k(\MM{l}))_{,t}{\frac{\partial }{\partial l_{k,t}}} + (\xi_k(\MM{l}))_{,i}{\frac{\partial }{\partial l_{k,i}}} \\ & & - \pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}{\frac{\partial }{\partial \pi_j}} - \left(\pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\right)_{,t}{\frac{\partial }{\partial \pi_{j,t}}} - \left(\pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\right)_{,i}{\frac{\partial }{\partial \pi_{j,i}}}.\end{aligned}$$ The relabelling symmetries are variational, because $$XL = \pi_k(\xi_k(\MM{l}))_{,t} + \pi_ku_i(\xi_k(\MM{l}))_{,i} - \pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\left(l_{j,t}+u_il_{j,i} \right)=0.$$ Noether’s theorem then gives the conservation law $$\left(\pi_k\xi_k(\MM{l})\right)_{,t}+\left(\pi_ku_j\xi_k(\MM{l})\right)_{,j}=0.$$ A conservation law exists for each element $\MM{\xi}$ of $\mathfrak{X}(\Omega)$, so particle relabelling generates an infinite space of conservation laws. Circulation theorem ------------------- To see how the particle relabelling conservation laws relate to conservation of circulation, note that if $\rho$ is any density that satisfies $$\rho_{,t} + (\rho u_j)_{,j} = 0,$$ then $$\left( \frac{\pi_k\xi_k(\MM{l})}{\rho}\right)_{,t} + u_j \left( \frac{\pi_k\xi_k(\MM{l})}{\rho}\right)_{,j}=0.$$ If we pick a loop $C(t)$ which is advected with the flow, then $${\frac{d }{d t}}\oint_{C(t)} \frac{\pi_k\xi_k(\MM{l})}{\rho}{\mathrm{d}}x = 0.$$ For a vector field $\MM{\xi}$ which is tangent to the loop at time $0$, and satisfies $|\MM{\xi}|=1$ on the loop, then $$\MM{\xi}{\mathrm{d}}x = (\nabla\MM{l})\cdot{\mathrm{d}}{\MM{x}}$$ for all times $t$, and one finds $$\label{circulation} {\frac{d }{d t}}\oint_{C(t)}\frac{\MM{\pi}\cdot(\nabla\MM{l})}{\rho} \cdot{\mathrm{d}}\MM{x}=0,$$ The momentum formula (\[momentum map\]) gives $${\frac{d }{d t}}\oint_{C(t)}\frac{(1-\lambda^2\nabla^2)\MM{u}}{\rho}\cdot{\mathrm{d}}\MM{x}= {\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=0,$$ which is the circulation theorem for EPDiff. Inverse map multisymplectic formulation for Euler-Poincaré equation with advected quantities {#advected} ============================================================================================ To extend this method to more general equations with advected quantities is very simple: take the Lagrangian obtained from equation (\[inverse map sec\]) and add variables to represent higher-order derivatives. For the sake of brevity we shall compute one example, the incompressible Euler equations, and briefly discuss the implications for the circulation theorem. Multisymplectic form of incompressible Euler equations ------------------------------------------------------ We start with the reduced Lagrangian $$\ell[\MM{u},p,\rho] = \int_{\Omega}\frac{1}{2}\rho u_iu_i + p(1-\rho){\mathrm{d}}V(\MM{x}),$$ where $p$ is the pressure and $\rho$ is the relative density, and add dynamical constraints to form the Lagrangian: $$L = \frac{1}{2}\rho u_iu_i + p(1-\rho) + \pi_k\left(l_{k,t}+u_il_{k,i}\right) +\phi\left(\rho_{,t}+(\rho u_i)_{,i}\right).$$ This Lagrangian is already affine in the first-order derivatives, so the Euler-Lagrange equations are automatically multisymplectic in these variables: $$\begin{pmatrix} 0 & 0 & \pi_k\partial_i & 0 & -\rho\partial_i & 0 \\ 0 & 0 & 0 & 0 & -\partial_t -u_i\partial_i & 0 \\ -\pi_k\partial_i & 0 & 0 & -\partial_t-u_i\partial_i & 0 & 0 \\ 0 & 0 & \partial_t+u_i\partial_i & 0 & 0 & 0 \\ \rho\partial_i & \partial_t+u_i\partial_i & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} u_i \\ \rho \\ l_k \\ \pi_k \\ \phi \\ p \\ \end{pmatrix} = \nabla H,$$ where the quantity $$H = -\left(\frac{1}{2}\rho u_iu_i + p(1-\rho)\right)$$ is negative of the Hamiltonian density. Circulation theorem for advected quantities ------------------------------------------- The conservation law for particle-relabelling follows exactly as in Section \[inverse map EPDiff\], and we obtain equation (\[circulation\]) as before. The difference is that now the momentum formula (momentum map) is $$\MM{m} = {\frac{\partial \ell}{\partial \MM{u}}} = -\pi_k\nabla l_k -\phi\diamond a$$ and so one obtains $${\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}= \oint_{C(t)}\frac{1}{\rho}{\frac{\partial \ell}{\partial a}}\diamond a\cdot {\mathrm{d}}\MM{x}.$$ For the incompressible Euler equations, $a$ is the relative density $\rho$, so $${\frac{\partial \ell}{\partial a}}\diamond a = \rho\nabla{\frac{\partial \ell}{\partial \rho}},$$ which leads to the circulation theorem $${\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=\oint_{C(t)}\nabla {\frac{\partial \ell}{\partial \rho}}\cdot{\mathrm{d}}\MM{x}=0.$$ A note on multisymplectic integrators {#numerics} ===================================== In this section we discuss briefly how to produce multisymplectic numerical integrators, using the inverse map formulation given in this paper. We note in particular that the multisymplectic method will satisfy a discrete form of the particle-relabelling symmetry and hence we will obtain a method that has discrete conservation laws for $-\MM{\pi}\cdot\nabla\MM{l}$. Variational integrators ----------------------- A multisymplectic integrator for a PDE is a numerical method which preserves a discrete conservation law for the two-form $\kappa$ given in equation (\[kappa\]) (Bridges & Reich, 2001). As described in (Hydon, 2005), a discrete variational principle with a Lagrangian that is affine in first-order differences automatically leads to a set of difference equations which are multisymplectic. This now makes it very simple to construct multisymplectic integrators for fluid dynamics using the inverse map formulation: one simply replaces the spatial and time integrals in the action with numerical quadratures, replaces the first-order derivatives by differences, and takes variations following the standard variational integrator approach (Lew *et al.*, 2003). Whilst the method will preserve the discrete conservation law for the two-form $\kappa$, the one-form quasi-conservation law will not be preserved in general, and hence the other conservation laws will not be exactly preserved. Discrete relabelling symmetry ----------------------------- As $\MM{\pi}$ and $\MM{l}$ are still continuous in the discretised equations, the multisymplectic integrator will have a discrete particle-relabelling symmetry analogous to the one given in Section \[inverse map EPDiff\], with the only difference being the discretisation of the cotangent lift. Following the variational integrator programme described in Lew *et al.* (2003), the discrete form of Noether’s theorem will give rise to discrete conservation laws for the multisymplectic method. Remapping labels ---------------- If this approach is to be applied to numerical solutions with intense vorticity then one needs to address the problem that eventually the numerical discretisation of the labels $\MM{l}$ will become very poor due to tangling, and hence the approximation to the momentum $$\label{mom} \MM{m} = -(\nabla\MM{l})^T\MM{\pi} = - \pi_k\nabla l_k \,,$$ will degrade with time. One possible approach would be to apply discrete particle-relabelling, mapping the labels back to the Eulerian grid in such a way that the momentum (\[mom\]) stays fixed. This transformation is exactly the relabelling given in Section \[inverse map EPDiff\]. Numerically, one could construct a transformation (using a generating function for example) which satisfies $$\MM{l} \mapsto \MM{X} + \mathcal{O}(\Delta x^p,\Delta t^p), \qquad \MM{\pi}(\nabla\MM{l})\mapsto \MM{\pi}(\nabla\MM{l}) + \mathcal{O}(\Delta x^p,\Delta t^p),$$ where $p$ is the order of the method. For instance, one might use a variational discretisation of the relabelling transformation, which is generated by a symplectic vector field whose Hamiltonian is $\MM{\pi}\cdot\MM{\xi}(\MM{l})=\pi_k\xi_k(\MM{l})$. In this way, one may still retain some of the conservative properties of the method. Summary and Outlook {#summary} =================== Summary ------- This paper describes a multisymplectic formulation of Euler-Poincaré equations (which are, in essence, fluid dynamical equations with a particle-relabelling symmetry). We have used the inverse map to obtain a canonical variational principle, following Holm and Kupershmidt (1983). As noted in Hydon (2005), a multisymplectic formulation can be obtained by choosing variables such that the Lagrangian at most linear in the first-order derivatives, and contains no higher-order derivatives. We have shown how to construct the multisymplectic formulation for the Euler-Poincaré equations for diffeomorphisms, using the example of the EPDiff($H^1$) equations, and how to extend the method to the Euler-Poincaré equations with advected quantities. These equations encompass many fluid systems, including incompressible Euler, shallow-water, Euler-alpha, Green-Naghdi, perfect complex fluids, inviscid magnetohydrodynamics, *etc.* The techniques of Hydon (2005) have led to conservation laws for these systems, including the usual multisymplectic conservation laws for energy and momentum plus an infinite set of conservation laws which arise from the particle-relabelling symmetry of fluid dynamics. We have highlighted the connection between these latter conservation laws and Kelvin’s circulation theorem, and showed that multisymplectic integrators based on this formulation will have discrete conservation laws associated with this symmetry. Outlook ------- In the last section of this paper we have discussed the possibility of developing multisymplectic integrators for fluids using this framework. It is undoubtedly simple to construct such integrators, but the issue of accuracy with time arises whenever the flow is strongly mixing and numerical errors make the label field $\MM{l}$ very noisy. A discretisation of the relabelling map discussed in this paper could provide a way to prevent this problem whilst retaining some of the geometric properties of the method. These ideas may aid the future development of integrators that have conservation laws for vorticity and circulation, which are desirable for numerical weather prediction and other applications. In a different direction, we believe that multisymplectic integrators would be especially apt for applications of EPDiff to template-matching in computational anatomy (Holm *et al*., 2004). The matching problem is an initial-final value problem. In such problems, space and time may be treated on an equal footing, just as in the multisymplectic formulation. Acknowledgements ---------------- The work of DDH was partially supported by the Royal Society of London Wolfson Award and the US Department of Energy Office of Science ASCR. Bridges, T. J. Multi-symplectic structures and wave propagation. [*Math. Proc. Camb. Philos. Soc.*]{}, 121:147–190, 1997. Bridges, T. J. & Reich, S. Multi-symplectic integrators: numerical schemes for [H]{}amiltonian [PDE]{}s that preserve symplecticity. [*Phys. Lett.*]{}, 284A:184–193, 2001. Camassa, R. & Holm, D. D. An integrable shallow-water equation with peaked solitons. [*Physical Review Letters*]{}, 71:1661–1664, 1993. Holm, D. D. & Kupershmidt, B. [P]{}oisson brackets and [C]{}lebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity. [*Physica D*]{}, 6:347–363, 1983. Holm, D. D. & Marsden, J. E. Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the [E]{}uler-[P]{}oincaré equations for the diffeomorphism group. In J. E. Marsden & T. S. Ratiu, editors, [*The Breadth of Symplectic and [P]{}oisson Geometry: A Festshrift for Alan Weinstein*]{}, Birkhäuser, Boston, MA., 2004, pp. 203–235. http://arxiv.org/abs/nlin.CD/0312048. Holm, D. D., Marsden, J. E. & Ratiu, T. S., The Hamiltonian Structure of Continuum Mechanics in Material, Inverse Material, Spatial and Convective Representations. In [*Hamiltonian Structure and Lyapunov Stability for Ideal Continuum Dynamics*]{}, Univ. Montreal Press, 1986, pp. 1–124. Holm, D. D., Marsden, J. E., & Ratiu, T. S. The [E]{}uler–[P]{}oincaré equations and semidirect products with applications to continuum theories. [*Adv. in Math.*]{}, 137:1–81, 1998. http://arxiv.org/abs/chao-dyn/9801015. Holm, D. D., Rananather, J. T., Trouvé, A., & Younes, L. Soliton dynamics in computational anatomy. [*NeuroImage*]{}, 23:170–178, 2004. http://arxiv.org/abs/nlin.SI/0411014. Hydon, P. E. Multisymplectic conservation laws for differential and differential-difference equations. [*Proc. Roy. Soc. LOnd. A*]{}, 461:1627–1637, 2005. Lew, A., Marsden, J. E., Ortiz, M. & West, M. An overview of variational integrators. In L. P. Franca, T. E. Tezduyar & A. Masud, editors, [*Finite Element Methods: 1970s and Beyond*]{}. CIMNE, Barcelona, Spain, 2004, pp. 98–115. Lin, C. C. Liquid Helium. In [*Proceedings of the Enrico Fermi International School of Physics, Course XXI*]{}, G. Careri, editor, Academic Press, New York, 1963, pp. 93–146. Marsden, J. E. & Weinstein, A. Coadjoint orbits, vortices, and [C]{}lebsch variables for incompressible fluids. [*Physica D*]{}, 7:305–323, 1983. Miller, M. I., Trouvé, A. & Younes, L. On the metrics and [E]{}uler-[L]{}agrange equations of computational anatomy. [*Ann. Rev. Biomed. Engrg.*]{}, 4:375–405, 2002. Seliger, R. L. & Whitham, G. B. Variational principles in continuum mechanics. [*Proc. Roy. Soc. Lond. A*]{} 305: 1–25, 1968. Serrin, J. Mathematical principles of classical fluid mechanics. In S. Flügge & C. Truesdell, editors, [*Handbuch der Physik*]{} VIII/I. Springer-Verlag, Berlin, 1959, pp. 125–263. [^1]: This is only done for clarity and the equations are easily extended to the case when the domain $\Omega$ is a curved manifold. [^2]: As we discuss later, this right action contrasts with fluid particle relabelling, which arises by the left action of the diffeomorphisms on the inverse map.

ArXiv

--- abstract: 'We investigate the onset of parity-time ($\mathcal{PT}$) symmetry breaking in non-Hermitian tight-binding lattices with spatially-extended loss/gain regions in presence of an advective term. Similarly to the instability properties of hydrodynamic open flows, it is shown that $\mathcal{PT}$ symmetry breaking can be either absolute or convective. In the former case, an initially-localized wave packet shows a secular growth with time at any given spatial position, whereas in the latter case the growth is observed in a reference frame moving at some drift velocity while decay occurs at any fixed spatial position. In the convective unstable regime, $\mathcal{PT}$ symmetry is restored when the spatial region of gain/loss in the lattice is limited (rather than extended). We consider specifically a non-Hermitian extension of the Rice-Mele tight binding lattice model, and show the existence of a transition from absolute to convective symmetry breaking when the advective term is large enough. An extension of the analysis to ac-dc-driven lattices is also presented, and an optical implementation of the non-Hermitian Rice-Mele model is suggested, which is based on light transport in an array of evanescently-coupled optical waveguides with a periodically-bent axis and alternating regions of optical gain and loss.' author: - Stefano Longhi title: 'Convective and absolute $\mathcal{PT}$ symmetry breaking in tight-binding lattices' --- Introduction ============ Non-Hermitian Hamiltonian models are often encountered in a wide class of quantum and classical systems [@Moiseyev]. They are introduced, for example, to model open systems and dissipative phenomena in quantum mechanics (see, for instance, [@Moiseyev; @Rotter; @ob1; @ob2; @vi; @kor]). In optics, non-Hermitian models naturally arise owing to the presence of optical gain and loss regions in dielectric or metal-dielectric structures [@Siegman]. A special class of non-Hermitian Hamiltonians is provided by complex potentials having parity-time ($\mathcal{PT}$) symmetry [@Bender_PRL_98; @Bender_RPP_2007], that is invariance under simultaneous parity transform ($\mathcal{P}$: $\hat{p} \rightarrow -\hat{p}$, $\hat{x} \rightarrow -\hat{x}$, where $\hat{p}$ and $\hat{x}$ stand for momentum and position operators, respectively) and time reversal ($\mathcal{T}$: $\hat{p} \rightarrow -\hat{p}$, $\hat{x} \rightarrow \hat{x}$, $i\rightarrow-i$). An important property of $\mathcal{PT}$ Hamiltonians is to admit of an entirely real-valued energy spectrum below a phase transition symmetry-breaking point, a property that attracted great attention in earlier studies on the subject owing to the possibility to formulate a consistent quantum mechanical theory in a non-Hermitian framework [@Bender_PRL_98; @Bender_RPP_2007; @Mos1; @Ben]. Indeed, $\mathcal{PT}$-symmetric Hamiltonians are a special case of pseudo-Hermitian Hamiltonians, which can be mapped into Hermitian ones [@Mos2]. $\mathcal{PT}$-symmetric Hamiltonians have found interest and applications in several physical fields, including magnetohydrodynamics [@Guenther_JMP_2005], cavity quantum electrodynamics [@Plenio_RMP_1998], quantum-field-theories [@Ben; @BenQFT], and electronics [@Kottos]. More recently, great efforts have been devoted to the study and the experimental implementation of optical structures possessing $\mathcal{PT}$ symmetry (see, for instance, [@Muga; @El-Ganainy_OL_07; @Makris_PRL_08; @Klaiman_PRL_08; @Mostafazadeh_PRL_09; @Longhi_PRL_09; @Guo09; @Ruter_NP_10; @Longhi10; @Feng2011; @Kivshar12; @Regensburger_Nature_12; @Feng12; @uffa] and references therein). The huge interest raised by the introduction of $\mathcal{PT}$ optical media is mainly motivated by their rather unique properties to mold the flow of light in non-conventional ways, with the possibility to observe, for example, double refraction and nonreciprocal diffraction patterns [@Makris_PRL_08], unidirectional Bragg scattering and invisibility [@Longhi_PRL_09; @Lin_PRL_2011; @Regensburger_Nature_12; @LonghiPRA10; @Longhi_JPA_2011; @Graefe11; @Feng12; @MosIN], non-reciprocity [@nonlinearPT], giant Goos-Hänchen shift [@Longhi_PRA_2011], and simultaneous perfect absorption and laser behaviour [@Longhi_PRA_2010; @Chong_PRL_2011]. So far, $\mathcal{PT}$ quantum and classical systems have been mainly investigated in the unbroken $\mathcal{PT}$ phase, where the energies are real-valued, or at the symmetry breaking point, where exceptional points or spectral singularities appear in the underlying Hamiltonian (see, for instance, [@Klaiman_PRL_08; @Longhi10; @MosRes]). In the broken $\mathcal{PT}$ phase, complex-conjugate energies appear. In the context of spatially-extended dissipative dynamical systems and hydrodynamic flows [@Cross], breaking of the $\mathcal{PT}$ phase indicates a bifurcation from a marginally-stable phase to an unstable phase. This means that, while an initially localized wave packet can not secularly grow in the unbroken $\mathcal{PT}$ phase, it does in the broken $\mathcal{PT}$ phase owing to the emergence of modes with complex energies. In hydrodynamics, an unstable open flow can be classified as either [*absolutely*]{} or [*convectively*]{} unstable [@rev1; @flow1; @flow2]. A one-dimensional flow described by an order parameter $\psi(x,t)$ is unstable if, for any given localized perturbation $\psi(x,0)$ at initial time $t=0$, $\psi(x,t) \rightarrow \infty$ as $t \rightarrow \infty$ along at least one ray $x/t=v={\rm const}$. The instability is said to be absolute if $\psi(x,t) \rightarrow \infty$ along the ray $x/t=0$, whereas it is convective if $\psi(x,t) \rightarrow 0$ along the ray $x/t=0$ [@rev1]. Physically, in the convectively unstable regime the initial perturbation grows when observed along the trajectory $x=vt$ at some drift velocity $v$, whereas it decays when observed at a fixed position. Convectively unstable flows generally arise in the presence of an advective (drift) term in the system, in such a way that the growing perturbation drifts in the laboratory reference frame and eventually escapes from the system. Originally introduced in hydrodynamic contexts, the concepts of convective and absolute instabilities have found interest and applications in other physical fields, for example in the study of dissipative optical patterns and noise-sustained structures in nonlinear optics [@Santa]. Inspired by the properties of hydrodynamic unstable flows [@rev1], in this work we introduce the concepts of convective and absolute $\mathcal{PT}$ symmetry breaking for spatially-extended Hamiltonian systems. Specifically, we investigate the symmetry-breaking properties of a tight-binding lattice model with spatially-extended alternating gain and loss regions, and show that the presence of an advective term can change the symmetry breaking from absolute to convective. The lattice model that we consider is a non-Hermitian extension of the famous Rice-Mele Hamiltonian, originally introduced to model conjugated diatomic polymers [@Rice]. In the convectively $\mathcal{PT}$ symmetry breaking regime, the $\mathcal{PT}$ symmetry can be restored when the gain/loss region becomes spatially confined. A physical implementation of the non-Hermitian Rice-Mele lattice model is proposed using arrays of coupled optical waveguides in a zig-zag geometry with periodically-bent axis and alternating optical gain and loss. The paper is organized as follows. In Sec.II the Rice-Mele tight-binding lattice model with non-Hermitian and advective terms is presented, and a physical implementation based on light transport in arrays of coupled optical waveguides is suggested. In Sec.III the concepts of absolute and convective $\mathcal{PT}$ symmetry breaking are introduced for periodic potentials, and the transition from absolute to convective symmetry breaking for the Rice-Mele lattice model is studied by application of asymptotic (saddle point) methods. The concepts of convective and absolute symmetry breaking are also discussed for ac-dc driven lattice models, where the quasi-energy bands . In Sec.IV the main conclusions and future developments are outlined. Finally, in two Appendixes some technical details on Floquet analysis of the ac-dc driven lattice model and saddle point calculations for the Rice-Mele Hamiltonian are presented. The model ========= Extended Rice-Mele Hamiltonian and $\mathcal{PT}$ symmetry breaking ------------------------------------------------------------------- We consider transport of classical or quantum waves on a $\mathcal{PT}$-invariant tight-binding dimerized superlattice with nearest and next-nearest neighborhood hopping schematically shown in Fig.1(a). The evolution of the amplitude probabilities $a_n(t)$, $b_n(t)$ at the two sites of the $n$-th unit cell in the lattice is governed by the following coupled-mode equations $$\begin{aligned} i \frac{d a_n}{d t} & = & -\kappa b_n-\sigma b_{n-1}-\rho \exp(i \varphi) a_{n+1} \nonumber \\ & - & \rho \exp(-i \varphi) a_{n-1}+ig a_n \\ i \frac{d b_n}{d t} & = & -\kappa a_n-\sigma a_{n+1}-\rho \exp(i \varphi) b_{n+1} \nonumber \\ & - & \rho \exp(-i \varphi) b_{n-1}-ig b_n \end{aligned}$$ ![(Color online). (a) Schematic of the non-Hermitian extension of the Rice-Mele tight-binding lattice model. The lattice unit cell contains two sites (a dimer), one with gain and the other with loss. Next-nearest neighborhood hopping occurs at a rate $\rho \exp(i \varphi)$. Convective transport at the $\mathcal{PT}$ symmetry breaking point is obtained for $\rho \neq 0$ and $\varphi \neq 0,\pi$. (b) Optical realization of the Rice-Mele model in a zig-zag array of optical waveguides with alternating optical gain and loss \[cross section in the transverse $(x,y)$ plane\]. The optical axis of the array is bent along the paraxial propagation distance $t$. Axis bending realizes an effective combined ac-dc driving of the lattice with forces $F_x(t)$ and $F_y(t)$ along the two transverse directions $x$ and $y$.](Fig1){width="8cm"} where $\kappa, \sigma >0$ are the nearest-neighborhood modulated hopping rates within the unit cell, $\rho \exp(i \varphi)$ is the complex-valued hopping rate of next nearest sites with controlled phase $\varphi$, and $g$ is the gain/loss rate at alternating sites. The coupled-mode equations (1) and (2) are derived from the tight-binding Hamiltonian $$\begin{aligned} \hat{H} & = & -\sum_n \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_n \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& ig \sum_n \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ which is Hermitian in the limiting case $g=0$ or after replacing $g \rightarrow ig$. The lattice Hamiltonian (3) is invariant under simultaneous parity transformation and time reversal, and can be regarded as a non-Hermitian extension of the Rice-Mele Hamiltonian [@Rice; @note], originally introduced to model conjugated diatomic polymers [@Rice] and found in other physical systems as well, for example in cold atoms moving in one-dimensional optical superlattices [@Bloch]. A possible physical implementation of this Hamiltonian will be discussed in the following subsection. We note that tight-binding lattice models with non-Hermitian terms have been introduced and studied in several recent works [@TB1; @TB2; @TB3; @TB4]. In particular, the limiting case $\rho=0$ and $\kappa_1=\kappa_2$ of the Hamiltonian (3) was previously considered in Refs.[@Longhi_PRL_09; @TB1], where for the infinitely-extended system $\mathcal{PT}$ symmetry breaking was shown to occur at $g=g_{th}=0$. As discussed in the next section, this kind of $\mathcal{PT}$ symmetry breaking is always absolute.\ For the general case $\kappa_1 \neq \kappa_2$ and $\rho \neq 0$, the onset of $\mathcal{PT}$ symmetry breaking can be readily determined by analytical calculation of the energy spectrum of the Hamiltonian (3). To this aim, let us search for a solution to Eqs.(1) and (2) in the form of Bloch-Floquet states $$\left( \begin{array}{c} a_{n}(t) \\ b_{n}( t) \end{array} \right) = \left( \begin{array}{c} A \\ B \end{array} \right) \exp(-iEt+iqn) \;\;\;$$ where $q$ is the quasi-momentum, which is assumed to vary in the interval $(0, 2 \pi)$, and $E=E(q)$ the corresponding energy. Substitution of the Ansatz (4) into Eqs.(1) and (2) yields the following homogeneous linear system for the complex amplitudes $A=A(q)$ and $B=B(q)$ $$\begin{aligned} \left[ E+2 \rho \cos( q+\varphi) -ig \right] A +[\kappa + \sigma \exp(-iq)] B & = & 0 \nonumber \\ \left[ \kappa+\sigma \exp(iq) \right] A+ \left[ E+2 \rho \cos( q+\varphi) +ig \right] B & = & 0 \;\;\;\;\end{aligned}$$ which is solvable provided that the determinantal equation $$\left| \begin{array}{cc} E + 2 \rho \cos (q+\varphi)-ig & \kappa+\sigma \exp(-iq ) \\ \kappa+\sigma \exp(iq ) & E + 2 \rho \cos (q+\varphi) +ig \end{array} \right|=0 \;\;\;$$ is satisfied. This yields the following dispersion relations $E=E_{\pm}(q)$ for the two superlattice minibands $$E_{\pm}(q)=-2 \rho \cos(q+\varphi) \pm \sqrt{-g^2+\kappa^2+\sigma^2+2 \kappa \sigma \cos q }$$ and the following expressions for the amplitudes $A$, $B$ of Bloch-Floquet eigenmodes $$\left( \begin{array}{c} A_{\pm}(q) \\ B_{\pm}(q) \end{array} \right)= \left( \begin{array}{c} \kappa+\sigma \exp(-iq) \\ ig-E_{\pm}(q)-2 \rho \cos(q+\varphi) \end{array} \right).$$ From Eq.(7) it follows that the energy spectrum is entirely real-valued for $g<g_{th}$ with $g_{th} \equiv |\sigma-\kappa|$. In this case, corresponding to the unbroken $\mathcal{PT}$ phase, the energy spectrum comprises two minibands which do not cross. In particular, at $q=\pi$ the two minibands are separated by an energy gap of width $2 \sqrt{g_{th}^2-g^2}$. As $g \rightarrow g_{th}^-$ the gap at $q= \pi$ shrinks and the two minibands touch at $q=\pi$; as $g$ overcomes $g_{th}$, complex-conjugate energies appear near $q=\pi$, which is the signature of $\mathcal{PT}$ symmetry breaking; see Fig.2. It is worth noticing that the group velocity $v_g$ of Bloch modes near $q= \pi$ at the symmetry breaking point, defined by $v_g=(d {\rm {Re}} (E_{\pm})/dq)$, is given by $$v_g=-2 \rho \sin \varphi$$ which does not vanish provided that $\rho \neq 0$, i.e. in the presence of next-nearest neighborhood hopping, and $\varphi \neq 0, \pi$. As it will be shown in Sec.III.B, a non-vanishing and sufficiently large group velocity can cause the $\mathcal{PT}$ symmetry breaking to change from absolute to convective. {width="14cm"} Ac-dc driven lattice model and optical realization of the non-Hermitian Rice-Mele Hamiltonian --------------------------------------------------------------------------------------------- Before discussing the nature of the $\mathcal{PT}$ symmetry breaking for the extended non-Hermitian Rice-Mele Hamiltonian (3), it is worth suggesting possible physical implementations of this model. To realize the Hamiltonian (3), in addition to the non-Hermitian (gain and loss) terms one needs to implement next-nearest neighborhood hoppings with controlled phase $\varphi$. Rather generally, tight-binding lattice models with controlled phase of hopping rates can be realized by combined ac-dc forcing. Here we briefly propose a photonic realization of the extended Rice-Mele model, based on light transport in a superlattice of evanescently-coupled optical waveguides. The Rice-Mele Hamiltonian (3) can be basically obtained as a limiting case of an ac-driven tight-binding lattice at high modulation frequencies. Another possible physical system where the combined ac-dc driven lattice model could be implemented is provided by cold atoms trapped in optical superlattices [@Bloch], where gain is introduced via atom injection at alternating sites [@kor; @BEC]. However, in spite of several theoretical proposals, experimental realizations of $\mathcal{PT}$-symmetric Hamiltonians using ultracold atoms is still missing, and hence we limit here to briefly discuss the photonic system. The optical structure that we consider is shown in Fig.1(b) and is basically composed by a sequence of evanescently-coupled optical waveguides in a zig-zag geometry with alternating optical amplification (gain) and loss. The waveguides are displaced in the horizontal ($x$) and vertical ($y$) directions by the distances $d_x$ and $d_y$, respectively. In the zig-zag geometry, non-negligible evanescent coupling occurs for nearest and next-nearest waveguides [@Felix], with coupling constants (hopping rates) $\kappa_1$, $\kappa_2$ for adjacent guides and $\kappa_3$ for next-nearest guides, as indicated in Fig.1(b). The values of the coupling constants $\kappa_1$, $\kappa_2$ and $\kappa_3$ are determined by certain overlapping integrals of the optical modes trapped in the waveguides, and they are usually exponentially-decaying functions of waveguide separation. For dielectric waveguides, the coupling constants take real and positive values. The difference of couplings $\kappa_1$ and $\kappa_2$ can be controlled by changing the horizontal ($d_x$) and vertical ($d_y$) distances of waveguides, with $\kappa_1=\kappa_2$ for $d_x \simeq d_y$. For straight waveguides, the array of Fig.1(b) thus realizes the extended Rice-Mele model of Fig.1(a) with $\kappa=\kappa_1$, $\sigma=\kappa_2$, $\rho=\kappa_3$ and $\varphi=0$. To realize an effective complex-valued amplitude for the hopping rate between next-nearest neighborhood guides, i.e. $\varphi \neq 0$, we bend the waveguide axis in both $x$ and $y$ directions along the paraxial propagation distance $t$, so that the optical axis of the array describes a curved path with parametric equations $x=x_0(t)$ and $y=y_0(t)$. Arrays of waveguides with arbitrarily curved axis in three-dimensions can be realized, for example, by the technique of femtosecond laser writing in optical glasses (see, for instance, [@Crespi]). In the tight-binding and paraxial approximations, light transport in the superlattice with a bent axis is governed by the following coupled-mode equations (see, for instance, [@LonghRev]) $$\begin{aligned} i \frac{dA_n}{dt} & = & -\kappa_1 B_n -\kappa_2 B_{n-1}-\kappa_3 (A_{n+1}+A_{n-1}) \nonumber \\ & - & [F_x(t)+F_y(t)]n A_n+igA_n \\ i \frac{dB_n}{dt} & = & -\kappa_1 A_n -\kappa_2 A_{n+1}-\kappa_3 (B_{n+1}+B_{n-1}) \nonumber \\ & - & [F_x(t)+F_y(t)]nB_n-F_x(t) B_n-igB_n \;\;\;\end{aligned}$$ where $A_n$, $B_n$ are the mode amplitudes of light trapped in the alternating waveguides with optical gain and loss, respectively, $g$ is the optical gain/loss coefficient, and $$F_x(t)=-\frac{2 \pi n_sd_x}{ \lambda} \frac{d^2x_0}{dt^2} \;, \; \; F_y(t)=\frac{2 \pi n_sd_y}{ \lambda} \frac{d^2y_0}{dt^2}. \;\;\;$$ account for the axis bending in the horizontal ($x$) and vertical ($y$) directions [@LonghRev; @LonghiPRL06]. In Eq.(12), $\lambda$ is the wavelength of the propagating light and $n_s$ is the substrate refractive index at wavelength $\lambda$. Note that Eqs.(10) and (11) describe a dimerized lattice with external forcing, with $F_x(t)$ and $F_y(t)$ playing the role of the external forces. Note also that, in the absence of axis bending, i.e. for $F_x=F_y=0$, Eqs.(10) and (11) reproduce the extended Rice-Mele model \[Eqs.(1) and (2)\] with $\varphi=0$. The equivalence of the driven lattice model \[Eqs.(10) and (11)\] with the static Rice-Mele lattice model \[Eqs.(1) and (2)\] with $\varphi \neq 0$ can be established as follows. Let us tailor the axis bending profiles $x_0(t)$ and $y_0(t)$ in the horizonatl and vertical directions to realize the following ac-dc forces $F_x(t)$ and $F_y(t)$ $$\begin{aligned} F_x(t) & = & U-(\Gamma \omega) \cos(\omega t + \phi) \nonumber \\ F_y(t) & = & -U-(\Gamma \omega) \cos(\omega t-\phi), \end{aligned}$$ where $U$, $\Gamma$ and $\omega$ are real-valued positive parameters. In our optical waveguide system, the combined ac-dc forcing corresponds to a sinusoidal axis bending with spatial frequency $\omega$ superimposed to a parabolic path [@Dignam]. Note that the sinusoidal bending is not in phase for the horizontal and vertical directions owing to the phase term $\phi$. Let us further assume that the following resonance condition $$M \omega=U$$ is satisfied for some integer $M$, and let us introduce the amplitudes $a_n$, $b_n$ via the gauge transformation $$\begin{aligned} A_n (t)& = & a_n (t) \exp[i \varphi n+i n \Phi(t)] \\ B_n (t) & = & b_n (t) \exp[i \varphi n +i \beta + i n \Phi(t)+i \Theta(t)]\end{aligned}$$ where we have set $$\Phi (t)=\int_0^t dt' [F_x(t')+F_y(t')] \; , \; \; \Theta(t)=\int_0^t dt' F_x(t'),$$ $\beta=M \phi-\Gamma \sin \phi$, and $$\varphi=2 M \phi + M \pi$$ Substitution of Eqs.(15) and (16) into Eqs.(10,11) yields a system of coupled-equations for the amplitudes $a_n(t)$ and $b_n(t)$ with time-periodic coefficients of period $T=2 \pi / \omega$. As shown in the Appendix A, if the system is observed at discrete times $\tau=0,T,2T,3T,...$, the evolution of the amplitudes $a_n(\tau)$, $b_n(\tau)$ can be mapped into the dynamics of an effective static lattice (i.e. with time-independent hopping rates) which sustains two minibands with dispersion relations $E_{\pm}(q)$ given by the quasi-energies of the original time-periodic system. In particular, in the large modulation limit $\omega \gg \kappa_{1}, \kappa_2, \kappa_3,g$, i.f. for $T \rightarrow 0$, it can be shown (see Appendix A) that the a-dc driven lattice model exactly reproduces the Rice-Mele static model \[Eqs.(1) and (2)\] with effective hopping rates given by $$\begin{aligned} \kappa & = & \kappa_1J_M(\Gamma) \\ \sigma & = & \kappa_2 J_M(\Gamma) \\ \rho & = & \kappa_3 J_0(2 \Gamma \cos \phi)\end{aligned}$$ and with the phase $\varphi$ given by Eq.(18), where $J_n$ is the Bessel function of first kind and order $n$. Therefore, the zig-zag waveguide array of Fig.1(b) with alternating optical gain and loss and with a suitable axis bending effectively realizes the extended Rice-Mele lattice model of Fig.1(a) with a non-vanishing advective term $\varphi$ and with controlled hopping rates $\kappa$, $\sigma$, $\rho$. Convective and absolute $\mathcal{PT}$ symmetry breaking ======================================================== In this section we introduce the notion of convective and absolute $\mathcal{PT}$ symmetry breaking, inspired by the concepts of convective and absolute unstable flows in hydrodynamics [@rev1; @flow1; @flow2], and then we apply such concepts to the non-Hermtiian Rice-Mele and ac-dc driven models presented in Sec.II. Definition of absolute and convective $\mathcal{PT}$ symmetry breaking for a periodic potential ----------------------------------------------------------------------------------------------- In this subsection we present the rather general definition of convective and absolute $\mathcal{PT}$ symmetry breaking for a continuous system in one spatial dimension $x$, described by a $\mathcal{PT}$-invariant Hamiltonian $\hat{H}=-\partial^2_x+V(x)$ with a potential $V(x)=V_R(x)+i g V_I(x)$, where $V_R(-x)=V_R(x)$ and $V_I(-x)=-V_I(x)$ are the real and imaginary parts of the potential and $g \geq 0$ is a real-valued parameter that measures the strength of the non-Hermitian part of the potential. The concept of convective and absolute $\mathcal{PT}$ symmetry breaking is meaningful in case where at the symmetry breaking point complex-conjugate energies emanate from the continuous spectrum of $\hat{H}$, i.e. the corresponding eigenstates are not normalizable. In fact, if the symmetry breaking arises because of the appearance of pairs of normalizable states with complex-conjugate energies, the $\mathcal{PT}$ symmetry breaking is always absolute and can not be convective, according to the hydrodynamic definitions of absolute and convective unstable flows briefly mentioned in the introduction section and formally defined below. An important case where $\mathcal{PT}$ symmetry breaking arises because of the emergence of extended (non-normalizable) states with complex conjugate energies is the one of a periodic potential, $V(x+d)=V(x)$. In this case, the energy spectrum is absolutely continuous and composed by energy bands. We assume that the energy spectrum of $\hat{H}$ is entirely real-valued for $g \leq g_{th}$, corresponding to the unbroken $\mathcal{PT}$ phase, whereas complex-conjugate energies appear for $g>g_{th}$, where $g_{th} \geq 0$ determines the symmetry breaking point. For example, for the potential $V_R(x)=\cos(2 \pi x/d)$ and $V_I(x)=\sin (2 \pi x /d)$ $\mathcal{PT}$ symmetry breaking is attained at $g_{th}=1$ [@Makris_PRL_08; @LonghiPRA10; @Longhi_JPA_2011; @Graefe11]. Let us then consider an initial wave packet $\psi(x,0)$ at time $t=0$, and let $\psi(x,t)=\exp(-i \hat{H}t) \psi(x,0)$ be the evolved wave packet at successive time $t$. In the unbroken $\mathcal{PT}$ phase, one has $\psi(x,t) \rightarrow 0$ as $t \rightarrow \infty$ at any fixed position $x$ owing to delocalization of the wave packet in the lattice. However, in the broken $\mathcal{PT}$ phase, i.e. for $g>g_{th}$, owing to the appearance of complex energies the wave packet $\psi(x,t)$ is expected to secularly grow as $t \rightarrow \infty$. According to the definitions of unstable flows in hydrodynamic systems [@rev1; @flow1], the $\mathcal{PT}$ symmetry breaking is said to be [*absolute*]{} if $\psi(x,t) \rightarrow \infty$ at $x=0$ (or at any fixed position $x=x_0$), whereas it is said to be convective if $\psi(x,t) \rightarrow \infty$ along the ray $x=vt$ for some drift velocity $v$, but $\psi(x,t) \rightarrow 0$ at $x=0$ (or at any fixed position $x=x_0$). The physics behind the definition of absolute and convective unstable flows is rather simple and is visualized in Fig.3. In the convectively unstable regime, an initial wave packet (perturbation) drifts in the laboratory reference frame with some velocity $v$, and along the ray $x=vt$, i.e. in the reference frame moving with the wave packet, the perturbation secularly grows with time. The drift velocity $v$ is basically determined by the wave packet group velocity at the quasi-momentum $k=k_s$ where the maximum growth rate (i.e. largest imaginary part of the energy) occurs. However, at a fixed position $x=x_0$ (e.g. $x_0=0$), the perturbation $\psi(x_0,t)$ can grow only transiently, but finally it vanishes as $t \rightarrow \infty$ owing to the (possibly fast) drift of the growing wave packet \[see Fig.3(a)\]. Conversely, in the absolutely unstable regime the perturbation grows so fast that, even in the presence of an advective term (a drift), at a fixed spatial position $x_0$ the perturbation $\psi(x_0,t)$ grows indefinitely with time \[see Fig.3(b)\]. To determine whether the $\mathcal{PT}$ symmetry breaking is convective or absolute, let us consider the Hamiltonian $\hat{H}$ with $g>g_{th}$, and let us consider an initial wave packet given by a superposition of Bloch-Floquet modes $\phi_k(x)=u_k(x) \exp(ikx)$ with energy $E=E(k)$, i.e. $\hat{H} \phi_k(x)=E(k) \phi_k(x)$, with $u_k(x+d)=u_k(x)$ and which the quasi-momentum $k$ that varies from $-\infty$ to $\infty$ to account for all the lattice bands (extended band representation). The wave packet then evolves according to the relation $$\psi(x,t)= \int_{-\infty}^{\infty} dk F(k) u_k(x) \exp[ikx-iE(k)t]$$ where $F(k)$ is the spectrum of excited Bloch-Floquet modes. Along the ray $x=vt$ one has $$\psi(t)= \int_{-\infty}^{\infty} dk F(k) u_k(vt) \exp[ikvt-iE(k)t].$$ The determination of the nature (absolute or convective) of the $\mathcal{PT}$ symmetry breaking entails the estimation of the asymptotic behavior of $\psi(t)$ as $ t \rightarrow \infty$. Since $u_k(x)$ is a limited and periodic function of $x$, we can study the asymptotic behavior of the associated wave packet $$\psi_1(t)= \int_{-\infty}^{\infty} dk F(k) \exp[ikvt-iE(k)t]$$ obtained by dropping the term $u_k(vt)$ under the integral in Eq.(23). In fact, it can be readily shown that ${\rm lim \; sup}_{t \rightarrow \infty } |\psi (t)| \rightarrow \infty$ ($ \rightarrow 0$) if and only if ${\rm lim \; sup}_{t \rightarrow \infty } |\psi_1(t)| \rightarrow \infty$ ($ \rightarrow 0$). Note that for the determination of the asymptotic behavior of $\psi_1(t)$ we only need to evaluate the integral on the right hand side of Eq.(24) for those values of $k$ for which ${\rm Im} \{ E(k) \} \geq 0$, the other modes giving no contribution (they are surely decaying).The asymptotic behavior of $\psi_1(t)$ as $t \rightarrow \infty$ can be determined, under certain conditions which are generally satisfied, by the saddle-point (or steepest descend) method [@rev1]. This entails analytic continuation of the function $E(k)$ is the complex $k$ plane and, using the Cauchy theorem, the deformation of the path of the integral along a suitable contour which crosses a (dominant) saddle point $k_s$ of $E(k)-kv$ in the complex plane, along the direction of the steepest descent [@rev1; @flow1; @flow2]. The asymptotic behavior of the integral is then given by the value of the exponential part of the integrand calculated at the saddle point. More precisely, for a saddle point of order $n \geq 2$, i.e. for which $E(k)=E(k_s)+v(k-k_s)+(d^n E/dk^n)_{k_s}(k-k_s)^n+o((k-k_s)^n)$, for $t \rightarrow \infty$ one has [@steep] $$\begin{aligned} \psi_1(t) & \sim & \frac{F(k_s)}{|t (d^n E/dk^n)_{k_s}|^{1/n}} (n!)^{1/n} \Gamma \left( \frac{1}{n} \right) \nonumber \\ & \times & \exp [it vk_s \pm i \pi /(2n) ] \exp[-it E(k_s)]\end{aligned}$$ where the saddle point $k_s$ in the complex plane is determined from the equation $$\left( \frac{dE}{dk} \right)_{k_s}=v.$$ The decay or secular growth of $\psi_1(t)$ thus depends on the sign of the imaginary part of $E(k)$ at the saddle point $k=k_s$. It can be readily shown that, for $g>g_{th}$, there is always a velocity $v=v_s$ for which the solution $k_s$ to Eq.(26) is real-valued and corresponds to the maximum growth rate \[i.e. the maximum of ${\rm Im} (E(k))>0$\], so that along the ray $x=v_s t$ the amplitude $\psi_1(t)$ shows a secular growth. To determine whether the symmetry breaking is either convective or absolute, we should consider the asymptotic behavior of $\psi_1(t)$ for $v=0$, which is determined by the sign of the imaginary part of $E(k)$ at the saddle point $k=k_s$ obtained from Eq.(26) with $v=0$. Hence, the $\mathcal{PT}$ symmetry breaking is absolute if ${\rm Im} \{ E(k_s) \}>0$, whereas it is convective if ${\rm Im} \{ E(k_s) \} \leq 0$, where the saddle point $k_s$ is determined from the equation $(dE/dk)_{k_s}=0$. As a general rule of thumb, for $g$ larger but close the $\mathcal{PT}$ symmetry breaking threshold, indicating by $k_s$ the quasi momentum on the real axis with maximum growth rate, i.e. that maximizes ${\rm Im}(E(k))$ for $k$ real, the $\mathcal{PT}$ symmetry breaking is absolute if the group velocity $v_g$ at $k=k_s$, given by $v_s=(d {\rm Re}(E) /dk)_{k_s}$, vanishes, whereas is it expected to be convective for a nonvanishing (and possibly large) value of $v_s$. Physically, the latter regime corresponds to the case where, owing to a non-vanishning group velocity, the unstable growing Bloch-Floquet mode is advected away, for an observer at rest, fast enough that it decays in time when observed at a fixed spatial position. {width="8.3cm"} {width="8.3cm"} Absolute and convective $\mathcal{PT}$ symmetry breaking for the non-Hermitian Rice-Mele Hamiltonian ---------------------------------------------------------------------------------------------------- In this subsection we describe in details the nature of the $\mathcal{PT}$ symmetry breaking for the extended Rice-Mele Hamiltonian defined by Eq.(3). As shown in Sec.II.A, the superlattice comprises two minibands, with dispersion relations $E_{\pm}(q)$ and corresponding Bloch-Floquet modes defined by Eqs.(7) and (4,8), respectively. After setting $\psi(n,t)=(a_n(t),b_n(t))^T$, let us consider the propagation of an initial wave packet $\psi(n,0)$ in the lattice, which is assumed to be given by a superposition of Bloch-Floquet modes belonging to the two minibands with spectral functions $F_{\pm}(q)$. The evolved wave packet at time $t$ is then given by $$\begin{aligned} \psi(n,t) & = & \int_{0}^{2 \pi} dq F_+(q) \phi_+(q) \exp[iqn-iE_{+}(q)t] \nonumber \\ & + & \int_{0}^{2 \pi} dq F_-(q) \phi_-(q) \exp[iqn-iE_{-}(q)t] \;\;\;\;\;\;\end{aligned}$$ where we have set $\phi_{\pm}(q)=(A_{\pm}(q),B_{\pm}(q))^T$. As shown in Sec.II.A, $\mathcal{PT}$ symmetry breaking occurs when the gain/loss parameter $g$ is increased to overcome the threshold value $g_{th}=|\kappa-\sigma|$. Correspondingly, complex conjugate energies appear for a wave number $q$ close to $q_0=\pi$ \[see Fig.2(c)\]. Note that, since ${\rm Im} \{ E(q) \} \geq 0$ for one miniband and ${\rm Im} \{ E(q) \} \leq 0$ for the other miniband, one of the two integrals on the right hand side of Eq.(27) decays toward zero as $t \rightarrow \infty$, and therefore we can limit to consider the contribution arising from the other integral involving unstable modes. Assuming, for the sake of definiteness, ${\rm Im} \{ E_+(q) \} \geq 0$ and ${\rm Im} \{ E_-(q) \} \leq 0$, one has $$\psi(n,t) \sim \int_{0}^{2 \pi} dq F_+(q) \phi_+(q) \exp[iqn-iE_{+}(q)t]$$ as $t \rightarrow \infty$. The asymptotic form of the integral on the right hand side of Eq.(28) along the ray $n=vt$ can be estimated by the saddle point method and takes a form similar to the one given by Eq.(25). According to the analysis presented in Sec.III.A, the $\mathcal{PT}$ symmetry breaking is thus convective if ${\rm{Im}} \{ E_+(q_s)\} \leq 0$ , whereas it is absolute for ${\rm{Im}} \{ E_+(q_s)\}>0$, where $q_s$ is the dominant saddle point obtained from the equation $(dE_+/dq)_{q_s}=0$, i.e. $$\frac{2 \rho}{\kappa \sigma} \left (\cos \varphi \sin q_s \sin \varphi \cos q_s \right)=\frac{\sin q_s}{-\epsilon^2+2 \kappa \sigma(1+ \cos q_s)}.$$ In Eq.(29) we have set $\epsilon^2=g^2-g_{th}^2$, which provides a measure of the distance from the $\mathcal{PT}$ symmetry breaking point. To simplify our analysis, let us consider the case where the gain/loss parameter $g$ is larger but close to its threshold value $g_{th}$, so that $\epsilon^2$ is a small quantity. In this case the solutions to Eq.(29) can be determined analytically by an asymptotic analysis in the small parameter $\epsilon$. The calculations are detailed in the Appendix B. The main result of the calculations is that the $\mathcal{PT}$ symmetry breaking is [*convective*]{} for $$|v_g|> \sqrt{\sigma \kappa}$$ ![(Color online). Numerically-computed quasi-energy minibands $E_{\pm}(q)$ for the ac-dc driven lattice model \[Eqs.(10) and (11)\] for increasing values of the modulation frequency $\omega$: (a) $\omega=6$, (b) $\omega=15$, and (c) $\omega=150$. The other parameter values are given in the text. In (d) the energy minibands of the static Rice-Mele lattice are shown, that correspond to the asymptotic limit $\omega \rightarrow \infty$. Solid curves refer to the real part of $ E_{\pm}(q)$, whereas the thin dotted curves to the imaginary part of $ E_{\pm}(q) $. For the sake of clearness, the imaginary part of $ E_{\pm}(q) $ has been multiplied by a factor of 10.](Fig5){width="8.3cm"} {width="8.3cm"} whereas it is [*absolute*]{} in the opposite case $|v_g| \leq \sqrt{\sigma \kappa}$, where $v_g=-2 \rho \sin \varphi$ is the group velocity at the symmetry breaking point of the most unstable mode with wave number $q=\pi$ \[see Eq.(9)\]. Hence, as expected, a sufficiently large advective term in the extended Rice-Mele Hamiltonian can change the $\mathcal{PT}$ symmetry breaking from absolute to convective. Note that for $\rho=0$ or $\rho \neq$ but real-valued, the symmetry breaking is always absolute. As discussed in the next subsection, an important physical implication of the convective (rather than absolute) $\mathcal{PT}$ symmetry breaking is that the unbroken $\mathcal{PT}$ phase can be restored in the convectively regime by making the region of alternating gain and loss sites in the lattice [*spatially limited*]{} rather than extended. Numerical results ----------------- We checked the predictions of the theoretical analysis and the transition form absolute to convective $\mathcal{PT}$ symmetry breaking induced by advection for both the static Rice-Mele lattice of Fig.1(a) and the ac-dc driven lattice of Fig.1(b) by direct numerical simulations. As an example, in Fig.4 we depict the evolution of a wave packet in the Rice-Mele lattice with advective term ($\rho \neq 0$, $\varphi \neq 0, \pi$), showing the transition from convective \[Fig.4(a)\] to absolute \[Fig.4(b)\] $\mathcal{PT}$ symmetry breaking. The numerical results are obtained by solving the coupled-mode equations (1) and (2) using an accurate fourth-order variable-step Runge-Kutta method assuming as an initial condition a Gaussian wave packet with carrier wave number $q_0=\pi$ at lattice sites $a_n$ solely, namely $a_n(0)=\exp[-2(n/w)^2+iq_0n] $ and $b_n(0)=0$, where $w$ is the size of the wave packet. Such an initial condition mainly excites (unstable) Bloch-Floquet modes with imaginary energy at wave numbers $q$ close to the most critical one $q=q_0=\pi$. Parameter values used in the simulations are $\kappa=\sigma=1$ (corresponding to $g_{th}=0$), $\varphi=\pi/2$, $g=0.05$ and $\rho=0.7$ in Fig.4(a), and $\rho=0.3$ in Fig.4(b). In Fig.4(a), the condition $|v_g|>\sqrt{\sigma \kappa}$ is satisfied and, according to the analysis of Sec.III.B, the symmetry breaking is of convective nature. In fact, while the wave packet $|\psi (t)|^2$ secularly grows along the ray $n=v_g t$, it decays when observed at a fixed spatial position (e.g. $n=0$), as shown in the lower panel of Fig.4(a). Conversely, in Fig.4(b) the advective term in the Rice-Mele Hamiltonian is lowered so that $|v_g|$ is smaller than $\sqrt{\sigma \kappa}$: in this case the symmetry breaking is absolute, as clearly shown in the lower panel of Fig.4(b).\ A similar transition from absolute to convective $\mathcal{PT}$ symmetry-breaking for increasing advection is observed in the ac-dc driven lattice model of Fig.1(b) presented in Sec.II.B. As shown in the Appendix A, the dynamical properties of the ac-dc driven lattice at discretized times $\tau=0,T,2T,...$ can be mapped into the ones of a static lattice with an energy band structure that is determined by the quasi-energy spectrum $E(q)$ of the ac-dc driven lattice. In particular, at large modulation frequencies the driven lattice model, defined by Eqs.(A1) and (A2), exactly reproduces the Rice-Mele model with effective hopping rates $\kappa$, $\sigma$, $\rho$ and phase $\varphi$ given by Eqs.(18-21). As an example, in Figs.5(a-c) we show the numerically-computed quasi energies of the two minibands (real and imaginary parts) for the ac-dc driven lattice above the $\mathcal{PT}$ symmetry breaking point for parameter values $\kappa_1=\kappa_2=2.1124$, $\kappa_3=1.4784$, $M=1$, $\Gamma=1.109$, $\phi=-\pi/4$, $g=0.05$ and for increasing values of the modulation frequency $\omega$. Parameter values have been chosen such as to reproduce, at large modulations frequencies, the static Rice-Mele lattice with parameters as in Fig.4(b). The quasi-energies have been obtained by numerical computation of the Floquet exponents for the eigenvalue problem defined by Eqs.(A6) and (A7) given in the Appendix. For comparison, in Fig.5(d) the minibands of the static Rice-Mele lattice with parameters of Fig.4(b) are also depicted. According to the theoretical analysis, in the high modulation regime the quasi-energy spectrum of the driven lattice asymptotically reproduces the spectrum of the static Rice-Mele model \[compare Fig.5(c) and (d)\]. At low or moderate values of the modulation frequency $\omega$, deviations from the two models can be clearly appreciated \[compare Figs.5(a) and (b) with Fig.5(d)\]. In particular, the driven lattice model at low modulation frequencies shows a wider range of wave numbers with complex energies, and the real part of the quasi energies for the two minibands are not degenerate. Nevertheless, the transition from convective to absolute symmetry breaking, which is basically related to the value of the group velocity (the derivative of the real part of the quasi-energy) of the unstable mode at the symmetry breaking point, can be observed even at moderate modulation frequencies. This is shown, as an example, in Fig.6, where we depict the numerically-computed evolution of the same initial Gaussian wave packet as in Fig.4 but in the ac-dc driven lattice for a modulation frequency $\omega=15$ and for $\kappa_3=1.4784$ \[Fig.6(a)\], corresponding to a convective symmetry breaking, and $\kappa_3=0.6336$ \[Fig.6(b)\], corresponding to absolute $\mathcal{PT}$ symmetry breaking. {width="8.3cm"} As a final comment, it is worth discussing a physically relevant implication of convective versus absolute $\mathcal{PT}$ symmetry breaking. In the convectively unstable regime, the growing wave packet drifts in the laboratory reference frame fast enough that locally (i.e. at a fixed spatial position) it is observed to decay in spite of its growth in a moving reference frame (see Figs.3 and 4). Let us now consider a lattice with a spatially confined (rather than infinitely extended) region of unit cells with gain and loss. In the convective regime, advection pushes the wave packet far from the “non-Hermitian” region of the lattice, and hence after a transient the wave packet ceases to grow. Conversely, in the absolute symmetry breaking regime it is expected to grow indefinitely even for a spatially-finite extension of unit cells with gain and loss. Such a simple physical picture suggests that in the convectively unstable regime $\mathcal{PT}$ symmetry (i.e. an entirely real-valued energy spectrum) may be restored when the gain/loss region in the lattice is spatially limited. We checked such a prediction by considering the Rice-Mele lattice Hamiltonian (3) with a spatially-dependent gain/loss term vanishing at infinity, namely $$\begin{aligned} \hat{H} & = & -\sum_{n=-\infty}^{\infty} \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_n \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& i \sum_{n=-\infty}^{\infty}g_n \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ where $g_n \rightarrow 0$ as $n \rightarrow \infty$. In particular, we numerically computed the energy spectrum of $\hat{H}$ by considering a square-wave profile of $g_n$, i.e. $g_n=g$ for $|n| \leq N_g/2$ and $g_n=0$ otherwise. This case corresponds to a central lattice section comprising $(N_g+1)$ dimers with gain and loss (i.e. locally non-Hermitian), and and abrupt transition to two outer lattice sections with locally-Hermitian dimers (i.e. $g_n=0$). In the numerical simulations, the total number of unit cells $(N+1)$ is finite, corresponding to truncation of the outer lattice sections. As an example, in Fig.7 we show the numerically-computed energies of the truncated lattice described by the Hamiltonian $$\begin{aligned} \hat{H} & = & -\sum_{n=-N/2}^{N/2} \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_{n=-N/2}^{N/2} \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& ig \sum_{n=-N_g/2}^{N_g/2} \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ for $N_g=10$, $N=300$ and for parameter values corresponding to absolute \[Fig.7(a)\] and convective \[Fig.7(b)\] $\mathcal{PT}$ symmetry breaking in the extended (i.e. $N,N_g \rightarrow \infty$) limit. Note that, within numerical accuracy, the energy spectrum is entirely real-valued in the convective regime \[Fig.7(b)\], whereas pairs of complex-conjugate energies persist in the absolute regime \[Fig.7(a)\]. It should be noted, however, that restoring of the $\mathcal{PT}$ symmetry in the convective regime is not a strict rule, since the interfaces from the outer lattice regions to the inner (non-Hermitian) lattice section might sustain localized (interface) modes with imaginary energies, which can not be predicted by our simple picture. Moreover, it is expected that restoring of the $\mathcal{PT}$ symmetry depends on the choice of the profile $g_n$; for example a smooth (rather than sharp) transition from the inner (locally non-Hermitian) to the outer (locally Hermitian) regions is expected to avoid the appearance of interface states. Symmetry breaking in case of inhomogeneous gain/loss parameter $g_n$ would require a further study, however this goes beyond the scope of the present work. Conclusions =========== In this work we have introduced the concepts of convective and absolute $\mathcal{PT}$ symmetry breaking for wave transport in periodic complex potentials, inspired by the hydrodynamic concepts of convective and absolute instabilities in open flows. In particular, we have investigated analytically and numerically the transition from absolute to convective $\mathcal{PT}$ symmetry breaking in two tight-binding lattice models: a non-Hermitian extension of the Rice-Mele dimerized lattice, originally introduced to model conjugated diatomic polymers, and an ac-dc driven lattice, which reproduces the Rice-Mele model in the large modulation frequency limit. In the context of spatially-extended dissipative dynamical systems, $\mathcal{PT}$ symmetry breaking can be viewed as a phase transition from a marginally stable state (the unbroken $\mathcal{PT}$ phase) to an unstable state (the broken $\mathcal{PT}$ phase). The instability arises because of the appearance of pairs of complex-conjugate energies in the broken $\mathcal{PT}$ phase. The distinction between convective and absolute $\mathcal{PT}$ symmetry breaking arises when considering the evolution of a wave packet in the broken $\mathcal{PT}$ phase: while in the absolute symmetry breaking case the wave packet amplitude observed at a fixed spatial position secularly grows in time, in the convective symmetry breaking case the amplitude grows in a reference frame moving at some drift velocity, however it decays when observed at a fixed spatial position, i.e. for an observer at rest. A convective regime is generally found when the unstable modes have a group (drift) velocity large enough that at a fixed spatial position the wave packet decay due to the drift overcomes the growth due to the instability. The nature (either absolute or convective) of the $\mathcal{PT}$ symmetry breaking is basically determined by the sign of the imaginary part of the energy (for static lattices) or quasi-energy (for periodically-driven lattices) at the dominant band saddle point in complex plane. An interesting application of the concepts of convective and absolute symmetry breaking is found when considering a spatially-limited region of gain/loss in the system, i.e when the periodicity of the system is broken and the imaginary part of the potential is confined to a limited region of space. Owing to the fast drift of a wave packet in the convective regime, after a transient the wave packet escapes from the imaginary potential region and thus it ceases to grow. This means that the instability is only transient, i.e. we expect that $\mathcal{PT}$ symmetry is restored in the convective regime when the imaginary potential is spatially confined. This is not the case of the absolute symmetry breaking regime, where the broken $\mathcal{PT}$ phase is expected to persist even for a spatially-limited imaginary potential. Other possible applications and developments of the hydrodynamic concepts of convective and absolute instabilities can be foreseen into the rapidly growing field of wave transport in $\mathcal{PT}$-symmetric quantum and classical systems. For example, like for hydrodynamic and dissipative optical systems [@rev1; @Santa], interesting effects (like the appearance of noise-sustained structures [@Santa]) might be envisaged for convective $\mathcal{PT}$ symmetry breaking in presence of classical or quantum noise [@uffa]. Floquet analysis of the ac-dc driven lattice and effective static lattice model =============================================================================== In this Appendix we present a Floquet analysis of the driven lattice model defined by Eqs.(10) and (11) with time-periodic coefficients and show that, at discretized times, it behaves like an effective static lattice with a band structure that is determined by the quasi-energy spectrum of the driven lattice. To this aim, let us note that, after the gauge transformation defined by Eqs.(15) and (16) given in the text, the evolution of the amplitudes $a_n(t)$, $b_n(t)$ is governed by the following linear system of equations $$\begin{aligned} i \frac{da_n}{dt} & = & -\kappa_1 F(t) b_{n}-\kappa_2 G(t) b_{n-1}-\kappa_3H(t) a_{n+1} \nonumber \\ & - & \kappa_3 H^*(t) a_{n-1}+iga_n \\ i \frac{db_n}{dt} & = & -\kappa_1 F^*(t) a_{n}-\kappa_2 G^*(t) a_{n+1}-\kappa_3H(t) b_{n+1} \nonumber \\ & - & \kappa_3 H^*(t) b_{n-1}-igb_n\end{aligned}$$ with time-dependent coefficients $F(t)$, $G(t)$ and $H(t)$ given by $$\begin{aligned} F(t) & = & \exp \left[ i \beta +i \Theta (t) \right] \nonumber \\ G(t) & = & \exp \left[ i \beta -i \varphi +i \Theta(t)-i \Phi(t) \right] \\ H(t) & = & \exp \left[ i \varphi + i \Phi(t) \right]. \nonumber\end{aligned}$$ In the previous equations, the functions $\Theta(t)$ and $\Phi(t)$ and constant parameters $\varphi$ and $\beta$ are defined by Eqs.(17) and (18) given in the text. For the driving terms $F_x$, $F_y$ defined by Eq.(13), one has explicitly $$\begin{aligned} F(t) & = & \exp \left[ iM \phi +i M \omega t -i \Gamma \sin(\omega t + \phi) \right] \nonumber \\ G(t) & = & \exp \left[ -i M (\phi+ \pi) +iM \omega t +i \Gamma \sin (\omega t - \phi) \right] \; \; \; \; \; \; \; \\ H(t) & = & \exp \left[ i M ( 2 \phi+\pi) -2 i \Gamma \cos \phi \sin (\omega t) \right]. \nonumber\end{aligned}$$ where we assumed the resonance condition $U= M \omega$. Since the coefficients $F(t)$, $G(t)$ and $H(t)$ are periodic in time with period $T=2 \pi / \omega$, the solution to Eqs.(A1) and (A2) can be obtained from Floquet theory of liner periodic systems. Specifically, the general solution to Eqs.(A1) and (A2) is given by an arbitrary superposition of Bloch-Floquet states $$\left( \begin{array}{c} a_n(q,t) \\ b_n(q,t) \end{array} \right)= \left( \begin{array}{c} A(q,t) \\ B(q,t) \end{array} \right) \exp \left[ iqn-i E(q) t \right]$$ where $q$ is the wave number (quasi-momentum), which varies in the range $(0, 2 \pi)$, $E(q)$ is the quasi-energy, with $-\omega/2 \leq {\rm {E}}(q)< \omega/2$, and $A(q,t)$, $B(q,t)$ are periodic in time with period $T$. The quasi-energy $E(q)$ and corresponding Floquet states $(A(q,t),B(q,t))^T$ are found by solving the eigenvalue problem $$\begin{aligned} E(q) A & = & -i \frac{dA}{dt}-\kappa_3[H \exp(iq)+H^* \exp(-iq)]A \nonumber \\ & + & igA-[\kappa_1 F + \kappa_2 G \exp(-iq)]B \\ E (q) B & = & -i \frac{dB}{dt}-\kappa_3[H \exp(iq)+H^* \exp(-iq)]B \nonumber \\ & - & igB-[\kappa_1 F^* + \kappa_2 G^* \exp(iq)]A \end{aligned}$$ in the interval $(0,T)$ with the periodic boundary conditions $A(q,T)=A(q,0)$ and $B(q,T)=B(q,0)$. Floquet theorem ensures that the quasi-energy spectrum comprises two branches $E(q)= E_{\pm}(q)$, like for the static lattice model discussed in Sec.II.A, with corresponding Floquet states $ \phi_{\pm}(q,t)=(A_{\pm}(q,t), B_{\pm}(q,t))^T$. Note that, if the dynamics of the system defined by Eqs.(A1) and (A2) is observed at discretized times $\tau=0,T,2T,...$, from Eq.(A5) and owing to the periodicity of the functions $A(q,t)$ and $B(q,t)$ it follows that it is equivalent to the dynamics of a static lattice with two minibands whose dispersion relations $E_{\pm}(q)$ are given by the quasi-energies of the periodic system. In fact, after setting $\psi(n,t)=(a_n(t),b_n(t))^T$, an initial wave packet, obtained from the superposition of Bloch-Floquet states with arbitrary spectra $F_{\pm}(q)$, evolves in time according to the relation $$\begin{aligned} \psi(n,t) & = & \int_{0}^{2 \pi}dq F_+(q) \phi_+(q,t) \exp[iqn-iE_{+}(q)t] \nonumber \\ & + & \int_{0}^{2 \pi}dq F_-(q) \phi_-(q,t) \exp[iqn-iE_{-}(q)t]. \;\;\;\;\;\end{aligned}$$ If the evolution of the wave packet is observed at discretized times $\tau =l T$ with $l=0,1,2,3,...$, since $\phi_{\pm}(q, l T)=\phi_{\pm}(q, 0)$ is independent of $\tau$, from Eq.(A8) it follows that $\psi(n,\tau)$ shows the same evolution as the one of a wave packet in a static dimerized lattice with energy band dispersion given by the quasi-energies $E_{\pm}(q)$ of the time-periodic lattice \[compare Eq.(A8) with Eq.(27) given in the text\]. Therefore, all the dynamical aspects of the time-periodic system defined by Eqs.(A1) and (A2), including the onset of $\mathcal{PT}$ symmetry breaking and its convective or absolute nature, can be derived from an equivalent static lattice with a band structure given by the quasi-energy band structure of the original time-periodic system. In particular, as discussed in Sec.III the convective or absolute nature of the symmetry breaking will be determined by the imaginary part of the quasi-energies at the dominant saddle point. The determination of the quasi-energy spectrum $E_{\pm}(q)$ generally requires to resort to a numerical analysis of Eqs.(A6) and (A7). An approximate analytical form the quasi energies can be obtained, however, in the large frequency limit. In fact, assuming $\kappa_{1}, \kappa_2, \kappa_3,g \ll \omega$, the change of the amplitudes $A$ and $B$ over one oscillation cycle are small, so that in Eqs.(A6) and (A7) we may neglect the derivative terms $(dA/ dt)$, $(dB/dt)$ and replace the functions $F(t)$, $G(t)$, $H(t)$ with their average values over the oscillation cycle (rotating-wave approximation), namely one can set $$\begin{aligned} E(q) A & \simeq & -\kappa_3[ \langle H \rangle \exp(iq)+\langle H^* \rangle \exp(-iq)]A \nonumber \\ & + & igA-[\kappa_1 \langle F \rangle + \kappa_2 \langle G \rangle \exp(-iq)]B \\ E (q) B & \simeq & -\kappa_3[\langle H \rangle \exp(iq)+\langle H^* \rangle \exp(-iq)]B \nonumber \\ & - & igB-[\kappa_1 \langle F^* \rangle + \kappa_2 \langle G^* \rangle \exp(iq)] A . \; \; \; \; \end{aligned}$$ where $\langle ... \rangle$ denotes the time average. Using the identity of Bessel functions $\exp(i \Gamma \sin x)=\sum_{n=-\infty}^{\infty}J_n(\Gamma) \exp(inx)$, from Eq.(A4) one readily obtains $$\langle F \rangle = \langle G \rangle =J_M(\Gamma), \;\; \langle H \rangle =J_0(2 \Gamma \cos \phi) \exp(i \varphi) \;\;\;\;$$ where $\varphi=M(\pi+2 \phi)$. A comparison of Eqs.(A9),(A10) with Eq.(5) given in the text shows that, in the large modulation limit, the ac-dc driven lattice described by Eqs.(A1) and (A2) effectively describes the static Rice-Mele lattice \[Eqs.(1) and (2) given in the text\], where the effective hopping rate $\kappa$, $\sigma$ and $\rho$ are given by Eqs.(19-21) and the phase $\varphi$ by Eq.(18). Determination of the saddle point for the extended Rice-Mele lattice model ========================================================================== In this Appendix we calculate the saddle points $q_s$, i.e. the roots $q_s$ of Eq.(29) given in the text, which determine the convective or absolute nature of the $\mathcal{PT}$ symmetry breaking for the non-Hermitian Rice-Mele Hamiltonian (3). To this aim, it is worth introducing the variables $X_s= \cos q_s$ and $Y_s=\sin q_s$. After some algebra, from Eq.(29) it follows that $X_s$ and $Y_s$ are the roots (in the complex plane) of following system of algebraic equations $$\begin{aligned} X_s^2+Y_s^2 & = & 1 \\ \left( \frac{\kappa \sigma}{2 \rho} \right)^2 Y_s^2 & = & \left( \cos^2 \varphi Y_s^2+ \sin^2 \varphi X_s^2-2 \cos \varphi \sin \varphi X_s Y_s\right) \;\;\; \nonumber \\ & \times & (-\epsilon^2+2 \kappa \sigma +2 \kappa \sigma X_s).\end{aligned}$$ To simplify the analysis, let us consider the case where the gain/loss parameter $g$ is larger but close to its threshold value $g_{th}$, so that $g^2-g_{th}^2=\epsilon^2$ is a small quantity. Note that, for $\epsilon \rightarrow 0$, a solution to Eqs.(B1) and (B2) is $X_s=-1$ and $Y_s=0$, corresponding to $q_s=\pi$, i.e. to the wave number where the most unstable mode arises at the $\mathcal{PT}$ symmetry breaking threshold. For $\epsilon^2>0$, we look for a solution to Eqs.(B1) and (B2) in the form of power series $$\begin{aligned} X_s & = & -1+\frac{\alpha^2}{2}-\frac{\alpha^4}{4 !} + ... \\ Y_s & = & -\alpha +\frac{\alpha^3}{3 !} -\frac{\alpha^5}{5!}+ ... \; ,\end{aligned}$$ where $\alpha=q_s-\pi$ is a small amplitude of order $\epsilon^ \gamma$ with $\gamma>0$ to be determined. Note that, at leading order in $\alpha$, the energy $E_{+}(q_s)$ is given by $$E_+(q_s) \simeq -2 \rho \cos \varphi - 2 \rho \sin \varphi \alpha +\sqrt{\kappa \sigma \alpha^2-\epsilon^2}.$$ Note also that, with the Ansatz (B3) and (B4), Eq.(B1) is automatically satisfied for any $\alpha$. The small complex amplitude $\alpha$ can be determined by substitution of Eqs.(B3) and (B4) into Eq.(B2) and letting equal the terms of lowest order on the left and right hands of the equations so obtained. Three cases should be distinguished.\ \ (1) $|v_g| \neq \sqrt{\sigma \kappa}$, where $v_g=-2 \rho \cos \varphi$.\ In this case Eq.(B2) is satisfied at leading order for $\alpha \sim \epsilon $ (i.e. $ \gamma=1$), namely one obtains $$\alpha^2=\frac{\epsilon^2 v_g^2}{\kappa \sigma (v_g^2-\kappa \sigma)}.$$ For $v_g^2 > \sigma \kappa$, the two roots $ \alpha$ of Eq.(B6) are real-valued, and correspondingly the imaginary part of $E_{+}(q_s)$, with $q_s=\pi+\alpha$, vanishes \[see Eq.(B5)\]. Therefore, for $v_g^2 > \sigma \kappa$ one has $\psi(n,t) \rightarrow 0$ as $t \rightarrow \infty$ along the ray $n/t=0$, i.e. the $\mathcal{PT}$ symmetry breaking is convective. Conversely, for $v_g^2 < \sigma \kappa$ according to Eq.(B6) the amplitude $\alpha$ is purely imaginary, and correspondingly for one of the two roots the imaginary part of $E_{+}(q_s)$ is positive according to Eq.(B5). In this case $|\psi(n,t)| \rightarrow \infty $ as $t \rightarrow \infty$ along the ray $n/t=0$, i.e. the $\mathcal{PT}$ symmetry breaking is absolute.\ \ (2) $|v_g| = \sqrt{\sigma \kappa}$ and $\varphi \neq \pm \pi/2$. In this case one obtains $\alpha \sim \epsilon^{2/3}$, i.e. $\gamma=2/3$, and $\alpha$ satisfies the cubic equation $$\alpha^3= - \frac{\sin \varphi \epsilon^2}{2 \kappa \sigma \cos \varphi}.$$ Two of the three roots of such an equation are complex-valued, and correspondingly one can readily shown from Eq.(B5) that a positive imaginary part for the energy $E_+(q_s)$ arises from one of the two complex roots. In fact, since $\epsilon^2$ is of higher order than $\alpha^2$ and $2 \rho \cos \varphi = \sqrt{\kappa \sigma}$, from Eq.(B5) one has ${\rm Im} \{ E_+(q_s)\} \simeq 2 v_g {\rm Im} (\alpha)$. Therefore in this case the $\mathcal{PT}$ symmetry breaking is absolute.\ \ (3) $|v_g| = \sqrt{\sigma \kappa}$ and $\varphi = \pm \pi/2$. In this case one has $\alpha \sim \epsilon^{1/2}$, i.e. $\gamma=1/2$, and $\alpha$ satisfies the quartic equation $$\alpha^4=-\frac{\epsilon^2}{\kappa \sigma}.$$ The four roots of such equation are complex-valued, two with positive and two with negative imaginary parts. Correspondingly, like in the previous case a positive imaginary part for the energy $E_+(q_s)$ does appear because ${\rm Im} \{ E_+(q_s)\} \simeq 2 v_g {\rm Im} (\alpha)$. Therefore the $\mathcal{PT}$ symmetry breaking is absolute like in the previous case. [99]{} N. Moiseyev, [ *Non-Hermitian Quantum Mechanics*]{} (Cambridge University Press, London, Cambridge, 2011). N. Moiseyev, Phys. Rep. [**302**]{}, 212 (1998); J.G. Muga, J.P. Palao, B. Navarro, and I.L. Egusquiza Phys. Rep. [**395**]{}, 357 (2004); I. Rotter, J. Phys. A: Math. Theor. [**42**]{}, 153001 (2009). C. Keller, M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, Phys. Rev. Lett. [**79**]{}, 3327 (1997). R. Stützle, M.C. Göbel, T. Hörner, E. Kierig, I. Mourachko, M.K. Oberthaler, M.A. Efremov, M.V. Fedorov, V.P. Yakovlev, K.A.H. van Leeuwen, and W.P. Schleich, Phys. Rev. Lett. [**95**]{}, 110405 (2005). M.S. Rudner and L.S. Levitov, Phys. Rev. Lett. [**102**]{}, 065703 (2009). E. M. Graefe, H. J. Korsch, and A. E. Niederle, Phys. Rev. Lett. [**101**]{}, 150408 (2008); Phys. Rev. A [**82**]{}, 013629 (2010). A. Kostenbauder, Y. Sun, and A. E. Siegman, J. Opt. Soc. Am. A [**14**]{}, 1780 (1997). C. M. Bender, Rep. Prog. Phys. [**70**]{}, 957 (2007). C. M. Bender and S. Boettcher, Phys. Rev. Lett. [**80**]{}, 5243 (1998). A. Mostafazadeh, J. Phys. A [**36**]{}, 7081 (2003). C.M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. [**93**]{}, 251601 (2004). A. Mostafazadeh, J. Math. Phys. [**43**]{}, 205 (2002). U. Guenther, F. Stefani, and M. Znojil, J. Math. Phys. [**46**]{}, 063504 (2005); U. Guenther, B. F. Samsonov, and F. Stefani, J. Phys. A: Math. Theor. [**40**]{}, F169 (2007). M. B. Plenio and P. L. Knight, Rev. Mod. Phys. [**70**]{}, 101 (1998). C. M. Bender, S. F. Brandt, J.-H. Chen, and Q.Wang, Phys. Rev. D [**71**]{}, 025014 (2005). J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, and T. Kottos, Phys. Rev. A [**84**]{}, 040101 (2011); H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, Phys. Rev. A [**85**]{}, 062122 (2012); Z. Lin, J. Schindler, F.M. Ellis, and T. Kottos, Phys. Rev. A [**85**]{}, 050101 (2012). A. Ruschhaupt, F. Delgado, and J.G. Muga, J. Phys. A: Math. Gen. [**38**]{}, L171 (2005). R. El-Ganainy, K.G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. [**32**]{}, 2632 (2007). S. Klaiman, U. Günther, and N. Moiseyev, Phys. Rev. Lett. [**101**]{}, 080402 (2008). K. G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. [**100**]{}, 103904 (2008). S. Longhi, Phys. Rev. Lett. [**103**]{}, 123601 (2009). A. Mostafazadeh, Phys. Rev. Lett. [**102**]{}, 220402 (2009). A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. [**103**]{}, 093902 (2009). C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, Nat. Phys. [**6**]{}, 192 (2010). S. Longhi, Phys. Rev. Lett. [**105**]{}, 013903 (2010). L. Feng, M. Ayache, J. Huang, Y.L. Xu, M.H. Lu, Y.F. Chen, Y. Fainman, and A. Scherer, Science [**333**]{}, 729 (2011). S. V. Dmitriev, A. A. Sukhorukov, and Yu. S. Kivshar, Opt. Lett. [**35**]{}, 2976(2010); A.A. Sukhorukov, Z. Xu, and Y.S. Kivshar, Phys. Rev. A [**82**]{}, 043818 (2010); N. V. Alexeeva, I. V. Barashenkov, A.A. Sukhorukov, and Y.S. Kivshar, Phys. Rev. A [**85**]{}, 063837 (2012). A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature [**488**]{}, 167 (2012). L. Feng, Y.-L. Xu, W.S. Fegadolli, M.-H. Lu, J.E.B. Oliveira, V.R. Almeida, Y.-F. Chen, and A. Scherer, Nature Mat. [**12**]{}, 108 (2013). H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010); G.S. Agarwal and K. Qu, Phys. Rev. A 85, 031802(R) (2012); G. Yoo, H.-S. Sim, and H. Schomerus, Phys. Rev. A [**84**]{}, 063833 (2011). A. Mostafazadeh, Phys. Rev. A [**87**]{}, 012103 (2013). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. [**106**]{}, 213901 (2011). S. Longhi, Phys. Rev. A [**81**]{}, 022102 (2010). S. Longhi, J. Phys. A: Math. Theor. [**44**]{}, 485302 (2011). E.-M. Graefe and H. F. Jones, Phys. Rev. A [**84**]{}, 013818 (2011). A. Mostafazadeh, Phys. Rev. A [**87**]{}, 012103 (2013). H. Ramezani, T. Kottos, R. El-Ganainy, and D.N. Christodoulides, Phys. Rev. A [**82**]{}, 043803 (2010). S. Longhi, G. Della Valle and K. Staliunas, Phys. Rev. A [**84**]{}, 042119 (2011). S. Longhi, Phys. Rev. A [**82**]{}, 031801(R) (2010). Y.D. Chong, L. Ge, and A.D. Stone, Phys. Rev. Lett. [**106**]{}, 093902 (2011). A. Mostafazadeh, Phys. Rev. Lett. [**102**]{}, 220402 (2009). M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. [**65**]{}, 851 (1993). P. Huerre and P.A. Monkewitz, Annu. Rev. Fluid Mech. [**22**]{}, 473 (1990). L. S. Hall and W. Heckrotte, Phys. Rev. [**166**]{}, 120 (1968); R. J. Deissler, J. Stat. Phys. [**40**]{}, 376 (1985); Physica (Amsterdam) [**56D**]{}, 303 (1992). G. Dee and J.S. Langer, Phys. Rev. Lett. [**50**]{}, 383 (1983). M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. [**79**]{}, 3633 (1997); M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E [**58**]{}, 3843 (1998); M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E [**61**]{}, 2133 (2000). M. J. Rice and E. J. Mele, Phys. Rev. Lett. [**49**]{}, 1455 (1982). The Hamiltonian (3) reduces to the Rice-Mele model for $\rho=0$ and after the substitution $g \rightarrow ig$, which makes the Hamiltonian Hermitian. M. Atala, M. Aidelsburger, J.T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, arXiv:1212.0572 Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A [**41**]{}, 244019 (2008); S.V. Suchkov, S.V. Dmitriev, B.A. Malomed, and Y.S. Kivshar, Phys. Rev. A [**85**]{}, 033825 (2012); G. Della Valle and S. Longhi, Phys. Rev. A [**87**]{}, 022119 (2013); I. V. Barashenkov, L. Baker, and N. V. Alexeeva, Phys. Rev. A [**87**]{}, 033819 (2013). O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. [**103**]{}, 030402 (2009); S. Longhi, Phys. Rev. B [**80**]{}, 165125 (2009); G.L. Giorgi, Phys. Rev. B [**82**]{}, 052404 (2010); M.C. Zheng, D.N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A [**82**]{}, 010103 (2010); S. Longhi, Phys. Rev. A [**82**]{}, 032111 (2010); A. Szameit, M.C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A [**84**]{}, 021806 (2011); H. Ramezani, T. Kottos, V. Kovanis, and D.N. Christodoulides, Phys. Rev. A [**85**]{}, 013818 (2012). L. Jin and Z. Song, Phys. Rev. A [**80**]{}, 052107 (2009); L. Jin and Z. Song, Phys. Rev. A [**84**]{}, 042116 (2011) ; X. Z. Zhang, L. Jin, and Z. Song, Phys. Rev. A [**85**]{}, 012106 (2012); L. Jin and Z. Song, Phys. Rev. A [**85**]{}, 012111 (2012); W. H. Hu, L. Jin, Y. Li, and Z. Song Phys. Rev. A [**86**]{}, 042110 (2012). Y.N. Joglekar, C. Thompson, and G. Vemuri, Phys. Rev. A [**83**]{}, 063817 (2011); D.D. Scott and Y.N. Joglekar, Phys. Rev. A [**83**]{}, 050102 (2011); Y.N. Joglekar and J.L. Barnett, Phys. Rev. A [**84**]{}, 024103 (2011); D.D. Scott and Y.N. Joglekar, Phys. Rev. A [**85**]{}, 062105 (2012); H. Vemuri and Y.N. Joglekar, Phys. Rev. A [**87**]{}, 044101 (2013). H. Cartarius and G. Wunner, Phys. Rev. A [**86**]{}, 013612 (2012). F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. [**33**]{}, 2689 (2008). A. Crespi, G. Corrielli, G. Della Valle, R. Osellame, and S. Longhi, New J. Phys. [**15**]{}, 013012 (2013). S. Longhi, Laser Photon. Rev. [**3**]{}, 243 (2009); I.L. Garanovich, S. Longhi, A.A. Sukhorukov, and Y.S. Kivshar, Phys. Rep. [**518**]{}, 1 (2012). S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. [**96**]{}, 243901 (2006). A. Joushaghani, R. Iyer, J.K.S. Poon, J.S. Aitchison, C.M. de Sterke, J. Wan, and M.M. Dignam, Phys. Rev. Lett. [**103**]{}, 143903 (2009). See, for instance: F. W. J. Olver, [*Asymptotics and Special Functions*]{} (Academic, New York, 1974).

ArXiv

--- abstract: 'We theoretically investigate heat transport in temperature-biased Josephson tunnel junctions in the presence of an in-plane magnetic field. In full analogy with the Josephson critical current, the phase-dependent component of the heat flux through the junction displays *coherent diffraction*. Thermal transport is analyzed in three prototypical junction geometries highlighting their main differences. Notably, minimization of the Josephson coupling energy requires the quantum phase difference across the junction to undergo $\pi$ *slips* in suitable intervals of magnetic flux. An experimental setup suited to detect thermal diffraction is proposed and analyzed.' author: - 'F. Giazotto' - 'M. J. Martínez-Pérez' - 'P. Solinas' title: Coherent diffraction of thermal currents in Josephson tunnel junctions --- Introduction {#intro} ============ The impressive advances achieved in nanoscience and technology are nowadays enabling the understanding of one central topic in science, i.e., *thermal flow* in solid-state nanostructures [@Giazotto2006; @Dubi2011]. Control and manipulation [@heattransistor; @ser] of thermal currents in combination with the investigation of the origin of dissipative phenomena are of particular relevance at such scale where heat deeply affects the properties of the systems, for instance, from *coherent caloritronic* circuits, which allow enhanced operation thanks to the quantum phase [@Meschke2006; @Vinokur2003; @Eom1998; @Chandrasekhar2009; @Ryazanov1982; @Panaitov1984; @virtanen2007; @Martinez2013], to more developed research fields such as ultrasensitive radiation detectors [@Giazotto2006; @Giazotto2008] or cooling applications [@Giazotto2006; @Giazotto2002]. In this context it has been known for more than $40$ years that heat transport in Josephson junctions can be, in principle, phase-dependent [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013]. The first ever Josephson thermal interferometer has been, however, demonstrated only very recently [@giazotto2012; @martinez2012; @giazottoexp2012; @simmonds2012], therefore proving that phase coherence extends to thermal currents as well. The heat interferometer of Ref. [@giazottoexp2012] might represent a prototypical circuit to implement novel-concept coherent caloritronic devices such as heat transistors [@martinez2012], thermal splitters and rectifiers [@Martinez2013]. In the present work we theoretically analyze heat transport in temperature-biased extended Josephson tunnel junctions showing that the phase-dependent component of thermal flux through the weak-link interferes in the presence of an in-plane magnetic field leading to *heat diffraction*, in analogy to what occurs for the Josephson critical current. In particular, thermal transport is investigated in three prototypical *electrically-open* junctions geometries showing that the quantum phase difference across the junction undergoes $\pi$ *slips* in order to minimize the Josephson coupling energy. These phase slips have energetic origin and are not related to fluctuations as conventional phase slips in low-dimensional superconducting systems [@Langer1967; @Zaikin1997; @astafiev2012]. We finally propose how to demonstrate thermal diffraction in a realistic microstructure, and to prove such $\pi$ slips exploiting an uncommon observable such as the heat current. The paper is organized as follows: In Sec. \[model\] we describe the general model used to derive the behavior of the heat current in a temperature-biased extended Josephson tunnel junction. In Sec. \[results\] we obtain the conditions for the quantum phase difference across an electrically-open short Josephson junction in the presence of an in-plane magnetic field, and the resulting behavior of the phase-dependent thermal current. In particular, we shall demonstrate the occurrence of phase-slips of $\pi$, independently of the junction geometry, in order to minimize the Josephson coupling energy. The phase-dependent heat current in three specific junction geometries is further analyzed in Sec. \[differentgeo\], where we highlight their main differences. In Sec. \[experiment\] we suggest and analyze a possible experimental setup suited to detect heat diffraction through electronic temperature measurements in a microstructure based on an extended Josephson junction, and to demonstrate the existence of $\pi$ slips. Finally, our results are summarized in Sec. \[summary\]. ![(Color online) (a) Cross section of a temperature-biased extended S$_1$IS$_2$ Josephson tunnel junction in the presence of an in-plane magnetic field $H$. The heat current $J_{S_1\rightarrow S_2}$ flows along the $z$ direction whereas $H$ is applied in the $x$ direction, i.e., parallel to a symmetry axis of the junction. Dashed line indicates the closed integration contour, $T_i$, $t_i$ and $\lambda_i$ represent the temperature, thickness and London penetration depth of superconductor S$_i$, respectively, and $d$ is the insulator thickness. $\Phi$ denotes the magnetic flux piercing the junction. Prototypical junctions with rectangular, circular, and annular geometry are shown in panel (b), (c) and (d), respectively. $L$, $W$, $R$ and $r$ represent the junctions geometrical parameters. []{data-label="fig1"}](fig1.pdf){width="\columnwidth"} Model ===== Our system is schematized in Fig. \[fig1\](a), and consists of an extended Josephson tunnel junction composed of two superconducting electrodes S$_1$ and S$_2$ in thermal steady-state residing at different temperatures $T_1$ and $T_2$, respectively. We shall focus mainly on symmetric Josephson junctions in the *short* limit, i.e., with lateral dimensions much smaller than the Josephson penetration depth \[see Fig. \[fig1\](b,c,d)\], $L,W,R,r\ll \lambda_J=\sqrt{\frac{\pi \Phi_0}{\mu_0i_ct_H}}$, where $\Phi_0=2.067\times 10^{-15}$ Wb is the flux quantum, $\mu_0$ is vacuum permeability, $i_c$ is the critical current areal density of the junction, and $t_H$ is the junction effective magnetic thickness to be defined below. In such a case the self-field generated by the Josephson current in the weak-link can be neglected with respect to the externally applied magnetic field, and no traveling solitons can be originated. $t_i$ and $\lambda_i$ denote the thickness and London penetration depth of superconductor S$_i$, respectively, whereas $d$ labels the insulator thickness. We choose a coordinate system such that the applied magnetic field ($H$) lies parallel to a symmetry axis of the junction and along $x$, and that the junction electrodes planes are parallel to the $xy$ plane. Furthermore, the junction lateral dimensions are assumed to be much larger than $d$ so that we can neglect the effects of the edges, and each superconducting layer is assumed to be thicker than its London penetration depth (i.e., $t_i>\lambda_i$) so that $H$ will penetrate the junction in the $z$ direction within a thickness $t_H=\lambda_1+\lambda_2+d$ [@magneticlength]. For definiteness, we assume $T_1\geq T_2$ so that the Josephson junction is temperature biased only, and no electric current flows through it. If $T_1\neq T_2$ there is a finite electronic heat current $J_{S_1\rightarrow S_2}$ flowing through the junction from S$_1$ to S$_2$ \[see Fig. \[fig1\](a)\] which is given by [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{S_1\rightarrow S_2}(T_1,T_2,\varphi)=J_{qp}(T_1,T_2)-J_{int}(T_1,T_2)\textrm{cos}\varphi. \label{heatcurrent}$$ Equation (\[heatcurrent\]) describes the oscillatory behavior of the thermal current flowing through a Josephson tunnel junction as a function of $\varphi$ predicted by Maki and Griffin [@Maki1965], and experimentally verified in Ref. [@giazottoexp2012]. In Eq. (\[heatcurrent\]), $J_{qp}$ is the usual heat flux carried by quasiparticles [@Giazotto2006; @Frank1997], $J_{int}$ is the phase-dependent part of the heat current which is peculiar of Josephson tunnel junctions, and $\varphi$ is the macroscopic quantum phase difference between the superconductors. By contrast, the Cooper pair condensate does not contribute to heat transport in a static situation [@Maki1965; @Golubev2013; @giazottoexp2012]. The two terms appearing in Eq. (\[heatcurrent\]) read [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{qp}=\frac{1}{e^2 R_J} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)\mathcal{N}_2 (\varepsilon,T_2)[f(\varepsilon,T_2)-f(\varepsilon,T_1)],$$ and $$J_{int}=\frac{1}{e^2 R_J}\int^{\infty}_{0}d\varepsilon \varepsilon \mathcal{M}_{1}(\varepsilon,T_1)\mathcal{M}_{2}(\varepsilon,T_2)[f(\varepsilon,T_2)-f(\varepsilon,T_1)],$$ where $\mathcal{N}_{i}(\varepsilon,T_i)=|\varepsilon|/\sqrt{\varepsilon^2-\Delta_{i}(T_i)^2}\Theta[\varepsilon^2-\Delta_{i}(T_i)^2]$ is the BCS normalized density of states in S$_{_i}$ at temperature $T_i$ ($i=1,2$), $\mathcal{M}_{i}(\varepsilon,T_i)=\Delta_{i}(T_i)/\sqrt{\varepsilon^2-\Delta_{i}(T_i)^2}\Theta[\varepsilon^2-\Delta_{i}(T_i)^2]$, and $\varepsilon$ is the energy measured from the condensate chemical potential. Furthermore, $\Delta_i(T_i)$ is the temperature-dependent superconducting energy gap, $f(\varepsilon,T_i)=\text{tanh}(\varepsilon/2 k_BT_i)$, $\Theta(x)$ is the Heaviside step function, $k_B$ is the Boltzmann constant, $R_J$ is the junction normal-state resistance, and $e$ is the electron charge. In the following analysis we neglect any contribution to thermal transport through the Josephson junction arising from lattice phonons. Results ======= In order to discuss the effect of the applied magnetic field on the heat current we shall focus first of all onto the phase-dependent component. To this end we need to determine the phase gradient $\varphi (x,y)$ induced by the application of the external magnetic flux. By choosing the closed integration contour indicated by the dashed line depicted in Fig. \[fig1\](a) it can be shown [@Tinkham; @Barone] that, neglecting screening induced by the Josephson current, $\varphi (x,y)$ obeys the equations $\partial \varphi/\partial x=0$ and $\partial \varphi/\partial y=2\pi \mu_0 t_H H/\Phi_0$. The latter equation can be easily integrated to yield $$\varphi (y)=\kappa y+\varphi_0,$$ where $\kappa \equiv 2\pi \mu_0 t_H H/\Phi_0$ and $\varphi_0$ is the phase difference at $y=0$. The phase-dependent component of the heat current can then be written as $$J_{H}(T_1,T_2,H)=\int\int dxdy J_A(x,y,T_1,T_2)\textrm{cos}(\kappa y+\varphi_0), \label{phasea}$$ where the integration is performed over the junction area, and $J_A(x,y,T_1,T_2)$ is the heat current density per unit area. We note that the integrand of Eq. (\[phasea\]) oscillates sinusoidally along the $y$ direction with period given by $\Phi_0(\mu_0t_H H)^{-1}$. After integration over $x$ we can write Eq. (\[phasea\]) as $$\begin{aligned} J_{H}(T_1,T_2,H)&=&\int dy \mathcal{J}(y,T_1,T_2)\textrm{cos}(\kappa y+\varphi_0)\nonumber\\ &=&\textrm{Re}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{J}(y,T_1,T_2)e^{i\kappa y}\right\}, \label{phaseb}\end{aligned}$$ where $\mathcal{J}(y,T_1,T_2)\equiv \int dx J_A(x,y,T_1,T_2)$ is the heat current density per unit length along $y$. In writing second equality in Eq. (\[phaseb\]) we have replaced the integration limits by $\pm \infty$ since the thermal current is zero outside the junction. Equation (\[phaseb\]) for $J_{H}(T_1,T_2,H)$ resembles the expression for the Josephson current, $I_{H}(T_1,T_2,H)$, which is given by [@Tinkham; @Barone] $$\begin{aligned} I_{H}(T_1,T_2,H)&=&\textrm{Im}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{I}(y,T_1,T_2)e^{i\kappa y}\right\} \nonumber\\ &=&\textrm{sin}\varphi_0\int^{\infty}_{-\infty} dy\mathcal{I}(y)\textrm{cos}\kappa y, \label{critical}\end{aligned}$$ where $\mathcal{I}(y,T_1,T_2)$ is the supercurrent density integrated along $x$, and second equality in Eq. (\[critical\]) follows from the assumed junctions *symmetry*, i.e., $\mathcal{I}(y,T_1,T_2)=\mathcal{I}(-y,T_1,T_2)$. In the actual configuration of electrically-open junction, the condition of *zero* Josephson current for any given value of $H$ yields the solution $\varphi_0=m\pi$, with $m=0,\pm 1,\pm2\ldots$. On the other hand, the Josephson coupling energy of the junction ($E_J$) can be expressed as $$\begin{aligned} E_J(T_1,T_2,H)&=&E_{J,0}-\frac{\Phi_0}{2\pi}\textrm{Re}\left\{e^{i\varphi_0}\int_{-\infty}^{\infty} dy \mathcal{I}(y,T_1,T_2)e^{i\kappa y}\right\}\nonumber\\ &=&E_{J,0}-\frac{\Phi_0}{2\pi}\textrm{cos}\varphi_0\int_{-\infty}^{\infty} dy\mathcal{I}(y)\textrm{cos}\kappa y \label{energy}\end{aligned}$$ where $E_{J,0}=\Phi_0I_c/2\pi$, $I_c$ is the zero-field critical supercurrent, and in writing the second equality we have used the symmetry property of $\mathcal{I}(y,T_1,T_2)$. Minimization of $E_J$ for any applied $H$ imposes the second term on rhs of Eq. (\[energy\]) to be always negative, so that $\varphi_0$ will undergo a $\pi$ *slip* whenever the integral does contribute to $E_J$ with negative sign. As a result, the Josephson coupling energy turns out to be written as $$E_J(T_1,T_2,H)=E_{J,0}-\frac{\Phi_0}{2\pi}\left|\int_{-\infty}^{\infty} dy\mathcal{I}(y,T_1,T_2)\textrm{cos}\kappa y\right|. \label{energybis}$$ We also assume that the symmetry of the junction and of the electric current density are reflected in an analogous symmetry in the heat current, i.e., $\mathcal{J}(y,T_1,T_2) = \mathcal{J}(-y,T_1,T_2)$. It therefore follows from Eq. (\[phaseb\]) that $J_H$ can be written as $$J_{H}(T_1,T_2,H)=\left|\int^{\infty}_{-\infty} dy \mathcal{J}(y,T_1,T_2)\textrm{cos}\kappa y\right|. \label{heatcurrfin}$$ Equation (\[heatcurrfin\]) is the main result of the paper. The above results hold for symmetric Josephson junctions under in-plane magnetic field parallel to a symmetry axis, and only occur without any electrical bias. The discussed phase slips, however, exists for any arbitrary junction geometry. If the junction has not a symmetric geometry with respect to the magnetic field direction, the constraints on vanishing Josephson current and minimization of coupling energy are translated in a more complex condition for the phase $\varphi_0$. As we shall demonstrate below, the phase undergoes nevertheless a $\pi$ slip as well. We choose the $x$ axis of the coordinate system parallel to the magnetic field \[as in Fig. \[fig1\](a)\]. For a junction with arbitrary symmetry, we split $\mathcal{I}(y,T_1,T_2)$ in its symmetric, $ \mathcal{I}_s(y)$, and antisymmetric part, $ \mathcal{I}_a(y)$. We thus have $$\begin{aligned} I_H&=&\textrm{Im}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{I}(y)e^{i\kappa y}\right\} \nonumber\\ &=& \cos \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_a (y) \sin \kappa y + \sin \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_s (y) \cos \kappa y\nonumber\\ &=& \cos \varphi_0 I_a + \sin \varphi_0 I_s, \label{eq:zero_current}\end{aligned}$$ where we have denoted the symmetric and antisymmetric integrals as $I_s$ and $I_a$, respectively. The case of symmetric junctions has already been discussed above so that in the following we shall focus on junctions with no symmetry, i.e., with $I_s\neq0$ [*and*]{} $I_a\neq0$. Notice that this already has consequences on the values of $\varphi_0$. In fact, if $I_s\neq0$ [*and*]{} $I_a\neq0$ we must have $\cos \varphi_0 \neq 0$ [*and*]{} $\sin \varphi_0 \neq 0$ to satisfy the zero-current condition. The Josephson coupling energy can be written as $$\begin{aligned} E_J&=&E_{J,0} - \frac{\Phi_0}{2 \pi} \Big [\cos \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_s (y) \cos \kappa y\nonumber\\ &-& \sin \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_a (y) \sin \kappa y \Big ]\nonumber\\ &=&E_{J,0} + \frac{\Phi_0}{2 \pi} \Big [ -\cos \varphi_0 I_s + \sin \varphi_0 I_a \Big].\end{aligned}$$ To find the energy minima, we differentiate twice with respect to $\varphi_0$ and impose the condition $\partial^2 E_J / \partial \varphi_0^2 > 0$. We therefore obtain $$\frac{\partial^2 E_J}{ \partial \varphi_0^2} = \frac{\Phi_0}{2 \pi} \Big [ \cos \varphi_0 I_s - \sin \varphi_0 I_a \Big] >0. \label{eq:energy_min}$$ Assuming that that $I_s \neq 0$, the condition of vanishing $I_H$ for any applied $H$ from Eq. (\[eq:zero\_current\]) reads $$\sin \varphi_0 = - \cos \varphi_0 \frac{I_a}{I_s} ~ {\rm or }~ \tan \varphi_0 = - \frac{I_a}{I_s}. \label{eq:sin_varphi_0}$$ By using the first of Eqs. (\[eq:sin\_varphi\_0\]), the condition to have minima in Eq. (\[eq:energy\_min\]) gives $$\frac{\partial^2 E_J}{ \partial \varphi_0^2} = \frac{\Phi_0}{2 \pi} \cos \varphi_0 \Big(I_s + \frac{I_a^2}{I_s} \Big) = \frac{\Phi_0}{2 \pi} \frac{\cos \varphi_0}{I_s} \left(I_s^2 + I_a^2 \right) > 0 \label{eq:min_condition}$$ that depends on the signs of $I_s$ and $\cos \varphi_0$. Now we turn to equations (\[eq:sin\_varphi\_0\]) that impose a constraint on $\varphi_0$ as a function of $I_s$ and $I_a$. To simplify the discussion we denote as $\phi_0 = - \arctan (I_a/I_s)$, and consider only solutions within a $2\pi$ variation from the latter. We have two solutions of Eqs. (\[eq:sin\_varphi\_0\]): $\varphi_{0,1} = \pi + \phi_0$ and $\varphi_{0,2} = \phi_0$ which correspond to cosine function $$\begin{aligned} \cos \varphi_{0,1} &=& - \cos \phi_0 = - \frac{1}{\sqrt{1+ \left(\frac{I_a}{I_s} \right)^2}} \nonumber \\ \cos \varphi_{0,2} &=& \cos \phi_0 = \frac{1}{\sqrt{1+ \left(\frac{I_a}{I_s} \right)^2}},\end{aligned}$$ where we have used the relation $\cos( \arctan x)= 1/\sqrt{1+x^2}$. As we can see, the first solution gives a negative $\cos \varphi_{0,1}$ while the second one corresponds to positive $\cos \varphi_{0,2}$. Going back to the inequality (\[eq:min\_condition\]), if $I_s>0$ we need to choose the solution $\varphi_{0,2} = \phi_0$ (for which $ \cos \varphi_{0,2}>0$) to minimize the Josephson coupling energy. By contrast, if $I_s<0$, we must choose the solution $\varphi_{0,1} = \pi + \phi_0$ (for which $ \cos \varphi_{0,1}<0$). Therefore, we get that the superconducting phase must undergo a $\pi$ slip to minimize the Josephson coupling energy whenever the integral $I_s$ changes sign as a function of the magnetic field. We shall conclude by discussing the pure *antisymmetric* junction case, i.e., $I_a\neq0$ and $I_s =0$. Because of the zero current condition the only values that the phase $\varphi_0$ can assume are $\pi/2$ or $3\pi/2 $. Equation (\[eq:energy\_min\]) implies that, if $I_a >0$, $\varphi_0=3\pi /2$ and, if $I_a <0$, $\varphi_0=\pi/2$. Therefore, also in this case, the phase $\varphi_0$ undergoes a $\pi$ slip when $I_a$ changes sign. We remark that the discussed phase slips differ from those present in low dimensional superconductors, caused by thermal [@Langer1967] and quantum [@Zaikin1997; @astafiev2012] fluctuations. In those cases, the phase slips are generated when, because of fluctuations, the modulus of the complex order parameter goes to zero, the phase becomes unrestricted and jumps of $2 \pi$ [@arutyunov08]. By contrast, the phase slips discussed above have an energetic origin and they occur when the system passes from one energetically stable configuration to another one [@kuplevakhsky06]. This transition takes place when the magnetic flux crosses one of the critical points and therefore can be experimentally induced by changing the magnetic flux. The different origin of the slips is exemplified by the fact that the fluctuation-induced phase slips are always of $2 \pi$ while in the present case we have slips of $\pi$. The identification of this effect is possible only in the electrically-open junctions. In fact, the presence of an electric current or a voltage bias would destroy or hide the original effect. ![(Color online) Normalized phase-dependent component of the heat current $J_{H}$ versus magnetic flux $\Phi$ calculated for a rectangular \[(a)\], circular \[(b)\], and annular \[(c)\] Josephson tunnel junction. In the curves of panel (c) we set $\alpha=0.9$, and $n$ indicates the number of fluxons trapped in the junction barrier. []{data-label="fig2"}](fig2.pdf){width="\columnwidth"} Heat current in Josephson junctions with different geometries {#differentgeo} ============================================================= With the help of Eq. (\[heatcurrfin\]) we can now determine the behavior of $J_{H}(T_1,T_2,H)$ for the three prototypical junction geometries sketched in Fig. (\[fig1\]). In particular, we shall consider two well-known examples such as the *rectangular* \[see Fig. \[fig1\](b)\] and *circular* \[see Fig. \[fig1\](c)\] junction, and the more exotic *annular* one \[see Fig. \[fig1\](d)\]. Annular junctions offer the possibility to investigate fluxons dynamics due to the absence of collisions with boundaries; yet, they provide fluxoid quantization thanks to their geometry which allows fluxons trapping. We assume that the total phase-dependent heat current is characterized by a uniform distribution, i.e., by a constant thermal current areal density $J_A(x,y,T_1,T_2)$ in Eq. (\[phasea\]). $J_{H}$ can therefore be calculated for the three considered geometries by following, for instance, Refs. [@Tinkham; @Barone]. In particular, for the rectangular junction, the absolute value of the sine cardinal function is obtained, $$J_{H}^{rect}(T_1,T_2,\Phi)=J_{int}(T_1,T_2)\left|\frac{\textrm{sin}(\pi\Phi/\Phi_0)}{(\pi\Phi/\Phi_0)}\right|,$$ where $J_{int}(T_1,T_2)=WLJ_A(T_1,T_2)$, $\Phi=\mu_0 HLt_H$, $L$ is the junction length and $W$ its width. For the circular geometry one gets the Airy diffraction pattern, $$J_{H}^{circ}(T_1,T_2,\Phi)=J_{int}(T_1,T_2)\left|\frac{J_1(\pi\Phi/\Phi_0)}{(\pi\Phi/2\Phi_0)}\right|,$$ where $J_{int}(T_1,T_2)=\pi R^2 J_A(T_1,T_2)$, $J_1(y)$ is the Bessel function of the first kind, $\Phi=2\mu_0 HRt_H$, and $R$ is the junction radius. Finally, for the annular junction [@Martucciello1996; @Nappi1997], the phase-dependent component of the heat current takes the form $$J_{H}^{ann}(T_1,T_2,\Phi)=\frac{2J_{int}(T_1,T_2)}{1-\alpha^2}\left|\int^1_\alpha dxxJ_n(x\pi \Phi/\Phi_0)\right|,$$ where $J_{int}(T_1,T_2)=\pi(R^2-r^2)J_A(T_1,T_2)$, $\Phi=2\mu_0 HRt_H$, $\alpha=r/R$, $J_n(y)$ is the $n$th Bessel function of integer order, $R$ ($r$) is the external (internal) radius, and $n=0,1,2,...$ is the number of $n$ trapped fluxons in the junction barrier. Figure \[fig2\] illustrates the behavior of $J_{H}$ for the three geometries. In particular, the curve displayed in Fig. \[fig2\](a) for the rectangular case shows the well-known Fraunhofer diffraction pattern analogous to that produced by light diffraction through a rectangular slit. In such a case, the heat current $J_{H}$ vanishes when the applied magnetic flux through the junction equals integer multiples of $\Phi_0$. Furthermore, the heat current is rapidly damped by increasing the magnetic field falling asymptotically as $\Phi^{-1}$.[@Tinkham] The behavior for a circular junction is displayed in Fig. \[fig2\](b). Here, the flux values where $J_{H}$ vanishes do not coincide anymore with multiples of $\Phi_0$, and $J_H$ falls more rapidly than in the rectangular junction case, i.e., as $\Phi^{-3/2}$.[@Tinkham] Figure \[fig2\](c) shows $J_H$ for an annular junction. In particular, the heat current diffraction pattern is strongly $n$-dependent and, differently from the rectangular and circular case, $J_H$ decays in general more slowly. It is apparent that annular junctions may provide, in principle, enhanced flexibility to tailor the heat current response. ![(Color online) (a) Possible experimental setup to demonstrate heat diffraction in a temperature-biased rectangular Josephson junction. Source and drain normal-metal electrodes are tunnel-coupled to one of the junction electrodes (S$_1$). Superconducting tunnel junctions operated as heaters and thermometers are connected to source and drain. A static in-plane magnetic field $H$ is applied perpendicular to the S$_1$IS$_2$ junction. (b) Thermal model describing the main heat exchange mechanisms existing in the structure shown in (a).[]{data-label="fig3"}](fig3.pdf){width="\columnwidth"} Proposed experimental setup {#experiment} =========================== Demonstration of diffraction of thermal currents could be achieved in the setup shown in Fig. \[fig3\](a). It consists of two normal-metal source and drain electrodes tunnel-coupled via resistances $R_t$ to one electrode (S$_1$) of a Josephson junction which, for the sake of clarity, is assumed to be *rectangular*. An in-plane static magnetic field $H$ is applied perpendicular to the Josephson weak-link. Furthermore, superconducting probes tunnel-coupled to both source and drain electrodes either implement heaters or allow accurate measurement of the electronic temperature in the leads [@Giazotto2006]. By intentionally heating electrons in the source up to $T_{src}$ yields a quasiparticle temperature $T_1>T_2$ in S$_1$, therefore leading to a finite heat current $J_{S_1\rightarrow S_2}$. Yet, the latter can be modulated by the applied magnetic field. Measurement of the drain electron temperature ($T_{dr}$) would thus allow to assess heat diffraction. Drain temperature can be predicted by solving a couple of thermal balance equations accounting for the main heat exchange mechanisms existing in the structure, according to the model shown in Fig. \[fig3\](b). In particular, S$_1$ exchanges heat with source electrons at power $J_{src\rightarrow S_1}$, with drain at power $J_{S_1\rightarrow drain}$, and with quasiparticles in S$_2$ at power $J_{S_1 \rightarrow S_2}$. Furthermore, electrons in the structure exchange heat with lattice phonons residing at bath temperature $T_{bath}$, in particular, at power $J_{e-ph, S_1}$ in S$_1$, and at power $J_{e-ph,src}$ and $J_{e-ph,dr}$ in source and drain electrodes, respectively. Finally, we assume S$_2$ to be large enough to provide substantial electron-phonon coupling $J_{e-ph, S_2}$ so that its quasiparticles will reside at $T_{bath}$. The electronic temperatures $T_1$ and $T_{dr}$ can therefore be determined under given conditions by solving the following system of thermal balance equations $$\begin{aligned} -J_{src\rightarrow S_1}+J_{S_1 \rightarrow S_2}+J_{S_1\rightarrow drain}+J_{e-ph, S_1}&=&0\\ -J_{S_1\rightarrow drain}+J_{e-ph,dr}&=&0\nonumber\end{aligned}$$ for S$_1$ and drain, respectively. In the above expressions, $J_{S_1 \rightarrow S_2}(T_1,T_{bath},\Phi)=J_{qp}(T_1,T_{bath})-J_{H}^{rect}(T_1,T_{bath},\Phi)$, $J_{src\rightarrow S_1}(T_{src},T_1)=\frac{1}{e^2 R_t} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)[f(\varepsilon,T_1)-f(\varepsilon,T_{src})]$, $J_{S_1\rightarrow drain}(T_1,T_{dr})=\frac{1}{e^2 R_t} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)[f(\varepsilon,T_{dr})-f(\varepsilon,T_{1})]$, and $J_{e-ph,dr}=\Sigma_{dr}\mathcal{V}_{dr}(T^5_{dr}-T^5_{bath})$ [@Giazotto2006], $\Sigma_{dr}$ and $\mathcal{V}_{dr}$ being the electron-phonon coupling constant and the volume of drain, respectively. Furthermore [@Timofeev2009], $$\begin{aligned} J_{e-ph,S_1}&=&-\frac{\Sigma_{S_1} \mathcal{V}_{S_1}}{96\zeta (5)k_B^5}\int^{\infty}_{-\infty}dEE\int^{\infty}_{-\infty}d\varepsilon \varepsilon^2\text{sign}(\varepsilon)M_{E,E+\varepsilon}\nonumber\\ &\times& [\text{coth}(\frac{\varepsilon}{2k_B T_{bath}})(f_E-f_{E+\varepsilon})-f_Ef_{E+\varepsilon}+1], \label{eph}\end{aligned}$$ where $f_E(T_1)=\text{tanh}(E/2k_B T_1)$, $M_{E,E'}(T_1)=\mathcal{N}_1(E,T_1)\mathcal{N}_1(E',T_1)[1-\Delta_1^2(T_1)/EE']$, $\Sigma_{S_1}$ is the electron-phonon coupling constant, and $\mathcal{V}_{S_1}$ is the volume of S$_1$. As a set of parameters representative for a realistic microstructure we choose $R_t=2\,\text{k}\Omega$, $R_J=500\,\Omega$, $\mathcal{V}_{dr}=10^{-20}$ m$^{-3}$, $\Sigma_{dr}=3\times 10^9$ WK$^{-5}$m$^{-3}$ (typical of Cu) [@Giazotto2006], $\mathcal{V}_{S_1}=10^{-18}$ m$^{-3}$, $\Sigma_{S_1}=3\times 10^8$ WK$^{-5}$m$^{-3}$ and $\Delta_1(0)=\Delta_2(0)=200\,\mu$eV, the last two parameters typical of aluminum (Al) [@Giazotto2006]. Finally, our thermal model neglects both heat exchange with photons, owing to poor matching impedance [@schmidt; @Meschke2006], and pure phononic heat conduction [@Maki1965; @giazottoexp2012]. ![(Color online) (a) Drain temperature $T_{dr}$ vs $\Phi$ calculated at $T_{bath}=250$ mK for several values of source temperature $T_{src}$ for a structure based on a *rectangular* Josephson junction. (b) Flux-to temperature transfer function $\mathcal{T}$ vs $\Phi$ calculated at 250 mK for a few selected values of $T_{src}$. (c) $T_{dr}$ vs $\Phi$ calculated for a few values of $T_{bath}$ at $T_{src}=1$ K. (d) $\mathcal{T}$ vs $\Phi$ at a few selected $T_{bath}$ calculated for $T_{src}=1$ K.[]{data-label="fig4"}](fig4.pdf){width="\columnwidth"} The results of thermal balance equations for drain temperature are shown in Fig. \[fig4\] [@Gamma]. In particular, panel (a) displays $T_{dr}$ vs $\Phi$ for different values of $T_{src}$ at $T_{bath}=250$ mK. As expected, $T_{dr}$ shows a response to magnetic flux resembling a Fraunhofer-like diffraction pattern. The minima appearing at integer values of $\Phi_0$ are the inequivocal manifestation of the above-described phase-slips. Increasing $T_{src}$ leads to a monotonic enhancement of the maximum of $T_{dr}$ at $\Phi=0$ which stems from an increased heat current flowing into drain electrode. Furthermore, the amplitude of $T_{dr}$ lobes follows a non-monotonic beahavior, initially increasing with source temperature, being maximized at intermediate temperatures, and finally decreasing at higher $T_{src}$ values. With the above-given structure parameters one would obtain a maximum peak-to-valley amplitude exceeding $\sim 60$ mK at $T_{src}\sim 700$ mK. By defining a figure of merit in the form of flux-to-temperature transfer coefficient, $\mathcal{T}=\partial T_{dr}/\partial \Phi$, we get that $\mathcal{T}$ as large as $\sim 90$ mK$/\Phi_0$ could be achieved at $T_{src}=600$ mK in the present structure \[see Fig. \[fig4\](b)\]. Moreover, the transfer coefficient clearly demonstrates the non-monotonicity of the amplitude of drain temperature lobes as a function of $T_{src}$. The impact of bath temperature on the structure response is shown in Fig. \[fig4\](c) where $T_{dr}$ is plotted against $\Phi$ for a few $T_{bath}$ values at fixed $T_{src}=1$ K. In particular, by increasing $T_{bath}$ leads to a smearing of drain temperature joined with a reduction of the lobes amplitude. This originates from both reduced temperature drop across the Josephson junction and enhanced electron-phonon relaxation in S$_1$ and drain at higher $T_{bath}$. We notice that already at 550 mK the temperature diffraction pattern is somewhat suppressed for a structure realized according to the chosen parameters. The drain temperature behavior as a function of $T_{bath}$ directly reflects on the transfer coefficient $\mathcal{T}(\Phi)$ \[see Fig. \[fig4\](d)\] which is calculated for a few selected values of $T_{bath}$. We finally notice that the temperature diffraction patterns shown in Figs. \[fig4\] implicitly assume the presence of the $\pi$ slips and, therefore, the same heat diffraction measure can be considered as a proof of the existence of such phase slips. Summary ======= In summary, we have investigated thermal transport in temperature-biased extended Josephson tunnel junctions under the influence of an in-plane magnetic field. We have shown, in particular, that the heat current through the junction displays *coherent diffraction*, in full analogy with the Josephson critical current. In an electrically-open junction configuration, minimization of the Josephson coupling energy imposes the quantum phase difference across the junction to undergo $\pi$ slips in suitable magnetic flux intervals, the latter depending on the specific junction geometry. Finally, we have proposed and analyzed a hybrid superconducting microstructure, easily implementable with current technology, which would allow to demonstrate diffraction of thermal currents. We wish further to stress that the described temperature detection is uniquely suited to reveal the hidden physical properties of the quantum phase in electrically-open tunnel junctions of whatever geometry otherwise more difficult to access with electric-type transport measurement. The effects here predicted could serve to enhance the flexibility to master thermal currents in emerging coherent caloritronic nanocircuitry. We would like to thanks C. Altimiras for useful discussions. F.G. and M.J.M.-P. acknowledges the FP7 program No. 228464 “MICROKELVIN”, the Italian Ministry of Defense through the PNRM project “TERASUPER”, and the Marie Curie Initial Training Action (ITN) Q-NET 264034 for partial financial support. P.S. acknowledges financial support from FIRB - Futuro in Ricerca 2012 under Grant No. RBFR1236VV HybridNanoDev. [99]{} F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, and J. P. Pekola, Rev. Mod. Phys. **78**, 217 (2006). Y. Dubi and M. Di Ventra, Rev. Mod. Phys. **83**, 131 (2011). O.-P. Saira, M. Meschke, F. Giazotto, A. M. Savin, M. Möttönen, and J. P. Pekola, Phys. Rev. Lett. **99**, 027203 (2007). J. P. Pekola, F. Giazotto, and O.-P. Saira, Phys. Rev. Lett. **98**, 037201 (2007). M. Meschke, W. Guichard, and J. P. Pekola, Nature **444**, 187 (2006). E. V. Bezuglyi and V. Vinokur, Phys. Rev. Lett. **91**, 137002 (2003). J. Eom, C.-J. Chien, and V. Chandrasekhar, Phys. Rev. Lett. **81**, 437 (1998). V. Chandrasekhar, Supercond. Sci. Technol. **22**, 083001 (2009). V. V. Ryazanov and V. V. Schmidt, Solid State Commun. **42**, 733 (1982). G. I. Panaitov, V. V. Ryazanov, and V. V. Schmidt, Phys. Lett. **100**, 301 (1984). P. Virtanen and T. T. Heikkilä, Appl. Phys. A **89**, 625 (2007). M. J. Martínez-Pérez and F. Giazotto, Appl. Phys. Lett. **102**, 182602 (2013). F. Giazotto, T. T. Heikkilä, G. P. Pepe, P. Helisto, A. Luukanen, and J. P. Pekola, Appl. Phys. Lett. **92**, 162507 (2008). F. Giazotto, F. Taddei, R. Fazio, and F. Beltram, Appl. Phys. Lett. **80**, 3784 (2002). K. Maki and A. Griffin, Phys. Rev. Lett. [**15**]{}, 921 (1965). G. D. Guttman, B. Nathanson, E. Ben-Jacob, and D. J. Bergman, Phys. Rev. B **55**, 3849 (1997). G. D. Guttman, E. Ben-Jacob, and D. J. Bergman, Phys. Rev. B **57**, 2717 (1998). E. Zhao, T. Löfwander, and J. A. Sauls, Phys. Rev. Lett. **91**, 077003 (2003). E. Zhao, T. Löfwander, and J. A. Sauls, Phys. Rev. B **69**, 134503 (2004). D. Golubev, T. Faivre, and J. P. Pekola, Phys. Rev. B **87**, 094522 (2013). F. Giazotto and M. J. Martínez-Pérez, Appl. Phys. Lett. **101**, 102601 (2012). M. J. Martínez-Pérez and F. Giazotto, Appl. Phys. Lett. **102**, 092602 (2013). F. Giazotto and M. J. Martínez-Pérez, Nature **492**, 401 (2012). R. W. Simmonds, Nature **492**, 358 (2012). J.S. Langer, V. Ambegaokar, Phys. Rev. [**164**]{}, 498 (1967). A.D. Zaikin, D.S. Golubev, A. van Otterlo, G.T. Zimanyi, Phys. Rev. Lett. [**78**]{}, 1552 (1997). O. V. Astafiev, L. B. Ioffe, S. Kafanov, Yu. A. Pashkin, K. Yu. Arutyunov, D. Shahar, O. Cohen, and J. S. Tsai, Nature **484**, 355 (2012). K.Yu. Arutyunov, D.S. Golubev and A.D. Zaikin, Physics Reports **464**, 1 (2008). When this condition is no longer satisfied the total magnetic penetration depth has to be replaced with an effective thickness ($\tilde{t}_H$) given by $\tilde{t}_H=\lambda_1\textrm{tanh}(t_1/2\lambda_1)+\lambda_2\textrm{tanh}(t_2/2\lambda_2)+d$ [@Weihnacht1969]. M. Weihnacht, Phys. Status Solidi **32**, K169 (1969). B. Frank and W. Krech, Phys. Lett. A **235**, 281 (1997). M. Tinkham, *Introduction to Superconductivity 2nd Edn.* (McGraw-Hill, New York, 1996). A. Barone and G. Paternó, *Physics and Applications of the Josephson Effect* (Wiley, New York, 1982). N. Martucciello and R. Monaco, Phys. Rev. B **53**, 3471 (1996). S. V. Kuplevakhsky and A. M. Glukhov, Phys. Rev. B **73**, 024513 (2006). C. Nappi, Phys. Rev. B **55**, 82 (1997). Throughout our analysis we added a small imaginary part to the energy in $\mathcal{N}_i(\varepsilon)$ and $\mathcal{M}_i(\varepsilon)$ to account for smearing, i.e., $\varepsilon \rightarrow \varepsilon+i\gamma$, where $\gamma=10^{-5}\Delta_1(0)$ [@Martinez2013; @Dynes1984; @Pekola2004; @Pekola2010]. R. C. Dynes, J. P. Garno, G. B. Hertel, and T. P. Orlando, Phys. Rev. Lett. **53**, 2437 (1984). J. P. Pekola, T. T. Heikkilä, A. M. Savin, J. T. Flyktman, F. Giazotto, and F. W. J. Hekking, Phys. Rev. Lett. **92**, 056804 (2004). J. P. Pekola, V. F. Maisi, S. Kafanov, N. Chekurov, A. Kemppinen, Yu. A. Pashkin, O.-P. Saira, M. Möttönen, and J. S. Tsai, Phys. Rev. Lett. **105**, 026803 (2010). A. V. Timofeev, C. P. Garcia, N. B. Kopnin, A. M. Savin, M. Meschke, F. Giazotto, and J. P. Pekola, Phys. Rev. Lett. **102**, 017003 (2009). D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland, Phys. Rev. Lett. **93**, 045901 (2004).

ArXiv

--- abstract: 'We study algebraically infinitely many infinitray extensions of predicate intuitionistic logic. We prove several representation theorems that reflect a (weak) Robinson’s joint consistency theorem for the extensions studied with and without equality. In essence a Henkin-Gabbay construction, our proof uses neat embedding theorems and is purely algebraic. Neat embedding theorems, are an algebraic version of Henkin constructions that apply to various infinitary extensions of predicate first order logics; to the best of our knowledge, they were only implemented in the realm of intuitionistic logic in the article ’Amalgamation of polyadic Heyting algebras’ Studia Math Hungarica, in press. [^1]' author: - Tarek Sayed Ahmed title: 'Representability, and amalgamation for various reducts of Heyting polyadic algebras' --- Introduction ============ Background and History ---------------------- It often happens that a theory designed originally as a tool for the study of a problem, say in computer science, came subsequently to have purely mathematical interest. When such a phenomena occurs, the theory is usually generalized beyond the point needed for applications, the generalizations make contact with other theories (frequently in completely unexpected directions), and the subject becomes established as a new part of pure mathematics. The part of pure mathematics so created does not (and need not) pretend to solve the problem from which it arises; it must stand or fall on its own merits. A crucial addition to the collection of mathematical catalysts initiated at the beginning of the 20 century, is formal logic and its study using mathematical machinery, better known as metamathematical investigations, or simply metamathematics. Traced back to the works of Frege, Hilbert, Russel, Tarski, Godel and others; one of the branches of pure mathematics that metamathematics has precipitated to is algebraic logic. Algebraic logic is an interdisciplinary field; it is the art of tackling problems in formal logic using universal algebraic machinery. It is similar in this respect to several branches in mathematics, like algebraic geometry, where algebraic machinery is used guided by geometric intuition. In algebraic logic, the intuition underlying its constructions is inspired from (mathematical) logic. The idea of solving problems in various branches of logic by first translating them to algebra, then using the powerful methodology of algebra for solving them, and then translating the solution back to logic, goes back to Leibnitz and Pascal. Such a methodology was already fruitfully applied back in the 19th century with the work of Boole, De Morgan, Peirce, Schröder, and others on classical logic. Taking logical equivalence rather than truth as the primitive logical predicate and exploiting the similarity between logical equivalence and equality, those pioneers developed logical systems in which metalogical investigations take on a plainly algebraic character. The ingenious transfer of ”logical equivalence“ to ” equations” turned out immensely useful and fruitful. In particular, Boole’s work evolved into the modern theory of Boolean algebras, and that of De Morgan, Peirce and Schröder into the well-developed theory of relation algebras, which is now widely used in such diverse areas, ranging from formalizations of set theory to applications in computer science. From the beginning of the contemporary era of logic, there were two approaches to the subject, one centered on the notion of logical equivalence and the other, reinforced by Hilbert’s work on metamathematics, centered on the notions of assertion and inference. It was not until much later that logicians started to think about connections between these two ways of looking at logic. Tarski gave the precise connection between Boolean algebra and the classical propositional calculus, inspired by the impressive work of Stone on Boolean algebras. Tarski’s approach builds on Lindenbaum’s idea of viewing the set of formulas as an algebra with operations induced by the logical connectives. When the Lindenbaum-Tarski method is applied to the predicate calculus, it lends itself to cylindric and polyadic algebras rather than relation algebras. In the traditional mid -20th century approach, algebraic logic has focused on the algebraic investigation of particular classes of algebras like cylindric, polyadic and relation algebras. When such a connection could be established, there was interest in investigating the interconnections between various metalogical properties of the logical system in question and the algebraic properties of the coresponding class of algebras (obtaining what are sometimes called “bridge theorems”). This branch has now evolved into the relatively new field of universal algebraic logic, in analogy to the well established field of universal algebra. For example, it was discovered that there is a natural relation between the interpolation theorems of classical, intuitionistic, intermediate propositional calculi, and the amalgamation properties of varieties of Heyting algebras, which constitute the main focus of this paper. The variety of Heyting algebras is the algebraic counterpart of propositional intuitionistic logic. We shall deal with Heyting algebras with extra (polyadic) operations reflecting quantifiers. Those algebras are appropriate to study (extensions) of predicate intuitionistic logic. Proving various interpolation theorems for such extensions, we thereby extend known amalgamation results of Heyting algebras to polyadic expansions. A historic comment on the development of intuitioinistic logic is in order. It was Brouwer who first initiated the programme of intuitionism and intuitionistic logic is its rigorous formalization developed originaly by Arend Heyting. Brouwer rejected formalism per se but admitted the potential usefulness of formulating general logical principles expressing intuitionistically correct constructions, such as modus ponens. Heyting realized the importance of formalization, being fashionable at his time, with the rapid development of mathematics. Implementing intuitionistic logic, turned out useful for diffrent forms of mathematical constructivism since it has the existing property. Philosophically, intuitionism differs from logicism by treating logic as an independent branch of mathematics, rather than as the foundations of mathematics, from finitism by permitting intuitionistic reasoning about possibly infinite collections; and from platonism by viewing mathematical objects as mental constructs rather than entities with an independent objective existence. There is also analogies between logisicm and intuitionism; in fact Hilbert’s formalist program, aiming to base the whole of classical mathematics on solid foundations by reducing it to a huge formal system whose consistency should be established by finitistic, concrete (hence constructive) means, was the most powerful contemporary rival to Brouwer’s and Heyting’s intuitionism. Subject Matter -------------- Connections between interpolation theorems in the predicate calculus and amalgamation results in varieties of cylindric and polyadic algebras, were initiated mainly by Comer, Pigozzi, Diagneault and Jonsson. As it happened, during the course of the development of algebraic logic, dating back to the work of Boole, up to its comeback in the contemporary era through the pioneering work of Halmos, Tarski, Henkin, Monk, Andréka, and Németi, it is now established that the two most famous widely used algebraisations of first order logic are Tarski’s cylindric algebras [@HMT1], [@HMT2], and Halmos’ polyadic algebras [@Halmos]. Each has its advantages and disadvantages. For example, the class of representable cylindric algebras, though a variety, is not finitely axiomatizable, and this class exhibits an inevitable degree of complexity in any of its axiomatizations [@Andreka]. However, its equational theory is recursive. On the other hand, the variety of (representable) polyadic algebras is axiomatized by a finite schema of equations but its equational theory is not recursively enumerable [@NS]. There have been investigations to find a class of algebras that enjoy the positive properties of both. The key idea behind such investigations is to look at (the continuum many) reducts of polyadic algebras [@AUamal], [@S] searching for the desirable finitely axiomatizable variety among them. Indeed, it is folkore in algebraic logic that cylindric algebras and polyadic algebras belong to different paradigms, frequently manifesting contradictory behaviour. The paper [@S] is a unification of the positive properties of those two paradigms in the Boolean case, and one of the results of this paper can be interpreted as a unification of those paradigms when the propositional reducts are Heyting algebras. A polyadic algebra is typically an instance of a transformation system. A transformation system can be defined to be a quadruple of the form $(\A, I, G, {\sf S})$ where $\A$ is an algebra of any similarity type, $I$ is a non empty set (we will only be concerned with infinite sets), $G$ is a subsemigroup of $(^II,\circ)$ (the operation $\circ$ denotes composition of maps) and ${\sf S}$ is a homomorphism from $G$ to the semigroup of endomorphisms of $\A$ $(End(\A))$. Elements of $G$ are called transformations. The set $I$ is called the dimension of the algebra, for a transformation $\tau$ on $I$, ${\sf S}({\tau})\in End(\A)$ is called a substitution operator, or simply a substitution. Polyadic algebras arise when $\A$ is a Boolean algebra endowed with quantifiers and $G={}^II$. There is an extensive literature for polyadic algebras dating back to the fifties and sixties of the last century, [@Halmos], [@J70], [@D], [@DM], [@AUamal], [@S]. Introduced by Halmos, the theory of polyadic algebras is now picking up again; indeed it’s regaining momentum with pleasing progress and a plathora of results, see the references [@MLQ], [@Fer1], [@Fer2], [@Fer3], [@Fer4], [@ANS], [@trans], to name just a few. In recent times reducts of polyadic algebras of dimension $I$ were studied [@S], [@AUamal]; these reducts are obtained by restricting quantifiers to involve only quantification on finitely many variables and to study (proper) subsemigroups of $^II$ The two extremes are the semigroup of finite transformations (a finite transformation is one that moves only finitely many points) and all of $^II$ but there are infinitely many semigroups in between. In this paper, we study reducts of polyadic algebras by allowing (proper) subsemigroups of $^II$, but we also weaken the Boolean structure to be a Heyting algebra. Thus we approach the realm of intuitionistic logic. We shall study the cases when $G$ consists of all finite transformations, when $G$ is a proper subsemigroup satisfying certain properties but essentially containing infinitary transformations, that is, transformations that move infinitely many points (this involves infinitely many cases), and when $G$ is the semigroup of all transformations. Our investigations will address the representation of such algebras in a concrete sense where the operations are interpreted as set-theoretic operations on sets of sequences, and will also address the amalgamation property and variants thereof of the classes in question. In all the cases we study, the scope of quantifiers are finite, so in this respect our algebras also resemble cylindric algebras. The interaction between the theories of Boolean cylindric algebras and Boolean polyadic algebras is extensively studied in algebraic logic, see e.g [@ANS], with differences and similarities illuminating both theories. In fact, the study of $G$ Boolean polyadic algebras ($G$ a semigroup) by Sain in her pioneering paper [@S], and its follow up [@AUamal], is an outcome, or rather a culmination, of such research; it’s a typical situation in which the positive properties of both theories amalgamate. Boolean polyadic algebras, when $G$ is the set of finite transformations of $I$ or $G={}^II$ are old [@Halmos], [@D] [@DM]. In the former case such algebras are known as quasipolyadic algebras, and those are substantially different from full polyadic algebras (in the infinite dimensional case), as is commonly accepted, quasipolyadic algebras belong to the cylindric paradigm; they share a lot of properties of cylindric algebras. While the substitution operators in full Heyting polyadic algebras are uncountable, even if both the algebra and its dimension are countable, the substitution operators for quasipolyadic equality algebras of countable dimension are countable. Unlike full polyadic algebras, quasipolyadic algebras can be formulated as what is known in the literature as a system of varieties definable by schemes making them akin to universal algebraic investigations in the spirit of cylindric algebras. Though polyadic algebras can be viewed as a system of varieties, this system cannot be definable by schemes due to the presence of infinitary substitutions. Studying reducts of polyadic algebras by allowing only those substitutions coming from an arbitrary subsemigroup of $^II$ is relatively recent starting at the turn of the last century [@S]. Such algebras (of which we study their Heyting reducts) also provide a possible solution to a central problem in algebraic logic, better known as the finitizability problem, which asks for a simple (hopefully) finite axiomatization for several classes of representable algebras that abound in algebraic logic. [^2] The finitizability problem is not easy, and has been discussed at length in the literature [@Bulletin]. Being rather a family of problems, the finitizability problem has several scattered reincarnations in the lierature, and in some sense is still open. The finitizability problem also has philosophical implications, repercussions, connotations, concerning reasoning about reasoning, and can, in so many respects, be likened to Hilbert’s programe of proving mathematics consistent by concrete finitistic methods. In fact, our results show that, when $G$ satisfies some conditions that are not particularly complicated, provides us with an algebraisable extension of predicate first order intuitionistic logic, whose algebraic counterpart is a variety that is finitely axiomatizable. An algebraisable extension is an extension of ordinary predicate intuitionistic logic (allowing formulas of infinite length), whose algebraic counterpart consisting of subdirect products of set algebras based on (Kripke) models, is a finitely based variety (equational class). This gives a clean cut solution to the analogue of the finitizability problem for ordinary predicate intuitionistic logic. Formal systems for intuitionistic propositional and predicate logic and arithmetic were developed by Heyting [@H],[@Hy] Gentzen [@G] and Kleene [@K]. Godel [@Godel] proved the equiconsistency of intuitionistic and classical theories. Kripke [@Kripke] provided a semantics with respect to which intuitionistic logic is sound and complete. We shall use a modified version of Kripke semantics below to prove our representability results. The algebraic counterparts of predicate intuitionistic logic, namely, Heyting polyadic algebras were studied by Monk [@Monk], Georgescu [@G] and the present author [@Hung]. Algebraically, we shall prove that certain reducts of algebras of full polyadic Heyting algebras (studied in [@Hung]) consist solely of representable algebras (in a concrete sense) and have the superamalgamation property (a strong form of amalgamation). Such results are essentially proved in Part 1, with the superamalgmation property deferred to part 3. We also present some negative results for other infinitary intiutionistic logics, based on non finite axiomatizability results proved in part 2, using bridge theorems. Indeed, in part 3, among other things, we show that the minimal algebraisable extension of predicate intuitionistic logic, in a sense to be made precise, is essentially incomplete, and fail to have the interpolation property. Roughly, minimal extension here means this (algebraisable) logic corresponding to the variety generated by the class of algebras arising from ordinary intuitionistic predicate logic. Such algebras are locally finite, reflecting the fact that formulas contain only finitely many variables. This correspondence is taken in the sense of Blok and Pigozzi associating quasivarieties to algebraisable logics. Algebraising here essentially means that we drop the condition of local finiteness, (hence alowing formulas of infinite length); this property is not warranted from the algebraic point of view because it is a poperty that cannot be expressed by first order formulas, let alone equations or quasiequations In fact, we show that all positive results in this paper extend to the classical case, reproving deep results in [@S], [@AUamal], and many negative results that conquer the cylindric paradigm, extend in some exact sense, to certain infinitary extensions of predicate intuitionistic logic, that arise naturally from the process of algebraising the intuitionistic predicate logic ( with and without equality). Such results are presented in the context of clarifying one facet of the finitizability problem for predicate intuionistic logic, namely that of drawing a line between positive and negative results. The techniques used in this paper intersects those adopted in our recent paper on Heyting polyadic algebras; it uses this part of algebraic logic developed essentially by Henkin, Monk, Tarski and Halmos - together with deep techniques of Gabbay - but there are major differences. We mention two Whereas the results in [@Hung] address full Heyting polyadic algebras where infinitary cylindrifications and infinitary substitutions are available; this paper, among many other things, shows that the proof survives when we restrict our attention to finitely generated semigroups still containing infinitary substitutions, and finite cylindrifiers. The algebras in [@Hung] have an axiomatization that is highly complex from the recursion theoretic point of view. The reducts studied here have recursive axiomatizations. We allow diagonal elements in our algebras (these elements reflect equality), so in fact, we are in the realm of infinitary extensions of intuitionistic predicate logic [*with*]{} equality. The interaction between algebraic logic and intuitionistic logic was developed in the monumental work of the Polish logicians Rasiowa and Sikorski, and the Russian logician Maksimova, but apart from that work, to the best of our knowledge, the surface of predicate intuitionistic logic was barely scratched by algebraic machinery. While Maksimova’s work [@b] is more focused on propositional intuitionistic logic, Rasiowa and Sikorski did deal with expansions of Heyting algebras, to reflect quantifiers, but not with polyadic algebras per se. Besides, Rasiowa and Sikorski, dealt only with classical predicate intuitionistic logic. In this paper, we continue the trend initiated in [@Hung], by studying strict reducts of full fledged infinitary logics, which are still infinitary, together with their expansions by the equality symbol, proving completenes theorems and interpolation properties, and we also maintain the borderline where such theorems cease to hold. Organization {#organization .unnumbered} ------------ In the following section we prepare for our algebraic proof, by formulating and proving the necessary algebraic preliminaries (be it concepts or theorems) addressing various reducts of Heyting polyadic algebras, possibly endowed with diagonal elements. Our algebraic proof of the interpolation property for infinitary extensions of predicate intuitionistic logic (with and without equality) are proved in section 3, which is the soul and heart of this part of the paper. This is accomplished using the well developed methodology of algebraic logic; particularly so-called neat embedding theorems, which are algebraic generalizations of Henkin constructions. On the notation {#on-the-notation .unnumbered} --------------- Throughout the paper, our notation is fairly standard, or self explanatory. However, we usually distinguish notationally between algebras and their domains, though there will be occasions when we do not distinguish between the two. Algebras will be denoted by Gothic letters, and when we write $\A$ for an algebra, then it is to be tacitly assumed that the corresponding Roman letter $A$ denotes its domain. Unfamiliar notation will be introduced at its first occurrence in the text. We extensively use the axiom of choice (any set can be well ordered, so that in many places we deal with ordinals or order types, that is, we impose well orders on arbitrary sets). For a set $X$, $Id_X$, or simply $Id$, when $X$ is clear from context, denotes the identity function on $X$. The set of all functions from $X$ to $Y$ is denoted by $^XY$. If $f\in {}^XY$, then we write $f:X\to Y$. The domain $X$ of $f$, will be denoted by $Dof$, and the range of $f$ will be denoted by $Rgf$. Composition of functions $f\circ g$ is defined so that the function at the right acts first, that is $(f\circ g)(x)=f(g(x))$, for $x\in Dog$ such that $g(x)\in Dof$. Algebraic Preliminaries ======================= In this section, we define our algebras and state and prove certain algebraic notions and properties that we shall need in our main (algebraic) proof implemented in the following section. Other results, formulated in lemmata \[dl\] and \[cylindrify\] in this section are non-trivial modifications of existing theorems for both cylindric algebras and polyadic algebras; we give detailed proofs of such results, skipping those parts that can be found in the literature referring to the necessary references instead. These lemmata address a very important and key concept in both cylindric and polyadic theories, namely, that of forming dilations and neat reducts (which are, in fact, dual operations.) The algebras ------------ For an algebra $\A$, $End(\A)$ denotes the set of endomorphisms of $\A$ (homomorphisms of $\A$ into itself), which is a semigroup under the operation $\circ$ of composition of maps. A transformation system is a quadruple $(\A, I, G, {\sf S})$ where $\A$ is an algebra, $I$ is a set, $G$ is a subsemigroup of $(^II,\circ)$ and ${\sf S}$ is a homomorphism from $G$ into $End(\A).$ Throughout the paper, $\A$ will always be a Heyting algebra. If we want to study predicate intuitionistic logic, then we are naturally led to expansions of Heyting algebras allowing quantification. But we do not have negation in the classical sense, so we have to deal with existential and universal quantifiers each separately. Let $\A=(A, \lor, \land,\rightarrow,0)$ be a Heyting algebra. An existential quantifier $\exists$ on $A$ is a mapping $\exists:\A\to \A$ such that the following hold for all $p,q\in A$: $\exists(0)=0,$ $p\leq \exists p,$ $\exists(p\land \exists q)=\exists p\land \exists q,$ $\exists(\exists p\rightarrow \exists q)=\exists p\rightarrow \exists q,$ $\exists(\exists p\lor \exists q)=\exists p\lor \exists q,$ $\exists\exists p=\exists p.$ Let $\A=(A, \lor, \land,\rightarrow,0)$ be a Heyting algebra. A universal quantifier $\forall$ on $A$ is a mapping $\forall:\A\to \A$ such that the following hold for all $p,q\in A$: $\forall 1=1,$ $\forall p\leq p,$ $\forall(p\rightarrow q)\leq \forall p\rightarrow \forall q,$ $\forall \forall p=\forall p.$ Now we define our algebras. Their similarity type depends on a fixed in advance semigroup. We write $X\subseteq_{\omega} Y$ to denote that $X$ is a finite subset of $Y$. Let $\alpha$ be an infinite set. Let $G\subseteq {}^{\alpha}\alpha$ be a semigroup under the operation of composition of maps. An $\alpha$ dimensional polyadic Heyting $G$ algebra, a $GPHA_{\alpha}$ for short, is an algebra of the following form $$(A,\lor,\land,\rightarrow, 0, {\sf s}_{\tau}, {\sf c}_{(J)}, {\sf q}_{(J)})_{\tau\in G, J\subseteq_{\omega} \alpha}$$ where $(A,\lor,\land, \rightarrow, 0)$ is a Heyting algebra, ${\sf s}_{\tau}:\A\to \A$ is an endomorphism of Heyting algebras, ${\sf c}_{(J)}$ is an existential quantifier, ${\sf q}_{(J)}$ is a universal quantifier, such that the following hold for all $p\in A$, $\sigma, \tau\in [G]$ and $J,J'\subseteq_{\omega} \alpha:$ ${\sf s}_{Id}p=p.$ ${\sf s}_{\sigma\circ \tau}p={\sf s}_{\sigma}{\sf s}_{\tau}p$ (so that ${\sf s}:\tau\mapsto {\sf s}_{\tau}$ defines a homomorphism from $G$ to $End(\A)$; that is $(A, \lor, \land, \to, 0, G, {\sf s})$ is a transformation system). ${\sf c}_{(J\cup J')}p={\sf c}_{(J)}{\sf c}_{(J')}p , \ \ {\sf q}_{(J\cup J')}p={\sf q}_{(J)}{\sf c}_{(J')}p.$ ${\sf c}_{(J)}{\sf q}_{(J)}p={\sf q}_{(J)}p , \ \ {\sf q}_{(J)}{\sf c}_{(J)}p={\sf c}_{(J)}p.$ If $\sigma\upharpoonright \alpha\sim J=\tau\upharpoonright \alpha\sim J$, then ${\sf s}_{\sigma}{\sf c}_{(J)}p={\sf s}_{\tau}{\sf c}_{(J)}p$ and ${\sf s}_{\sigma}{\sf q}_{(J)}p={\sf s}_{\tau}{\sf q}_{(J)}p.$ If $\sigma\upharpoonright \sigma^{-1}(J)$ is injective, then ${\sf c}_{(J)}{\sf s}_{\sigma}p={\sf s}_{\sigma}{\sf c}_{\sigma^{-1}(J)}p$ and ${\sf q}_{(J)}{\sf s}_{\sigma}p={\sf s}_{\sigma}{\sf q}_{\sigma^{-1}(J)}p.$ Let $\alpha$ and $G$ be as in the prevoius definition. By a $G$ polyadic equality algebra, a $GPHAE_{\alpha}$ for short, we understand an algebra of the form $$(A,\lor,\land,\rightarrow, 0, {\sf s}_{\tau}, {\sf c}_{(J)}, {\sf q}_{(J)}, {\sf d}_{ij})_{\tau\in G, J\subseteq_{\omega} \alpha, i,j\in \alpha}$$ where $(A,\lor,\land,\rightarrow, 0, {\sf s}_{\tau}, {\sf c}_{(J)}, {\sf q}_{(J)})_{\tau\in G\subseteq {}^{\alpha}\alpha, J\subseteq_{\omega} \alpha}$ is a $GPHA_{\alpha}$ and ${\sf d}_{ij}\in A$ for each $i,j\in \alpha,$ such that the following identities hold for all $k,l\in \alpha$ and all $\tau\in G:$ ${\sf d}_{kk}=1$ ${\sf s}_{\tau}{\sf d}_{kl}={\sf d}_{\tau(k), \tau(l)}.$ $x\cdot {\sf d}_{kl}\leq {\sf s}_{[k|l]}x$ Here $[k|l]$ is the replacement that sends $k$ to $l$ and otherwise is the identity. In our definition of algebras, we depart from [@HMT2] by defining polyadic algebras on sets rather than on ordinals. In this manner, we follow the tradition of Halmos. We refer to $\alpha$ as the dimension of $\A$ and we write $\alpha=dim\A$. Borrowing terminology from cylindric algebras, we refer to ${\sf c}_{(\{i\})}$ by ${\sf c}_i$ and ${\sf q}_{(\{i\})}$ by ${\sf q}_i.$ However, we will have occasion to impose a well order on dimensions thereby dealing with ordinals. When $G$ consists of all finite transformations, then any algebra with a Boolean reduct satisfying the above identities relating cylindrifications, diagonal elements and substitutions, will be a quasipolyadic equality algebra of infinite dimension. Besides dealing with the two extremes when $G$ consists only of finite transformations, supplied with an additional condition, and when $G$ is $^{\alpha}\alpha$, we also consider cases when $G$ is a possibly proper subsemigroup of $^{\alpha}\alpha$ (under the operation of composition). We need some preparations to define such semigroups. [Notation.]{} For a set $X$, $|X|$ stands for the cardinality of $X$. For functions $f$ and $g$ and a set $H$, $f[H|g]$ is the function that agrees with $g$ on $H$, and is otherwise equal to $f$. Recall that $Rgf$ denotes the range of $f$. For a transformation $\tau$ on $\alpha$, the support of $\tau$, or $sup(\tau)$ for short, is the set: $$sup(\tau)=\{i\in \alpha: \tau(i)\neq i\}.$$ Let $i,j\in \omega$, then $\tau[i|j]$ is the transformation on $\alpha$ defined as follows: $$\tau[i|j](x)=\tau(x)\text { if } x\neq i \text { and }\tau[i|j](i)=j.$$ Recall that the map $[i|j]$ is the transformation that sends $i$ to $j$ and is the equal to the identity elsewhere. On the other hand, the map denoted by $[i,j]$ is the transpostion that interchanges $i$ and $j$. For a function $f$, $f^n$ denotes the composition $f\circ f\ldots \circ f$ $n$ times. We extend the known definition of (strongly) rich semigroups [@S], [@AUamal], allowing possibly uncountable sets and semigroups. This will be needed when $G={}^\alpha\alpha$, cf. lemma \[cylindrify\]. However, throughout when we mention rich semigroups, then we will be tacitly assuming that both the dimension of the algebra involved and the semigroup are countable, [*unless*]{} otherwise explicity mentioned. \[rich\]Let $\alpha$ be any set. Let $T\subseteq \langle {}^{\alpha}\alpha, \circ \rangle$ be a semigroup. We say that $T$ is [*rich* ]{} if $T$ satisfies the following conditions: 1. $(\forall i,j\in \alpha)(\forall \tau\in T) \tau[i|j]\in T.$ 2. There exist $\sigma,\pi\in T$ such that $(\pi\circ \sigma=Id,\ Rg\sigma\neq \alpha), $ satisfying $$(\forall \tau\in T)(\sigma\circ \tau\circ \pi)[(\alpha\sim Rg\sigma)|Id]\in T.$$ \[stronglyrich\] Let $T\subseteq \langle {}^{\alpha}\alpha, \circ\rangle$ be a rich semigroup. Let $\sigma$ and $\pi$ be as in the previous definition. If $\sigma$ and $\pi$ satisfy: 1. $(\forall n\in \omega) |supp(\sigma^n\circ \pi^n)|<\alpha, $ 2. $(\forall n\in \omega)[supp(\sigma^n\circ \pi^n)\subseteq \alpha\smallsetminus Rng(\sigma^n)];$ then we say that $T$ is [*a strongly rich*]{} semigroup. Examples of rich semigroups of $\omega$ are $(^{\omega}\omega, \circ)$ and its semigroup generated by $\{[i|j], [i,j], i, j\in \omega, suc, pred\}$. Here $suc$ abbreviates the successor function on $\omega$ and $pred$ acts as its right inverse, the predecessor function, defined by $pred(0)=0$ and for other $n\in \omega$, $pred(n)=n-1$. In fact, both semigroups are strongly rich, in the second case $suc$ plays the role of $\sigma$ while $pred$ plays the role of $\pi$. Rich semigroups were introduced in [@S] (to prove a representability result) and those that are strongly rich were intoduced in [@AUamal] (to prove an amalgamation result). Next, we collect some properties of $G$ algebras that are more handy to use in our subsequent work. In what follows, we will be writing $GPHA$ ($GPHAE$) for all algebras considered. \[axioms\] Let $\alpha$ be an infinite set and $\A\in GPHA_{\alpha}$. Then $\A$ satisfies the following identities for $\tau,\sigma\in G$ and all $i,j,k\in \alpha$. 1. $x\leq {\sf c}_ix={\sf c}_i{\sf c}_ix,\ {\sf c}_i(x\lor y)={\sf c}_ix\lor {\sf c}_iy,\ {\sf c}_i{\sf c}_jx={\sf c}_j{\sf c}_ix$. That is ${\sf c}_i$ is an additive operator (a modality) and ${\sf c}_i,{\sf c}_j$ commute. 2. ${\sf s}_{\tau}$ is a Heyting algebra endomorphism. 3. ${\sf s}_{\tau}{\sf s}_{\sigma}x={\sf s}_{\tau\circ \sigma}x$ and ${\sf s}_{Id}x=x$. 4. ${\sf s}_{\tau}{\sf c}_ix={\sf s}_{\tau[i|j]}{\sf c}_ix$. Recall that $\tau[i|j]$ is the transformation that agrees with $\tau$ on $\alpha\smallsetminus\{i\}$ and $\tau[i|j](i)=j$. 5. ${\sf s}_{\tau}{\sf c}_ix={\sf c}_j{\sf s}_{\tau}x$ if $\tau^{-1}(j)=\{i\}$, ${\sf s}_{\tau}{\sf q}_ix={\sf q}_j{\sf s}_{\tau}x$ if $\tau^{-1}(j)=\{i\}$. 6. ${\sf c}_i{\sf s}_{[i|j]}x={\sf s}_{[i|j]}x$,  ${\sf q}_i{\sf s}_{[i|j]}x={\sf s}_{[i|j]}x$ 7. ${\sf s}_{[i|j]}{\sf c}_ix={\sf c}_ix$,   ${\sf s}_{[i|j]}{\sf q}_ix={\sf q}_ix$. 8. ${\sf s}_{[i|j]}{\sf c}_kx={\sf c}_k{\sf s}_{[i|j]}x$,  ${\sf s}_{[i|j]}{\sf q}_kx={\sf q}_k{\sf s}_{[i|j]}x$ whenever $k\notin \{i,j\}$. 9. ${\sf c}_i{\sf s}_{[j|i]}x={\sf c}_j{\sf s}_{[i|j]}x$,   ${\sf q}_i{\sf s}_{[j|i]}x={\sf q}_j{\sf s}_{[i|j]}x$. [Proof]{} The proof is tedious but fairly straighforward. Obviously the previous equations hold in $GPHAE_{\alpha}$. Following cylindric algebra tradition and terminology, we will be often writing ${\sf s}_j^i$ for ${\sf s}_{[i|j]}$. For $GPHA_{\alpha}$ when $G$ is rich or $G$ consists only of finite transformation it is enough to restrict our attenstion to replacements. Other substitutions are definable from those. Neat reducts and dilations -------------------------- Now we recall the important notion of neat reducts, a central concept in cylindric algebra theory, strongly related to representation theorems. This concept also occurs in polyadic algebras, but unfortunately under a different name, that of compressions. Forming dilations of an algebra, is basically an algebraic reflection of a Henkin construction; in fact, the dilation of an algebra is another algebra that has an infinite number of new dimensions (constants) that potentially eliminate cylindrifications (quantifiers). Forming neat reducts has to do with restricting or compressing dimensions (number of variables) rather than increasing them. (Here the duality has a precise categorical sense which will be formulated in the part 3 of this paper as an adjoint situation). Let $ \alpha\subseteq \beta$ be infinite sets. Let $G_{\beta}$ be a semigroup of transformations on $\beta$, and let $G_{\alpha}$ be a semigroup of transformations on $\alpha$ such that for all $\tau\in G_{\alpha}$, one has $\bar{\tau}=\tau\cup Id\in G_{\beta}$. Let $\B=(B, \lor, \land, \to, 0, {\sf c}_i, {\sf s}_{\tau})_{i\in \beta, \tau\in G_{\beta}}$ be a $G_{\beta}$ algebra. We denote by $\Rd_{\alpha}\B$ the $G_{\alpha}$ algebra obtained by dicarding operations in $\beta\sim \alpha$. That is $\Rd_{\alpha}\B=(B, \lor, \land, \to, 0, {\sf c}_i, {\sf s}_{\bar{\tau}})_{i\in \alpha, \tau\in G_{\alpha}}$. Here ${\sf s}_{\bar{\tau}}$ is evaluated in $\B$. For $x\in B$, then $\Delta x,$ the dimension set of $x$, is defined by $\Delta x=\{i\in \beta: {\sf c}_ix\neq x\}.$ Let $A=\{x\in B: \Delta x\subseteq \alpha\}$. If $A$ is a subuniverse of $\Rd_{\alpha}\B$, then $\A$ (the algebra with universe $A$) is a subreduct of $\B$, it is called the [*neat $\alpha$ reduct*]{} of $\B$ and is denoted by $\Nr_{\alpha}\B$. If $\A\subseteq \Nr_{\alpha}\B$, then $\B$ is called a [*dilation*]{} of $\A$, and we say that $\A$ [*neatly embeds*]{} in $\B$. if $A$ generates $\B$ (using all operations of $\B$), then $\B$ is called a [*minimal dilation*]{} of $\A$. The above definition applies equally well to $GPHAE_{\alpha}$. In certain contexts minimal dilations may not be unique (up to isomorphism), but what we show next is that in all the cases we study, they are unique, so for a given algebra $\A$, we may safely say [*the*]{} minimal dilation of $\A$. For an algebra $\A$, and $X\subseteq \A$, $\Sg^{\A}X$ or simply $\Sg X$, when $\A$ is clear from context, denotes the subalgebra of $\A$ generated by $X.$ The next theorems apply equally well to $GPHAE_{\alpha}$ with easy modifications which we state as we go along. \[dl\] Let $\alpha\subseteq \beta$ be countably infinite sets. If $G$ is a strongly rich semigroup on $\alpha$ and $\A\in GPHA_{\alpha}$, then there exists a strongly rich semigroup $T$ on $\beta$ and $\B\in TPHA_{\beta},$ such that $\A\subseteq \Nr_{\alpha}\B$ and for all $X\subseteq A,$ one has $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X$. Let $G_{I}$ be the semigroup of finite transformations on $I$. Let $\A\in G_{\alpha}PHA_{\alpha}$ be such that $\alpha\sim \Delta x$ is infinite for every $x\in A$. Then for any set $\beta$, such that $\alpha\subseteq \beta$, there exists $\B\in G_{\beta}PHA_{\beta},$ such that $\A\subseteq \Nr_{\alpha}\B$ and for all $X\subseteq A$, one has $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X.$ Let $G_I$ be the semigroup of all transformations on $I$. Let $\A\in G_{\alpha}PHA_{\alpha}$. Then for any set $\beta$ such that $\alpha\subseteq \beta$, there exists $\B\in G_{\beta}PHA_{\beta},$ such that $\A\subseteq \Nr_{\alpha}\B$ and for all $X\subseteq A,$ one has $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X.$ [Proof]{} cf. [@AUamal]. We assume that $\alpha$ is an ordinal; in fact without loss of generality we can assume that it is the least infinite ordinal $\omega.$ We also assume a particular strongly rich semigroup, that namely that generated by finite transformations together with $suc$, $pred$. The general case is the same [@AUamal] Remark 2.8 p. 327. We follow [@AUamal] p. 323-336. For $n\leq \omega$, let $\alpha_n=\omega+n$ and $M_n=\alpha_n\sim \omega$. Note that when $n\in \omega$, then $M_n=\{\omega,\ldots,\omega+n-1\}$. Let $\tau\in G$. Then $\tau_n=\tau\cup Id_{M_n}$. $T_n$ denotes the subsemigroup of $\langle {}^{\alpha_n}\alpha_n,\circ \rangle$ generated by $\{\tau_n:\tau\in G\} \cup \cup_{i,j\in \alpha_n}\{[i|j],[i,j]\}$. For $n\in \omega$, we let $\rho_n:\alpha_n\to \omega$ be the bijection defined by $\rho_n\upharpoonright \omega=suc^n$ and $\rho_n(\omega+i)=i$ for all $i<n$. Let $n\in \omega$. For $v\in T_n,$ let $v'=\rho_n\circ v\circ \rho_n^{-1}$. Then $v'\in G$. For $\tau\in T_{\omega}$, let $D_{\tau}=\{m\in M_{\omega}:\tau^{-1}(m)=\{m\}=\{\tau(m)\}\}$. Then $|M_{\omega}\sim D_{\tau}|<\omega.$ Let $\A$ be a given countable $G$ algebra. Let $\A_n$ be the algebra defined as follows: $\A_n=\langle A,\lor, \land, \to, 0, {\sf c}_i^{\A_n},{\sf s}_v^{\A_n}\rangle_{i\in \alpha_n,v\in T_n}$ where for each $i\in \alpha_n$ and $v\in T_n$, ${\sf c}_i^{\A_n}:= {\sf c}_{\rho_n(i)}^{\A} \text { and }{\sf s}_v^{\A_n}:= {\sf s}_{v'}^{\A}.$ Let $\Rd_{\omega}\A_n$ be the following reduct of $\A_n$ obtained by restricting the type of $\A_n$ to the first $\omega$ dimensions: $\Rd_{\omega}\A_n=\langle A_n,\lor,\land, \to,0, {\sf c}_i^{\A_n}, {\sf s}_{\tau_n}^{\A_n}\rangle_{i\in \omega,\tau\in G}.$ For $x\in A$, let $e_n(x)={\sf s}_{suc^n}^{\A}(x)$. Then $e_n:A\to A_n$ and $e_n$ is an isomorphism from $\A$ into $\Rd_{\omega}\A_n$ such that $e_n(\Sg^{\A}Y)=\Nr_{\omega}(\Sg^{\A_n}e_n(Y))$ for all $Y\subseteq A$, cf. [@AUamal] claim 2.7. While $\sigma$ and condition (2) in the definition of \[rich\] are needed to implement the neat embedding, the left inverse $\pi$ of $\sigma$ is needed to show that forming neat reducts commute with froming subalgebras; in particular $\A$ is the full $\omega$ neat reduct of $\A_n$. To extend the neat embedding part to infinite dimensions, we use a fairly straightforward construction involving an ultraproduct of exapansions of the algebras $\A_n$, on any cofinite ultrafilter on $\omega$. For the sake of brevity, let $\alpha=\alpha_{\omega}=\omega+\omega$. Let $T_{\omega}$ is the semigroup generated by the set $\{\tau_{\omega}: \tau\in G\}\cup_{i,j\in \alpha}\{[i|j],[i,j]\}.$ For $\sigma\in T_{\omega}$, and $n\in \omega$, let $[\sigma]_n=\sigma\upharpoonright \omega+n$. For each $n\in \omega,$ let $\A_n^+=\langle A,\lor,\land, \to, 0, {\sf c}_i^{\A_n^+}, {\sf s}_{\sigma}^{\A_n^+}\rangle_{i\in \alpha, \sigma\in T_{\omega}}$ be an expansion of $\A_n$ such that there Heyting reducts coincide and for each $\sigma\in T_{\omega}$ and $i\in \alpha,$ ${\sf s}_{\sigma}^{\A_n^+}:={\sf s}_{[\sigma]_n}^{\A_n} \text { iff } [\sigma]_n\in T_n,$ and ${\sf c}_i^{\A_n^+}:={\sf c}_i^{\A_n}\text { iff }i<\omega+n.$ Let $F$ to be any non-principal ultrafilter on $\omega$. Now forming the ultraproduct of the $\A_n^+$’s relative to $F$, let $\A^+=\prod_{n\in \omega}\A_n^+/F.$ For $x\in A$, let $e(x)=\langle e_n(x):n\in \omega\rangle/F.$ Let $\Rd_{\omega}A^+=\langle A^+, \lor, \land, \to, 0, {\sf c}_i^{\A^+}, {\sf s}_{\tau_{\omega}}^{\A^+} \rangle_{i<\omega,\tau\in T}.$ Then $e$ is an isomorphism from $\A$ into $\Rd_{\omega}\A^+$ such that $e(\Sg^{\A}Y)=\Nr_{\omega}\Sg^{\A^+}e(Y)$ for all $Y\subseteq A.$ We have shown that $\A$ neatly embeds in algebras in finite extra dimensions and in $\omega$ extra dimension. An iteration of this embedding yields the required result. In the presence of diagonals one has to check that homomorphisms defined preserve diagonal elements. But this is completely straightforward using properties of substitutions when applied to diagonal elements. Let $\alpha\subseteq \beta$. We assume, loss of generality, that $\alpha$ and $\beta$ are ordinals with $\alpha<\beta$. The proof is a direct adaptation of the proof of Theorem 2.6.49(i) in [@HMT1]. First we show that there exists $\B\in G_{\alpha+1}PHA_{\alpha+1}$ such that $\A$ embeds into $\Nr_{\alpha}\B,$ then we proceed inductively. Let $$R = Id\upharpoonright (\alpha\times A) \cup \{ ((k,x), (\lambda, y)) : k, \lambda < \alpha, x, y \in A, \lambda \notin \Delta x, y = {\mathsf s}_{[k|\lambda]} x \}.$$ It is easy to see that $R$ is an equivalence relation on $\alpha \times A$. Define the following operations on $(\alpha\times A)/R$ with $\mu, i, k\in \alpha$ and $x,y\in A$ : $$\label{l5} \begin{split} (\mu, x)/R \lor (\mu, y)/R = (\mu, x \lor y)/R, \end{split}$$ $$\label{l6} \begin{split} (\mu, x)/R\land (\mu, y)/R = (\mu, x\land y)/R, \end{split}$$ $$\label{l7} \begin{split} (\mu, x)/R\to (\mu, y)/R = (\mu, x\to y)/R, \end{split}$$ $$\label{l10} \begin{split} {\mathsf c}_i ((\mu, x)/R) = (\mu, {\mathsf c}_i x )/R, \quad \mu \in \alpha \smallsetminus \{i\}, \end{split}$$ $$\label{l11} \begin{split} {\mathsf s}_{[j|i]} ((\mu, x)/R) = (\mu, {\mathsf s}_{[j|i]} x )/R, \quad \mu \in \alpha \smallsetminus \{i, j\}. \end{split}$$ It can be checked that these operations are well defined. Let $$\C=((\alpha\times A)/R, \lor, \land, \to, 0, {\sf c_i}, {\sf s}_{i|j]})_{i,j\in \alpha},$$ and let $$h=\{(x, (\mu,x)/R): x\in A, \mu\in \alpha\sim \Delta x\}.$$ Then $h$ is an isomorphism from $\A$ into $\C$. Now to show that $\A$ neatly embeds into $\alpha+1$ extra dimensions, we define the operations ${\sf c}_{\alpha}, {\sf s}_{[i|\alpha]}$ and ${\sf s}_{[\alpha|i]}$ on $\C$ as follows: $${\mathsf c}_\alpha = \{ ((\mu, x)/R, (\mu, {\mathsf c}_\mu x)/R) : \mu \in \alpha, x \in B \},$$ $${\mathsf s}_{[i|\alpha]} = \{ ((\mu, x)/R, (\mu, {\mathsf s}_{[i|\mu]} x)/R) : \mu \in \alpha \smallsetminus \{i\}, x \in B \},$$ $${\mathsf s}_{[\alpha|i]} = \{ ((\mu, x)/R, (\mu, {\mathsf s}_{[\mu|i]} x)/R) : \mu \in \alpha \smallsetminus \{i\}, x \in B \}.$$ Let $$\B=((\alpha\times A)/R, \lor,\land, \to, {\sf c}_i, {\sf s}_{[i|j]})_{i,j\leq \alpha}.$$ Then $$\B\in G_{\alpha+1}PA_{\alpha+1}\text{ and }h(\A)\subseteq \Nr_{\alpha}\B.$$ It is not hard to check that the defined operations are as desired. We have our result when $G$ consists only of replacements. But since $\alpha\sim \Delta x$ is infinite one can show that substitutions corresponding to all finite transformations are term definable. For a finite transformation $\tau\in {}^{\alpha}\alpha$ we write $[u_0|v_0, u_1|v_1,\ldots, u_{k-1}|v_{k-1}]$ if $sup\tau=\{u_0,\ldots ,u_{k-1}\}$, $u_0<u_1 \ldots <u_{k-1}$ and $\tau(u_i)=v_i$ for $i<k$. Let $\A\in GPHA_{\alpha}$ be such that $\alpha\sim \Delta x$ is infinite for every $x\in A$. If $\tau=[u_0|v_0, u_1|v_1,\ldots, u_{k-1}|v_{k-1}]$ is a finite transformation, if $x\in A$ and if $\pi_0,\ldots ,\pi_{k-1}$ are in this order the first $k$ ordinals in $\alpha\sim (\Delta x\cup Rg(u)\cup Rg(v))$, then $${\mathsf s}_{\tau}x={\mathsf s}_{v_0}^{\pi_0}\ldots {\mathsf s}_{v_{k-1}}^{\pi_{k-1}}{\mathsf s}_{\pi_0}^{u_0}\ldots {\mathsf s}_{\pi_{k-1}}^{u_{k-1}}x.$$ The ${\sf s}_{\tau}$’s so defined satisfy the polyadic axioms, cf [@HMT1] Theorem 1.11.11. Then one proceeds by a simple induction to show that for all $n\in \omega$ there exists $\B\in G_{\alpha+n}PHA_{\alpha+n}$ such that $\A\subseteq \Nr_{\alpha}\B.$ For the transfinite, one uses ultraproducts [@HMT1] theorem 2.6.34. For the second part, let $\A\subseteq \Nr_{\alpha}\B$ and $A$ generates $\B$ then $\B$ consists of all elements ${\sf s}_{\sigma}^{\B}x$ such that $x\in A$ and $\sigma$ is a finite transformation on $\beta$ such that $\sigma\upharpoonright \alpha$ is one to one [@HMT1] lemma 2.6.66. Now suppose $x\in \Nr_{\alpha}\Sg^{\B}X$ and $\Delta x\subseteq \alpha$, then there exist $y\in \Sg^{\A}X$ and a finite transformation $\sigma$ of $\beta$ such that $\sigma\upharpoonright \alpha$ is one to one and $x={\sf s}_{\sigma}^{\B}y.$ Let $\tau$ be a finite transformation of $\beta$ such that $\tau\upharpoonright \alpha=Id \text { and } (\tau\circ \sigma) \alpha\subseteq \alpha.$ Then $x={\sf s}_{\tau}^{\B}x={\sf s}_{\tau}^{\B}{\sf s}_{\sigma}y= {\sf s}_{\tau\circ \sigma}^{\B}y={\sf s}_{\tau\circ \sigma\upharpoonright \alpha}^{\A}y.$ In the presence of diagonal elements, one defines them in the bigger algebra (the dilation) precisely as in [@HMT1], theorem 2.6.49(i). Here we extensively use the techniques in [@DM], but we have to watch out, for we only have finite cylindrifications. Let $(\A, \alpha,S)$ be a transformation system. That is to say, $\A$ is a Heyting algebra and $S:{}^\alpha\alpha\to End(\A)$ is a homomorphism. For any set $X$, let $F(^{\alpha}X,\A)$ be the set of all functions from $^{\alpha}X$ to $\A$ endowed with Heyting operations defined pointwise and for $\tau\in {}^\alpha\alpha$ and $f\in F(^{\alpha}X, \A)$, ${\sf s}_{\tau}f(x)=f(x\circ \tau)$. This turns $F(^{\alpha}X,\A)$ to a transformation system as well. The map $H:\A\to F(^{\alpha}\alpha, \A)$ defined by $H(p)(x)={\sf s}_xp$ is easily checked to be an isomorphism. Assume that $\beta\supseteq \alpha$. Then $K:F(^{\alpha}\alpha, \A)\to F(^{\beta}\alpha, \A)$ defined by $K(f)x=f(x\upharpoonright \alpha)$ is an isomorphism. These facts are straighforward to establish, cf. theorem 3.1, 3.2 in [@DM]. $F(^{\beta}\alpha, \A)$ is called a minimal dilation of $F(^{\alpha}\alpha, \A)$. Elements of the big algebra, or the cylindrifier free dilation, are of form ${\sf s}_{\sigma}p$, $p\in F(^{\beta}\alpha, \A)$ where $\sigma$ is one to one on $\alpha$, cf. [@DM] theorem 4.3-4.4. We say that $J\subseteq I$ supports an element $p\in A,$ if whenever $\sigma_1$ and $\sigma_2$ are transformations that agree on $J,$ then ${\sf s}_{\sigma_1}p={\sf s}_{\sigma_2}p$. $\Nr_JA$, consisting of the elements that $J$ supports, is just the neat $J$ reduct of $\A$; with the operations defined the obvious way as indicated above. If $\A$ is an $\B$ valued $I$ transformaton system with domain $X$, then the $J$ compression of $\A$ is isomorphic to a $\B$ valued $J$ transformation system via $H: \Nr_J\A\to F(^JX, \A)$ by setting for $f\in\Nr_J\A$ and $x\in {}^JX$, $H(f)x=f(y)$ where $y\in X^I$ and $y\upharpoonright J=x$, cf. [@DM] theorem 3.10. Now let $\alpha\subseteq \beta.$ If $|\alpha|=|\beta|$ then the the required algebra is defined as follows. Let $\mu$ be a bijection from $\beta$ onto $\alpha$. For $\tau\in {}^{\beta}\beta,$ let ${\sf s}_{\tau}={\sf s}_{\mu\tau\mu^{-1}}$ and for each $i\in \beta,$ let ${\sf c}_i={\sf c}_{\mu(i)}$. Then this defined $\B\in GPHA_{\beta}$ in which $\A$ neatly embeds via ${\sf s}_{\mu\upharpoonright\alpha},$ cf. [@DM] p.168. Now assume that $|\alpha|<|\beta|$. Let $\A$ be a given polyadic algebra of dimension $\alpha$; discard its cylindrifications and then take its minimal dilation $\B$, which exists by the above. We need to define cylindrifications on the big algebra, so that they agree with their values in $\A$ and to have $\A\cong \Nr_{\alpha}\B$. We let (\*): $${\sf c}_k{\sf s}_{\sigma}^{\B}p={\sf s}_{\rho^{-1}}^{\B} {\sf c}_{\rho(\{k\}\cap \sigma \alpha)}{\sf s}_{(\rho\sigma\upharpoonright \alpha)}^{\A}p.$$ Here $\rho$ is a any permutation such that $\rho\circ \sigma(\alpha)\subseteq \sigma(\alpha.)$ Then we claim that the definition is sound, that is, it is independent of $\rho, \sigma, p$. Towards this end, let $q={\sf s}_{\sigma}^{\B}p={\sf s}_{\sigma_1}^{\B}p_1$ and $(\rho_1\circ \sigma_1)(\alpha)\subseteq \alpha.$ We need to show that (\*\*) $${\sf s}_{\rho^{-1}}^{\B}{\sf c}_{[\rho(\{k\}\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho\circ \sigma\upharpoonright \alpha)}^{\A}p= {\sf s}_{\rho_1{^{-1}}}^{\B}{\sf c}_{[\rho_1(\{k\}\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho_1\circ \sigma\upharpoonright \alpha)}^{\A}p.$$ Let $\mu$ be a permutation of $\beta$ such that $\mu(\sigma(\alpha)\cup \sigma_1(\alpha))\subseteq \alpha$. Now applying ${\sf s}_{\mu}$ to the left hand side of (\*\*), we get that $${\sf s}_{\mu}^{\B}{\sf s}_{\rho^{-1}}^{\B}{\sf c}_{[\rho(\{k\})\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho\circ \sigma|\alpha)}^{\A}p ={\sf s}_{\mu\circ \rho^{-1}}^{\B}{\sf c}_{[\rho(\{k\})\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho\circ \sigma|\alpha)}^{\A}p.$$ The latter is equal to ${\sf c}_{(\mu(\{k\})\cap \sigma(\alpha))}{\sf s}_{\sigma}^{\B}q.$ Now since $\mu(\sigma(\alpha)\cap \sigma_1(\alpha))\subseteq \alpha$, we have ${\sf s}_{\mu}^{\B}p={\sf s}_{(\mu\circ \sigma\upharpoonright \alpha)}^{\A}p={\sf s}_{(\mu\circ \sigma_1)\upharpoonright \alpha)}^{\A}p_1\in A$. It thus follows that $${\sf s}_{\rho^{-1}}^{\B}{\sf c}_{[\rho(\{k\})\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho\circ \sigma\upharpoonright \alpha)}^{\A}p= {\sf c}_{[\mu(\{k\})\cap \mu\circ \sigma(\alpha)\cap \mu\circ \sigma_1(\alpha))}{\sf s}_{\sigma}^{\B}q.$$ By exactly the same method, it can be shown that $${\sf s}_{\rho_1{^{-1}}}^{\B}{\sf c}_{[\rho_1(\{k\})\cap \sigma(\alpha)]}^{\A}{\sf s}_{(\rho_1\circ \sigma\upharpoonright \alpha)}^{\A}p ={\sf c}_{[\mu(\{k\})\cap \mu\circ \sigma(\alpha)\cap \mu\circ \sigma_1(\alpha))}{\sf s}_{\sigma}^{\B}q.$$ By this we have proved (\*\*). Furthermore, it defines the required algebra $\B$. Let us check this. Since our definition is slightly different than that in [@DM], by restricting cylindrifications to be olny finite, we need to check the polyadic axioms which is tedious but routine. The idea is that every axiom can be pulled back to its corresponding axiom holding in the small algebra $\A$. We check only the axiom $${\sf c}_k(q_1\land {\sf c}_kq_2)={\sf c}_kq_1\land {\sf c}_kq_2.$$ We follow closely [@DM] p. 166. Assume that $q_1={\sf s}_{\sigma}^{\B}p_1$ and $q_2={\sf s}_{\sigma}^{\B}p_2$. Let $\rho$ be a permutation of $I$ such that $\rho(\sigma_1I\cup \sigma_2I)\subseteq I$ and let $$p={\sf s}_{\rho}^{\B}[q_1\land {\sf c}_kq_2].$$ Then $$p={\sf s}_{\rho}^{\B}q_1\land {\sf s}_{\rho}^{\B}{\sf c}_kq_2 ={\sf s}_{\rho}^{\B}{\sf s}_{\sigma_1}^{\B}p_1\land {\sf s}_{\rho}^{\B}{\sf c}_k {\sf s}_{\sigma_2}^{\B}p_2.$$ Now we calculate ${\sf c}_k{\sf s}_{\sigma_2}^{\B}p_2.$ We have by (\*) $${\sf c}_k{\sf s}_{\sigma_2}^{\B}p_2= {\sf s}^{\B}_{\sigma_2^{-1}}{\sf c}_{\rho(\{k\}\cap \sigma_2I)} {\sf s}^{\A}_{(\rho\sigma_2\upharpoonright I)}p_2.$$ Hence $$p={\sf s}_{\rho}^{\B}{\sf s}_{\sigma_1}^{\B}p_1\land {\sf s}_{\rho}^{\B}{\sf s}^{\B}_{\sigma^{-1}}{\sf c}_{\rho(\{k\}\cap \sigma_2I)} {\sf s}^{\A}_{(\rho\sigma_2\upharpoonright I)}p_2.$$ $$\begin{split} &={\sf s}^{\A}_{\rho\sigma_1\upharpoonright I}p_1\land {\sf s}_{\rho}^{\B}{\sf s}^{\A}_{\sigma^{-1}}{\sf c}_{\rho(\{k\}\cap \sigma_2I)} {\sf s}^{\A}_{(\rho\sigma_2\upharpoonright I)}p_2,\\ &={\sf s}^{\A}_{\rho\sigma_1\upharpoonright I}p_1\land {\sf s}_{\rho\sigma^{-1}}^{\A} {\sf c}_{\rho(\{k\}\cap \sigma_2I)} {\sf s}^{\A}_{(\rho\sigma_2\upharpoonright I)}p_2,\\ &={\sf s}^{\A}_{\rho\sigma_1\upharpoonright I}p_1\land {\sf c}_{\rho(\{k\}\cap \sigma_2I)} {\sf s}^{\A}_{(\rho\sigma_2\upharpoonright I)}p_2.\\ \end{split}$$ Now $${\sf c}_k{\sf s}_{\rho^{-1}}^{\B}p={\sf c}_k{\sf s}_{\rho^{-1}}^{\B}{\sf s}_{\rho}^{\B}(q_1\land {\sf c}_k q_2)={\sf c}_k(q_1\land {\sf c}_kq_2)$$ We next calculate ${\sf c}_k{\sf s}_{\rho^{-1}}p$. Let $\mu$ be a permutation of $I$ such that $\mu\rho^{-1}I\subseteq I$. Let $j=\mu(\{k\}\cap \rho^{-1}I)$. Then applying (\*), we have: $$\begin{split} &{\sf c}_k{\sf s}_{\rho^{-1}}p={\sf s}^{\B}_{\mu^{-1}}{\sf c}_{j}{\sf s}_{(\mu\rho^{-1}|I)}^{\A}p,\\ &={\sf s}^{\B}_{\mu^{-1}}{\sf c}_{j}{\sf s}_{(\mu\rho^{-1}|I)}^{\A} {\sf s}^{\A}_{\rho\sigma_1\upharpoonright I}p_1\land {\sf c}_{(\rho\{k\}\cap \sigma_2I)} {\sf s}^{\B}_{(\rho\sigma_2\upharpoonright I)}p_2,\\ &={\sf s}^{\B}_{\mu^{-1}}{\sf c}_{j}[{\sf s}_{\mu \sigma_1\upharpoonright I}p_1\land r].\\ \end{split}$$ where $$r={\sf s}_{\mu\rho^{-1}}^{\B}{\sf c}_j {\sf s}_{\rho \sigma_2\upharpoonright I}^{\A}p_2.$$ Now ${\sf c}_kr=r$. Hence, applying the axiom in the small algebra, we get: $${\sf s}^{\B}_{\mu^{-1}}{\sf c}_{j}[{\sf s}_{\mu \sigma_1\upharpoonright I}^{\A}p_1]\land {\sf c}_k q_2 ={\sf s}^{\B}_{\mu^{-1}}{\sf c}_{j}[{\sf s}_{\mu \sigma_1\upharpoonright I}^{\A}p_1\land r].$$ But $${\sf c}_{\mu(\{k\}\cap \rho^{-1}I)}{\sf s}_{(\mu\sigma_1|I)}^{\A}p_1= {\sf c}_{\mu(\{k\}\cap \sigma_1I)}{\sf s}_{(\mu\sigma_1|I)}^{\A}p_1.$$ So $${\sf s}^{\B}_{\mu^{-1}}{\sf c}_{k}[{\sf s}_{\mu \sigma_1\upharpoonright I}^{\A}p_1]={\sf c}_kq_1,$$ and we are done. To show that neat reducts commute with forming subalgebras, we proceed as in the previous proof replacing finite transformation by transformation. When we have diagonal elements, we first discard them, obtaining a $GPHA_{\alpha}$ then form the diagonal free dilation of this algebra, and finally define the diagonal elements in the dilation as in [@HMT2], theorem 5.4.17, p.233. The next lemma formulated only for $GPHA_{\alpha}$ will be used in proving our main (algebraic) result. The proof works without any modifications when we add diagonal elements. The lemma says, roughly, that if we have an $\alpha$ dimensional algebra $\A$, and a set $\beta$ containing $\alpha$, then we can find an extension $\B$ of $\A$ in $\beta$ dimensions, specified by a carefully chosen subsemigroup of $^{\beta}\beta$, such that $\A=\Nr_{\alpha}\B$ and for all $b\in B$, $|\Delta b\sim \alpha|<\omega$. $\B$ is not necessarily the minimal dilation of $\A$, because the large subsemigroup chosen maybe smaller than the semigroup used to form the unique dilation. It can happen that this extension is the minimal dilation, but in the case we consider all transformations, the constructed algebra is only a proper subreduct of the dilation obtained basically by discarding those elements $b$ in the original dilation for which $\Delta b\sim \alpha$ is infinite. \[cylindrify\] For a set $I,$ let $G_I$ be the semigroup of all finite transformations on $I$. Let $\alpha\subseteq \beta$ be infinite sets. Let $\A\in G_{\alpha}PHA_{\alpha}$ and $\B\in G_{\beta}PHA_{\beta}.$ If $\A\subseteq \Nr_{\alpha}\B$ and $X\subseteq A$, then for any $b\in \Sg^{\B}X,$ one has $|\Delta b\sim \alpha|<\omega.$ In particular, the cylindrifier ${\sf c}_{(\Delta\sim\alpha)}b$, for any such $b$ is meaningful. Let $\alpha<\beta$ be countable ordinals and let $G_{\alpha}$ and $G_{\beta}$ be strongly rich semigroups on $\alpha$ and $\beta$, respectivey. Let $\A\in G_{\alpha}PHA_{\alpha}$ and $\B\in G_{\beta}PHA_{\beta}.$ If $\A\subseteq \Nr_{\alpha}\B$ and $X\subseteq A$, then for any $b\in \Sg^{\B}X,$ we have $|\Delta b\sim \alpha|<\omega.$ For a set $I$, let $G_I$ denote the set of all transformations on $I$. Let $\alpha\subseteq \beta$ be infinite sets, such that $|\alpha|<|\beta|$. Let $\A\in G_{\alpha}PHA_{\alpha}$. Then there exist a semigroup $S$ of $G_{\beta}$ and $\B\in SPHA_{\beta},$ such that $\A=\Nr_{\alpha}\B$, $S$ contains elements $\pi$, $\sigma$ as in definition \[stronglyrich\], and for all $X\subseteq A$, one has $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X$. Furthermore, for all $b\in B$, $|\Delta b\sim \alpha|<\omega.$ In this case we say that $\B$ is a minimal extension of $\A$. Let $\alpha\subseteq \beta$ be infinite sets, and assume that $|\alpha|=|\beta|$. Let $S\subseteq {}^{\alpha}\alpha$ be a semigroup that contains all finite transformations, and two infinitary ones $\pi$ and $\sigma$ as in the definition \[stronglyrich\]. Let $\A\in SPHA_{\alpha}$. Then there exist a semigroup $T\subseteq {}^{\beta}\beta$, such that $\B\in TPHA_{\beta}$, $\A=\Nr_{\alpha}\B,$ and for all $X\subseteq A$, one has $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X$. Furthermore, for all $b\in B$, $|\Delta b\sim \alpha|<\omega.$ [Proof]{} This trivially holds for elements of $\A$. The rest follows by an easy inductive argument, since substitution can move only finitely many points. This part is delicate because we have infinitary substitutions, so, in principal, it can happen that $|\Delta x\sim \alpha|<\omega$ and $|\Delta({\sf s}_{\tau}x)\sim \alpha|\geq \omega$, when $\tau$ moves infinitely many points. We show that in this particular case, this cannot happen. Let $M=\beta\sim \alpha$. We can well assume that $\beta=\omega+\omega$ and $\alpha=\omega$. Then since $M\cap \Delta x=\emptyset$ for all $x\in \A$, it suffices to show inductively that for any $x\in \B$ and any (unary) operation $f$ of $\B$, the following condition holds: $$\text {If }|M\cap \Delta x|<\omega\text { then }|M\cap \Delta (fx)|<\omega.$$ Of course, we should check that the above holds for the Heyting operations as well, but this is absolutely straightforward. Assume that $f$ is a substitution. So let $\tau\in G_{\beta}$, such that $f={\sf s}_{\tau}$. Let $D_{\tau}=\{m\in M:\tau^{-1}(m)=\{m\}=\{\tau(m)\}\}.$ Then, it is easy to check that $|M\sim D_{\tau}|<\omega$. For the sake of brevity, let $C_{\tau}$ denote the finite set $M\sim D_{\tau}.$ By $|M\cap \Delta x|<\omega$, we have $|(M\cap \Delta x)\cup C_{\tau}|<\omega$. We will show that $M\cap \Delta ({\sf s}_{\tau}x)\subseteq (M\cap \Delta x)\cup C_{\tau}$ by which we will be done. So assume that $i\in M\sim\omega$, and that $i\notin (M\cap \Delta x)\cup C_{\tau}$. Then $i\in D_{\tau}\sim \Delta x$, so $\{\tau(i)\}=\{i\}=\tau^{-1}(i)$. Thus we get that ${\sf c}_i{\sf s}_{\tau}x={\sf s}_{\tau}{\sf c}_ix$ by item (5) in theorem \[axioms\], proving that $i\notin M\cap \Delta {\sf s}_{\tau}x.$ Now assume that $f={\sf c}_j$ with $j\in \alpha$. If $i\in M$ and $i\notin \Delta x,$ then we have ${\sf c}_i{\sf c}_jx={\sf c}_j{\sf c}_ix={\sf c}_jx$ and we are done in this case, too. We note that the condition (2) in the definition of richness suffices to implement the neat embeding, while strong richness is needed so that $\A$ exhausts the ful neat reduct. Let $\A$ and $\beta$ be given. Choose $\pi$ and $\sigma$ in ${}^{\beta}\beta$ satisfying (3) and (4) in definition \[rich\]. Let $H_{\beta}=\{\rho\in {}^{\beta}\beta: |\rho(\alpha)\cap (\beta\sim \alpha)|<\omega\}\cup \{\sigma, \pi\}$. Let $S$ be the semigroup generated by $H_{\beta}.$ Let $\B'\in G_{\beta}PHA_{\beta}$ be an ordinary dilation of $\A$ where all transformations in $^{\beta}\beta$ are used. Exists by \[dl\]. Then $\A=\Nr_{\alpha}\B'$. We take a suitable reduct of $\B'$. Let $\B$ be the subalgebra of $\B'$ generated from $A$ be all operations except for substitutions indexed by transformations not in $S$. Then, of course $A\subseteq \B$; in fact, $\A=\Nr_{\alpha}\B$, since for each $\tau\in {}^{\alpha}\alpha$, $\tau\cup Id\in S.$ We check that $\B$ is as required. It suffices to show inductively that for $b\in B$, if $|\Delta b\sim \alpha|<\omega$, and $\rho\in S$, then $|\rho(\Delta b)\sim \alpha|<\omega$. For $\rho \in H_{\beta}\sim \{\pi, \sigma\}$, this easily follows from how $\rho$ is defined, otherwise the proof is as in the previous item. We can obviously write $\beta$ as a sum of ordinals $\alpha+\omega$, so that $\beta$ itself is an ordinal, and iterate $\sigma$ as in theorem \[dl\] (1), by noting that the proof does not depend on the countability of $\A,$ but rather on that of $\beta\sim \alpha$. In more detail, for $n\leq \omega$, let $\alpha_n=\alpha+n$ and $M_n=\beta\sim \alpha_n$. For $\tau\in S$, let $\tau_n=\tau\cup Id_{M_n}$. Let $T_n$ be subsemigroup of $\langle {}^{\alpha_n}\alpha_n,\circ \rangle$ generated by $\{\tau_n:\tau\in G\} \cup \cup_{i,j\in \alpha_n}\{[i|j],[i,j]\}$. For $n\in \omega$, we let $\rho_n:\alpha_n\to \alpha$ be the bijection defined by $\rho_n\upharpoonright \alpha=\sigma^n$ and $\rho_n(\alpha+i)=i$ for all $i<n$. (Here $\sigma$ is as in the fdefinition of \[rich\]. For $n\in \omega$, for $v\in T_n$ let $v'=\rho_n\circ v\circ \rho_n^{-1}$. Then $v'\in S$. For $\tau\in T_{\omega}$, let $D_{\tau}=\{m\in M_{\omega}:\tau^{-1}(m)=\{m\}=\{\tau(m)\}\}$. Then $|M_{\omega}\sim D_{\tau}|<\omega.$ Let $\A$ is an $S$ algebra. Let $\A_n$ be the algebra defined as follows: $\A_n=\langle A,\lor, \land, \to, 0, {\sf c}_i^{\A_n},{\sf s}_v^{\A_n}\rangle_{i\in \alpha_n,v\in T_n}$ where for each $i\in \alpha_n$ and $v\in T_n$, ${\sf c}_i^{\A_n}:= {\sf c}_{\rho_n(i)}^{\A} \text { and }{\sf s}_v^{\A_n}:= {\sf s}_{v'}^{\A.}$ Then continue as in the proof of the above theorem \[dl\], by taking the ultraproduct of the $\A_n$’s relative to a cofinite ultrafilter, one then gets a dilation in $\beta$ dimensions in which $\A$ neatly embeds satisfying the required. If $\A \in GPHA_{\alpha}$ where $G_{\alpha}$ is the semigroup of all transformations on $\alpha$ and $\alpha\subseteq \beta$, there are two kinds of extensions of $\A$ to $\beta$ dimensions. The minimal dilation of $\A$ which uses all substitutions in $G_{\beta}$, and a minimal extension of $\A$ which is can be a proper subreduct of the minimal dilation, using operations in a rich subsemigroup of $G_{\beta}.$ Algebraic Proofs of main theorems ================================= Henceforth, when we write $GPHA_{\alpha}$ without further specification, we understand that we simultaneously dealing with all possibilities of $G$, and that whatever we are saying applies equaly well to all cases considered. We could also say $\A$ is a $G$ algebra without further notice; the same is to be understood. Throughout the paper dimensions will be specified by [*infinite*]{} sets or ordinals. Our work in this section is closely related to that in [@Hung]. Our main theorem is a typical representabilty result, where we start with an abstract (free) algebra, and we find a non-trivial homomorphism from this algebra to a concrete algebra based on Kripke systems (an algebraic version of Kripke frames). The idea (at least for the equality-free case) is that we start with a theory (which is defined as a pair of sets of formulas, as is the case with classical intuitionistic logic), extend it to a saturated one in enough spare dimensions, or an appropraite dilation (lemma \[t2\]), and then iterate this process countably many times forming consecutive (countably many) dilations in enough spare dimensions, using pairs of pairs (theories), cf. lemma \[t3\]; finally forming an extension that will be used to construct desired Kripke models (theorem \[main\]). The extensions constructed are essentially conservative extensions, and they will actually constitute the set of worlds of our desired Kripke model. The iteration is done by a subtle zig-zag process, a technique due to Gabbay [@b]. When we have diagonal elements (equality), constructing desired Kripke model, is substantialy different, and much more intricate. All definitions and results up to lemma \[main1\], though formulated only for the diagonal-free case, applies equally well to the case when there are diagonal elements, with absolutely no modifications. (The case when diagonal elements are present will be dealt with in part 2). Let $\A\in GPHA_{\alpha}$. A theory in $\A$ is a pair $(\Gamma, \Delta)$ such that $\Gamma, \Delta\subseteq \A$. A theory $(\Gamma, \Delta)$ is consistent if there are no $a_1,\ldots a_n\in \Gamma$ and $b_1,\ldots b_m\in \Delta$ ($m,n\in \omega$) such that $$a_1\land\ldots a_n\leq b_1\lor\ldots b_m.$$ Not that in this case, we have $\Gamma\cap \Delta=\emptyset$. Also if $F$ is a filter (has the finite intersection property), then it is always the case that $(F, \{0\})$ is consistent. A theory $(\Gamma, \Delta)$ is complete if for all $a\in A,$ either $a\in \Gamma$ or $a\in \Delta$. A theory $(\Gamma, \Delta)$ is saturated if for all $a\in A$ and $j\in \alpha$, if ${\sf c}_ja\in \Gamma$, then there exists $k\in \alpha\sim \Delta a$, such that ${\sf s}^j_ka\in \Gamma$. Note that a saturated theory depends only on $\Gamma$. \[t1\]Let $\A\in GPHA_{\alpha}$ and $(\Gamma,\Delta)$ be a consistent theory. For any $a\in A,$ either $(\Gamma\cup \{a\}, \Delta)$ or $(\Gamma, \Delta\cup\{a\})$ is consistent. $(\Gamma,\Delta)$ can be extended to a complete theory in $\A.$ [Proof]{} Cf. [@Hung]. Suppose for contradiction that both theories are inconsistent. Then we have $\mu_1\land a\leq \delta_1$ and $\mu_2\leq a\land \delta_2$ where $\mu_1$ and $\mu_2$ are some conjunction of elements of $\Gamma$ and $\delta_1$, $\delta_2$ are some disjunction of elements of $\Delta$. But from $(\mu_1\land a\to \delta_1)\land (\mu_2\to a\lor \delta_2)\leq (\mu_1\land \mu_2\to \delta_1\lor \delta_2),$ we get $\mu_1\land \mu_2\leq \delta_1\lor \delta_2,$ which contradicts the consistency of $(\Gamma, \Delta)$. Cf. [@Hung]. Assume that $|A|=\kappa$. Enumerate the elements of $\A$ as $(a_i:i<\kappa)$. Then we can extend $(\Gamma, \Delta)$ consecutively by adding $a_i$ either to $\Gamma$ or $\Delta$ while preserving consistency. In more detail, we define by transfinite induction a sequence of theories $(\Gamma_i,\Delta_i)$ for $i\in \kappa$ as follows. Set $\Gamma_0=\Gamma$ and $\Delta_0=\Delta$. If $\Gamma_i,\Delta_i$ are defined for all $i<\mu$ where $\mu$ is a limit ordinal, let $\Gamma_{\mu}=(\bigcup_{i\in \mu} \Gamma_i, \bigcup_{i\in \mu} \Delta_i)$. Now for successor ordinals. Assume that $(\Gamma_i, \Delta_i)$ are defined. Set $\Gamma_{i+1}=\Gamma_i\cup \{a_i\}, \Delta_{i+1}=\Delta_i$ in case this is consistent, else set $\Gamma_{i+1}=\Gamma_i$ and $\Delta_{i+1}=\Delta_i\cup \{a_i\}$. Let $T=\bigcup_{i\in \kappa}T_i$ and $F= \bigcup_{i\in \kappa} F_i$, then $(T, F)$ is as desired. \[t2\] Let $\A\in GPHA_{\alpha}$ and $(\Gamma,\Delta)$ be a consistent theory of $\A$. Let $I$ be a set such that $\alpha\subseteq I$ and let $\beta=|I\sim \alpha|=\max(|A|, |\alpha|).$ Then there exists a minimal dilation $\B$ of $\A$ of dimension $I$, and a theory $(T,F)$ in $\B$, extending $(\Gamma,\Delta)$ such that $(T,F)$ is saturated and complete. [Proof]{} Let $I$ be provided as in the statement of the lemma. By lemma \[dl\], there exists $\B\in GPHA_I$ such that $\A\subseteq \Nr_{\alpha}\B$ and $\A$ generates $\B$. We also have for all $X\subseteq \A$, $\Sg^{\A}X=\Nr_{\alpha}\Sg^{\B}X$. Let $\{b_i:i<\kappa\}$ be an enumeration of the elements of $\B$; here $\kappa=|B|.$ Define by transfinite recursion a sequence $(T_i, F_i)$ for $i<\kappa$ of theories as follows. Set $T_0=\Gamma$ and $F_0=\Delta$. We assume inductively that $$|\beta\sim \bigcup_{x\in T_i} \Delta x\cup \bigcup_{x\in F_i}\Delta x|\geq \omega.$$ This is clearly satisfied for $F_0$ and $T_0$. Now we need to worry only about successor ordinals. Assume that $T_i$ and $F_i$ are defined. We distinguish between two cases: 1. $(T_i, F_i\cup \{b_i\})$ is consistent. Then set $T_{i+1}=T_i$ and $F_{i+1}=F_i\cup \{b_i\}.$ 2. If not, that is if $(T_i, F_i\cup \{b_i\})$ is inconsistent. In this case, we distinguish between two subcases: \(a) $b_i$ is not of the form ${\sf c}_jp.$ Then set $T_{i+1}=T_i\cup \{b_i\}$ and $F_{i+1}=F_i$. \(b) $b_i={\sf c}_jp$ for some $j\in I$. Then set $T_{i+1}=T_i\cup \{{\sf c}_jp, {\sf s}_u^jp\}$ where $u\notin \Delta p\cup \bigcup_{x\in T_i}\cup \bigcup_{x\in F_i}\Delta x$ and $F_{i+1}=F_i$. Such a $u$ exists by the inductive assumption. Now we check by induction that each $(T_i, F_i)$ is consistent. The only part that needs checking, in view of the previous lemma, is subcase (b). So assume that $(T_i,F_i)$ is consistent and $b_i={\sf c}_jp.$ If $(T_{i+1}, F_{i+1})$ is inconsistent, then we would have for some $a\in T_i$ and some $\delta\in F_i$ that $a\land {\sf c}_jp\land {\sf s}_u^jp\leq \delta.$ From this we get $a\land {\sf c}_jp\leq \delta,$ because ${\sf s}_u^jp\leq {\sf c}_jp.$ But this contradicts the consistency of $(T_i\cup \{{\sf c}_jp\}, F_i)$. Let $T=\bigcup_{i\in \kappa}T_i$ and $F=\bigcup_{i\in \kappa} F_i$, then $(T,F)$ is consistent. We show that it is saturated. If ${\sf c}_jp\in T$, then ${\sf c}_jp\in T_{i+1}$ for some $i$, hence ${\sf s}_u^jp\in T_{i+1}\subseteq T$ and $u\notin \Delta p$. Now by lemma \[t1\], we can extend $(T,F)$ is $\B$ to a complete theory, and this will not affect saturation, since the process of completion does not take us out of $\B$. The next lemma constitutes the core of our construction; involving a zig-zag Gabbay construction, it will be used repeatedly, to construct our desired representation via a set algebra based on a Kripke system defined in \[Kripke\] \[t3\] Let $\A\in GPHA_{\alpha}$ be generated by $X$ and let $X=X_1\cup X_2$. Let $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ be two consistent theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively such that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$, $\Gamma_0\subseteq \Gamma_0^*$. Assume further that $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. Suppose that $I$ is a set such that $\alpha\subseteq I$ and $|I\sim \alpha|=max (|A|,|\alpha|)$. Then there exist a dilation $\B\in GPHA_I$ of $\A$, and theories $T_1=(\Delta_{\omega}, \Gamma_{\omega})$, $T_2=(\Theta_{\omega}, \Gamma_{\omega}^*)$ extending $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$, such that $T_1$ and $T_2$ are consistent and saturated in $\Sg^{\B}X_1$ and $\Sg^{\B}X_2,$ respectively, $(\Delta_{\omega}\cap \Theta_{\omega}, \Gamma_{\omega})$ is complete in $\Sg^{\B}X_1\cap \Sg^{\B}X_2,$ and $\Gamma_{\omega}\subseteq \Gamma_{\omega}^*$. [Proof]{} Like the corresponding proof in [@Hung], we will build the desired theories in a step-by-step zig-zag manner in a large enough dilation whose dimension is specified by $I$. The spare dimensions play a role of added witnesses, that will allow us to eliminate quantifiers, in a sense. Let $\A=\A_0\in GPHA_{\alpha}$. The proof consists of an iteration of lemmata \[t1\] and \[t2\]. Let $\beta=max(|A|, |\alpha|)$, and let $I$ be such that $|I\sim \alpha|=\beta$. We distinguish between two cases: Assume that $G$ is strongly rich or $G$ contains consists of all finite transformations. In this case we only deal with minimal dilations. We can write $\beta = I\sim \alpha$ as $\bigcup_{n=1}^{\infty}C_n$ where $C_i\cap C_j=\emptyset$ for distinct $i$ and $j$ and $|C_i|=\beta$ for all $i$. Then iterate first two items in lemma \[dl\]. Let $\A_1=\A(C_1)\in G_{\alpha\cup C_1}PHA_{\alpha\cup C_1}$ be a minimal dilation of $\A$, so that $\A=\Nr_{\alpha}\A_1$. Let $\A_2=\A(C_1)(C_2)$ be a minimal dilation of $\A_1$ so that $\A_1=\Nr_{\alpha\cup C_1}\A_2$. Generally, we define inductively $\A_n=\A(C_1)(C_2)\ldots (C_n)$ to be a minimal dilation of $\A_{n-1}$, so that $\A_{n-1}=\Nr_{\alpha\cup C_1\cup \ldots C_{n-1}}\A_n$. Notice that for $k<n$, $\A_n$ is a minimal dilation of $\A_k$. So we have a sequence of algebras $\A_0\subseteq \A_1\subseteq \A_2\ldots.$ Each element in the sequence is the minimal dilation of its preceding one. $G$ contains all transformations. Here we shall have to use minimal extensions at the start, i.e at the first step of the iteration. We iterate lemma \[dl\], using items (3) and (4) in lemma \[cylindrify\] by taking $|C_1|=\beta$, and $|C_i|=\omega$ for all $i\geq 2$; this will yield the desired sequence of extensions. Now that we have a sequence of extensions $\A_0\subseteq \A_1\ldots$ in different increasing dimensions, we now form a limit of this sequence in $I$ dimensions. We can use ultraproducts, but instead we use products, and quotient algebras. First form the Heyting algebra, that is the product of the Heyting reducts of the constructed algebras, that is take $\C=\prod_{n=0}^{\infty}\Rd A_n$, where $\Rd \A_n$ denotes the Heyting reduct of $\A_n$ obtained by discarding substitutions and cylindrifiers. Let $$M=\{f\in C: (\exists n\in \omega)(\forall k\geq n) f_{k}=0\}.$$ Then $M$ is a Heyting ideal of $\C$. Now form the quotient Heyting algebra $\D=\C/M.$ We want to expand this Heyting algebra algebra by cylindrifiers and substitutions, i.e to an algebra in $GPHA_{I}$. Towards this aim, for $\tau\in {}G,$ define $\phi({\tau})\in {} ^CC$ as follows: $$(\phi(\tau)f)_n={\sf s}_{\tau\upharpoonright dim \A_n}^{\A_n}f_n$$ if $\tau(dim(\A_n))\subseteq dim (\A_n)$. Otherwise $$(\phi(\tau)f)_n=f_n.$$ For $j\in I$, define $${\sf c}_jf_n={\sf c}_{(dim \A_n\cap \{j\})}^{\A_n}f_n,$$ and $${\sf q}_jf _n={\sf q}_{(dim \A_n\cap \{j\})}^{\A_n}f_n.$$ Then for $\tau\in G$ and $j\in I$, set $${\sf s}_{\tau}(f/M)=\phi({\tau})f/M,$$ $${\sf c}_{j}(f/M)=({\sf c}_j f)/M,$$ and $${\sf q}_{j}(f/M)=({\sf q}_j f)/M.$$ Then, it can be easily checked that, $\A_{\infty}=(\D, {\sf s}_{\tau}, {\sf c}_{j}, {\sf q}_{j})$ is a $GPHA_I$, in which every $\A_n$ neatly embeds. We can and will assume that $\A_n=\Nr_{\alpha\cup C_1\ldots \cup C_n}\A_{\infty}$. Also $\A_{\infty}$ is a minimal dilation of $\A_n$ for all $n$. During our ’zig-zagging’ we shall be extensively using lemma \[cylindrify\]. From now on, fix $\A$ to be as in the statement of lemma \[t3\] for some time to come. So $\A\in GPHA_{\alpha}$ is generated by $X$ and $X=X_1\cup X_2$. $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ are two consistent theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively such that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$, $\Gamma_0\subseteq \Gamma_0^*$. Finally $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2.$ Now we have: $$\Delta_0\subseteq \Sg^{\A}X_1\subseteq \Sg^{\A(C_1)}X_1\subseteq \Sg^{\A(C_1)(C_2)}X_1\subseteq \Sg^{\A(C_1)(C_2)(C_3)}X_1 \ldots\subseteq \Sg^{\A_{\infty}}X_1.$$ $$\Theta_0\subseteq \Sg^{\A}X_2\subseteq \Sg^{\A(C_1)}X_2\subseteq \Sg^{\A(C_1)(C_2)}X_2\subseteq \Sg^{\A(C_1)(C_2)(C_3)}X_2 \ldots\subseteq \Sg^{\A_{\infty}}X_2.$$ In view of lemmata \[t1\], \[t2\], extend $(\Delta_0, \Gamma_0)$ to a complete and saturated theory $(\Delta_1, \Gamma_1')$ in $\Sg^{\A(C_1)}X_1$. Consider $(\Delta_1, \Gamma_0)$. Zig-zagging away, we extend our theories in a step by step manner. The proofs of the coming Claims, 1, 2 and 3, are very similar to the proofs of the corresponding claims in [@Hung], which are in turn an algebraic version of lemmata 4.18-19-20 in [@b], with one major difference from the former. In our present situation, we can cylindrify on only finitely many indices, so we have to be careful, when talking about dimension sets, and in forming neat reducts (or compressions). Our proof then becomes substantially more involved. In the course of our proof we use extensively lemmata \[dl\] and \[cylindrify\] which are not formulated in [@Hung] because we simply did not need them when we had cylindrifications on possibly infinite sets. [Claim 1]{} The theory $T_1=(\Theta_0\cup (\Delta_1\cap \Sg^{\A(C_1)}X_2), \Gamma_0^*)$ is consistent in $\Sg^{\A(C_1)}X_2.$ [Proof of Claim 1]{} Assume that $T_1$ is inconsistent. Then for some conjunction $\theta_0$ of elements in $\Theta_0$, some $E_1\in \Delta_1\cap \Sg^{\A(C_1)}X_2,$ and some disjunction $\mu_0^*$ in $\Gamma_0^*,$ we have $\theta_0\land E_1\leq \mu_0^*,$ and so $E_1\leq \theta_0\rightarrow \mu_0^*.$ Since $\theta_0\in \Theta_0\subseteq \Sg^{\A}X_2$ and $\mu_0^*\in \Gamma_0^*\subseteq \Sg^{\A}X_2\subseteq \Nr_{\alpha}^{\A(C_1)}\A$, therefore, for any finite set $D\subseteq C_1\sim \alpha$, we have ${\sf c}_{(D)}\theta_0=\theta_0$ and ${\sf c}_{(D)}\mu_0^*=\mu_0^*$. Also for any finite set $D\subseteq C_1\sim \alpha,$ we have ${\sf c}_{(D)}E_1\leq {\sf c}_{(D)}(\theta_0\to \mu_0^*)=\theta_0\to \mu^*.$ Now $E_1\in \Delta_1$, hence $E_1\in \Sg^{\A(C_1)}X_1$. By definition, we also have $E_1\in \Sg^{\A(C_1)}X_2.$ By lemma \[cylindrify\] there exist finite sets $D_1$ and $D_2$ contained in $C_1\sim \alpha,$ such that $${\sf c}_{(D_1)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_1$$ and $${\sf c}_{(D_2)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_2.$$ Le $D=D_1\cup D_2$. Then $D\subseteq C_1\sim \alpha$ and we have: $${\sf c}_{(D)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_1=\Sg^{\Nr_{\alpha}\A(C_1)}X_1=\Sg^{\A}X_1$$ and $${\sf c}_{(D)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_2=\Sg^{\Nr_{\alpha}\A(C_1)}X_2=\Sg^{\A}X_2,$$ that is to say $${\sf c}_{(D)}E_1\in \Sg^{\A}X_1\cap \Sg^{\A}X_2.$$ Since $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2,$ we get that ${\sf c}_{(D)}E_1$ is either in $\Delta_0\cap \Theta_0$ or $\Gamma_0$. We show that either way leads to a contradiction, by which we will be done. Suppose it is in $\Gamma_0$. Recall that we extended $(\Delta_0, \Gamma_0)$ to a complete saturated extension $(\Delta, \Gamma')$ in $\Sg^{\A(C_1)}X_1$. Since $\Gamma_0\subseteq \Gamma_1',$ we get that ${\sf c}_{(D)}E_1\in \Gamma_1'$ hence ${\sf c}_{(D)}E_1\notin \Delta_1$ because $(\Delta_1,\Gamma_1')$ is saturated and consistent. But this contradicts that $E_1\in \Delta_1$ because $E_1\leq {\sf c}_{(D)}E_1.$ Thus, we can infer that ${\sf c}_{(D)}E_1\in \Delta_0\cap \Theta_0$. In particular, it is in $\Theta_0,$ and so $\theta_0\rightarrow \mu_0^*\in \Theta_0$. But again this contradicts the consistency of $(\Theta_0, \Gamma_0^*)$. Now we extend $T_1$ to a complete and saturated theory $(\Theta_2, \Gamma_2^*)$ in $\Sg^{\A(C_1)(C_2)}X_2$. Let $\Gamma_2=\Gamma_2^*\cap \Sg^{\A(C_1)(C_2)}X_1$. [Claim 2]{} The theory $T_2=(\Delta_1\cup (\Theta_2\cap \Sg^{\A(C_1)(C_2))}X_1), \Gamma_2)$ is consistent in $\Sg^{\A(C_1)(C_2)}X_1$. [Proof of Claim 2]{} If the Claim fails to hold, then we would have some $\delta_1\in \Delta_1$, $E_2\in \Theta_2\cap \Sg^{\A(C_1)(C_2)}X_1,$ and a disjunction $\mu_2\in \Gamma_2$ such that $\delta_1\land E_2\rightarrow \mu_2,$ and so $\delta_1\leq (E_2\rightarrow \mu_2)$ since $\delta_1\in \Delta_1\subseteq \Sg^{\A(C_1)}X_1$. But $\Sg^{\A(C_1)}X_1\subseteq \Nr_{\alpha\cup C_1}^{\A(C_1)(C_2)}X_1$, therefore for any finite set $D\subseteq C_2\sim C_1,$ we have ${\sf q}_{(D)}\delta_1=\delta_1.$ The following holds for any finite set $D\subseteq C_2\sim C_1,$ $$\delta_1\leq {\sf q}_{(D)}(E_2\rightarrow \mu_2).$$ Now, by lemma \[cylindrify\], there is a finite set $D\subseteq C_2\sim C_1,$ satisfying $$\begin{split} \delta_1\to {\sf q}_{(D)}(E_2\rightarrow \mu_2) &\in \Nr_{\alpha\cup C_1}\Sg^{\A(C_1)(C_2)}X_2,\\ &=\Sg^{\Nr_{\alpha\cup C_1}\A(C_1)(\A(C_2)}X_2,\\ &=\Sg^{\A(C_1)}X_2.\\ \end{split}$$ Since $\delta_1\in \Delta_1$, and $\delta_1\leq {\sf q}_{(D)}(E_2\to \mu_2)$, we get that ${\sf q}_{(D)}(E_2\rightarrow \mu_2)$ is in $\Delta_1\cap \Sg^{\A(C_1)}X_2$. We proceed as in the previous claim replacing $\Theta_0$ by $\Theta_2$ and the existental quantifier by the universal one. Let $E_1= {\sf q}_{(D)}(E_2\to \mu_2)$. Then $E_1\in \Sg^{\A(C_1)}X_1\cap \Sg^{\A(C_2)}X_2$. By lemma \[cylindrify\] there exist finite sets $D_1$ and $D_2$ contained in $C_1\sim \alpha$ such that $${\sf q}_{(D_1)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_1,$$ and $${\sf q}_{(D_2)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_2.$$ Le $J=D_1\cup D_2$. Then $J\subseteq C_1\sim \alpha,$ and we have: $${\sf q}_{(J)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_1=\Sg^{\Nr_{\alpha}\A(C_1)}X_1=\Sg^{\A}X_1$$ and $${\sf q}_{(J)}E_1\in \Nr_{\alpha}\Sg^{\A(C_1)}X_2=\Sg^{\Nr_{\alpha}\A(C_1)}X_2=\Sg^{\A}X_2.$$ That is to say, $${\sf q}_{(J)}E_1\in \Sg^{\A}X_1\cap \Sg^{\A}X_2.$$ Now $(\Delta_0\cap \Theta_2\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2,$ we get that ${\sf q}_{(J)}E_1$ is either in $\Delta_0\cap \Theta_2$ or $\Gamma_0$. Suppose it is in $\Gamma_0$. Since $\Gamma_0\subseteq \Gamma_1'$, we get that ${\sf q}_{(J)}E_1\in \Gamma_1'$, hence ${\sf q}_{(J)}E_1\notin \Delta_1,$ because $(\Delta_1,\Gamma_1')$ is saturated and consistent. Here, recall that, $(\Delta, \Gamma')$ is a saturated complete extension of $(\Gamma, \Delta)$. But this contradicts that $E_1\in \Delta_1$. Thus, we can infer that ${\sf q}_{(J)}E_1\in \Delta_0\cap \Theta_2$. In particular, it is in $\Theta_2$. Hence ${\sf q}_{(D\cup J)}(E_2\to \mu_2)\in \Theta_2$, and so $E_2\to \mu_2\in \Theta_2$ since ${\sf q}_{(D\cup J)}(E_2\to \mu_2)\leq E_1\to \mu_2$. But this is a contradiction, since $E_2\in \Theta_2$, $\mu_2\in \Gamma_2^*$ and $(\Theta_2,\Gamma_2^*)$ is consistent. Extend $T_2$ to a complete and saturated theory $(\Delta_3, \Gamma_3')$ in $\Sg^{\A(C_1)(C_2)(C_3)}X_1$ such that $\Gamma_2\subseteq \Gamma_3'$. Again we are interested only in $(\Delta_3, \Gamma_2)$. [Claim 3 ]{} The theory $T_3=(\Theta_2\cup \Delta_3\cap \Sg^{\A(C_1)(C_2)(C_3)}X_2, \Gamma_2^*)$ is consistent in $\Sg^{\A(C_1)(C_2)(C_3)}X_2.$ [Proof of Claim 3]{} Seeking a contradiction, assume that the Claim does not hold. Then we would get for some $\theta_2\in \Theta_2$, $E_3\in \Delta_3\cap \Sg^{\A(C_1)(C_2)(C_3)}X_2$ and some disjunction $\mu_2^*\in \Gamma_2^*,$ that $\theta_2\land E_3\leq \mu_2^*.$ Hence $E_3\leq \theta_2\rightarrow \mu_2^*.$ For any finite set $D\subseteq C_3\sim (C_1\cup C_2),$ we have ${\sf c}_{(D)}E_3\leq \theta_2\rightarrow \mu_2^*$. By lemma \[cylindrify\], there is a finite set $D_3\subseteq C_3\sim (C_1\cup C_2),$ satisfying $$\begin{split} {\sf c}_{(D_3)}E_3 &\in \Nr_{\alpha\cup C_1\cup C_2}\Sg^{\A(C_1)(C_2)(C_3)}X_1\\ &=\Sg^{\Nr_{\alpha\cup C_1\cup C_2}\A(C_1)C_2)(C_3)}X_1\\ &= \Sg^{\A(C_1)(C_2)}X_1.\\ \end{split}$$ If ${\sf c}_{(D_3)}E_3\in \Gamma_2^*$, then it in $\Gamma_2$, and since $\Gamma_2\subseteq \Gamma_3'$, it cannot be in $\Delta_3$. But this contradicts that $E_3\in \Delta_3$. So ${\sf c}_{(D_3)}E_3\in \Theta_2,$ because $E_3\leq {\sf c}_{(D_3)}E_3,$ and so $(\theta_2\rightarrow \mu_2^*)\in \Theta_2,$ which contradicts the consistency of $(\Theta_2, \Gamma_2^*).$ Likewise, now extend $T_3$ to a complete and saturated theory $(\Delta_4, \Gamma_4')$ in $\Sg^{\A(C_1)(C_2)(C_3)(C_4)}X_2$ such that $\Gamma_3\subseteq \Gamma_4'.$ As before the theory $(\Delta_3, \Theta_4\cap \Sg^{\A(C_1)(C_2)(C_3)(C_4)}X_1, \Gamma_4)$ is consistent in $\Sg^{\A(C_1)(C_2)(C_3)(C_4)}X_1$. Continue, inductively, to construct $(\Delta_5, \Gamma_5')$, $(\Delta_5, \Gamma_4)$ and so on. We obtain, zigzaging along, the following sequences: $$(\Delta_0, \Gamma_0), (\Delta_1, \Gamma_0), (\Delta_3, \Gamma_2)\ldots$$ $$(\Theta_0, \Gamma_0^*), (\Theta_2, \Gamma_2^*), (\Theta_4, \Gamma_4^*)\ldots$$ such that $(\theta_{2n}, \Gamma_{2n}^*)$ is complete and saturated in $\Sg^{\A(C_1)\ldots (C_{2n})}X_2,$ $(\Delta_{2n+1}, \Gamma_{2n})$ is a saturated theory in $\Sg^{\A(C_1)\ldots (C_{2n+1})}X_1,$ $\Theta_{2n}\subseteq \Theta_{2n+2}$, $\Gamma_{2n}^*\subseteq \Gamma_{2n+2}^*$ and $\Gamma_{2n}=\Gamma_{2n}^*\cap \Sg^{\A(C_1)\ldots \A(C_{2n})}X_1,$ $\Delta_0\subseteq \Delta_1\subseteq \Delta_3\subseteq \ldots .$ Now let $\Delta_{\omega}=\bigcup_{n}\Delta_n$, $\Gamma_{\omega}=\bigcup_{n}\Gamma_n$, $\Gamma_{\omega}^*=\bigcup_{n}\Gamma_n^*$ and $\Theta_{\omega}=\bigcup_n\Theta_n$. Then we have $T_1=(\Delta_{\omega}, \Gamma_{\omega})$, $T_2=(\Theta_{\omega}, \Gamma_{\omega}^*)$ extend $(\Delta, \Gamma)$, $(\Theta, \Gamma^*)$, such that $T_1$ and $T_2$ are consistent and saturated in $\Sg^{\B}X_1$ and $\Sg^{\B}X_2,$ respectively, $\Delta_{\omega}\cap \Theta_{\omega}$ is complete in $\Sg^{\B}X_1\cap \Sg^{\B}X_2,$ and $\Gamma_{\omega}\subseteq \Gamma_{\omega}^*$. We check that $(\Delta_{\omega}\cap \Theta_{\omega},\Gamma_{\omega})$ is complete in $\Sg^{\B}X_1\cap \Sg^{\B}X_2$. Let $a\in \Sg^{\B}X_1\cap \Sg^{\B}X_2$. Then there exists $n$ such that $a\in \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2$. Now $(\Theta_{2n}, \Gamma_{2n}^*)$ is complete and so either $a\in \Theta_{2n}$ or $a\in \Gamma_{2n}^*$. If $a\in \Theta_{2n}$ it will be in $\Delta_{2n+1}$ and if $a\in \Gamma_{2n}^*$ it will be in $\Gamma_{2n}$. In either case, $a\in \Delta_{\omega}\cap \Theta_{\omega}$ or $a\in \Gamma_{\omega}$. Let $\A$ be an algebra generated by $X$ and assume that $X=X_1\cup X_2$. A pair $((\Delta,\Gamma)$ $(T,F))$ of theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2$ is a matched pair of theories if $(\Delta\cap T\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma\cap F\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. A theory $(T, F)$ extends a theory $(\Delta, \Gamma)$ if $\Delta\subseteq T$ and $\Gamma\subseteq F$. A pair $(T_1, T_2)$ of theories extend another pair $(\Delta_1, \Delta_2)$ if $T_1$ extends $\Delta_1$ and $T_2$ extends $\Delta_2.$ The following Corollary follows directly from the proof of lemma \[t3\]. \[main1\] Let $\A\in GPHA_{\alpha}$ be generated by $X$ and let $X=X_1\cup X_2$. Let $((\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*))$ be a matched pair in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively. Let $I$ be a set such that $\alpha \subseteq I$, and $|I\sim \alpha|=max(|A|, |\alpha|)$. Then there exists a dilation $\B\in GPHA_I$ of $\A$, and a matched pair, $(T_1, T_2)$ extending $((\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*))$, such that $T_1$ and $T_2$ are saturated in $\Sg^{\B}X_1$ and $\Sg^{\B}X_2$, respectively. We next define set algebras based on Kripke systems. We stipulate that ubdirect products (in the univerasl algebraic sense) are the representable algebras, which the abstract axioms formulated in ? aspire to capture. Here Kripke systems (a direct generalization of Kripke frames) are defined differently than those defined in [@Hung], because we allow [*relativized*]{} semantics. In the clasical case, such algebras reduce to products of set algebras. [^3] Let $\alpha$ be an infinite set. A Kripke system of dimension $\alpha$ is a quadruple $\mathfrak{K}=(K, \leq \{X_k\}_{k\in K}, \{V_k\}_{k\in K}),$ such that $V_k\subseteq {}^{\alpha}X_k,$ and $(K,\leq)$ is preordered set, For any $k\in K$, $X_k$ is a non-empty set such that $$k\leq k'\implies X_k\subseteq X_{k'}\text { and } V_k\subseteq V_{k'}.$$ \[Kripke\] Let $\mathfrak{O}$ be the Boolean algebra $\{0,1\}$. Now Kripke systems define concrete polyadic Heyting algebras as follows. Let $\alpha$ be an infinite set and $G$ be a semigroup of transformations on $\alpha$. Let $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}, \{V_k\}_{k\in K})$ be a Kripke system. Consider the set $$\mathfrak{F}_{\mathfrak{K}}=\{(f_k:k\in K); f_k:V_k\to \mathfrak{O}, k\leq k'\implies f_k\leq f_{k'}\}.$$ If $x,y\in {}^{\alpha}X_k$ and $j\in \alpha$ we write $x\equiv_jy$ if $x(i)=y(i)$ for all $i\neq j$. We write $(f_k)$ instead of $(f_k:k\in K)$. In $\mathfrak{F}_{\mathfrak{K}}$ we introduce the following operations: $$(f_k)\lor (g_k)=(f_k\lor g_k)$$ $$(f_k)\land (g_k)=(f_k\land g_k.)$$ For any $(f_k)$ and $(g_k)\in \mathfrak{F}$, define $$(f_k)\rightarrow (g_k)=(h_k),$$ where $(h_k)$ is given for $x\in V_k$ by $h_k(x)=1$ if and only if for any $k'\geq k$ if $f_{k'}(x)=1$ then $g_{k'}(x)=1$. For any $\tau\in G,$ define $${\sf s}_{\tau}:\mathfrak{F}\to \mathfrak{F}$$ by $${\sf s}_{\tau}(f_k)=(g_k)$$ where $$g_k(x)=f_k(x\circ \tau)\text { for any }k\in K\text { and }x\in V_k.$$ For any $j\in \alpha$ and $(f_k)\in \mathfrak{F},$ define $${\sf c}_{j}(f_k)=(g_k),$$ where for $x\in V_k$ $$g_k(x)=\bigvee\{f_k(y): y\in V_k,\ y\equiv_j x\}.$$ Finally, set $${\sf q}_{j}(f_k)=(g_k)$$ where for $x\in V_k,$ $$g_k(x)=\bigwedge\{f_l(y): k\leq l, \ y\in V_k, y\equiv_j x\}.$$ The diagonal element ${\sf d}_{ij}$ is defined to be the tuple $(f_k:k\in K)$ where for $x\in V_k$, $f_k(x)=1$ iff $x_i=x_j.$ The algebra $\F_{\bold K}$ is called the set algebra based on the Kripke system $\bold K$. Diagonal Free case ------------------ Our next theorem addresses the cases of $GPHA_{\alpha}$ with $G$ a rich semigroup, and everything is countable, and the case when $G={}^{\alpha}\alpha$ with no restrictions on cardinality. It is an algebraic version of a version of Robinson’s joint consistency theorem: A pair of consistent theories that agree on their common part can be amalgamated by taking their union to form a consistent extension of both; however, we stipulate that the second component of the first theory is included in the second component of the second theory. We will provide examples showing that we cannot omit this condition. The case when $G$ consists of finite transformations will be dealt with separately. It also says, that the results in [@Hung] proved for full polyadic Heyting algebras remains valid when we restrict cylindrifications to be finite, possibly add diagonal elements, and consider semigroups that could be finitely generated, showing that the presence of all infinitary substitutions and infinitary cylindrifications is somewhat of an overkill. Indeed, the axiomatization of full polyadic Heyting algebras studied in [@Hung] is extremely complex from the recursion theoretic point of view [@NS], while the axiomatizations studied here are far less complex; indeed they are recursive. This is definitely an acet from the algebraic point of view. \[main\] Let $\alpha$ be an infinite set. Let $G$ be a semigroup on $\alpha$ containing at least one infinitary transformation. Let $\A$ be the free $G$ algebra generated by $X$, and suppose that $X=X_1\cup X_2$. Let $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ be two consistent theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively. Assume that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$ and $\Gamma_0\subseteq \Gamma_0^*$. Assume, further, that $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. Then there exist a Kripke system $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}\{V_k\}_{k\in K}),$ a homomorphism $\psi:\A\to \mathfrak{F}_{\mathfrak K},$ $k_0\in K$, and $x\in V_{k_0}$, such that for all $p\in \Delta_0\cup \Theta_0$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=1$ and for all $p\in \Gamma_0^*$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=0$. [Proof]{} We use lemma \[t3\], extensively. Assume that $\alpha$, $G$, $\A$ and $X_1$, $X_2$ and everything else in the hypothesis are given. Let $I$ be a set containing $\alpha$ such that $\beta=|I\sim \alpha|=max(|A|, |\alpha|).$ If $G$ is strongly rich, let $(K_n:n\in \omega)$ be a family of pairwise disjoint sets such that $|K_n|=\beta.$ Define a sequence of algebras $\A=\A_0\subseteq \A_1\subseteq \A_2\subseteq \A_2\ldots \subseteq \A_n\ldots,$ such that $\A_{n+1}$ is a minimal dilation of $\A_n$ and $dim(\A_{n+1})=\dim\A_n\cup K_n$. If $G={}^{\alpha}\alpha$, then let $(K_n:n\in \omega\}$ be a family of pairwise disjoint sets, such that $|K_1|=\beta$ and $|K_n|=\omega$ for $n\geq 1$, and define a sequence of algebras $\A=\A_0\subseteq \A_1\subseteq \A_2\subseteq \A_2\ldots \subseteq \A_n\ldots,$ such that $\A_1$ is a minimal extension of $\A$, and $\A_{n+1}$ is a minimal dilation of $\A_n$ for $n\geq 2$, with $dim(\A_{n+1})=\dim\A_n\cup K_n$. We denote $dim(\A_n)$ by $I_n$ for $n\geq 1$. Recall that $dim(\A_0)=\dim\A=\alpha$. We interrupt the main stream of the proof by two consecutive claims. Not to digress, it might be useful that the reader at first reading, only memorize their statements, skip their proofs, go on with the main proof, and then get back to them. The proofs of Claims 1 and 2 to follow are completely analogous to the corresponding claims in [@Hung]. The only difference is that we deal with only finite cylindrifiers, and in this respect they are closer to the proofs of lemmata 4.22-23 in [@b]. Those two claims are essential in showing that the maps that will be defined shortly into concrete set algebras based on appropriate Kripke systems, defined via pairs of theories, in increasing extensions (dimensions), are actually homomorphisms. In fact, they have to do with the preservation of the operations of implication and universal quantification. The two claims use lemma \[t3\]. [Claim 1]{} Let $n\in \omega$. If $((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, then the following hold. For any $a,b\in \Sg^{\A_n}X_1$ if $a\rightarrow b\notin \Delta$, then there is a matched pair $((\Delta',\Gamma'), (T', F'))$ of saturated theories in $\Sg^{\A_{n+1}}X_1$ and $\Sg^{\A_{n+1}}X_2,$ respectively, such that $\Delta\subseteq \Delta'$, $T\subseteq T',$ $a\in \Delta'$ and $b\notin \Delta'$. [Proof of Claim 1]{} Since $a\rightarrow b\notin \Delta,$ we have $(\Delta\cup\{a\}, b)$ is consistent in $\Sg^{\A_n}X_1$. Then by lemma \[t1\], it can be extended to a complete theory $(\Delta', T')$ in $\Sg^{\A_n}X_1$. Take $$\Phi=\Delta'\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2,$$ and $$\Psi=T'\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2.$$ Then $(\Phi, \Psi)$ is complete in $\Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2.$ We shall now show that $(T\cup \Phi, \Psi)$ is consistent in $\Sg^{\A_n}X_2$. If not, then there is $\theta\in T$, $\phi\in \Phi$ and $\psi\in \Psi$ such that $\theta\land \phi\leq \psi$. So $\theta\leq \phi\rightarrow \psi$. Since $T$ is saturated, we get that $\phi\rightarrow \psi$ is in $T$. Now $\phi\rightarrow \psi\in \Delta\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2\subseteq \Delta'\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2=\Phi$. Since $\phi\in \Phi$ and $\phi\rightarrow \psi\in \Phi,$ we get that $\psi\in \Phi\cap \Psi$. But this means that $(\Phi, \Psi)$ is inconsistent which is impossible. Thus $(T\cup \Phi, \Psi)$ is consistent. Now the pair $((\Delta', T') (T\cup \Phi, \Psi))$ satisfy the conditions of lemma \[t3\]. Hence this pair can be extended to a matched pair of saturated theories in $\Sg^{\A_{n+1}}X_1$ and $\Sg^{\A_{n+1}}X_2$. This pair is as required by the conclusion of lemma \[t3\]. [Claim 2]{} Let $n\in \omega$. If $((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, then the following hold. For $x\in \Sg^{\A_n}X_1$ and $j\in I_n=dim\A_n$, if ${\sf q}_{j}x\notin \Delta$, then there is a matched pair $((\Delta',\Gamma'), (T', F'))$ of saturated theories in $\Sg^{\A_{n+2}}X_1$ and $\Sg^{\A_{n+2}}X_2$ respectively, $u\in I_{n+2}$ such that $\Delta\subseteq \Delta'$, $T\subseteq T'$ and ${\sf s}_u^j x\notin \Delta'$. [Proof]{} Assume that $x\in \Sg^{\A_n}X_1$ and $j\in I_n$ such that ${\sf q}_{j}x\notin \Sg^{\A_n}X_1$. Then there exists $u\in I_{n+1}\sim I_n$ such that $(\Delta, {\sf s}_u^j x)$ is consistent in $\Sg^{\A_{n+1}}X_1$. So $(\Delta, {\sf s}_u^jx)$ can be extended to a complete theory $(\Delta', T')$ in $\Sg^{\A_{n+1}}X_1$. Take $$\Phi=\Delta'\cap \Sg^{\A_{n+1}}X_1\cap \Sg^{\A_{n+1}}X_2,$$ and $$\Psi=T'\cap \Sg^{\A_{n+1}}X_1\cap \Sg^{\A_{n+1}}X_2.$$ Then $(\Phi,\Psi)$ is complete in $\Sg^{\A_{n+1}}X_1\cap \Sg^{\A_{n+1}}X_2$. We shall show that $(T\cup \Phi,\Psi)$ is consistent in $\Sg^{\A_{n+1}}X_2$. If not, then there exist $\theta\in T,$ $\phi\in \Phi$ and $\psi\in \Psi,$ such that $\theta\land \phi\leq \psi$. Hence, $\theta\leq \phi\rightarrow \psi$. Now $$\theta={\sf q}_j(\theta)\leq {\sf q}_{j}(\phi\rightarrow \psi).$$ Since $(T,F)$ is saturated in $\Sg^{\A_n}X_2,$ it thus follows that $${\sf q}_{j}(\phi\rightarrow \psi) \in T\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2=\Delta\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2.$$ So ${\sf q}_{j}(\phi\rightarrow \psi)\in \Delta'$ and consequently we get ${\sf q}_{j}(\phi\rightarrow \psi)\in \Phi$. Also, we have, $\phi\in \Phi$. But $(\Phi, \Psi)$ is complete, we get $\psi\in \Phi$ and this contradicts that $\psi\in \Psi$. Now the pair $((\Delta', \Gamma'), (T\cup \Phi, \Psi))$ satisfies the hypothesis of lemma \[t3\] applied to $\Sg^{\A_{n+1}}X_1, \Sg^{\A_{n+1}}X_2$. The required now follows from the concusion of lemma \[t3\]. Now that we have proved our claims, we go on with the proof. We prove the theorem when $G$ is a strongly rich semigroup, because in this case we deal with relativized semantics, and during the proof we state the necessary modifications for the case when $G$ is the semigroup of all transformations. Let $$K=\{((\Delta, \Gamma), (T,F)): \exists n\in \omega \text { such that } (\Delta, \Gamma), (T,F)$$ $$\text { is a a matched pair of saturated theories in } \Sg^{\A_n}X_1, \Sg^{\A_n}X_2\}.$$ We have $((\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*))$ is a matched pair but the theories are not saturated. But by lemma \[t3\] there are $T_1=(\Delta_{\omega}, \Gamma_{\omega})$, $T_2=(\Theta_{\omega}, \Gamma_{\omega}^*)$ extending $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$, such that $T_1$ and $T_2$ are saturated in $\Sg^{\A_1}X_1$ and $\Sg^{\A_1}X_2,$ respectively. Let $k_0=((\Delta_{\omega}, \Gamma_{\omega}), (\Theta_{\omega}, \Gamma_{\omega}^*)).$ Then $k_0\in K.$ If $i=((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, let $M_i=dim \A_n$, where $n$ is the least such number, so $n$ is unique to $i$. Before going on we introduce a piece of notation. For a set $M$ and a sequence $p\in {}^{\alpha}M$, $^{\alpha}M^{(p)}$ is the following set $$\{s\in {}^{\alpha}M: |\{i\in \alpha: s_i\neq p_i\}|<\omega\}.$$ Let $${\mathfrak{K}}=(K, \leq, \{M_i\}, \{V_i\})_{i\in \mathfrak{K}}$$ where $V_i=\bigcup_{p\in G_n}{}^{\alpha}M_i^{(p)}$, and $G_n$ is the strongly rich semigroup determining the similarity type of $\A_n$, with $n$ the least number such $i$ is a saturated matched pair in $\A_n$. The order $\leq $ is defined as follows: If $i_1=((\Delta_1, \Gamma_1)), (T_1, F_1))$ and $i_2=((\Delta_2, \Gamma_2), (T_2,F_2))$ are in $\mathfrak{K}$, then define $$i_1\leq i_2\Longleftrightarrow M_{i_1}\subseteq M_{i_2}, \Delta_1\subseteq \Delta_2, T_1\subseteq T_2.$$ This is, indeed as easily checked, a preorder on $K$. We define two maps on $\A_1=\Sg^{\A}X_1$ and $\A_2=\Sg^{\A}X_2$ respectively, then those will be pasted using the freeness of $\A$ to give the required single homomorphism, by noticing that they agree on the common part, that is on $\Sg^{\A}(X_1\cap X_2).$ Set $\psi_1: \Sg^{\A}X_1\to \mathfrak{F}_{\mathfrak K}$ by $\psi_1(p)=(f_k)$ such that if $k=((\Delta, \Gamma), (T,F))\in K$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, and $M_k=dim \A_n$, then for $x\in V_k=\bigcup_{p\in G_n}{}^{\alpha}M_k^{(p)}$, $$f_k(x)=1\Longleftrightarrow {\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p\in \Delta\cup T.$$ To avoid tiresome notation, we shall denote the map $x\cup Id_{M_k\sim \alpha}$ simply by $\bar{x}$ when $M_k$ is clear from context. It is easily verifiable that $\bar{x}$ is in the semigroup determining the similarity type of $\A_n$ hence the map is well defined. More concisely, we we write $$f_k(x)=1\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}p\in \Delta\cup T.$$ The map $\psi_2:\Sg^{\A}X_2\to \mathfrak{F}_{\mathfrak K}$ is defined in exactly the same way. Since the theories are matched pairs, $\psi_1$ and $\psi_2$ agree on the common part, i.e. on $\Sg^{\A}(X_1\cap X_2).$ Here we also make the tacit assumption that if $k\leq k'$ then $V_k\subseteq V_{k'}$ via the embedding $\tau\mapsto \tau\cup Id$. When $G$ is the semigroup of all transformations, with no restrictions on cardinalities, we need not relativize since $\bar{\tau}$ is in the big semigroup. In more detail, in this case, we take for $k=((\Delta,\Gamma), (T,F))$ a matched pair of saturated theories in $\Sg^{\A_n}X_1,\Sg^{\A_n}X_2$, $M_k=dim\A_n$ and $V_k={}^{\alpha}M_k$ and for $x\in {}^{\alpha}M_k$, we set $$f_k(x)=1\Longleftrightarrow {\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p\in \Delta\cup T.$$ Before proving that $\psi$ is a homomorphism, we show that $$k_0=((\Delta_{\omega},\Gamma_{\omega}), (\Theta_{\omega}, \Gamma^*_{\omega}))$$ is as desired. Let $x\in V_{k_0}$ be the identity map. Let $p\in \Delta_0\cup \Theta_0$, then ${\sf s}_xp=p\in \Delta_{\omega}\cup \Theta_{\omega},$ and so if $\psi(p)=(f_k)$ then $f_{k_0}(x)=1$. On the other hand if $p\in \Gamma_0^*$, then $p\notin \Delta_{\omega}\cup \Theta_{\omega}$, and so $f_{k_0}(x)=0$. Then the union $\psi$ of $\psi_1$ and $\psi_2$, $k_0$ and $Id$ are as required, modulo proving that $\psi$ is a homomorphism from $\A$, to the set algebra based on the above defined Kripke system, which we proceed to show. We start by $\psi_1$. Abusing notation, we denote $\psi_1$ by $\psi$, and we write a matched pair in $\A_n$ instead of a matched pair of saturated theories in $\Sg^{\A_n}X_1$, $\Sg^{\A_n}X_2$, since $X_1$ and $X_2$ are fixed. The proof that the postulated map is a homomorphism is similar to the proof in [@Hung] baring in mind that it is far from being identical because cylindrifiers and their duals are only finite. We prove that $\psi$ preserves $\land$. Let $p,q\in A$. Assume that $\psi(p)=(f_k)$ and $\psi(q)=(g_k)$. Then $\psi(p)\land \psi(q)=(f_k\land g_k)$. We now compute $\psi(p\land q)=(h_k)$ Assume that $x\in V_k$, where $k=((\Delta,\Gamma), (T, F))$ is a matched pair in $\A_n$ and $M_k=dim\A_n$. Then $$h_k(x)=1\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}(p\land q)\in \Delta\cup T$$ $$\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}p\land {\sf s}_{\bar{x}}^{\A_n}q\in \Delta\cup T$$ $$\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}p\in T\cup \Delta\text { and }{\sf s}_{\bar{x}}^{\A_n}q\in \Delta\cup T$$ $$\Longleftrightarrow f_k(x)=1 \text { and } g_k(x)=1$$ $$\Longleftrightarrow (f_k\land g_k)(x)=1$$ $$\Longleftrightarrow (\psi(p)\land \psi(q))(x)=1.$$ $\psi$ preserves $\rightarrow.$ (Here we use Claim 1). Let $p,q\in A$. Let $\psi(p)=(f_k)$ and $\psi(q)=(g_k)$. Let $\psi(p\rightarrow q)=(h_k)$ and $\psi(p)\rightarrow \psi(q)=(h'_k)$. We shall prove that for any $k\in \mathfrak{K}$ and any $x\in V_k$, we have $$h_k(x)=1\Longleftrightarrow h'_k(x)=1.$$ Let $x\in V_k$. Then $k=((\Delta,\Gamma),(T,F))$ is a matched pair in $\A_n$ and $M_k=dim\A_n$. Assume that $h_k(x)=1$. Then we have $${\sf s}_{\bar{x}}^{\A_n}(p\rightarrow q)\in \Delta\cup T,$$ from which we get that $$(*) \ \ \ {\sf s}_{\bar{x}}^{\A_n}p\rightarrow {\sf s}_{\bar{x}}^{\A_n}q\in \Delta\cup T.$$ Let $k'\in K$ such that $k\leq k'$. Then $k'=((\Delta', \Gamma'), (T', F'))$ is a matched pair in $\A_m$ with $m\geq n$. Assume that $f_{k'}(x)=1$. Then, by definition we have (\*\*) $${\sf s}_{\bar{x}}^{\A_m}p\in \Delta'\cup T'.$$ But $\A_m$ is a dilation of $\A_n$ and so $${\sf s}_{\bar{x}}^{\A_m}p={\sf s}_{\bar{x}}^{\A_n}p\text { and } {\sf s}_{\bar{x}}^{\A_m}q={\sf s}_{\bar{x}}^{\A_n}q.$$ From (\*) we get that, $${\sf s}_{\bar{x}}^{\A_m}p\rightarrow {\sf s}_{\bar{x}}^{\A_m}q\in \Delta'\cup T'.$$ But, on the other hand, from (\*\*), we have ${\sf s}_{\bar{x}}^{\A_m}q\in \Delta'\cup T',$ so $$f_{k'}(x)=1\Longrightarrow g_{k'}(x)=1.$$ That is to say, we have $h_{k'}(x)=1$. Conversely, assume that $h_k(x)\neq 1,$ then $${\sf s}_{\bar{x}}^{\A_n}p\rightarrow {\sf s}_{\bar{x}}^{\A_n}q\notin \Delta\cup T,$$ and consequently $${\sf s}_{\bar{x}}^{\A_n}p\rightarrow {\sf s}_{\bar{x}}^{\A_n}q\notin \Delta.$$ From Claim 1, we get that there exists a matched pair $k'=((\Delta',\Gamma')((T',F'))$ in $\A_{n+2},$ such that $${\sf s}_{\bar{x}}^{\A_{n+2}}p\in \Delta'\text { and } {\sf s}_{\bar{x}}^{\A_{n+2}}q\notin \Delta'.$$ We claim that ${\sf s}_{\bar{x}}^{\A_{n+2}}q\notin T'$, for otherwise, if it is in $T'$, then we would get that $${\sf s}_{\bar{x}}^{\A_{n+2}}q\in \Sg^{\A_{n+2}}X_1\cap \Sg^{\A_{n+2}}X_2.$$ But $$(\Delta'\cap T'\cap \Sg^{\A_{n+2}}X_1\cap \Sg^{\A_{n+2}}X_2, \Gamma'\cap F'\cap\Sg^{\A_{n+2}}X_1\cap \Sg^{\A_{n+2}}X_2)$$ is complete in $\Sg^{\A_{n+2}}X_1\cap \Sg^{\A_{n+2}}X_2,$ and ${\sf s}_{\bar{x}}^{\A_{n+2}}q\notin \Delta'\cap T'$, hence it must be the case that $${\sf s}_{\bar{x}}^{\A_{n+2}}q\in \Gamma'\cap F'.$$ In particular, we have $${\sf s}_{\bar{x}}^{\A_{n+2}}q\in F',$$ which contradicts the consistency of $(T', F'),$ since by assumption ${\sf s}_x^{\A_{n+2}}q\in T'$. Now we have $${\sf s}_{\bar{x}}^{\A_{n+2}}q\notin \Delta'\cup T',$$ and $${\sf s}_{\bar{x}}^{\A_{n+2}}p\in \Delta'\cup T'.$$ Since $\Delta'\cup T'$ extends $\Delta\cup T$, we get that $h_k'(x)\neq 1$. $\psi$ preserves substitutions. Let $p\in \A$. Let $\sigma\in {}G$. Assume that $\psi(p)=(f_k)$ and $\psi({\sf s}_{\sigma}p)=(g_k).$ Assume that $M_k=\dim\A_n$ where $k=((\Delta,\Gamma),(T,F))$ is a matched pair in $\A_n$. Then, for $x\in V_k$, we have $$g_k(x)=1\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}{\sf s}_{\sigma}^{\A}p\in \Delta\cup T$$ $$\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}{\sf s}_{\bar{\sigma}}^{\A_n}p\in \Delta\cup T$$ $$\Longleftrightarrow {\sf s}_{\bar{x}\circ {\bar{\sigma}}}^{\A_n}p\in \Delta\cup T$$ $$\Longleftrightarrow {\sf s}_{\overline{x\circ \sigma}}^{\A_n}p\in \Delta\cup T$$ $$\Longleftrightarrow f_k(x\circ \sigma)=1.$$ $\psi$ preserves cylindrifications. Let $p\in A.$ Assume that $m\in I$ and assume that $\psi({\sf c}_{m}p)=(f_k)$ and ${\sf c}_m\psi(p)=(g_k)$. Assume that $k=((\Delta,\Gamma),(T,F))$ is a matched pair in $\A_n$ and that $M_k=dim\A_n$. Let $x\in V_k$. Then $$f_k(x)=1\Longleftrightarrow {\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\in \Delta\cup T.$$ We can assume that $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\in \Delta.$$ For if not, that is if $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\notin \Delta\text { and } {\sf s}_{\bar{x}}^{\A_n}{\sf c}_{(m)}p\in T,$$ then $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\in \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2,$$ but $$(\Delta\cap T\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2, \Gamma\cap F\cap \Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2)$$ is complete in $\Sg^{\A_n}X_1\cap \Sg^{\A_n}X_2$, and $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\notin \Delta\cap T,$$ it must be the case that $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\in \Gamma\cap F.$$ In particular, $${\sf s}_{\bar{x}}^{\A_n}{\sf c}_{m}p\in F.$$ But this contradicts the consistency of $(T,F)$. Assuming that ${\sf s}_x{\sf c}_mp\in \Delta,$ we proceed as follows. Let $$\lambda\in \{\eta\in I_n: x^{-1}\{\eta\}=\eta\}\sim \Delta p.$$ Let $$\tau=x\upharpoonright I_n\sim\{m, \lambda\}\cup \{(m,\lambda)(\lambda, m)\}.$$ Then, by item (5) in theorem \[axioms\], we have $${\sf c}_{\lambda}{\sf s}^{\A_n}_{\bar{\tau}}p={\sf s}_{\bar{\tau}}^{\A_n}{\sf c}_{m}p={\sf s}_{\bar{x}}^{\A_n}{\sf c}_mp\in \Delta.$$ We introduce a piece of helpful notation. For a function $f$, let $f(m\to u)$ is the function that agrees with $f$ except at $m$ where its value is $u$. Since $\Delta$ is saturated, there exists $u\notin \Delta x$ such that ${\sf s}_u^{\lambda}{\sf s}_xp\in \Delta$, and so ${\sf s}_{(x(m\to u))} p\in \Delta$. This implies that $x\in {\sf c}_mf(p)$ and so $g_k(x)=1$. Conversely, assume that $g_k(x)=1$ with $k=((\Gamma,\Delta))$, $(T,F))$ a matched pair in $\A_n$. Let $y\in V_k$ such that $y\equiv_m x$ and $\psi(p)y=1$. Then ${\sf s}_{\bar{y}}p\in \Delta\cup T$. Hence ${\sf s}_{\bar{y}}{\sf c}_mp\in \Delta\cup T$ and so ${\sf s}_{\bar{x}}{\sf c}_mp\in \Delta\cup T$, thus $f_k(x)=1$ and we are done. $\psi$ preserves universal quantifiers. (Here we use Claim 2). Let $p\in A$ and $m\in I$. Let $\psi(p)=(f_k)$, ${\sf q}_{m}\psi(p)=(g_k)$ and $\psi({\sf q}_{m}p)=(h_k).$ Assume that $h_k(x)=1$. We have $k=((\Delta,\Gamma), (T,F))$ is a matched pair in $\A_n$ and $x\in V_k$. Then $${\sf s}_{\bar{x}}^{\A_n}{\sf q}_{m}p\in \Delta\cup T,$$ and so $${\sf s}_{\bar{y}}^{\A_n}{\sf q}_{m}p\in \Delta\cup T \text{ for all } y\in {}^IM_k, y\equiv_m x.$$ Let $k'\geq k$. Then $k'=((\Delta',\Gamma'), (T',F'))$ is a matched pair in $\A_l$ $l\geq n$, $\Delta\subseteq \Delta'$ and $T\subseteq T'.$ Since $p\geq {\sf q}_{m}p$ it follows that $${\sf s}_{\bar{y}}^{\A_n}p\in \Delta'\cup T' \text{ for all } y\in {}^IM_k, y\equiv_mx.$$ Thus $g_k(x)=1$. Now conversely, assume that $h_k(x)=0$, $k=((\Delta,\Gamma), (T,F))$ is a matched pair in $\A_n,$ then, we have $${\sf s}_{\bar{x}}^{\A_n}{\sf q}_{m}p\notin \Delta\cup T,$$ and so $${\sf s}_{\bar{x}}^{\A_n}{\sf q}_{m}p\notin \Delta.$$ Let $$\lambda\in \{\eta\in I_n: x^{-1}\{\eta\}=\eta\}\sim \Delta p.$$ Let $$\tau=x\upharpoonright I_n\sim\{m, \lambda\}\cup \{(m,\lambda)(\lambda, m)\}.$$ Then, like in the existential case, using polyadic axioms, we get $${\sf q}_{\lambda}{\sf s}_{\tau}p={\sf s}_{\tau}{\sf q}_{m}p={\sf s}_{x}{\sf q}_mp\notin \Delta$$ Then there exists $u$ such that ${\sf s}_u^{\lambda}{\sf s}_xp\notin \Delta.$ So ${\sf s}_u^{\lambda}{\sf s}_xp\notin T$, for if it is, then by the previous reasoning since it is an element of $\Sg^{\A_{n+2}}X_1\cap \Sg^{\A_{n+2}}X_2$ and by completeness of $(\Delta\cap T, \Gamma\cap F)$ we would reach a contradiction. The we get that ${\sf s}_{(x(m\to u))}p\notin \Delta\cup T$ which means that $g_k(x)=0,$ and we are done. We now deal with the case when $G$ is the semigroup of all finite transformations on $\alpha$. In this case, we stipulate that $\alpha\sim \Delta x$ is infinite for all $x$ in algebras considered. To deal with such a case, we need to define certain free algebras, called dimension restricted. Those algebras were introduced by Henkin, Monk and Tarski. The free algebras defined the usual way, will have the dimensions sets of their elements equal to their dimension, but we do not want that. For a class $K$, ${\bf S}$ stands for the operation of forming subalgebras of $K$, ${\bf P}K$ that of forming direct products, and ${\bf H}K$ stands for the operation of taking homomorphic images. In particular, for a class $K$, ${\bf HSP}K$ stands for the variety generated by $K$. Our dimension restricted free algebbras, are an instance of certain independently generated algebras, obtained by an appropriate relativization of the universal algebraic concept of free algebras. For an algebra $\A,$ we write $R\in Con\A$ if $R$ is a congruence relation on $\A.$ Assume that $K$ is a class of algebras of similarity $t$ and $S$ is any set of ordered pairs of words of $\Fr_{\alpha}^t,$ the absolutely free algebra of type $t$. Let $$Cr_{\alpha}^{(S)}K=\cap \{R\in Con \Fr_{\alpha}^t, \Fr_{\alpha}^t/R\in SK, S\subseteq R\}$$ and let $$\Fr_{\alpha}^{(S)}K=\Fr_{\alpha}^t/Cr_{\alpha}^{(S)}K.$$ $\Fr_{\alpha}^{(S)}K$ is called the free algebra over $K$ with $\alpha$ generators subject to the defining relations $S$. As a special case, we obtain dimension restricted free algebra, defined next. Let $\delta$ be a cardinal. Let $\alpha$ be an ordinal, and let $G$ be the semigroup of finite transformations on $\alpha$. Let$_{\alpha} \Fr_{\delta}$ be the absolutely free algebra on $\delta$ generators and of type $GPHA_{\alpha}$. Let $\rho\in {}^{\delta}\wp(\alpha)$. Let $L$ be a class having the same similarity type as $GPHA_{\alpha}.$ Let $$Cr_{\delta}^{(\rho)}L=\bigcap\{R: R\in Con_{\alpha}\Fr_{\delta}, {}_{\alpha}\Fr_{\delta}/R\in \mathbf{SP}L, {\mathsf c}_k^{_{\alpha}\Fr_{\delta}}{\eta}/R=\eta/R \text { for each }$$ $$\eta<\delta \text { and each }k\in \alpha\smallsetminus \rho(\eta)\}$$ and $$\Fr_{\delta}^{\rho}L={}_{\alpha}\Fr_{\beta}/Cr_{\delta}^{(\rho)}L.$$ The ordinal $\alpha$ does not figure out in $Cr_{\delta}^{(\rho)}L$ and $\Fr_{\delta}^{(\rho)}L$ though it is involved in their definition. However, $\alpha$ will be clear from context so that no confusion is likely to ensue. Assume that $\delta$ is a cardinal, $L\subseteq GPHA_{\alpha}$, $\A\in L$, $x=\langle x_{\eta}:\eta<\beta\rangle\in {}^{\delta}A$ and $\rho\in {}^{\delta}\wp(\alpha)$. We say that the sequence $x$ $L$-freely generates $\A$ under the dimension restricting function $\rho$, or simply $x$ freely generates $\A$ under $\rho,$ if the following two conditions hold: $\A=\Sg^{\A}Rg(x)$ and $\Delta^{\A} x_{\eta}\subseteq \rho(\eta)$ for all $\eta<\delta$. Whenever $\B\in L$, $y=\langle y_{\eta}, \eta<\delta\rangle\in {}^{\delta}\B$ and $\Delta^{\B}y_{\eta}\subseteq \rho(\eta)$ for every $\eta<\delta$, then there is a unique homomorphism from $\A$ to $\B$, such that $h\circ x=y$. The second item says that dimension restricted free algebras has the universal property of free algebras with respect to algebras whose dimensions are also restricted. The following theorem can be easily distilled from the literature of cylindic algebra. Assume that $\delta$ is a cardinal, $L\subseteq GPHA_{\alpha}$, $\A\in L$, $x=\langle x_{\eta}:\eta<\delta\rangle\in {}^{\delta}A$ and $\rho\in {}^{\delta}\wp(\alpha).$ Then the following hold: $\Fr_{\delta}^{\rho}L\in GPHA_{\alpha}$ and $x=\langle \eta/Cr_{\delta}^{\rho}L: \eta<\delta \rangle$ $\mathbf{SP}L$- freely generates $\A$ under $\rho$. In order that $\A\cong \Fr_{\delta}^{\rho}L$ it is necessary and sufficient that there exists a sequence $x\in {}^{\delta}A$ which $L$ freely generates $\A$ under $\rho$. [Proof]{} [@HMT1] theorems 2.5.35, 2.5.36, 2.5.37. Note that when $\rho(i)=\alpha$ for all $i$ then $\rho$ is not restricting the dimension, and we recover the notion of ordinary free algebras. That is for such a $\rho$, we have $\Fr_{\beta}^{\rho}GPHA_{\alpha}\cong \Fr_{\beta}GPHA_{\alpha}.$ Now we formulate the analogue of theorem \[main\] for dimension restricted agebras, which adresses infinitely many cases, because we have infinitely many dimension restricted free algebras having the same number of generators. \[main2\] Let $G$ be the semigroup of finite transformations on an infinite set $\alpha$ and let $\delta$ be a cardinal $>0$. Let $\rho\in {}^{\delta}\wp(\alpha)$ be such that $\alpha\sim \rho(i)$ is infinite for every $i\in \delta$. Let $\A$ be the free $G$ algebra generated by $X$ restristed by $\rho$; that is $\A=\Fr_{\delta}^{\rho}GPHA_{\alpha},$ and suppose that $X=X_1\cup X_2$. Let $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ be two consistent theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively. Assume that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$ and $\Gamma_0\subseteq \Gamma_0^*$. Assume, further, that $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. Then there exist a Kripke system $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}\{V_k\}_{k\in K}),$ a homomorphism $\psi:\A\to \mathfrak{F}_K,$ $k_0\in K$, and $x\in V_{k_0}$, such that for all $p\in \Delta_0\cup \Theta_0$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=1$ and for all $p\in \Gamma_0^*$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=0$. [Proof]{} We state the modifications in the above proof of theorem \[main\]. Form the sequence of minimal dilations $(\A_n:n\in \omega)$ built on the sequence $(K_n:n\in \omega)$, with $|K_n|=\beta$, $\beta=|I\sim \alpha|=max(|A|, \alpha)$ with $I$ is a superset of $\alpha.$ If $i=((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, let $M_i=dim \A_n$, where $n$ is the least such number, so $n$ is unique to $i$. Define $K$ as in in the proof of theorem \[main\], that is, let $$K=\{((\Delta, \Gamma), (T,F)): \exists n\in \omega \text { such that } (\Delta, \Gamma), (T,F)$$ $$\text { is a a matched pair of saturated theories in } \Sg^{\A_n}X_1, \Sg^{\A_n}X_2\}.$$ Let $${\mathfrak{K}}=(K, \leq, \{M_i\}, \{V_i\})_{i\in \mathfrak{K}},$$ where now $V_i={}^{\alpha}M_i^{(Id)}=\{s\in {}^{\alpha}M: |\{i\in \alpha: s_i\neq i\}|<\omega\},$ and the order $\leq $ is defined by: If $i_1=((\Delta_1, \Gamma_1)), (T_1, F_1))$ and $i_2=((\Delta_2, \Gamma_2), (T_2,F_2))$ are in $\mathfrak{K}$, then $$i_1\leq i_2\Longleftrightarrow M_{i_1}\subseteq M_{i_2}, \Delta_1\subseteq \Delta_2, T_1\subseteq T_2.$$ This is a preorder on $K$. Set $\psi_1: \Sg^{\A}X_1\to \mathfrak{F}_{\mathfrak K}$ by $\psi_1(p)=(f_k)$ such that if $k=((\Delta, \Gamma), (T,F))\in \mathfrak{K}$ is a matched pair of saturated theories in $\Sg^{\A_n}X_1$ and $\Sg^{\A_n}X_2$, and $M_k=dim \A_n$, then for $x\in V_k={}^{\alpha}M_k^{(Id)}$, $$f_k(x)=1\Longleftrightarrow {\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p\in \Delta\cup T.$$ Define $\psi_2$ analogously. The rest of the proof is identical to the previous one. It is known that the condition $\Gamma\subseteq \Gamma^*$ cannot be omitted. On the other hand, to prove our completeness theorem, we need the following weaker version of theorem \[main\], with a slight modification in the proof, which is still a step-by-step technique, though, we do not ‘zig-zag’. \[rep\] Let $\A\in GPHA_{\alpha}$. Let $(\Delta_0, \Gamma_0)$ be consistent. Suppose that $I$ is a set such that $\alpha\subseteq I$ and $|I\sim \alpha|=max (|A|,|\alpha|)$. Then there exists a dilation $\B\in GPHA_I$ of $\A$, and theory $T=(\Delta_{\omega}, \Gamma_{\omega})$, extending $(\Delta_0, \Gamma_0)$, such that $T$ is consistent and saturated in $\B$. There exists $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}\{V_k\}_{k\in K}),$ a homomorphism $\psi:\A\to \mathfrak{F}_K,$ $k_0\in K$, and $x\in V_{k_0}$, such that for all $p\in \Delta_0$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=1$ and for all $p\in \Gamma_0$ if $\psi(p)=(g_k)$, then $g_{k_0}(x)=0.$ [Proof]{} We deal only with the case when $G$ is strongly rich. The other cases can be dealt with in a similar manner by undergoing the obvious modifications, as indicated above. As opposed to theorem \[main\], we use theories rather than pairs of theories, since we are not dealing with two subalgebras simultaneously. (i) follows from \[t2\]. Now we prove (ii). The proof is a simpler version of the proof of \[main\]. Let $I$ be a set such that $\beta=|I\sim \alpha|=max(|A|, |\alpha|).$ Let $(K_n:n\in \omega)$ be a family of pairwise disjoint sets such that $|K_n|=\beta.$ Define a sequence of algebras $\A=\A_0\subseteq \A_1\subseteq \A_2\subseteq \A_2\ldots \subseteq \A_n\ldots$ such that $\A_{n+1}$ is a minimal dilation of $\A_n$ and $dim(\A_{n+1})=\dim\A_n\cup K_n$. We denote $dim(\A_n)$ by $I_n$ for $n\geq 1$. If $(\Delta, \Gamma)$ is saturated in $\A_n$ then the following analogues of Claims 1 and 2 in theorem \[main\] hold: For any $a,b\in \A_n$ if $a\rightarrow b\notin \Delta$, then there is a saturated theory $(\Delta',\Gamma')$ in $\A_{n+1}$ such that $\Delta\subseteq \Delta'$ $a\in \Delta'$ and $b\notin \Delta'$. If $(\Delta, \Gamma)$ is saturated in $\A_n$ then for all $x\in \A_n$ and $j\in I_n$, if ${\sf q}_{j}x\notin \Delta,$ then there $(\Delta',\Gamma')$ of saturated theories in $\A_{n+2}$, $u\in I_{n+2}$ such that $\Delta\subseteq \Delta'$, and ${\sf s}_j^u x\notin \Delta'$. Now let $$K=\{(\Delta, \Gamma): \exists n\in \omega \text { such that } (\Delta,\Gamma) \text { is saturated in }\A_n.\}$$ If $i=(\Delta, \Gamma)$ is a saturated theory in $\A_n$, let $M_i=dim \A_n$, where $n$ is the least such number, so $n$ is unique to $i$. If $i_1=(\Delta_1, \Gamma_1)$ and $i_2=(\Delta_2, \Gamma_2)$ are in $K$, then set $$i_1\leq i_2\Longleftrightarrow M_{i_1}\subseteq M_{i_2}, \Delta_1\subseteq \Delta_2.$$ This is a preorder on $K$; define the kripke system ${\mathfrak K}$ based on the set of worlds $K$ as before. Set $\psi: \A\to \mathfrak{F}_{\mathfrak K}$ by $\psi_1(p)=(f_k)$ such that if $k=(\Delta, \Gamma)\in \mathfrak{K}$ is saturated in $\A_n$, and $M_k=dim \A_n$, then for $x\in V_k=\bigcup_{p\in G_n}{}^{\alpha}M_k^{(p)}$, $$f_k(x)=1\Longleftrightarrow {\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p\in \Delta.$$ Let $k_0=(\Delta_{\omega}, \Gamma_{\omega})$ be defined as a complete saturated extension of $(\Delta_0, \Gamma_0)$ in $\A_1$, then $\psi,$ $k_0$ and $Id$ are as desired. The analogues of Claims 1 and 2 in theorem \[main\] are used to show that $\psi$ so defined preserves implication and universal quantifiers. Presence of diagonal elements ============================= All results, in Part 1, up to the previous theorem, are proved in the absence of diagonal elements. Now lets see how far we can go if we have diagonal elements. Considering diagonal elements, as we shall see, turn out to be problematic but not hopeless. Our representation theorem has to respect diagonal elements, and this seems to be an impossible task with the presence of infinitary substitutions, unless we make a compromise that is, from our point of view, acceptable. The interaction of substitutions based on infinitary transformations, together with the existence of diagonal elements tends to make matters ‘blow up’; indeed this even happens in the classical case, when the class of (ordinary) set algebras ceases to be closed under ultraproducts [@S]. The natural thing to do is to avoid those infinitary substitutions at the start, while finding the interpolant possibly using such substitutions. We shall also show that in some cases the interpolant has to use infinitary substitutions, even if the original implication uses only finite transformations. So for an algebra $\A$, we let $\Rd\A$ denote its reduct when we discard infinitary substitutions. $\Rd\A$ satisfies cylindric algebra axioms. \[main3\] Let $\alpha$ be an infinite set. Let $G$ be a semigroup on $\alpha$ containing at least one infinitary transformation. Let $\A\in GPHAE_{\alpha}$ be the free $G$ algebra generated by $X$, and suppose that $X=X_1\cup X_2$. Let $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ be two consistent theories in $\Sg^{\Rd\A}X_1$ and $\Sg^{\Rd\A}X_2,$ respectively. Assume that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$ and $\Gamma_0\subseteq \Gamma_0^*$. Assume, further, that $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\Rd\A}X_1\cap \Sg^{\Rd\A}X_2$. Then there exist $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}\{V_k\}_{k\in K}),$ a homomorphism $\psi:\A\to \mathfrak{F}_K,$ $k_0\in K$, and $x\in V_{k_0}$, such that for all $p\in \Delta_0\cup \Theta_0$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=1$ and for all $p\in \Gamma_0^*$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=0$. [Proof]{} The first half of the proof is almost identical to that of lemma \[main\]. We highlight the main steps, for the convenience of the reader, except that we only deal with the case when $G$ is strongly rich. Assume, as usual, that $\alpha$, $G$, $\A$ and $X_1$, $X_2$, and everything else in the hypothesis are given. Let $I$ be a set such that $\beta=|I\sim \alpha|=max(|A|, |\alpha|).$ Let $(K_n:n\in \omega)$ be a family of pairwise disjoint sets such that $|K_n|=\beta.$ Define a sequence of algebras $\A=\A_0\subseteq \A_1\subseteq \A_2\subseteq \A_2\ldots \subseteq \A_n\ldots$ such that $\A_{n+1}$ is a minimal dilation of $\A_n$ and $dim(\A_{n+1})=\dim\A_n\cup K_n$.We denote $dim(\A_n)$ by $I_n$ for $n\geq 1$. The proofs of Claims 1 and 2 in the proof of \[main\] are the same. Now we prove the theorem when $G$ is a strongly rich semigroup. Let $$K=\{((\Delta, \Gamma), (T,F)): \exists n\in \omega \text { such that } (\Delta, \Gamma), (T,F)$$ $$\text { is a a matched pair of saturated theories in } \Sg^{\Rd\A_n}X_1, \Sg^{\Rd\A_n}X_2\}.$$ We have $((\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*))$ is a matched pair but the theories are not saturated. But by lemma \[t3\] there are $T_1=(\Delta_{\omega}, \Gamma_{\omega})$, $T_2=(\Theta_{\omega}, \Gamma_{\omega}^*)$ extending $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$, such that $T_1$ and $T_2$ are saturated in $\Sg^{\Rd\A_1}X_1$ and $\Sg^{\Rd\A_1}X_2,$ respectively. Let $k_0=((\Delta_{\omega}, \Gamma_{\omega}), (\Theta_{\omega}, \Gamma_{\omega}^*)).$ Then $k_0\in K,$ and $k_0$ will be the desired world and $x$ will be specified later; in fact $x$ will be the identity map on some specified domain. If $i=((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\Rd\A_n}X_1$ and $\Sg^{\Rd\A_n}X_2$, let $M_i=dim \A_n$, where $n$ is the least such number, so $n$ is unique to $i$. Let $${\bf K}=(K, \leq, \{M_i\}, \{V_i\})_{i\in \mathfrak{K}},$$ where $V_i=\bigcup_{p\in G_n, p\text { a finitary transformation }}{}^{\alpha}M_i^{(p)}$ (here we are considering only substitutions that move only finitely many points), and $G_n$ is the strongly rich semigroup determining the similarity type of $\A_n$, with $n$ the least number such $i$ is a saturated matched pair in $\A_n$, and $\leq $ is defined as follows: If $i_1=((\Delta_1, \Gamma_1)), (T_1, F_1))$ and $i_2=((\Delta_2, \Gamma_2), (T_2,F_2))$ are in $\bold K$, then set $$i_1\leq i_2\Longleftrightarrow M_{i_1}\subseteq M_{i_2}, \Delta_1\subseteq \Delta_2, T_1\subseteq T_2.$$ We are not yet there, to preserve diagonal elements we have to factor out $\bold K$ by an infinite family equivalence relations, each defined on the dimension of $\A_n$, for some $n$, which will actually turn out to be a congruence in an exact sense. As usual, using freeness of $\A$, we will define two maps on $\A_1=\Sg^{\Rd\A}X_1$ and $\A_2=\Sg^{\Rd\A}X_2$, respectively; then those will be pasted to give the required single homomorphism. Let $i=((\Delta, \Gamma), (T,F))$ be a matched pair of saturated theories in $\Sg^{\Rd\A_n}X_1$ and $\Sg^{\Rd\A_n}X_2$, let $M_i=dim \A_n$, where $n$ is the least such number, so $n$ is unique to $i$. For $k,l\in dim\A_n=I_n$, set $k\sim_i l$ iff ${\sf d}_{kl}^{\A_n}\in \Delta\cup T$. This is well defined since $\Delta\cup T\subseteq \A_n$. We omit the superscript $\A_n$. These are infinitely many relations, one for each $i$, defined on $I_n$, with $n$ depending uniquely on $i$, we denote them uniformly by $\sim$ to avoid complicated unnecessary notation. We hope that no confusion is likely to ensue. We claim that $\sim$ is an equivalence relation on $I_n.$ Indeed, $\sim$ is reflexive because ${\sf d}_{ii}=1$ and symmetric because ${\sf d}_{ij}={\sf d}_{ji};$ finally $E$ is transitive because for $k,l,u<\alpha$, with $l\notin \{k,u\}$, we have $${\sf d}_{kl}\cdot {\sf d}_{lu}\leq {\sf c}_l({\sf d}_{kl}\cdot {\sf d}_{lu})={\sf d}_{ku},$$ and we can assume that $T\cup \Delta$ is closed upwards. For $\sigma,\tau \in V_k,$ define $\sigma\sim \tau$ iff $\sigma(i)\sim \tau(i)$ for all $i\in \alpha$. Then clearly $\sigma$ is an equivalence relation on $V_k$. Let $W_k=V_k/\sim$, and $\mathfrak{K}=(K, \leq, M_k, W_k)_{k\in K}$, with $\leq$ defined on $K$ as above. We write $h=[x]$ for $x\in V_k$ if $x(i)/\sim =h(i)$ for all $i\in \alpha$; of course $X$ may not be unique, but this will not matter. Let $\F_{\mathfrak K}$ be the set algebra based on the new Kripke system ${\mathfrak K}$ obtained by factoring out $\bold K$. Set $\psi_1: \Sg^{\Rd\A}X_1\to \mathfrak{F}_{\mathfrak K}$ by $\psi_1(p)=(f_k)$ such that if $k=((\Delta, \Gamma), (T,F))\in K$ is a matched pair of saturated theories in $\Sg^{\Rd\A_n}X_1$ and $\Sg^{\Rd\A_n}X_2$, and $M_k=dim \A_n$, with $n$ unique to $k$, then for $x\in W_k$ $$f_k([x])=1\Longleftrightarrow {\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p\in \Delta\cup T,$$ with $x\in V_k$ and $[x]\in W_k$ is define as above. To avoid cumbersome notation, we write ${\sf s}_{x}^{\A_n}p$, or even simply ${\sf s}_xp,$ for ${\sf s}_{x\cup (Id_{M_k\sim \alpha)}}^{\A_n}p$. No ambiguity should arise because the dimension $n$ will be clear from context. We need to check that $\psi_1$ is well defined. It suffices to show that if $\sigma, \tau\in V_k$ if $\sigma \sim \tau$ and $p\in \A_n$, with $n$ unique to $k$, then $${\sf s}_{\tau}p\in \Delta\cup T\text { iff } {\sf s}_{\sigma}p\in \Delta\cup T.$$ This can be proved by induction on the cardinality of $J=\{i\in I_n: \sigma i\neq \tau i\}$, which is finite since we are only taking finite substitutions. If $J$ is empty, the result is obvious. Otherwise assume that $k\in J$. We recall the following piece of notation. For $\eta\in V_k$ and $k,l<\alpha$, write $\eta(k\mapsto l)$ for the $\eta'\in V$ that is the same as $\eta$ except that $\eta'(k)=l.$ Now take any $$\lambda\in \{\eta\in I_n: \sigma^{-1}\{\eta\}= \tau^{-1}\{\eta\}=\{\eta\}\}\smallsetminus \Delta x.$$ This $\lambda$ exists, because $\sigma$ and $\tau$ are finite transformations and $\A_n$ is a dilation with enough spare dimensions. We have by cylindric axioms (a) $${\sf s}_{\sigma}x={\sf s}_{\sigma k}^{\lambda}{\sf s}_{\sigma (k\mapsto \lambda)}p.$$ We also have (b) $${\sf s}_{\tau k}^{\lambda}({\sf d}_{\lambda, \sigma k}\land {\sf s}_{\sigma} p) ={\sf d}_{\tau k, \sigma k} {\sf s}_{\sigma} p,$$ and (c) $${\sf s}_{\tau k}^{\lambda}({\sf d}_{\lambda, \sigma k}\land {\sf s}_{\sigma(k\mapsto \lambda)}p)$$ $$= {\sf d}_{\tau k, \sigma k}\land {\sf s}_{\sigma(k\mapsto \tau k)}p.$$ and (d) $${\sf d}_{\lambda, \sigma k}\land {\sf s}_{\sigma k}^{\lambda}{\sf s}_{{\sigma}(k\mapsto \lambda)}p= {\sf d}_{\lambda, \sigma k}\land {\sf s}_{{\sigma}(k\mapsto \lambda)}p$$ Then by (b), (a), (d) and (c), we get, $${\sf d}_{\tau k, \sigma k}\land {\sf s}_{\sigma} p= {\sf s}_{\tau k}^{\lambda}({\sf d}_{\lambda,\sigma k}\cdot {\sf s}_{\sigma}p)$$ $$={\sf s}_{\tau k}^{\lambda}({\sf d}_{\lambda, \sigma k}\land {\sf s}_{\sigma k}^{\lambda} {\sf s}_{{\sigma}(k\mapsto \lambda)}p)$$ $$={\sf s}_{\tau k}^{\lambda}({\sf d}_{\lambda, \sigma k}\land {\sf s}_{{\sigma}(k\mapsto \lambda)}p)$$ $$= {\sf d}_{\tau k, \sigma k}\land {\sf s}_{\sigma(k\mapsto \tau k)}p.$$ The conclusion follows from the induction hypothesis. Now $\psi_1$ respects all quasipolyadic equality operations, that is finite substitutions (with the proof as before; recall that we only have finite substitutions since we are considering $\Sg^{\Rd\A}X_1$) except possibly for diagonal elements. We check those: Recall that for a concrete Kripke frame $\F_{\bold W}$ based on ${\bold W}=(W,\leq ,V_k, W_k),$ we have the concrete diagonal element ${\sf d}_{ij}$ is given by the tuple $(g_k: k\in K)$ such that for $y\in V_k$, $g_k(y)=1$ iff $y(i)=y(j)$. Now for the abstract diagonal element in $\A$, we have $\psi_1({\sf d}_{ij})=(f_k:k\in K)$, such that if $k=((\Delta, \Gamma), (T,F))$ is a matched pair of saturated theories in $\Sg^{\Rd\A_n}X_1$, $\Sg^{\Rd\A_n}X_2$, with $n$ unique to $i$, we have $f_k([x])=1$ iff ${\sf s}_{x}{\sf d}_{ij}\in \Delta \cup T$ (this is well defined $\Delta\cup T\subseteq \A_n).$ But the latter is equivalent to ${\sf d}_{x(i), x(j)}\in \Delta\cup T$, which in turn is equivalent to $x(i)\sim x(j)$, that is $[x](i)=[x](j),$ and so $(f_k)\in {\sf d}_{ij}^{\F_{\mathfrak K}}$. The reverse implication is the same. We can safely assume that $X_1\cup X_2=X$ generates $\A$. Let $\psi=\psi_1\cup \psi_2\upharpoonright X$. Then $\psi$ is a function since, by definition, $\psi_1$ and $\psi_2$ agree on $X_1\cap X_2$. Now by freeness $\psi$ extends to a homomorphism, which we denote also by $\psi$ from $\A$ into $\F_{\mathfrak K}$. And we are done, as usual, by $\psi$, $k_0$ and $Id\in V_{k_0}$. Theorem \[main2\], generalizes as is, to the expanded structures by diagonal elements. That is to say, we have: \[main4\] Let $G$ be the semigroup of finite transformations on an infinite set $\alpha$ and let $\delta$ be a cardinal $>0$. Let $\rho\in {}^{\delta}\wp(\alpha)$ be such that $\alpha\sim \rho(i)$ is infinite for every $i\in \delta$. Let $\A$ be the free $G$ algebra with equality generated by $X$ restristed by $\rho$; that is $\A=\Fr_{\delta}^{\rho}GPHAE_{\alpha},$ and suppose that $X=X_1\cup X_2$. Let $(\Delta_0, \Gamma_0)$, $(\Theta_0, \Gamma_0^*)$ be two consistent theories in $\Sg^{\A}X_1$ and $\Sg^{\A}X_2,$ respectively. Assume that $\Gamma_0\subseteq \Sg^{\A}(X_1\cap X_2)$ and $\Gamma_0\subseteq \Gamma_0^*$. Assume, further, that $(\Delta_0\cap \Theta_0\cap \Sg^{\A}X_1\cap \Sg^{\A}X_2, \Gamma_0)$ is complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. Then there exist a Kripke system $\mathfrak{K}=(K,\leq \{X_k\}_{k\in K}\{V_k\}_{k\in K}),$ a homomorphism $\psi:\A\to \mathfrak{F}_K,$ $k_0\in K$, and $x\in V_{k_0}$, such that for all $p\in \Delta_0\cup \Theta_0$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=1$ and for all $p\in \Gamma_0^*$ if $\psi(p)=(f_k)$, then $f_{k_0}(x)=0$. [Proof]{} $\Rd\A$ is just $\A$. Results in logical form ======================= We start by describing our necessary syntactical and semantical notions to follow. Informally a language is a triple $(V, P, G)$ where $V$ is a set providing an infinite supply of variables, $P$ is a another set of predicates disjoint from $V,$ and $G$ is a semigroup of transformations on $V$. There is no restriction on the arity of $p\in P$; sometimes referred to as the rank of $p$, that is the arity may be infinite. Formulas are defined recursively the usual way. Atomic formulas are of the form $p\bar{v}$, the length of $\bar{v}$ is equal to the arity of $p$. If $\phi, \psi$ are formulas and $v\in V,$ then $\phi\lor\psi$, $\phi\land \psi$, $\phi\to \psi$, $\exists v\phi,$ $\forall v\phi$ are formulas. For each $\tau\in G$, ${\sf S}({\tau})$ is a unary operation on formulas, that is, for any formula $\phi$, ${\sf S}{(\tau)}\phi$ is another formula, reflecting the metalogical operation of simultaneous substitution of variables (determined by $\tau$) for variables, such that the substitution is free. Notice that although we allow infinitary predicates, quantifications are defined only on finitely many variables, that is the scope of quantifiers is finite. We will also deal with the case when we have equality; for this purpose we add a newlogical symbol $=$ and we view it, as usual, as a binary relation. We recall some basic semantical notions for intuitionistic logic but adpated to the presence of atomic formulas possibly having infinite length. An intuitionistic or Kripke frame is a triple $\bold W=(W, R, \{D_w\}_{w\in W})$ where $W$ is a non-empty set called worlds, preordered by $R$ and $D_w$ is a non-empty subset of $D$ called the domain of $w$ for any $w$, and the monotoncity condition of domains is satisfied: $$(\forall w,w'\in W)[wRw'\implies D_w\subseteq D_{w'}.]$$ On the other hand, an intuitionistic or Kripke model is a quadruple $\bold M=(W, R, \{D_{w}\}_{w\in W} \models),$ where $(W, R, \{D_w\}_{w\in W})$ is an intuitionistic frame, $\models$ is a tenary relation between worlds, formulas, and assignments (maps from $V$ to $D$). We write $x\models \phi[s]$ if $(x, \phi ,s)\in \models$. This tenary relation $\models$ satisfies for any predicate $p$, any $s\in {}^{V}D$, any formulas $\phi$, $\psi$ and any $x\in W$ the following: $$\text { It is not the case that }x\models \bot,$$ $$(\forall y\in W)(x\models p[s]\land xRy\implies y\models p[s]),$$ $$x\models (\phi\land \psi)[s]\Longleftrightarrow x\models \phi[s] \text { and } x\models \psi[s],$$ $$x\models (\phi\lor \psi)[s]\Longleftrightarrow x\models \phi[s]\text { or }\phi\models \psi[s],$$ $$x\models (\phi\to \psi)[s]\Longleftrightarrow \forall y(xRy\implies(y\models \phi[s]\implies y\models \psi[s])).$$ For $s$ a function $s^k_a$ is the function defined by $s^k_a(i)=s(i)$ when $i\neq k$ and $s^k_a(k)=a$. Continuing the definition: $$x\models \forall v\phi[s]\Longleftrightarrow( (\forall y)(xRy\implies (\forall a\in D_y)y\models \phi[s^v_a]))),$$ $$x\models \exists v\phi[s]\Longleftrightarrow (\exists a\in D_x)(x\models \phi[s^v_a])),$$ $$x\models {\sf S}({\tau})\phi[s]\Longleftrightarrow x\models \phi[\tau\circ s].$$ Evidently the model is completely determined by the frame $(W, R, \{D_{w}\}_{w\in W})$ and by $\models$ on atomic formulas. That is for each for each $p\in P$ and each world $x$ and $s\in{}^VD_x$, $p$ determines a possibly infinitary relation $p_x\subseteq {}^VD_x$, and we stipulate that $x\models p[s]$ if $s\in p_x$. If we have equality, then for the world $x$ and $s\in {}^VD_x$, we add the clause $x\models v_1=v_2$ if $s(v_1)=s(v_2)$. We now define a calculas (in a usual underlying set theory $ZFC$, say) that we prove to be complete with respect to Kripke semantics; this will follow from our stronger proven result that such logics enjoy the interpolation property. We first deal with the equality free case. In such a case, our calculas is inspired by that of Keisler [@K]. Let $V$ and $P$ be disjoint sets of symbols, such that $V$ is infinite, $\rho$ a function with domain is $P$ and whose range is $Set$ (the class of all sets) or $Ord$ (the class of all ordinals). [^4] Let $G\subseteq {}^VV$. We define a logic $\mathfrak{L}_G$ in the following way. The symbols of $\mathfrak{L}_G$ consists of: The falsity symbol $\bot$. the disjunction $\lor$, conjunction $\land$, and the implication symbol $\to$. universal quantification symbol $\forall$. existential quantification symbol $\exists$. the individual variables $v\in V$ and predicates $p\in P.$ We assume that $\bot, \lor, \land,\to, \forall, \exists$ are not members of $V$ nor $P$. An atomic formula is an ordered pair $(p,x)$ where $p\in P$ and $x\in {}^{\rho(p)}V.$ We call $\rho(p)$ the rank of $p$. Formulas are defined the usual way by recursion; in this respect we regard $(\phi\to \phi)$ as an ordered triple and so are formulas involving other connectives including $\exists v\phi$ and $\forall v \phi.$ (In the former formula, the brackets are not syntactic brackets because we do no have brackets in our language.) The set $V_f(\phi)$ of free variables and the set $V_b(\phi)$ of bound variables in a formula $\phi$ are defined recursively the usual way. That is: If $\phi$ is an atomic formula $(p,x)$, then $V_f(\phi)$ is the range of $x$. if $\phi=\bot$, then $V_f(\bot)=\emptyset.$ If $\phi$ is $(\psi\lor \theta)$ or $(\psi\land \theta)$ or $(\psi\to \theta)$, then $V_f(\phi)=V_f(\psi)\cup V_f(\theta).$ If $\phi=(\forall v \psi)$ or $(\exists v\psi)$, then $V_f(\psi)=V_f(\phi)\sim \{v\}$. Now for the bound variables $V_b(\phi)$: If $\phi$ is an atomic formula $(p,x)$, then $V_b(\phi)=\emptyset.$ if $\phi=\bot$, then $V_b(\bot)=\emptyset.$ If $\phi$ is $(\psi\lor \theta)$ or $(\psi\land \theta)$ or $(\psi\to \theta),$ then $V_b(\phi)=V_b(\psi)\cup V_b(\theta).$ If $\phi=(\forall v \psi)$, then $V_b(\psi)=V_b(\phi)\cup \{v\}.$ If $\phi=(\exists v \psi)$, then $V_b(\psi)=V_b(\phi)\cup \{v\}.$ Note that the variables occurring in a formula $\phi$, denoted by $V(\phi)$ is equal to $V_f(\phi)\cup V_b(\phi)$ which could well be infinite. For $\tau\in G$ and $\phi$ a formula, ${\sf S}(\tau)\phi$ (the result of substituting each variable $v$ in $\phi$ by $\tau(v)$) is defined recursively and so is ${\sf S}_f(\tau)\phi$ (the result of substituting each free variable $v$ by $\tau(v)$). If $\phi$ is atomic formula $(p,x),$ then ${\sf S}(\tau)\phi=(p,\tau\circ x).$ if $\phi=\bot,$ then ${\sf S}(\tau)\bot=\bot$ If $\phi$ is $(\psi\lor \theta),$ then ${\sf S}(\tau)\phi=({\sf S}(\tau)\psi\lor {\sf S}(\tau)\theta).$ The same for other propositional connectives. If $\phi=(\forall v\phi),$ then ${\sf S}(\tau)\phi=(\forall\tau(v){\sf S}(\tau)\phi).$ If $\phi=(\exists v\phi),$ then ${\sf S}(\tau)\phi=(\exists\tau(v){\sf S}(\tau)\phi).$ To deal with free substitutions, that is when the resulted substituted variables remain free, we introduce a piece of notation that proves helpful. For any function $f\in {}^XY$ and any set $Z$, we let $$f|Z=\{(x, f(x)): x\in X\cap Z\}\cup \{(z,z)|z\in Z\sim X\}.$$ Then $f|Z$ always has domain $Z$ and $0|Z$ is the identity function on $Z$. If $\tau\in \bigcup\{^WV: W\subseteq V\}$, and $\phi$ is a formula, let ${\sf S}(\tau)\phi={\sf S}(\tau|V)\phi$ and ${\sf S}_f(\tau)\phi=S_f(\tau|V)\phi$. For free subtitution the first three clauses are the same, but if $\phi=(\forall v \psi)$, then ${\sf S}_f({\tau})\phi=(\forall v {\sf S}_f(\sigma)\psi)$ and if $\phi=(\exists v \psi),$ then ${\sf S}_f({\tau})\phi=(\exists v {\sf S}_f(\sigma)\psi)$ where $\sigma=\tau\upharpoonright (V\sim \{v\})\upharpoonright V$. Now we specify the axioms and the rules of inference. The axioms are: Axioms for propositional intuitionistic logic (formulated in our syntax). $((\forall v(\phi\to \psi)\to (\phi\to \forall v \psi)))$ where $v\in (V\sim V_{f} \phi)).$ $((\forall v(\phi\to \psi)\to (\exists v\phi\to \psi)))$ where $v\in (V\sim V_{f} \phi)).$ $(\forall v\phi\to {\sf S}_f(\tau)\phi)$, when $\tau(v) \notin (V\sim V_b(\phi)).$ $({\sf S}_f(\tau)\phi\to (\exists v \phi))$, when $\tau(v) \notin (V\sim V_b(\phi)).$ The rules of inference are: Form $\phi$, $(\phi\to \psi)$ infer $\psi.$ (Modus ponens.) From $\phi$ infer $(\forall v\phi).$ (Rule of generalization.) From ${\sf S}_f(\tau)\phi$ infer $\phi$ whenever $\tau\in {}^{V_f(\phi)} (V\sim V_b(\phi))$ and $\tau$ is one to one. (Free substitution.) From $\phi$ infer ${\sf S}(\tau)\phi$ whenever $\tau\in {}^{V(\phi)}V$ is one to one (Substitution). Now if we have $=$ as a primitive symbol, we add the following axioms (in this case no more rules of inference are needed): $v=v$ $v=w\to w=v$ If $\phi$ is a formula and $\tau, \sigma$ are substitutions that agree on the indices of the free variables occuring in $\phi$, then ${\sf S}_f(\tau)\phi={\sf S}_f(\sigma)\phi.$ We write ${\mathfrak L}_G$ for logics without equality, and we write ${\mathfrak L}_G^{=}$ for those with equality, when $G$ is specified in advance. Proofs are defined the usual way. For a set of formulas $\Gamma\cup \{\phi\}$, we write $\Gamma\vdash \phi$, if there is a proof of $\phi$ from $\Gamma$. To formulate the main results of this paper, we need some more basic definitions. Let $\bold M=(W,R, \{D_{w}\}_{w\in W}, \models)$ be a Kripke model over $D$ and let $s\in {}^VD,$ where $V$ is the set of all variables. A formula $\phi$ is satsifiable at $w$ under $s$ if $w\models \phi[s]$. The formula $\phi$ is satisfiable in $\bold M$ if there a $w\in W$ and $s\in {}^VD$ such that $w\models \phi[s].$ For a set of formulas $\Gamma$, we write $w\models \Gamma[s]$ if $w\models \phi[s]$ for every $\phi\in \Gamma$. The set of formulas $\Gamma$ is satisfiable in $\bold M$ if there is a $w\in W$ and $s\in {}^VD$ such that $w\models \Gamma[s]$. The formula $\phi$ is valid in $\bold M$ under $s$ if $w\models \phi[s]$ for all $w\in W$ and $s\in {}^{V}D_w$; $\phi$ is valid in $\bold M$ if it is valid for any $s\in {}^VD$. A formula $\phi$ is valid in a frame $(W, R, \{D_w\}_{w\in W})$ if it is valid in every model based on $W$ after specifying the semantical consequence relation $\models$. A set of formulas $\Gamma$ is consistent if no contradiction is derivable from $\Gamma$ relative to the proof system defined above, that is, it is not the case that $T\vdash \bot.$ The custom in intuitionistic logic is to deal with pairs of theories, the first component dealing with a set of formulas that are ’true’, and the second deals with a set formulas that are ‘false’, in the intended interpretaton. This is natural, since we do not have negation. So in fact, our algebraic counterpart proved in section 3, is in fact more general than the completeness theorem stated below; the latter follows from the special case when the second component of pairs is the theory $\{\bot\}$. The following theorems hold for logics without equality. In the presence of infinitary substitutions, we obtain a weaker result for logics with equality. The set $V$ denoting the set of variables in the next theorems is always infinite (which means that we will deal only with infinite dimensional algebras), however, $P$ (specifying the number of atomic formulas) could well be finite. \[com\] Let $V$ and $P$ be countable disjoint sets with $|V|\geq \omega$. When $G$ is a rich semigroup, then $\mathfrak{L}_G$ is strongly complete, that is if $\Gamma$ is a consistent set of formulas, then it is satisfiable at a world of some model based on a Kripke frame. For arbitrary (disjoint) sets $V$ and $P$ with $|V|\geq \omega$, when $G$ is the semigroup of finite transformations, and $\rho\in {}^{V}Ord$ is such that $V\sim \rho(p)$ is infinite for every $p\in P$, or $G={}^VV$ without any restrictions, then $\mathfrak{L}_G$ is strongly complete. [Proof]{} cf. Theorem \[complete\], item (1). We say that a logic $\mathfrak{L}$ has the Craig interpolation property if whenever $\models \phi\to \psi$ then there is a formula containing only symbols occurring in both $\phi$ and $\psi,$ $\theta$ say, such that $\models \phi\to \theta$ and $\models \theta\to \psi.$ (By the above completeness theorem, we can replace $\models$ by $\vdash$.) \[interpolation\] Let $\mathfrak{L}_G$ be as in the previous theorem, except that $G$ is assumed to be strongly rich. Then $\mathfrak{L}_G$ has the interpolation property [Proof]{} cf. Theorem \[complete\], item (2). In the case we have equality then we can prove a slightly weaker result when we have infinite substitutions. We say that the substitution operation ${\sf S}_{\tau}$ is finitary, if $\tau$ moves only finitely many points, otherwise, it is called infinitary. Now we have: \[interpolationeq\] For arbitrary (disjoint) sets $V$ and $P$, with $|V|\geq \omega$, when $G$ is the semigroup of finite transformations, and $\rho\in {}^{V}Ord$ is such that $V\sim \rho(p)$ is infinite for every $p\in P$, then $\mathfrak{L}_G^{=}$ is strongly complete, and has the interpolation property. When $G$ is rich or $G={}^VV,$ then ${\mathfrak L}_G^{=}$ is weakly complete, that is, if a formula is valid in all Kripke models, then it is provable. When $G$ is strongly rich or $G={}^VV$, the logic $\mathfrak{L}_G^{=}$ has the following weak interpolation property. If $\phi$ and $\psi$ are formulas such that only finitary substitutions were involved in their built up from atomic formulas, and $\models \phi\to \psi,$ then there is a formula containing only (atomic formulas, and possibly equality) occurring in both $\phi$ and $\psi,$ $\theta$ say, such that $\models \phi\to \theta$ and $\models \theta\to \psi$, and $\theta$ may involve infinitary substitutions during its formation from atomic formulas. (By weak completenes $\models$ can be replaced by $\vdash$.) [Proof]{} From theorems \[main3\], \[main4\]. \[negative\] For arbitrary (disjoint) sets $V$ and $P$, with $|V|\geq \omega$, when $G$ is the semigroup of finite transformations, and $\rho\in {}^{V}Ord$ is such that $\rho(p)=V$ for every $p\in P$, then both $\mathfrak{L}_G$ and $\mathfrak{L}_G^{=}$ are essentially incomplete, and fail to enjoy the interpolation property. [Proof]{} This is proved in \[Sayed\]. In intuitionistic ordinary predicate logic, interpolation theorems proved to hold for logic without equality, remain to hold when we add equality. This is reflected by item (1) in theorem \[interpolationeq\]. Indeed, in this case our logics are very close to ordinary ones. The sole difference is that atomic formulas could have infinite arity, but like the ordinary case, infinitely many variables lie outside (atomic) formulas. The next item in theorem \[interpolationeq\] shows that the situation is not as smooth nor as evident as the ordinary classical case. The presence of infinitary substitutions seems to make a drastic two-fold change. In the absence of diagonal elements, it turns negative results to positive ones, but it in the presence of diagonal elements the positive results obtained are weaker. Indeed, we do not know whether strong completeness or usual interpolation holds for such logics, but it seems unlikely that they do. We know that there will always be cases when infinitary substitutions are needed in the interpolant. Our logics manifest themselves as essentially infinitary in at least two facets. One is that the atomic formulas can have infinite arity and the other is that (infinitary) substitutions, when available, can move infinitely many points. But they also have a finitary flavour since quantification is taken only on finitely many variables. The classical counterpart of such logics has been studied frequently in algebraic logic, and they occur in the literature under the name of finitary logics of infinitary relations, or typless logics [@HMT2], though positive interpolation theorems for such logics are only rarely investigated \[IGPL\], for this area is dominated by negative results \[references\]. It is well known that first order predicate intuitionistic logic has the following two properties: (\*) Each proof involves finitely many formulas. (\*) A set of formulas is consistent if and only if it is satisfiable. In most cases, such as those logics which have infinitary propositional connectives, it is known to be impossible to define a notion of proof in such a way that both (\*) and (\*\*) are satisfied. We are thus confronted with the special situation that the logic ${\mathfrak L}_G$ behave like ordinary first order intuitionistic logic. In passing, we note that (infinitary) generalizations of the classical Lowenheim-Skolem Theorem and of the Compactness Theorem for ${\mathfrak L}_G$ without equality follows immediately from theorem \[com\]. Now we are ready to prove theorem \[interpolation\]. \[complete\] Let $G$ be a semigroup as in \[complete\] and \[interpolation\] $\mathfrak L_G$ is strongly complete $\mathfrak L_G$ has the interpolation property [Proof]{} We prove the theorem when $G$ is a strongly rich semigroup on $\alpha$, $\alpha$ a countable ordinal specifying the the number of variables in ${\mathfrak L}_G$. Let $\{R_i:i\in \omega\}$ be the number of relation symbols available in our language each of arity $\alpha$. We show that every consistent set of formulas $T$ is satisfiable at some world in a Kripke model. Assume that $T$ is consistent. Let $\A=\Fm/\equiv$ and let $\Gamma=\{\phi/\equiv: \phi\in T\}.$ Then $\Gamma$ generates a filter $F$. Then $\A\in GPHA_{\alpha}$ and $(F,\{0\})$ is consistent. By the above proof, it is satisfiable, that is there exists a Kripke system $\bold K=(K, \leq, M_k, \{V_k\}_{k\in K})$ a homomorphism $\psi:\A\to \mathfrak{F}_{\bold K}$ and an element $k_0\in \bold K$ and $x\in V_{k_0}$ such that for every $p\in \Gamma$, if $\psi(p)=(f_k)$ then $f_{k_0}(x)=1$. Define for $k\in K$, $R_i$ an atomic formula and $s\in {}^{\alpha}M_k$, $k\models R_i[s]$ iff $(\psi(R/\equiv))_k(s)=1.$ This defines the desired model. When $G$ is a strongly rich semigroup, or $G={}^{\alpha}\alpha$, we show that for any $\beta$, $\A=\Fr_{\beta}GPHA_{\alpha}$ has the interpolation property, that is if $a\in \Sg^{\A}X_1$ and $b\in \Sg^{\A}X_2,$ then there exists $c\in \Sg^{\A}(X_1\cap X_2)$ such that $a\leq c\leq b$. When $G$ is the semigroup of all finite transformations and $\rho\in {}^{\beta}\wp(\alpha)$ is dimension restricting, the algebra $\Fr_{\beta}^{\rho}(GPHA_{\alpha})$ can be shown to have the interpolation property in exactly the same manner. We use theorem \[main\] for the former case, while we use its analogue for dimension restricted free algebras, namely, theorem \[main2\] for the latter. Assume that $\theta_1\in \Sg^{\A}X_1$ and $\theta_2\in \Sg^{\A}X_2$ such that $\theta_1\leq \theta_2$. Let $\Delta_0=\{\theta\in \Sg^{\A}(X_1\cap X_2): \theta_1\leq \theta\}.$ If for some $\theta\in \Delta_0$ we have $\theta\leq \theta_2$, then we are done. Else $(\Delta_0, \{\theta_2\})$ is consistent. Extend this to a complete theory $(\Delta_2, \Gamma_2)$ in $\Sg^{\A}X_2$. Consider $(\Delta, \Gamma)=(\Delta_2\cap \Sg^{\A}(X_1\cap X_2), \Gamma_2\cap \Sg^{\A}(X_1\cap X_2))$. Then $(\Delta\cup \{\theta_1\}), \Gamma)$ is consistent. For otherwise, for some $F\in \Delta, \mu\in \Gamma,$ we would have $(F\land \theta_1)\to \mu$ and $\theta_1\to (F\to \mu)$, so $(F\to \mu)\in \Delta_0\subseteq \Delta_2$ which is impossible. Now $(\Delta\cup \{\theta_1\}, \Gamma)$ $(\Delta_2,\Gamma_2)$ are consistent with $\Gamma\subseteq \Gamma_2$ and $(\Delta,\Gamma)$ complete in $\Sg^{\A}X_1\cap \Sg^{\A}X_2$. So by theorem \[main\], $(\Delta_2\cup \{\theta_1\}, \Gamma_2)$ is satisfiable at some world in some set algbra based on a Kripke system, hence consistent. But this contradicts that $\theta_2\in \Gamma_2, $ and we are done. The logic ${\mathfrak L}_G^{=}$ has the weak interpolation property. [Proof]{} Assume that $\theta_1\in \Sg^{\Rd\A}X_1$ and $\theta_2\in \Sg^{\Rd\A}X_2$ such that $\theta_1\leq \theta_2$. Let $\Delta_0=\{\theta\in \Sg^{\A}(X_1\cap X_2): \theta_1\leq \theta\}.$ If for some $\theta\in \Delta_0$ we have $\theta\leq \theta_2$, then we are done. Else $(\Delta_0, \{\theta_2\})$ is consistent, hence $(\Delta_0\cap \Sg^{\Rd\A}X_2,\theta_2)$ is consistent. Extend this to a complete theory $(\Delta_2, \Gamma_2)$ in $\Sg^{\Rd\A}X_2$; this is possible since $\theta_2\in \Sg^{\Rd\A}X_2$. Consider $(\Delta, \Gamma)=(\Delta_2\cap \Sg^{\A}(X_1\cap X_2), \Gamma_2\cap \Sg^{\A}(X_1\cap X_2))$. It is complete in the ‘common language’, that is, in $\Sg^{\A}(X_1\cap X_2)$. Then $(\Delta\cup \{\theta_1\}), \Gamma)$ is consistent in $\Sg^{\Rd\A}X_1$ and $(\Delta_2, \Gamma_2)$ is consistent in $\Sg^{\Rd\A}X_2$, and $\Gamma\subseteq \Gamma_2.$ Applying the previous theorem, we get $(\Delta_2\cup \{\theta_1\}, \Gamma_2)$ is satisfiable. Let $\psi_1, \psi_2$ and $\psi$ and $k_0$ be as in the previous proof. Then $\psi\upharpoonright \Sg^{\Rd\A}X_1=\psi_1$ and $\psi\upharpoonright \Sg^{\Rd\A}X_2=\psi_2$. But $\theta_1\in \Sg^{\Rd\A}X_1$, then $\psi_1(\theta_1)=\psi(\theta_2)$. Similarly, $\psi_2(\theta_2)=\psi(\theta_2).$ So, it readily follows that $(\psi(\theta_1))_{k_0}(Id)=1$ and $(\psi(\theta_2))_{k_0}(Id)=0$. This contradicts that $\psi(\theta_1)\leq \psi(\theta_2),$ and we are done. When $G$ consists only of finite transformations and $v\sim \rho(p)$ is infinite, then ${\mathfrak L}_G^{=}$ has the interpolation property. In the next example, we show that the condition $\Gamma_0\subseteq \Gamma_0^*$ cannot be omitted. The example is an an algebraic version of theorem 4.31, p.121 in [@b], but modified appropriately to deal with infinitary languages. \[counter\] Let $G$ be a strongly rich semigroup on $\omega$. Let $\Lambda_{\omega}$ be a language with three predicate symbols each of arity $\omega$; this is a typless logic abstracting away from rank of atomic formulas, so that we might as well forget about the variables, since we allow them only in their natural order. The real rank of such relation symbols will be recovered from the semantics. Let $\bold M=(\N, \leq, \D_i)_{i\in \omega}$ be the Kripke frame with $D_i=\N$ for every $i$, and let $\N=\bigcup_{n\in \omega} B_n$, where $B_n$ is a sequence of pairwise disjoint infinite sets. We define the relation $\models$ on atomic formulas. Let $m\in \N$. If $m=2n+1$, and $s\in {}^{\omega}\N,$ then $m\models p_0[s]$ if $s_0\in \bigcup_{i\leq 2n+1}B_i$, $m\models p_1[s]$ if $s_0\in \bigcup_{i\leq 2n+1}B_i$ and $m\models p_3[s]$. If $m=2n$, and $s\in {}^{\omega}\N,$ then $m\models p_0[s]$ if $s_0\in \bigcup_{i\leq n}B_i$ and $m\models p_2[s]$ if $s_0\in \bigcup_{i\leq 2n+1}B_i$ and $m\models p_3[s]$. Let $\F_{\bold M}$ be the set algebra based on the defined above Kripke model $\bold M$. Let $\A=\Fr_3GPHA_{\omega}$ and let $x_1, x_2, x_3$ be its generators. Let $f$ be the unique map from $\A$ to $\F_{\bold M}$ such that for $i\in \{0,1,2\}$, $f(x_i)=p_i^{\bold M}$. We have $\A\cong \Fm/\equiv$. We can assume that the isomorphism is the identity map. Let $\Delta'=\{a\in A: f(a)=1\}$ and $\Theta'=\{a\in A: f(a)=0\}$. Let $\Delta=\{\phi:\phi/\equiv\in \Delta'\},$ and $\Theta=\{\phi:\phi/\equiv\in \Gamma'\}.$ Let $$\Delta_1=\Delta\cup \{{\sf q}_0(x_1\lor x_2), {\sf c}_0(x_2\land x_3)\}$$ $$\Theta_1=\Theta\cup \{{\sf c}_0(x_1\land x_3)\}$$ $$\Delta_2=\Delta\cup \{{\sf q}_0(x_1\lor x_3)\}$$ $$\Theta_2=\Theta\cup \{{\sf c}_0(x_1\land x_3), c_0(x_2\land x_3)\}.$$ Then by analogy to 4.30 in [@b], $(\Delta_1, \Theta_1)$, $(\Delta_2, \Theta_2)$ are consistent, but their union is not. \[mak\] If $G$ is strongly rich or $G={}^{\alpha}\alpha$, then $Var(\mathfrak{L}_G)$ has $SUPAP.$ In particular, $GPHA_{\alpha}$ has $SUPAP$. [Proof]{} Cf. [@b] p.174. Suppose that $\A_0, \A_1, \A_2\in Var(\mathfrak{L}_G)$. Let $i_1:\A_0\to \A_2$ and $i_2: \A_0\to \A_2$ be embeddings. We need to find an amalgam. We assume that $A_0\subseteq A_1\cap A_2$. For any $a\in A_i$, let $x_a^i$ be a variable such that $x_a^0=x_a^1=x_a^2$ for all $a\in A_0$ and the rest of the variables are distinct. Let $V_i$ be the set of variables corresponding to $\A_i$; then $|V_i|=|A_i|$. Let $V$ be the set of all variables, endowed with countably infinitely many if the algebras are finite. Then $|V|=\beta\geq \omega.$ We assume that the set of variables $V$ of ${\mathfrak L}_G$ is the same as the set variables of the equational theory of $Var(\mathfrak{L}_G).$ We fix an assignment $s_i$ for each $i\in \{0,1,2\}$ such that $s_i: V_i\to A_i$ and $s_i(x_a^i)=a$ and so $s_1\upharpoonright V_0=s_2\upharpoonright V_0=s_0$. In view of the correspondence established in \[terms\], we identify terms of the equational theory of $Var({\mathfrak L}_G)$ with formulas of $\mathfrak{L}_G$; which one we intend will be clear from context. Accordingly, we write $\A_i\models \psi\leftrightarrow \phi$ if $\bar{s}_i(\psi)=\bar{s}_i(\phi)$, where $\bar{s_i}$ is the unique extension of $s_i$ to the set all terms. Let $\Fm_i$ be the set of formulas of $\mathfrak{L}_{G}$ in the variables $x_a^i$, $a\in A_i$, and let $\Fm$ be the set of all formulas built up from the set of all variables. (Note that $Fm_i$ can be viewed as the set of terms built up from the variables $x_a^i$, and $Fm$ is the set of all terms built up from the set of all variables, defining operations corresponding to connectives turn them to absolutely free algebras.) For $i=1,2$, let $T_i=\{\psi\in \Fm_i: \A_i\models \psi=1\}$, and let $T=\{\psi\in \Fm: T_1\cup T_2\vdash \psi\}$. We will first prove (\*): For $\{i,j\}=\{1,2\},$ $\psi\in \Fm_i$ and $\phi\in \Fm_j,$ we have $T\vdash \psi\leftrightarrow \phi$ iff $(\exists c\in \Fm_0)(\A_i\models \psi\leq c\land \A_j\models c\leq \phi.)$ Only one direction is non trivial. Assume that $T\vdash \psi\leftrightarrow \phi.$ Then there exist finite subsets $\Gamma_i\subseteq T_i$ and $\Gamma_j\subseteq T_j$ such that $\Gamma_i\cup \Gamma_j\vdash \psi\leftrightarrow \phi.$ Then, by the deduction theorem for propositional intuitinistic logics, we get $${\mathfrak L}_G\vdash \bigwedge \Gamma_i\to (\bigwedge \Gamma_j\to (\psi\to \phi)),$$ and so $${\mathfrak L}_G\vdash (\bigwedge \Gamma_i\land \psi)\to (\bigwedge \Gamma_j\to \phi).$$ Notice that atomic formulas and variables occuring in the last deduction are finite. So the interpolation theorem formulated for $G$ countable algebras apply also, and indeed by this interpolation theorem \[interpolation\] for ${\mathfrak L}_G$, there is a formula $c\in \Fm_0$ such that such that $\vdash \bigwedge \Gamma_i\land \psi\to c$ and $\vdash c\to (\bigwedge \Gamma_j\to \phi.)$ Thus $\A_i\models \psi\leq c$ and $\A_j\models c\leq \phi$. We have proved (\*). Putting $\psi=1,$ we get $T\vdash \phi$ iff ($\exists c\in \Fm_0)(\A_i\models 1\leq c\land \A_j\models c\leq \phi)$ iff $\A_j\models \phi=1.$ Define on $\Fm$ the relation $\psi\sim \phi$ iff $T\vdash \alpha\leftrightarrow \beta$. Then $\sim$ is a congruence on $\Fm$. Also for $i=1,2$ and $\psi,\phi\in Fm_i$, we have $T\vdash \psi\sim \phi$ iff $\A_i\models \psi=\phi$. Let $\A=\Fm/\sim$, and $e_i=\A_i\to \A$ be defined by $e_i(a)=x_a^i/\sim$. Then clearly $e_i$ is one to one. If $a\in \A_0$, then $x_a^0=x_a^1=x_a^2$ hence $e_1(a)=e_2(a)$. Thus $\A$ is an amalgam via $e_1$ and $e_2.$ We now show that the superamalgamation property holds. Suppose $\{j,k\}=\{1,2\}$, $a\in \A_j$, $b\in \A_k$ and $e_j(a)\leq e_k(b)$. Then $(e_j(a)\to e_k(b))=1$, so $(x_a^j\to x_b^k)=1$, that is $T\vdash (x_a^j\to x_b^k)$. Hence there exists $c\in \Fm_0$ such that $(\A_j\models x_a\leq c\land \A_k\models c\leq x_b)$. Then $a\leq c$ and $c\leq b.$ By taking ${\mathfrak L}_G$ to be the logic based on $\alpha$ many variables, and ${\mathfrak L}_G$ has countably many atomic formulas each containing $\alpha$ many variables in their natural order, we get that $V=Var({\mathfrak L}_G)$, hence $V$ has $SUPAP$. [HMT85]{} Andréka, H. [*Complexity of equations valid in algebras of relations*]{}. Annals of Pure and Applied logic, [**89**]{} (1997), p. 149 - 209. Andreka, H., Nemeti I., Sayed Ahmed T. [*A non representable quasi-polyadic algebra with a representable cylindric reduct*]{} Studia Math Hungarica, in press Sayed Ahmed, T. [*On Amalgamation of Reducts of Polyadic Algebras.*]{} Algebra Universalis [**51**]{} (2004), p.301-359. Sayed Ahmed, T. [*Algebraic Logic, where does it stand today?*]{} Bulletin of Symbolic Logic, [**11**]{}(4) (2005), p. 465–516 Sayed Ahmed T, [*Some results about neat reducts*]{} Algebra Universalis, [**1**]{}(2010) p. 17-36. Sayed Ahmed , T. [*The class of polyadic algebras has the superamalgamation property*]{} Mathematical Logic Quarterly [**56**]{}(1)(2010)p.103-112 Sayed Ahmed T. [*The amalgmation property, and a problem of Henkin Monk and Tarski*]{} Journal of Algebra, number theory, advances and applications [**1**]{}(2)(2009) p. 127-141 Sayed Ahmed, T. [*On neat embeddings of cylindric algebras*]{} Mathematical Logic Quarterly [**55**]{}(6)(2009)p.666-668 Sayed Ahmed [*Classes of algebras without the amalgamation property*]{} Logic Journal of IGPL. [**1**]{} (2011) 87-104. Sayed Ahmed T. [*Amalgamation of polyadic Heyting algebras*]{} Studia Math Hungarica, in press. Daigneault, A., [*Freedom in polyadic algebras and two theorems of Beth and Craig*]{}. Michigan Math. J.[**11**]{} (1963), p. 129-135. Daigneault, A., and Monk,J.D., [*Representation Theory for Polyadic algebras*]{}. Fund. Math. [**52**]{}(1963) p.151-176. Ferenczi, M., [*On representation of neatly embeddable cylindric algebras*]{} Journal of Applied Non-classical Logics, [**10**]{}(3-4) (2000) p.34-56 Ferenczi, M., [*Finitary polyadic algebras from cylindric algebras.*]{} Studia Logica [**87**]{}(1)(2007) p.1-11 Ferenczi, M., [*On cylindric algebras satisfying the merry-go-round properties*]{} Logic Journal of IGPL, [**15**]{}(2) (2007), p. 183-199 Ferenczi, M., [*On representability of neatly embeddable cylindric algebras*]{} Journal of Appl. Non-classical Logic, 3-4, 10 (2000) 1-11 Ferenczi M, [*The polyadic representation* ]{} Transactions of Amer Math Society, to appear. Gabbay M.D., Maksimova L. [*Interpolation and Definability: Modal and Intuitionistic Logic*]{} Oxford Science Publications (2005) Gentzen, G., 1934-5, “Untersuchungen Über das logische Schliessen,” Math. Zeitschrift 39: 176-210, 405-431. Georgescu G. [*A representation theorem for polyadic Heyting algebras*]{} Algebra Universalis [**14**]{} (1982), 197-209. Gödel, K., 1933, “Zur intuitionistischen Arithmetik und Zahlentheorie,” Ergebnisse eines mathematischen Kolloquiums 4: 34-38. Halmos, P., [*Algebraic Logic.*]{} Chelsea Publishing Co., New York, (1962.) Henkin, L., [*An extension of the Craig-Lyndon interpolation theorem*]{} Journal of Symbolic Logic 28(3) (1963) p.201-216 Henkin, L., Monk, J.D., and Tarski, A., [*Cylindric Algebras Part I*]{}. North Holland, 1971. Henkin, L., Monk, J.D., and Tarski, A., [*Cylindric Algebras Part II*]{}. North Holland, 1985. Herrlich H, Strecker G. [*Category theory*]{} Allyn and Bacon, Inc, Boston (1973) Heyting, A., [*Die formalen Regeln der intuitionistischen Logik, in three parts*]{}, Sitzungsber. preuss. Akad. Wiss.: 42-71,(1930) 158-169. English translation of Part I in Mancosu 1998: 311-327. Heyting, A., 1956, Intuitionism: An Introduction, North-Holland Publishing, Amsterdam. Third Revised Edition (1971). Hodges, W. [*A shorter Model Theory*]{}. Cambridge. University Press. 1997. Johnson, J.S. [*Amalgamation of Polyadic Algebras*]{}. Transactions of the American Mathematical Society, [**149**]{}(1970) p.627-652 Pigozzi,D. [*Amalgamation, congruence extension, and interpolation properties in algebras.*]{} Algebra Universalis. [**1**]{}(1971), p.269-349. Keisler H.J., [*A complete first order logic with infinitary predicates*]{} Fund. Math [**52**]{}(1963) p.177-203 Kleene, S. C., 1952, Introduction to Metamathematics, Van Nostrand, Princeton. Kripke “Semantical analysis of intuitionistic logic,” in J. Crossley and M. A. E. Dummett, eds., 1965: 92-130. Madárasz J. and Sayed Ahmed T., [*Amalgamation, interpolation and epimorphisms.*]{} Algebra Universalis [**56**]{} (2) (2007) p. 179-210. Madárasz J. and Sayed Ahmed T. [*Neat reducts and amalgamation in retrospect, a survey of results and some methods. Part 1: Results on neat reducts*]{} Logic Journal of IGPL [**17**]{}(4)(2009) p.429-483 Madárasz J. and Sayed Ahmed T., [*Neat reducts and amalgamation in retrospect, a survey of results and some methods. Part 2: Results on amalgamation*]{} Logic Journal of IGPL (2009) doi: 10.1093/jigpal/jzp013 Monk J.D. [*Polyadic Heyting algebras*]{} Notices Amer Math Soc (1966) 735. Maksimova, L. [*Amalgamation and interpolation in normal modal logics*]{}. Studia Logica [**50**]{}(1991) p.457-471. Németi, I., Sági, G. [*On the equational theory of representable polyadic algebras*]{}. Journal of Symbolc Logic (65)(3) 2000, p. 1143-1167 Sagi, G, Ferenszi, M, [*On some developments in the representation theory of cylindric- like algebras*]{} Algebra Universalis, [**55**]{}(2-3)(2006) p.345-353 Sagi, G, Shelah S., [*Weak and strong interpolation for algebraic logics.*]{} Journal of Symbolic Logic [**71**]{}(2006) p.104-118. Sain, I. [*Searching for a finitizable algebraization of first order logic*]{}. 8, Logic Journal of IGPL. Oxford University, Press (2000) no 4, p.495–589. A. Tarski, *Grundz¨uge der Systemenkalk¨uls. Erster Teil*. Fundamenta Mathematica, Vol. **25**, (1935), p.503-526. English translation in \[A. Tarski, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, edited by J. Corcoran, Hackett Pub. Co., Indianapolis, Indiana, second edition, (1983)\]: Foundations of the calculus of systems, p.342-383. [^1]: 2000 [*Mathematics Subject Classification.*]{} Primary 03G15. [*Key words*]{}: algebraic logic, neat reducts, cylindric algebras, amalgamation [^2]: The class of representable algebras is given by specifying the universes of the algebras in the class, as sets of certain sets endowed with set theoretic concrete operations; thus representable algebras are completely determined once one specifies their universes. [^3]: The idea of relativization, similar to Henkin’s semantics for second order logic, has proved a very fruitful idea in the theory of cylindric algebras. [^4]: Strictly speaking, in $ZFC$ we cannot talk about classes, but classes can be stimulated rigorously with formulas; in our context we chose not to be pedantic about it. Alternatively, we could have replaced $Set$ ($Ord$) by a set of sets (ordinals), but the notation $Set$ ($Ord$) is more succint and ecconomic.

ArXiv